o Fas a 2S exer i a) Ath, OU A if iW ep WO. OEMCO O. E LIBRARY ee AWEH GI, DOCUMENT | COLLECTION / HENAN A ua : WOODS HOLE OCEANOGRAPHIC INSTITUTION QV D4y. G1 vf LABORATORY BOOK COLLECTION PURCHASE ORDER wo. A432/72L he On ane ion teh oy PHYSICS OF SOUND IN THE SEA ORIGINALLY ISSUED AS SUMMARY TECHNICAL REPORT OF DIVISION 6, NDRC VOLUME 8 WASHINGTON, D.C. 1946 REPRINTED BY DEPARTMENT OF THE NAVY HEADQUARTERS NAVAL MATERIAL COMMAND WASHINGTON, D.C. 20360 1969 All Navy Activities can obtain copies of this publication on MILSTRIP Form DD 1348 in accordance with NAVSANDA 2002. Re- quest should be addressed to: Navy Publications and Forms Center 5801 Tabor Avenue Philadelphia, Pennsylvania 19120 VIA Chief of Naval Material MAT-0325 Washington, D.C. 20360 For sale by the Superintendent of Documents, U.S. Government Printing Office Washington, D.C. 20402 - Price $7.00 NATIONAL DEFENSE RESEARCH COMMITTEE James B. Conant, Chairman Richard C. Tolman, Vice Chairman Roger Adams Frank B. Jewett Karl T. Compton Army Representative! Navy Representative? Commissioner of Patents? Irvin Stewart, Executive Secretary 1Army representatives in order of service: Maj. Gen. G. V. Strong Col. L. A. Denson Maj. Gen. R. C. Moore Col. P. R. Faymonville Maj. Gen. C. C. Williams Brig. Gen. E. A. Regnier Brig. Gen. W. A. Wood, Jr. Col. M. M. Irvine Col. E. A. Routheau * Navy representatives in order of service: Rear Adm. H. G. Bowen Rear Adm. J. A. Furer Capt. Lybrand P. Smith Rear Adm. A. H. Van Keuren Commodore H. A. Schade 3Commissioners of Patents in order of service: Conway P. Coe Casper W. Ooms NOTES ON THE ORGANIZATION OF NDRC The duties of the National Defense Research Committee were (1) to recommend to the Director of OSRD suitable projects and research programs on the instrumentalities of warfare, together with contract facilities for carrying out these projects and programs, and (2) to administer the tech- nical and scientific work of the contracts. More specifically, NDRC functioned by initiating research projects on re- quests from the Army or the Navy, or on requests from an allied government transmitted through the Liaison Office of OSRD, or on its own considered initiative as a result or the experience of its members. Proposals prepared by the Division, Panel, or Committee for research contracts for performance of the work involved in such projects were first reviewed by NDRC, and if approved, recommended to the Director of OSRD. Upon approval of a proposal by the Director, a contract permitting maximum flexibility of scientific effort was arranged. The business aspects of the contract, including such matters as materials, clearances, vouchers, patents, priorities, legal matters, and administra- tion of patent matters were handled by the Executive Sec- retary of OSRD. Originally NDRC administered its work through five divisions, each headed by one of the NDRC members. These were: Division A — Armor and Ordnance Division B — Bombs, Fuels, Gases, & Chemical Problems Division C — Communication and Transportation Division D — Detection, Controls, and Instruments Division E — Patents and Inventions In a reorganization in the fall of 1942, twenty-three ad- ministrative divisions, panels, or committees were created, each with a chief selected on the basis of his outstanding work in the particular field. The NDRC members then be- came a reviewing and advisory group to the Director of OSRD. The final organization was as follows: 1 — Ballistic Research 2 — Effects of Impact and Explosion 3 — Rocket Ordnance Division Division Division Division 4— Ordnance Accessories Division 5 — New Missiles Division 6 — Sub-Surface Warfare Division 7— Fire Control Division 8 — Explosives Division 9— Chemistry Division 10 — Absorbents and Aerosols Division 11 — Chemical Engineering Division 12 — Transportation Division 13 — Electrical Communication Division 14 — Radar Division 15 — Radio Coordination Division 16 — Optics and Camouflage Division 17 — Physics Division 18 — War Metallurgy Division 19 — Miscellaneous Applied Mathematics Panel Applied Psychology Panel Committee on Propagation Tropical Deterioration Administrative Committee iil NDRC FOREWORD AS EVENTS of the years preceding 1940 revealed more and more clearly the seriousness of the world situation, many scientists in this country came to realize the need of organizing scientific research for service in a national emergency. Recommendations which they made to the White House were given care- ful and sympathetic attention, and as a result the National Defense Research Committee [NDRC] was formed by Executive Order of the President in the summer of 1940. The members of NDRC, appointed by the President, were instructed to supplement the work of the Army and the Navy in the development of the instrumentalities of war. A year later, upon the establishment of the Office of Scientific Research and Development [OSRD], NDRC became one of its units. The Summary Technical Report of NDRC is a conscientious effort on the part of NDRC to sum- marize and evaluate its work and to present it in a useful and permanent form. It comprises some seventy volumes broken into groups corresponding to the NDRC Divisions, Panels, and Committees. The Summary Technical Report of each Division, Panel, or Committee is an integral survey of the work of that group. The first volume of each group’s re- port contains a summary of the report, stating the problems presented and the philosophy of attacking them and summarizing the results of the research, de- velopment, and training activities undertaken. Some volumes may be “‘state of the art” treatises covering subjects to which various research groups have con- tributed information. Others may contain descrip- tions of devices developed in the laboratories. A master index of all these divisional, panel, and com- mittee reports which together constitute the Sum- mary Technical Report of NDRC is contained in a separate volume, which also includes the index of a microfilm record of pertinent technical laboratory reports and reference material. Some of the NDRC-sponsored researches which had been declassified by the end of 1945 were of sufficient popular interest that it was found desirable to report them in the form of monographs, such as the series on radar by Division 14 and the monograph on sampling inspection by the Applied Mathematics Panel. Since the material treated in them is not dupli- iv cated in the Summary Technical Report of NDRC, the monographs are an important part of the story of these aspects of NDRC research. In contrast to the information on radar, which is of widespread interest and much of which is released to the public, the research on subsurface warfare is largely classified and is of general interest to a more restricted group. As a consequence, the report of Division 6 is found almost entirely in its Summary Technical Report, which runs to over twenty volumes. The extent of the work of a Division cannot therefore be judged solely by the number of volumes devoted to it in the Summary Technical Report of NDRC: account must be taken of the monographs and avail- able reports published elsewhere. Any great cooperative endeavor must stand or fall with the will and integrity of the men engaged in it. This fact held true for NDRC from its inception, and for Division 6 under the leadership of Dr. John T. Tate. To Dr. Tate and the men who worked with him — some as members of Division 6, some as representatives of the Division’s contractors — be- longs the sincere gratitude of the Nation for a diffi- cult and often dangerous job well done. Their efforts contributed significantly to the outcome of our naval operations during the war and richly deserved the warm response they received from the Navy. In ad- dition, their contributions to the knowledge of the ocean and to the art of oceanographic research will assuredly speed peacetime investigations in this field and bring rich benefits to all mankind. The Summary Technical Report of Division 6, prepared under the direction of the Division Chief and authorized by him for publication, not only presents the methods and results of widely varied re- search and development programs but is essentially a record of the unstinted loyal cooperation of able men linked in a common effort to contribute to the defense of their Nation. To them all we extend our deep appreciation. VaNNEvVAR Buss, Director Office of Scientific Research and Development J. B. Conant, Chairman National Defense Research Committee FOREWORD qIS VOLUME, together with Volumes 6, 7, and 9, T summarizes four years of research on underwater sound phenomena. The purpose of this research was to provide a firmer foundation for the most effective design and use of sonar gear. It is generally true that wide basic knowledge is an important element in en- gineering practice. In the development of sonar gear, knowledge of how sound is generated, transmitted, reflected, received, and detected is clearly useful both in the design of new equipment and in the most efficient utilization of existing gear. As a result of the time delay between the design of new equipment and its use in service, the most important application of this basic information during World War II has been in suggesting how existing equipment could best be operated and tactically used. The importance of basic information on under- water sound had been evident to both our own Navy and the British for some time. Practical experience had shown that the maximum distance at which a target could be detected with underwater sound was highly variable, even when the equipment was in good operating condition. Since it was realized that such variability might well be related to a variability in oceanographic conditions, the Navy brought this problem to the attention of the Woods Hole Oceano- graphic Institution and the Scripps Institute of Oceanography. To support an investigation, NDRC contracted in 1940 with the former institution to carry out studies and experimental investigations of the structure of the superficial layer of the ocean and its effect on the transmission of sonic and supersonic vibrations. The work carried out under this contract, together with supporting information obtained elsewhere, em- phasized the relation of such basic factors to the variable performance of sonar gear. Thus when some months later it was proposed to establish a section in NDRC to undertake research and development, re- lating to the detection of submerged submarines, plans were made to increase substantially this re- search effort. To this end, the plans which were for- mulated by NDRC and approved by the Navy in- cluded research on underwater sound phenomena at the proposed laboratory at San Diego, to be operated under a contract with the University of California Division of War Research. This step not only in- creased the number of personnel engaged in this re- search and facilitated study of oceanic conditions peculiar to the Pacific area, but also most fortunately made it possible for the San Diego Laboratory to recruit certain of its staff from the Scripps Institution of Oceanography and to draw upon the director and staff of the Scripps Institution for very pertinent background information in oceanography. While the major source of the experimental data continued to be the Woods Hole and San Diego Laboratories, very pertinent data were from time to time obtained from other laboratories, notably New London, Harvard, the Massachusetts Institute of Technology, and the Underwater Sound Reference Laboratories. Quite promptly, an analytical section, later known as the Sonar Analysis Group, was organized under a contract with Columbia University Division of War Research. The function of this group was to assist in the analyses of data being accumulated by Woods Hole, San Diego, and other laboratories, and, as it became possible to draw conclusions, to present these to other groups interested in operations or design. In this connection it should be emphasized that the seeming importance of this research to the Navy led to the assignment of naval personnel to follow the work actively. In particular, officers of the Sonar De- sign Section of the Bureau of Ships followed very closely the research of this analytical group, partici- pating directly in much of the work. The results obtained in this research and sum- marized in this and companion volumes found many important applications during World War II. The rules used for operating sonar gear were based in part on these results. Many tactical rules embodied in submarine and antisubmarine doctrine were directly based on information obtained in these basic studies of transmission, reflection, detection, and the like. As an example, the spacing between antisubmarine vessels in different tactical and oceanographic condi- tions was varied according to the measured tempera- ture gradients in the upper layers of the ocean. In addition, the choice of operating frequency, pulse length, size, and power for new equipment, especially for submarines, was considerably influenced by such basic knowledge. It can be stated with considerable confidence that a detailed basic knowledge of under- water sound phenomena will be of increasing help in Vv vi FOREWORD the design and operation of Navy sonar equip- ment. Only a few of the scientists and others contribut- ingto thiswar effort can be named. Mr.C.0’D. Iselin, Director of the Woods Hole Oceanographic Institu- tion, and his staff brought to this research, to which they ably contributed, a sound background knowl- edge of oceanography. Dr. V. O. Knudsen, Dean of the Graduate School of the University of California at Los Angeles and for some time the Director of the Division’s San Diego Laboratory, was one of this country’s foremost scientists in the field of acoustics. Dr. Knudsen played a prominent part in organizing the research program, and after leaving the San Diego Laboratory he contributed actively and ef- fectively to research work closely related to the sub- ject of this volume. Dr. G. P. Harnwell, Chairman of the Department of Physics at the University of Pennsylvania, who succeeded Dr. Knudsen as Di- rector at San Diego after having served some time as a technical aide to the Division, gave wise general direction to this research at San Diego. In operations at San Diego, Dr. Knudsen and Dr. Harnwell were ably supported by Dr. Carl Eckart, Professor of Theoretical Physics at the University of Chicago, who became Associate Director at San Diego, re- sponsible for the planning and execution of the basic research there. Dr. H. Sverdrup, Director of the Scripps Institu- tion of Oceanography, and his staff also contributed significantly to this work. The U.S. Navy Electronics Laboratory at San Diego collaborated most helpfully in much of this basic research. The task of organizing the very important analytical work was assumed by Dr. W. V. Houston, Professor of Physics at the Cali- fornia Institute of Technology and Director of the Special Studies Group; he delegated the very large part of the responsibility to Dr. Lyman Spitzer, Jr., an outstanding member of the Departments of Physics and Astronomy at Yale University, who be- came Director of the Sonar Analysis Group. As the reader will note, Dr. Spitzer undertook the responsibility for preparing this volume, and in this he had had the assistance not only of members of his own staff but also of naval personnel and of members of the Woods Hole and San Diego staffs. The Divi- sion appreciates the efforts of all those who have participated. This research project secured most effective sup- port from the Navy. The broad program of research and study which was proposed by Dr. Jewett and Dr. Bush, and which included this basic research on underwater sound, was supported by Rear Admiral S. M. Robinson, Chief of the Bureau of Ships, who took steps to provide facilities for this work at San Diego. Later, when Rear Admiral Van Keuren be- came Chief of the Bureau of Ships, he likewise strongly backed the program, which was still in its initial stages. Support of the program continued with Vice Admiral Cochrane as Chief of the Bureau of Ships, and most helpful liaison was provided by Captain Rawson Bennett, Jr.. Commander J. C. Myers, Commander Roger Revelle, and others in the Bureau. The Coordinator of Research and Develop- ment and his staff continually gave support to this research. The results of much of this work were of special interest to the Tenth Fleet and very close contact was accordingly maintained with its staff, particularly with the Operations Research Group. In presenting this volume the hope is expressed that research in this area will be energetically con- tinued. It is also hoped that general interest in this field may be maintained by the distribution to the widest possible audience of this volume and other volumes which have been written from the stand- point of basic science. Joun T. Tate Chief, Division 6 PREFACE | be THE COURSE Of prosubmarine and antisubmarine research carried out during World War II, a large amount of information was obtained on the propaga- tion of underwater sound. Much of this was gathered in fairly random ways, such as while testing under- water sound equipment. Most of the useful informa- tion, however, was obtained by groups devoted pri- marily to the problem of underwater sound propaga- tion. While valuable results had been found before World War II by the Naval Research Laboratory, the British, and other groups, most of the informa- tion on underwater sound transmission obtained during the war resulted from a program of studies organized by Division 6 of the National Defense Re- search Committee and carried out in collaboration with Navy laboratories at San Diego and elsewhere. It should be kept in mind that these so-called fundamental programs were not fundamental in the usual scientific sense. They were not aimed at isolat- ing and understanding the different factors at work, but were designed rather for the accumulation of in- formation which would be useful in antisubmarine and prosubmarine operations. Thus effort was con- centrated on the study of the transmission loss of sound generated with standard sonar gear under varying oceanographic conditions, rather than on a detailed study of each of the individual factors af- fecting underwater sound transmission. Similarly, the reflection of sound from actual submarines was studied rather than the individual mechanisms re- sponsible for the origin of echoes from underwater targets. During the war this approach was abundantly justified by its results. The information obtained on underwater sound propagation under different ocea- nographic and tactical situations was immediately ap- plied to the more effective use of existing underwater sound equipment in different situations. The results of transmission, reverberation and other studies were usually used operationally much more rapidly than the results of equipment development. Over a longer period, however, information on underwater sound can be most useful if the phenom- ena are not merely observed but also explained. An understanding of each of the basic factors affecting underwater sound propagation would make it possi- ble to predict the transmission and reflection to be expected under conditions widely different from those prevailing when the original measurements were taken. While the primarily experimental re- search carried out during the war could be immedi- ately applied to the gear then in existence, the de- velopment of new equipment for new and unforeseen tactical situations requires an understanding of the factors which influence underwater sound. The ulti- mate aim of basic underwater sound research, espe- cially during peacetime, should be to develop such an understanding. The present volume presents the essential results obtained in the studies of underwater sound up to the middle of 1945. This volume was written pri- marily from the fundamental viewpoint of scientific research; in other words, the data are presented against a framework of an attempted understanding of the factors involved rather than as an unadorned summary of the experimental results. Since the meas- urements were not carried out primarily to increase this understanding, this presentation of the subject leads to many obvious gaps. However, it is hoped that the overall scientific picture presented will be stimulating to any future research workers in this field. To aid those interested in application, practical summaries of the results are given at the end of each of the four parts comprising this volume. Since our understanding of the details of under- water sound has not been sufficient in most cases to allow an elaborate comparison between theory and experiment, it has been possible in most of this volume to write the text on the level of a senior engineering student. A deliberate effort has been made to keep to this level wherever possible in order to make the results available to the widest possible group of readers. However, more elaborate theoretical devel- opments have been included where it was believed that they were essential to an understanding of the full significance of current information. The first two parts of this volume deal with the propagation of sound in the absence of targets. Part I discusses the transmission loss of sound sent out from a projector, while Part II deals with sound which has Vil Viil been scattered back to the vicinity of the original sound source. Part III deals with the echoes returned from submarines and surface vessels. Part IV dis- cusses the transmission of sound through wakes and echoes received from wakes. It should be emphasized that this work is essen- tially a report of the work carried out by the Univer- sity of California Division of War Research in col- laboration with the U. S. Navy Electronics Labora- tory, formerly the U. S. Navy Radio and Sound Laboratory; and by the Woods Hole Oceanographic Institution. Both the Underwater Sound Reference Laboratories and the New London Laboratory of Columbia University Division of War Research, as well as the underwater sound laboratory of the Massachusetts Institute of Technology, have also made important contributions in special fields. All these groups, under contract with Division 6, have been very helpful in the preparation of this volume. They have at times supplied unpublished data and have made many helpful comments and suggestions for improving the presentation. The direct preparation of this volume has been largely a cooperative enterprise of the Sonar Analysis Group, operating under different auspices at different times. This work was initiated under the Special Studies Group of Columbia University Division of War Research, under Contract OKMsr-1131. Most of the writing was done by the Sonar Analysis Group PREFACE under Contract OEMsr-1483; during this time, the Group operated under the auspices of the Office of Field Service but under the cognizance of Section 940 of the Bureau of Ships. Final preparation of the manuscript was completed while the Group formed part of the Woods Hole Oceanographic Institution under Contract Nobs-2083 with the Bureau of Ships. The scientific staff of the Sonar Analysis Group engaged in this work were: P. G. Bergmann, E. Gerjuoy, P. G. Frank, A. N. Guthrie, Lieut. (jg) J. K. Major, USNR (Project Officer of the Group), J. J. Markham, L. Spitzer, Jr., R. Wildt, and A. Yaspan. The work was under the general supervision of the director of the Group, aided by A. N. Guthrie, Administrative Director. The editors were: Part I P.G. Bergmann and A. Yaspan Part II E. Gerjuoy and A. Yaspan Part III Lieut. (jg) J. K. Major, USNR Part IV R. Wildt Because Part I was largely the result of cooperative effort by many members of the Group, as well as by C. Herring of the Special Studies Group, names of individual chapter authors of Part I are listed in the Table of Contents. Final assembly of the material was under the supervision of Mrs. E. E. Wagner, and of H. Birnbaum and M. Klapper. Lyman Spitzer, JR. Director, Sonar Analysis Group CONTENTS CHAPTER oOo on OD oO SE 10 ia 12 13 14 15 16 17 Ail TRANSMISSION Introduction by P. G. Bergmann and L. Spitzer, Jr. Wave Acoustics by P. G. Frank and A. Yaspan . Ray Acoustics by P. G. Frank, P. G. Bergmann, and A. Yaspan Se a IL, ih Experimental Procedures by P. G. Bergmann . Deep-Water Transmission by L. Spitzer, Jr. . Shallow-Water Transmission by P. G. Bergmann Intensity Fluctuations by P. G. Bergmann Explosions as Sources of Sound by C. Herring ; Transmission of Explosive Sound in the Sea by C. Herring Summary by L. Spitzer, Jr. and P. G. Bergmann PART II REVERBERATION Introduction . See ae Theory of Reverberation Intensity Experimental Procedures Deep-Water Reverberation . Shallow-Water Reverberation . : Variability and Frequency Characteristics . Summary . PAGE 137 158 173 192 236 247 250 272 281 308 324 334 CONTENTS CHAPTER 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 PART III REFLECTION OF SOUND FROM SUBMARINES AND SURFACE VESSELS Introduction . Principles . Theory Direct Measurement Techniques . Indirect Measurement, Techniques Submarine Target Strengths Surface Vessel Target Strengths Summary . PART IV ACOUSTIC PROPERTIES OF WAKES Introduction . SERENE. Fol Monson ie Formation and Dissolution of Air Bubbles Acoustic Theory of Bubbles Velocity and Temperature Structure . Technique of Wake Measurements Wake Geometry ACs Sar Observed Transmission Through Wakes Observations of Wake Echoes . Role of Bubbles in Acoustic Wakes Summary . Bibliography . Contract Numbers . Service Project Numbers Index PAGE 343 345 352 363 379 388 422 434 44] 449 460 478 484 494 503 512 533 041 547 507 508 559 PART I TRANSMISSION Beweties. WTigyan twit Meh Soria) b ire cme Tigh aves bi obese) ab aay a Ue iim: a | ae We Weriarsdin! ae Chapter 1 INTRODUCTION 1.1 IMPORTANCE OF TRANSMISSION STUDIES Sx SOUND WAVES are transmitted through water very much more readily than radio and light waves, the use of underwater sound has become a basic part of subsurface warfare. There are always many different ways in which equipment can be de- signed and used. An intelligent choice between the different alternatives depends on accurate knowledge of the different factors affecting final performance. One of these factors is the extent to which sound is weakened in passing from one point to another; this weakening is called transmission loss. The present volume summarizes the information available in 1945 on transmission loss of underwater sound.* Much of the detailed discussion refers to a sound frequency of 24 ke since this is the frequency most commonly used in practical echo ranging, and most of the available data are at that frequency. This information, although incomplete, is useful in a variety of ways. In particular, it is helpful both in the design of gear and in the development of opera- tional doctrine. It is evident that the intelligent design of new equipment requires reliable information on under- water sound transmission as well as on a variety of other factors. For example, the choice of frequency in any device usually involves a compromise between high frequency for the sake of directivity and low frequency for the sake of good transmission. It is possible to arrive at a suitable compromise by trial- and-error methods. However, the choice is made more quickly if routine methods can be used to predict the transmission loss at each frequency, the directivity, » This volume includes primarily those data applicable in the frequency range above 200 cycles. Sound of lower fre- quencies has not been used in sonar equipment and its trans- mission has not been investigated by Division 6 of the NDRC, except occasionally in connection with the transmission of explosive pulses. and other factors, such as the noise level, which affect performance. These different predictions can then be combined to find which frequency gives the best results. The optimum frequency will, of course, de- pend on the purpose for which the equipment is de- signed, and on the limitations of size, available power, and other characteristics. Thus, in some types of echo-ranging equipment, low-frequency gear with a wide beam pattern and a long maximum range is used in searching for submarines, but tilting high-fre- quency gear is provided for tracking a submarine at close range during an attack. The development of operational doctrine for the gear already in use also depends on the results of transmission studies largely because of the wide variability of underwater sound transmission. If a pulse of sound is sent into the water and received near the surface 3,000 yd away, the signal energy received will sometimes be only a millionth of the signal energy received at other times. This enormous variation is due mainly to changes in the vertical temperature gradients present in the water. These changes have a direct effect on the maximum range at which sub- marines can be detected by echo-ranging gear. When the maximum range is known to be short, the gear can be operated most effectively with a short keying interval, since more rapid keying increases the chances of finding a submarine which happens to be within the maximum range. If a long keying interval were used under these conditions, time would be wasted in listening for echoes during periods when no echoes would be possible. Information on the change of sound transmission conditions with changing temperature conditions is useful in the choice of antisubmarine tactics as well as in the selection of rules for operating the sonar gear. When the transmission loss of sound is high and the maximum range of sonar gear is short, the spacing between surface vessels conducting an antisubmarine hunt must be reduced. Sharp temperature gradients at considerable depths may weaken sound passing 3 4 INTRODUCTION through them, and reduce the maximum range on a deep submarine to much less than the maximum range on the same submarine at periscope depth. The maximum range at which two surface ships can ob- tain echoes from each other gives, by itself, no infor- mation on the maximum range that can be expected on a deep submarine. Thus, use of the bathythermo- graph is required to estimate the approximate maxi- mum range obtainable on a submarine at evasive depths. Such an estimate is useful not only in the choice of spacing between antisubmarine vessels but also in evaluating the desirability of detaching escort vessels from a convoy to hunt a submarine reported sighted some distance away. When sound conditions are good, detaching antisubmarine vessels is less likely to endanger the convoy and more likely to sink the submarine than when sound conditions are bad. Information on sound transmission conditions is also useful in the choice of submarine tactics. A sub- mariner is free to choose his depth of operation, and one of the factors influencing this choice is the maxi- mum range at which he is likely to be detected at each depth. In any case, the behavior of the sub- marine may be influenced by knowledge of the maxi- mum range at which detection may be expected. At times, transmission conditions are so severe that the submarine cannot be detected even at 500 yd; such conditions, if they can be readily and reliably identi- fied, provide opportunity for unusually aggressive action. 1.2 NATURE OF SOUND Historically, the various types of physical phe- nomena were first defined in terms of the human senses. Physics was divided into the fields of (1) me- chanics (dealing with touch and displacements ef- fected by human muscle power), (2) light (dealing with the perception of objects by the eye), (3) sound (pertaining to hearing), (4) heat (dealing with the sensations of heat and cold), and other similar fields. Gradually, as the causes of the nerve stimuli became understood, the subject. matter of physics was re- grouped; classification in accordance with physiologi- cal perception was gradually replaced by classifica- tion according to the physical nature of the phenom- ena studied. Thus, optics became more and more a subdivision of the theory of electricity and magnet- ism, while heat and sound came to be treated as sub- divisions of mechanics. The theory of heat is con- cerned with random motions of many particles. In contrast, sound is concerned with the formation and propagation of vibrations, primarily in a fluid,» at frequencies both within and above the range of audi- bility. This definition is purely arbitrary, dictated by practical considerations, and may be ambiguous under certain circumstances. Nevertheless, it is gen- erally accepted. The physics of sound is usually called acoustics. Although a major part of the work in acoustics deals with sound perceptible by the human ear (the acous- tics of rooms, the physiology of sound, and similar subjects), inaudible sound, consisting of mechanical vibrations above the range of frequencies perceived by the ear, has come to play an important role in subsurface warfare. In this volume on the proper- ties of sound in the ocean, more than half of the dis- cussion will be devoted to the propagation of super- sonic sound, that is, sound at frequencies well above those which can be heard. 1.2.1. Sound as Mechanical Energy It must be understood that sound energy is a form of mechanical energy. The particles of a fluid in which sound is traveling are set in motion and tem- porary stresses are produced which increase and de- crease during each vibration. The motion of the indi- vidual particles gives the fluid kinetic energy while the stresses induce potential energy. In acoustics, the sum of these two kinds of energy is called sound energy or acoustic energy. It is not always easy to separate the acoustic energy from other forms of mechanical energy possessed by the fluid. A fluid obtains acoustic energy by some kind of energy transformation. As an illustration, consider a tuning fork in air. When this tuning fork is struck with a rubber hammer, its two prongs are set in rhythmic vibratory motion. The vibrating prongs of the tuning fork produce compressions and rarefac- tions in the surrounding air by pushing the adjacent air mass away and then permitting it to rush back. These alternating compressions and rarefactions are propagated through the air and may be detected as sound by a suitable instrument, such as the human ear or a microphone. The original source of energy was the rubber hammer, which had kinetic energy of translation. This energy was transformed, by means of a collision, to vibratory energy in the tuning fork, > The term fluid, as used in physics and chemistry, means any liquid or gaseous substance. Thus air and water are fluids, but steel is not. NATURE OF SOUND 5 which was communicated to the air as acoustic energy. The foregoing example illustrates the general proc- ess by which sound is generated and detected. A source of sound converts mechanical or electrical energy into energy of vibration and communicates this energy to the surrounding medium as acoustic energy. This acoustic energy travels through the medium to the receiving instrument where it is de- tected. 1.2.2 Production and Reception of Sound Most types of sonar gear produce sound by con- verting electrical energy into acoustic energy and de- tect sound by converting acoustic energy into elec- trical energy. They do this by making use of one of two effects, magnetostriction or the piezoelectric effect. When certain metals, such as nickel, are placed in a magnetic field, they contract (or expand) in the direc- tion of the field; conversely, when they are subjected to a contracting (or expanding) force they become partially magnetized. Thus, if a nickel rod is made the core of a solenoid and if it is given a permanent magnetization by means of a direct current, then an alternating current passed through the winding will cause the magnetization to increase and decrease with the frequency of the current. As a result, the rod will contract and expand or, in other words, vibrate with the frequency of the impressed current. In this arrangement, electrical energy is converted into acoustic energy which is passed into the surrounding medium. Conversely, if a sound wave hits this instru- ment and causes the nickel rod to alternately expand and contract, the rod will be magnetized and demag- netized rhythmically, thus inducing an electromotive force in the surrounding solenoid. The resulting alter- nating current may be amplified and ultimately re- corded in one form or another. Such a magnetostric- tion transducer may thus be used both as a source of sound and as a receiver of sound. Certain crystals, such as quartz, Rochelle salt, and ammonium dihydrogen phosphate, exhibit the piezo- electric effect. If a slice is cut from such a crystal and if an electric potential difference is applied across such a slice, the crystal will either contract or expand, depending on which of the two faces is electrically positive. Conversely, if such a slice is compressed or expanded mechanically, the two opposite faces will develop a potential difference. Thus, a piezoelectric crystal, or an array of such crystals, may be used as a transducer. If an alternating voltage is applied to the opposite sides of the crystal slice, it will vibrate with the frequency of the applied voltage; and if it is placed in a fluid where the pressure is fluctuating, it will develop a fluctuating emf across its faces. Other important sources of waterborne sound are underwater explosions, ships, submarines, waves, underwater ordnance, and biological sources. Wo Propagation of Sound Chapters 1 through 10 are concerned with the prop- agation of sound in the ocean. The complexity of this problem is due to the great variability of the mechani- cal properties of the medium in which the propaga- tion takes place, but the basic underlying physical concepts are fairly simple. These principles are dis- cussed in the following sections. DIRECTION OF PROPAGATION Sound energy is propagated away from the source into a medium. If a single pulse of sound is considered, such as that produced by a sudden explosion, the course of the sound energy in the medium can be fol- lowed by placing a large number of recording micro- phones in the general vicinity and by noting the times at which they show the first response. Each will respond at a slightly different time. Some, placed be- hind obstructions, may not respond at all. Byusing a sufficiently large number of such micro- phones, we can record all those points in space which are reached by the spreading sound pulse at the same time. We shall call the surface on which these points are located a sound front (a better expression will be introduced later). The progression of the pulse in space may then be described by a succession of sound fronts along with the statement of the time at which each front is activated. If the medium of propagation is homogeneous, the perpendicular distance between two sound fronts is proportional to the time it takes the sound pulse to travel from one to the other. In other words, in a homogeneous medium sound travels at a constant speed in a direction perpendicular to the sound front. This direction is called the direction of propagation. These simple rules apply only if the sound beam meets no obstructions. If an obstruction is placed be- tween source and microphone, the microphone usu- ally registers some sound, but with a delay indicating 6 INTRODUCTION that the sound pulse had to travel “around the cor- ner’ to reach the microphone. In that case, sound energy is obviously deflected around the obstruction; and it can be shown that this energy does not travel everywhere in a direction normal to the sound front. A rigorous treatment of these more involved cases shows, nevertheless, that a direction of propagation always can be defined in a natural and unique man- ner. INTENSITY OF THE SOUND FIELD Sound is weakened as it travels and at very great distances from the sound source cannot be detected. We specify the strength of the sound by its intensity. Sound intensity is defined as the rate at which sound energy passes through an area 1 centimeter square placed squarely in the path of the traveling sound. In theoretical studies sound intensity is usually expressed in units of ergsper squarecentimeter. In en- gineering work, on tke other hand, it is usually more practical to express intensities on a logarithmic scale both because of the very wide range of sound inten- sities in practice and because sound intensities are frequently the product of several factors. Use of the logarithmic scale narrows down the numerical range between very faint and very loud sounds and also simplifies the computation of many sound intensities by replacing multiplication by addition. The logarithmic scale in general use is the decibel scale. This scale may be explained as follows. Sup- pose we want to compare two sound intensities J; and I;. To find the decibel difference between J, and Iz, the common logarithm (base 10) of the ratio I,/I2 is multiplied by 10. As an example, suppose the inten- sity J; is 1,000,000 times the intensity J. The loga- rithm of 1,000,000, multiplied by 10, is 60. Thus the intensity J; is 60 db above the intensity J2. In many studies it is the decibel difference between two dif- ferent sounds rather than the absolute strength of any one sound, which is of most interest and can be most readily determined. The decibel scale is also suitable for expressing ab- solute sound intensities. For this purpose, a standard intensity is first selected, called the reference intensity or reference level, and then all other sound field in- tensities are expressed in terms of decibels above (or below) the standard. Unfortunately, different stand- ards have been used by different groups in under- water sound research. Sometimes, 10—* watt per sq cm has been used as the standard since this is the usu- ally accepted standard in air. More frequently, the reference level has been expressed in terms of the sound pressure. Since sound represents vibrations and since vibra- tions of a fluid (such as air or sea water) are associated with periodic changes in the local pressure, the devia- tion of instantaneous pressure from the hydrostatic or atmospheric pressure may be used as a measure of sound intensity. This excess pressure oscillates dur- ing each cycle; therefore, the intensity must be ex- pressed in terms of some averaged quantity. Since the excess pressure is positive during one half of the cycle and negative during the other, its arithmetic mean vanishes. It is possible to obtain a nonvanishing average quantity by considering the rms excess pres- sure. In the case of a sinusoidal vibration, the rms excess pressure is equal to 1/+/2, or 0.7 times the maximum value of the excess pressure. It will be shown in Chapter 2 that in a given medium the sound intensity is proportional to the mean square excess pressure. Two standards based on pressure have been used in underwater sound studies. One is a sound in- tensity corresponding to an rms excess pressure of 0.0002 dyne per sq cm. This standard has been re- cently replaced by that of an intensity corresponding to an rms excess pressure of 1 dyne per sq cm. When sound field intensities are expressed on a decibel scale relative to some standard intensity, they are usually referred to as sound levels. 1.3 PROPAGATION OF SOUND IN THE SEA When the propagation of sound in the sea first be- came a matter of prime military importance, it was hoped and expected that sound would travel along straight lines from the source and that the sound field intensity would decrease in accordance with the simple inverse square law. However, this hope was not realized. Because of the peculiar characteristics of the ocean as a sound-transmitting medium, marked deviations occur from both straight-line propagation and inverse square intensity decay. Straight-line propagation of sound is to be expected only if the velocity of propagation is constant throughout the medium. In the ocean this condition is usually violated primarily because of the variation of temperature with depth. There is almost always a layer in which the water temperature drops ap- preciably with increasing depth. This layer may begin right at the sea surface, or it may lie beneath a top layer of constant temperature. In such a region of PROPAGATION temperature change, the sound paths are bent in the direction of lower velocity of propagation, in other words, in the direction of lower temperature. Even though the changes in sound velocity are small (about 1 per cent for a temperature drop of 10 F), the result- ant bending of the sound path becomes appreciable over a distance of a few hundred yards. If, for in- stance, the drop in temperature begins directly at the surface of the water, and totals a degree or more in 80 ft of depth, most of the sound energy will travel along paths bent downward and will miss a shallow target at a range of 1,000 yd. Because of this bending of sound by temperature gradients, some departure of sound intensity from the inverse square law is to be expected. The amount of this departure can be calculated if the temperature distribution in the ocean is known. However, even this more complicated process for computing the in- tensity is too simple. The effects of the boundaries of the medium (ocean bottom and surface), and of the absorption and scattering of sound in the body of the ocean must, also, be considered. Both the sea surface and the sea bottom affect the sound field intensity. Some of the sound energy strikes these boundaries and is then partly reflected back into the ocean, partly permitted to pass into the adjoining medium (air or sea bottom). The portion of the energy which is reflected will return into the in- terior in a variety of directions. Also, little under- OF SOUND IN THE SEA 7 stood processes in the body of the ocean affect sound intensity. In some way, a certain amount of the pass- ing sound energy is converted into heat (absorption of sound); and chance impurities such as fish, sea- weed, plankton, and gas bubbles, tend to scatter a small amount of the passing sound energy in all direc- tions out of its principal path. For all these reasons, the propagation of under- water sound presents, at first, a rather confusing picture. Considerable progress has been made, how- ever, in understanding the behavior of underwater sound and in utilizing this partial understanding in the design and tactical use of sound gear. The results which have been achieved are due to a combination of theoretical and experimental investigations. Chap- ters 2 through 10 discuss the background and progress of these investigations. Chapters 2 and 3 lay the theoretical groundwork for the physics of underwater sound. Chapter 4 leads toward the experimental re- sults by reporting on the equipment and procedures employed in the experiments. Chapters 5 and 6 re- port experimental results on the propagation of sound, primarily sound generated by transducers. Chapter 7 is concerned with the observed short-term fluctuation of underwater sound intensity. Chapters 8 and 9 deal with the formation and transmission of explosive sound. Finally, Chapter 10 summarizes the results obtained to date and discusses possibilities for future research. Chapter 2 WAVE ACOUSTICS Goon ENERGY takes the form of disturbances of the pressure and density of some medium. There- fore, the basic relationships between impressed forces and resulting changes in pressure and density are use- ful in an understanding of sound transmission. In this chapter we shall derive several such relationships, and shall combine them into one differential equation relating the time derivatives and space derivatives of the pressure changes to several physical constants of the medium itself. This differential equation is the foundation for the mathematical treatment of sound transmission to which the rest of the chapter is devoted. We shall see that this mathematical approach can- not in itself furnish complete information on sound transmission in the ocean. The physical picture must necessarily be simplified to make mathematical de- scription possible — and even this simplified scheme does not yield explicit results for the sound intensity in all cases. However, it is valuable to know the mathematical theories even if they are partially un- successful in predicting the qualities of sound trans- mission. Tendencies predicted by a simplified theory are often verified qualitatively in practice. Also, there is always the hope that by changes and ampli- fications an incomplete theory can be made much more useful. 2.1 BASIC EQUATIONS In this section we shall derive the basic equations which will be put together to derive the fundamental differential equation of wave propagation, the wave equation. These equations are (1) the equation of continuity, which is the mathematical expression of the law of conservation of mass; (2) the equations of motion, which are merely Newton’s second law ap- plied to the small particles of a disturbed fluid; (3) force equations, which relate the fluid pressure inside a small volume of the fluid to the external forces acting on the periphery of the volume; (4) the equation of state, which relates the pressure changes inside a fluid to the density changes. 8 PeTlell Equation of Continuity The equation of continuity is simply a mathemati- cal statement of the law that no disturbance of a fluid can cause mass to be either created or destroyed. In particular, any difference between the amounts of fluid entering and leaving a region must be accom- panied by a corresponding change in the fluid density in the region. To express this law in mathematical terms we must first derive an expression for the mass of fluid which passes through a certain small area of a surface in one second. Let the small surface element have the area A, asin Figure 1, and let the fluid move in a direction t30 t= SSS Figure 1. Passage of fluid through area element A. perpendicular to A with the velocity u. In one second, a rectangular fluid element of base A and height u has passed through this element of area; that is, a volume of Au cubic units of fluid has traversed the area. The mass of fluid passing through the area per second will thus be pAw, where p is the density at the point and time in question. If the fluid is moving not perpen- dicular to the element A, but in some other direction, the mass passing A per second will still be given by pAu, if wis taken to be the velocity component in the direction perpendicular to A. Now consider a small hypothetical box-shaped volume inside the fluid, and examine the amounts of fluid entering and leaving this box (pictured in Figure 2). For simplicity, we can assume that the edges of the box are parallel to the coordinate axes. Let the dimensions of the box be 1.,l,,/., as shown in the diagram, and let the coordinates of the point H be (x,y,z). Let the components of the fluid velocity at the point H be u2,ty,Uz. BASIC EQUATIONS 9 The mass of fluid entering the face AHED in unit time is clearly the rate at which mass is moving in the x direction times the area of AHED, or puzl,l.. The mass of fluid leaving the box through BCGF is a A B E Fieure 2. Infinitesimal cube of fluid. similar expression, but with p and u, measured at (x + 1,,y,2). The value of pu, at (x + I,,y,z) is just its value at (x,y,z) increased by 1,0(puz)/dzx since I, is very small. That is, the mass leaving in one second through face BCGF is E a = (ou fae Then the net increase per second in the mass inside the box caused by the flow through the two faces perpendicular to the z axis is Co) = ay rue) Lelyle. Similarly, the net increase per second caused by the flow through the two faces perpendicular to the y axis is te) ae ay reall, and through the two faces perpendicular to the z axis, () — —(puz)lzl,l.. aa? ely The total time rate of increase in the mass con- tained in the box is simply the sum of these three quantities, or A(puz) _ A(puy) ats = ES aL ane omeea [ollil (1) Since no mass can be created or destroyed inside the box, this rate of deposit of mass must result in a corresponding change in the average density p inside the box. That is, a _ _ [A(pue) , A(pwy) ta | pclae] = | Ag ttl Mayenne renee Canceling out the constant factor I,l,1., we obtain the general equation of continuity OD we. EE 4. (ou) a @) ot Ox oy 0z This equation can be simplified if it is assumed that all displacements and changes of density are so small that second-order and higher products of them can be neglected. The actual density p, then, will not be very different from the constant equilibrium den- sity po. If o is defined by Chiara (3) Po then c, the fractional change in density caused by the displacement of the fluid from equilibrium, will be a very small number. Henceforth o will be called the fractional density change or condensation. With this understanding, it is clear that ) (pus) ) ax (0 + por) Uz | Ox Bee OUz = nO) = Pr since the second-order product cu, can be neglected. By substituting this value of 0(puz)/dx and similar expressions for 0(pu,)/dy and 0(pu,)/dz into equation (2), the following s¢mplified equation of continuity results: Oo = oy ae (4) Ox oy 0z Equation (4) is the form of the equation of con- tinuity which will be used in the derivation of the wave equation (27). For later reference, we shall note what happens to the volume occupied by an infinitesimal mass of the fluid when the fluid is given a small displacement from equilibrium. If 7 is the volume occupied by the small mass at equilibrium, and »v is the volume at time ¢, then a fractional volume change w can be de- fined by ig lames Los (5) 3) Uo 10 WAVE ACOUSTICS From equations (3) and (5), p = po(l +0), (6) vy = »(1 +). (7) Since the masses at equilibrium and at time ¢ are equal, (8) pv = poo. By combining equations (6), (7), and (8) (1+o)1+.6) =1. The product wo, a second-order term, can be neg- lected, giving @) — (5 (9) That is, under the assumption of small displacements and small density changes, the fractional volume change w is the negative of the fractional density change o. 2nle2 Equations of Motion In this section, we shall apply Newton’s second law of motion to the mass of fluid within the volume ele- ment v. This law states that the product of the mass of a particle by its acceleration in any direction is equal to the force acting on the particle in that direc- tion. Given the velocity distribution within the fluid as a function of the position coordinates and time, Uz = Uz(2,Y,2,t), etc.; (10) then the distribution of acceleration within the fluid is to be calculated, a, = a,(z,y,z,t), etc. (11) We cannot immediately say that a, = du-,/dt. For, in order to calculate the acceleration at a particular point and a particular time, we must focus attention on one particular particle. At the end of a time incre- ment dt, the particle has moved to a point (x + dz, y + dy,z + dz), where it has the velocity component uz(x + dz, y + dy, z + dz). The difference between its new velocity and its original velocity, divided by the time interval dt, gives the desired acceleration component du,/dt. This value is not exactly the same as the simple partial derivative du./dt, because the latter does not focus attention on the change of velocity of a single particle, but instead compares the velocity of a particle at the point (x,y,z) and time ¢ with the velocity of the particle which occupies the position (x,y,z) at the end of the time interval df. However, du,/dt and du,/dt will be almost equal under the assumptions that second-order products of displacements, particle velocities, and pressure changes are negligible. To show this, we note that Ouz dt —at gus dx dt Ou, dz dz dt’ OUz dy dy dt (12) which is the usual equation found in calculus texts re- lating the partial and total time derivatives of a function. The last three terms are second-order prod- ucts, since dz/dt, dy/dt, dz/dt are merely wz,Uy,Uz. Thus, the component of acceleration in the x direc- tion may be approximated by du,/dt, and similarly for a, and a,. That is, Ouz Ouz , = a, = ot ot ot az = (13) The mass of fluid within the volume element 1 is pvo. If F,,F,,/, are the components of the forces acting on the element, then the equations governing the motion of this element are, in view of equation (13), t) Ouz pu —; F,= pvo—- (14) Ou; PNR tot at ot It is desirable to make the equations governing the motion of the small element independent of the par- ticular value of the small volume v. For this reason, we rewrite equations (14) as OUz OuUy Ouz = = — 5 = rae z — 15 fz =p Fy fu =p y f-=p ae (15) where F, F F, fz = —; pes f= Vo Vo Vo The normalized force components may be regarded as the force components per unit volume acting on the small volume element. The next section is concerned with calculating Jz,fy,f2 in terms of the pressure or density changes oc- curring within the fluid. 2.1.3 Law of Forces in a Perfect Fluid A fluid is called perfect if the forces in its interior are solely forces of compression and expansion, in other words, if the fluid is incapable of shear stress. If a fluid is perfect, the force on any portion of its surface is perpendicular to the surface. Fluids which can exhibit shear stress in response to shear deforma- tion, in addition to responding to compressive and expansive forces, are called viscous. BASIC EQUATIONS 11 Before the equations of motion (15) can be used, expressions must be derived for the force components tz, fy, fz acting on the small box of Figure 2. Accord- ingly, we shall calculate these forces under the as- sumption that the fluid is perfect. According to this assumption, the box will move in the z direction if and only if the pressure on face ADEH is different from the pressure on face BCFG. Similarly, it will move in the z direction only if the pressures on faces ABCD and HFGH are unequal. Motion in the y (vertical) direction is not quite so simple because of the hydrostatic, or gravity-produced pressure dif- ferences, which do not of themselves cause motion. The box will move in the y direction if and only if the pressure on face DCFE is not exactly equal to the pressure on face ABGH plus the total weight of the box. If the corrected pressure p(,y,z,t) is defined as the total pressure P(x,y,z,t) minus the hydrostatic pressure at the point («,y,z) when the fluid is at equilibrium, then the criterion for motion in the y direction may be restated as follows. Motion will oc- cur in the y direction if and only if the corrected pres- sure at face A BGH differs from the corrected pressure at face DCFE. We shall have little occasion to use the total pressure P since the hydrostatic pressures are seldom important in sound propagation. ap 7°" ax 4, Ficure 3. Pressure on opposite faces of infinitesimal fluid element. Figure 3 is a duplication of the box of Figure 2 showing the forces acting in the x direction. If the pressure at the left-hand surface is p, the total force on that surface is pl,l.. The pressure at the right-hand surface is clearly p + (dp/dz)l.; and the total force on that surface is therefore [p + (dp/0zx) llzl,l.. Since the fluid is assumed to be perfect, these forces are parallel and their resultant can be obtained by simple subtraction. Thus, the total force on the volume 1% in the x direction is given by fo) Fe = fy = —— ldap (16) Ox Thus, the force per unit volume in the z direction f, is given by 0 fo=—2 (16a) Ox Similarly, dp =-—— 1 ia (16D) dp Phe re cee 1 hese (16c) From equations (15) and (16a, b, c) we obtain Op OUz Op OU, Op uz => =S= — 5 = = pasty ed . 1 Or Mente Pony alow aMarote sae Pape Ue) 2.1.4 Equation of State Our aim is to derive a differential equation which will relate certain properties of the disturbed fluid (pressure changes, density changes) to the independ- ent variables x,y,z,t. For effective use, this differen- tial equation should contain only one dependent variable. The basic equations derived up to this point — (4), (15), and (16) — contain the dependent vari- ables o, p, p, Uz, Uy, Uz. ¢ and p are one variable since they are related by equation (8). It will be seen in Section 2.2.1 that the velocity components can be easily eliminated by use of the equation of con- tinuity (4). However, we must consider the relation- ship between density and pressure before we obtain a differential equation for the propagation of sound. Such a relationship between density and pressure is obtained from the equation of state of the fluid. The equation of state of any fluid* is that equation which describes the pressure of the fluid as a function of its density and its temperature, iD (aH). This function P(,7) must be determined experi- mentally for each fluid separately. In the case of sea water, it depends on the percentage of dissolved salts. A relation between pressure and density is obtained from the equation of state by making two assump- tions. First, it is assumed that a passing sound wave ®We shall follow the common usage of physicists and use the term fluids to denote both liquids and gases. 12 WAVE ACOUSTICS causes the fluid to deviate so slightly from its state of equilibrium that the change in pressure is propor- tional to the fractional change in density. Second, it is assumed that the changes caused by the passing of the sound wave take place so rapidly that there is practically no conduction of heat. We shall denote the fractional change in density as heretofore by oc; the change in pressure will be called excess pressure and denoted by p. Thus we assume that the fractional change in density and the excess pressure caused by the passing sound wave are both small and that they are propor- tional to each other: (18) The constant of proportionality « is called the bulk modulus. It depends not only on the chemical nature of the fluid (such as the concentration of salts in the sea), but also on the equilibrium temperature T, the equilibrium pressure Py, and the equilibrium density p. The temperature in the ocean varies from point to point, usually decreasing with increasing depth. The equilibrium pressure increases rapidly with the depth, and the density increases very slightly with depth. As a result, the bulk modulus « is itself a function of all three coordinates z, y, and z, although its greatest changes take place in a vertical direction. To the ex- tent that the temperature distribution of the ocean is subject to diurnal and seasonal changes, « is also a function of the time ¢. However, in the following sections we shall usually simplify matters by disre- garding these variations in space and time and by treating x as a constant. p = ko. 2.2 WAVE EQUATION IN A PERFECT FLUID 2.2.1 Derivation If in a certain region of a fluid in equilibrium the pressure is changed from its equilibrium value, the fluid immediately produces forces which aim toward restoring the equilibrium state. Vibrations result, which are propagated as waves through the fluid. These waves are sound waves, and the fundamental differential equation governing their propagation will now be developed by using the basic equations de- rived in the preceding sections. The particular equa- tions used are the equation of continuity (4), the equations of motion (15), the law of forces (16), and the equation of state (18). From equation (18) we have Op 0a Op 0a Eg ea RO «| aca Og Oz = ox’”—Sss OY “Oy ” dz By putting these values for dc/dz, d0/dy, and 0c/0z in the law of forces (16) we obtain 0a 0a 0a fz = maaee f= aah f= SRD After these values for the components of the force on a small box are substituted into the equations of motion (15), we obtain the following relations. (19) OUz e 0c Pat ax OUy 0c Oras Wey (20) Ouz i 0a En PES Since we assume that density changes and velocity changes are all comparatively small, the expressions pdu;/dt and p,duz/dt will differ by a second-order term, and hence can be regarded as equal. Then equations (20) become Beg 0 Can SU OU, 0a pect = = (0) Coy TEES (21) due | 25 _y Lane Ro vais In order to apply the equation of continuity (4), we differentiate the first equation of (21) with respect to x, the second with respect to y, and the third with respect to z. The equations are added, leading to Ofdu, . du, Ouz cr Aa rol ee 2 a J+ PSS 0t\ 0x =) Oy a2 dx? ay? a eR) ~ ie (22) From the equation of continuity, the first parenthesis is —do/dt; and equation (22) reduces to a v: 2 hes =) dP pNast ay? | a] Since ¢ = p/kx, from equation (18), where x is con- stant, equation (23) implies ey , #9) Op Kk ( oy? = az? (23) 02 pp\ dz? ee) WAVE EQUATION IN A PERFECT FLUID 13 It can be shown that the velocity components Uz,Uy,Uz also satisfy a differential equation of the form of (23), provided the motion of the disturbed fluid is irrotational. That is, if sound is propagated in a perfect fluid in such a manner that no eddies are produced, ip =) 02? Pur (= Ox? with similar equations for u, and w;. Equations (23), (24), and (25) are equivalent; that is, the fundamental laws of sound propagation can be deduced from any one of them by using the known relationship between o, wz, and p. In the following sections, the most frequent reference is to equation (24). Variation in the excess pressure is the most familiar and probably the most intuitive change in the disturbed fluid ; also, the majority of hydrophones used in the reception of underwater sound respond directly to variations in excess pressure rather than to variations in particle velocity or condensation. It is convenient, in equations (23) to (25), to set Uz (25) ot? Po oy” @=s=, (26) Po so that the wave equation becomes 0p = 0p = — = ce| — + —4— }]. 27 BEM Noun Gy tne Ce It will be pointed out in Section 2.3.1 that c, defined by equation (26), has the general significance of sound velocity. The wave equation (27) gives the relationship be- tween the time derivatives and the space derivatives of the pressure in the fluid through which the sound is passing. Relationships of this sort have been used by generations of mathematicians as a starting point for the development of physical theory. In the field of sound, these mathematicians have explored the methods by which the future course of pressure in a fluid can be calculated if only the initial distribution of pressure is given. Mathematically, this amounts to solving the wave equation (27) with given initial and boundary conditions. Once the distribution of pres- sure is known, the sound intensity at any point and time can be calculated by the methods of acoustics. 2.2.2 Initial and Boundary Conditions The differential equation of a physical process gives a dynamical description of the process relating the various temporal and spatial rates of change, but does not of itself tell all we want to know. In the case of the wave equation, we desire knowledge of how the ex- cess pressure varies in space and time. This informa- tion is obtainable, not from the wave equation itself, but from its mathematical solution. The general solu- tion of a partial differential equation like the wave equation always contains arbitrary constants and even arbitrary functions. These arbitrary constants and functions are, in any individual problem, ad- justed to make the solution fit the special condi- tions of the problem. These special conditions are of two kinds: boundary conditions and initial conditions. In the problem of sound propagation, the two types of conditions can be defined as follows. Boundary conditions are fixed by the geometry of the medium itself. If the medium is finite, boundary conditions must always be con- sidered. The excess pressure must fulfill certain con- ditions at a boundary such as the sea surface, sea bottom, or internal obstacle. The pressure may have to be zero at one boundary, or a maximum at some other boundary, or may satisfy some other condi- tion.> Initial conditions are concerned not with the fixed geometry of the fluid and its surroundings, but with the special disturbances which cause sound to be propagated. One type of initial conditions specifies the pressure distribution at a certain instant of time, t = t, over the whole fluid. That is, we are given a function p(z,y,z), and are told that p(x,Y,2,to) = p(2,Y,2). (28) Another type of initial conditions specifies the pres- sure as a function of time at a fixed point (£0,Y0,Z0) of the fluid. That is, we are given a function p(t) and are told that P(x0,Yo,20,t) = p(t). (29) Every actual case of sound propagation involves both initial conditions and boundary conditions. However, in our mathematical approximations to reality it is best to start with the simplest case, that is, where sound is propagated into a medium that is infinite. Of course, in theory every problem can be regarded as a problem in an infinite medium. We can consider the sea and air together as one medium, whose physical properties at equilibrium (elasticity, bTt will be seen in Section 2.6.1 that the excess pressure must be nearly zero at the boundary separating water and air and a maximum at the solid bottom of the sea. 14 WAVE density, and other properties) suffer a sharp change at the separating surface. However, the mathematical treatment of a medium with strongly variable physi- cal properties is still more difficult than the treatment of boundary conditions. We are free to schematize the physical situation in the most convenient way. Accordingly, we shall start with the consideration of an infinite medium, where the elasticity and density at equilibrium are not strongly variable, and shall later specialize our treatment by the consideration of boundary conditions (Sections 2.6 and 2.7). Initial conditions must always be considered since without them no sound could possibly originate. Un- less the initial conditions are themselves very simple, the solution of problems even in an infinite medium is quite involved mathematically. The most practical procedure is to first find solutions under very simple initial conditions and use these solutions as building blocks for constructing solutions of problems with more complicated initial conditions. 2.3 SIMPLE TYPES OF SOUND WAVES 2.3.1 Plane Waves It is convenient to start our study of the solution of the wave equation with the assumption that the disturbance is propagated in layers. We assume that at any time ¢ the excess pressure p is a function of x only; that is, p is independent of y and z. With this understanding, the wave equation (27) reduces to 0p 0p ae SS ae” da? eo) The eighteenth century mathematicians knew that if p is an arbitrary function of either (t — x/c) or (¢ + x/c), or a sum of two such functions, then p satisfies equation (30). The proof is easy. Assume that p = f(t — z/c) where f is any function. Then,° op ( =) ep 1 ( | et = ; il ste RA SS |I\6 00? f Cc Ox? ol Cc The proof is identical for a function of the argument (t + x/c). The sum of two solutions will itself be a solution because of the general theorem that the sum of two solutions of a homogeneous linear differential equation will also satisfy the equation. ° In accordance with usual calculus notations, f’’(t — z/c) represents the second derivative of f(z) evaluated for z = t —2/c. ACOUSTICS Also, it can be shown that any function of x and t¢ which is not of the form s(t = *) ft nC f 2) C. C. cannot possibly satisfy equation (30). The proof is carried out as follows. Represent the unknown solu- tion of equation (30) in the form Pp = f(é,n), E=2—c, yn =x + ct. If the differential equation (30) is written in terms of the new variables £ and 7, it reduces to afem _ 5 aE On This equation implies that the first derivative df/dé must be a function of £ only and independent of 7, for otherwise the second derivative 0?f/d&dn could not vanish. Thus f itself must have the form f(én) = fil) + fon). Let us focus attention on all the solutions of equa- tion (30) which have the form Ain?) There are an infinite number of such solutions cor- responding to the infinite number of possible choices of f. However, no more than one of these solutions can fit the special conditions of a particular physical situation since the actual pressure at a specified point and specified time can have only one value. Suppose that there is one member of the family of functions, denoted by fi(t — x/c), which satisfies the given initial and boundary conditions. Then a fixed value of (t — x/c), say 4.13, will always be associated with some fixed value of the excess pressure, given by fi(4.13). If fi(4.13) is equal to 0.02, then the excess pressure will be 0.02 at those combinations of time and place where (¢ — x/c) = 4.13, that is, where x =ct — 4.138c. In other words, as the time increases, any fixed value of the excess pressure travels in the positive x direc- tion with the speed c. This result is clearly true no matter what the form of f, or the particular value of the excess pressure. Such a process, in which a given pressure change travels outward through a medium, is referred to as propagation of progressive waves. Similarly, if a function f.(¢ ++ x/c) is the sole mem- ber of the family of functions (31) which satisfies the imposed initial and boundary conditions, progressive waves will be propagated with the speed c in the (31) SIMPLE TYPES OF SOUND WAVES 15 negative x direction. If, however, an expression of the form (31) is the function describing the given physical situation, the situation is more complicated. The re- sulting distribution of pressure will be the mathe- matical resultant of the pressure distributions calcu- lated for fi and f2; and a given value of the excess pressure will no longer be propagated in a ‘ingle direction with the speed c. Discussion of this more complicated type of wave propagation will be de- ferred until Section 2.7. Now we consider two specific examples of the prop- agation of plane waves in an infinite homogeneous medium (no boundary conditions). In the first ex- ample, we assume as an initial condition that the exact pressure distribution is specified at the time instant ¢ = 0 between the planes x = 0 and x = %, by p(x,0) = p(x); and also that the excess pressure is zero at t = O for all values of x less than 0 and greater than x. We assume that this initial disturb- ance gives rise to progressive waves traveling in the positive x direction, that is, that the solution is of the form p = f(¢ — z/c). Then the solution of the wave equation with these conditions must be p(az,t) = p(t — ct) (82) since first, it satisfies the initial conditions p(x,0) = p(x); second, it is a function of (¢ — z/c) and there- fore satisfies the wave equation (30); and third, there can be only one solution to this physical problem. By the results of preceding paragraphs, we know that a given value of the excess pressure will be propagated in the x direction with the speed c. Thus, at the time ¢ the initial disturbance will be duplicated between the planes x = ct and x = x) + ct; and the excess pressure will be zero for z < ctand x > a + ct. The disturbance of the fluid remains of width 2p, remains unchanged in “shape,” and is propagated with the speed c. As another example, we suppose as an initial con- dition that the values of the excess pressure are Specified only for the plane x = 0, but for the total time interval between t = 0 andé = t, by the equa- tion p(0,t) = p(t) ard that the excess pressure at the plane x = 0 is zero for t < 0 and ¢ > t. Here, also, we assume that this disturbance causes progressive waves to be propagated in the positive x direction. Arguing as in the preceding example, the solution of the wave equation (30) with these imposed conditions 18 p(a,t) = AC - 2). (33) The expression (33) differs somewhat in form from equation (32) because the initial conditions are ex- tended in time instead of in space. In this example, it is known that the excess pres- sure will be the same at all combinations of space and time where x — ct = constant. Since x — ct equals zero when x = 0, t = 0, the value of the excess pres- sure corresponding to = 0,¢ = 0, must be assumed by the plane x = ct at the time ¢. Further, since x — ct equals —cty) at t = f), x = 0, the value of the excess pressure corresponding to ¢ = t, x = 0, will be assumed by the plane x = ct — ct) at the time instant ¢t. Thus, at time ¢ the disturbance is confined between the planes x = ct — ct) and x = ct; that is, the region of disturbance is always of width ct) and is propagated along the positive x axis with the speed c. SounpD VELOCITY We have seen that in some simple situations the quantity c may be regarded as the velocity with which the disturbance is propagated in the medium, or more simply, the velocity of sound in the medium. It will be recalled that c was defined in equation (26) by / GS \/—, Po where «x is the bulk modulus and py is the density of the fluid at equilibrium. If the medium is a perfect gas, the relation of the sound velocity to the temperature and pressure can be expressed in a simple formula. The pressure changes produced by sound in a fluid are usually so rapid that they are accomplished without appreciable heat transfer, that is, they are practically adiabatic. For a perfect gas suffering adiabatic pressure changes the bulk modulus is yP, where P is the total pressure and y is the ratio between the specific heat at con- stant pressure and the specific heat at constant volume. For a perfect gas suffering any kind of pres- sure change P = pRT. Thus, the simple result fol- lows that c= VyYRT. Hence, in air at normal pressure, which is not far from a perfect gas, the sound velocity increases with the square root of the absolute temperature. No such relationship can be derived for the velocity of sound in sea water since the pressure, density, and temperature of sea water are not related by any 16 WAVE ACOUSTICS zo Tw z 2) > UH SALINITY 35 PARTS PER THOUSAND j So ow ® ATMOSPHERIC PRESSURE 4 Fu ° On aw 1 rd a o 6 = Ot O 100 200 300 400 500 cn DEPTH IN FEET 85 aT ——] — 1,00 eons sibel el ee a} {SS || TEMPERATURE IN DEGREES F oy t=} I2SS2001F5 A EFFECT OF TEMPERATURE fee as SaaS See Sa Sas eS eS vee RANGE OF SALINITY, E> t = iio es Soe = : S= = SSSe= SS ‘ [pea | aa s oop op S 4 € SS ee ee eee ae SSS SALINITY IN PARTS PER THOUSAND 70 Ss Se B EFFECT OF SALINITY SSS Ssss = eS SS SSS E TEMPERATURE 50 F — a ee | — Sa -—_] SALINITY 35 PARTS PER THOUSAND SS aa Sa a ( > 65—— S455 ee eS water temperature, salinity, and depth. A. Effect of Wis5h=4900 Se a et temperature. B. Effect of salinity. C. Effect of depth. 2 a Se SS Sas Fi [ee ee ed F a RSE SS SSS SSS Sa simple formula. However, tables have been con- E —— SEs structed which show the velocity of sound as a func- 538 a | es eae a . . . G 50 a —— 2 SS =a tion of three variables which can be measured di a r— -&— — rectly: the water temperature, pressure, and salin- oo — ity. Although the relationship is not simple, these — = h iables determi isely both the bulk = three variables determine precisely both the bu == modulus and the density, from which the sound ln velocity can be calculated from equation (26). i l WET TO TU UIE Tau 32 33 x4 35 36 37 SALINITY IN PARTS PER THOUSAND Fiaure 4. Speed of sound in sea water. At 32 F, atmospheric pressure, and normal salinity (34 parts per thousand by weight), the velocity of sound in sea water is about 4,740 ft per sec. Increase of either temperature, pressure, or salinity causes the sound velocity to increase. The increase of sound velocity with temperature is about 8.5 ft per sec per degree F at 32 F, and about 4.0 ft per sec per degree F at 90 F. The increase of sound velocity with water depth, caused by the increase in pressure, is 1.82 ft per sec per 100 ft of depth. In the open ocean for the depths of interest in sonar operations the water temperature is the controlling factor in determining the velocity; since sonar gear is usually operated at SIMPLE TYPES OF SOUND WAVES 17 shallow depths, the pressure changes are relatively unimportant, and salinity changes in the open ocean are usually too small to matter much. Near the mouths of large rivers, however, where fresh water is continuously mixing with ocean water, the variations in sound velocity may be largely controlled by varia- tions in salinity. The quantitative dependence of sound velocity on temperature, pressure, and salinity is summarized in Figures 4 and 5. In Figure 4, obtained from a report by Woods Hole Oceanographic Institution [WHOT] ! the value of the sound velocity at zero depth can be read from the main charts for any given combination of temperature and salinity. This velocity can theh be corrected to the velocity at the actual depth by use of the curve in the small box. Figure 5 gives the per- centage changes in sound velocity caused by specified absolute changes in the three determining variables. It will be shown in Chapter 3 that it is the relative changes in sound velocity which determine whether sound transmission is expected to be good or bad rather than the absolute changes. The direct measurement of sound velocity in the ocean is very difficult. The intuitive method of di- viding distance traveled by time is difficult since sufficiently accurate measurement of distances at sea is usually not feasible. The U. S. Navy Electronics Laboratory at San Diego, formerly the U. 8. Navy Radio and Sound Laboratory [USNRSL], developed an acoustic interferometer for the determination of the wavelength of sound at a point in the ocean;? multiplication of this local wavelength by the known frequency gives the local sound velocity. This instru- ment was developed mainly for the purpose of check- ing whether the temperature changes indicated by the bathythermograph were correlated with the actual changes of sound velocity in the ocean. Good general agreement was observed between the velocity-depth plots obtained with the interferometer or velocity meter and those computed from bathythermograph observations. However, since the bathythermograph cannot followrapid changes in water temperature with the detail possible with the velocity meter, a velocity microstructure was frequently recorded with the meter which deviated as much as 0.1 per cent from the velocity calculated from the simultaneous bathy- thermograph reading. That these deviations were due to the slow response of the bathythermograph rather than to physical factors was verified by correlating the velocity microstructure with the temperature microstructure obtained by a thermocouple recorder. 2.3.2 Harmonic Waves Up to now we have allowed the initial disturbances p(a,y,z) or p(t) to be arbitrary functions. However, most initial disturbances which occur in practice are of a very special type that originate in the elastic vibration of some medium. Such disturbances are pro- duced by small displacements of some parts of the medium from their positions of equilibrium; these displacements in turn produce restoring forces which tend to restore the state of equilibrium. Such restor- ing forces are, in first approximation, proportional to the displacements. It is well known that under such conditions (re- storing forces proportional to displacements) the initial disturbance must be of the form of a harmonic vibration; that is, it must be representable by trigo- nometric functions of the time. In acoustics, such a vibration produces a pure tone of a definite frequency. Since echo-ranging pulses are very nearly pure tones, the importance of a study of harmonic vibrations is obvious. Also, harmonic vibrations are of crucial im- portance because they are the most convenient build- ing stones of the more general solutions of the wave equation (see Section 2.7). Suppose, in the second example under plane waves, that the initial disturbance of the plane z = 0 is a harmonic vibration. That is, p(t) = a cos 2nf(t — e) for values of ¢ between 0 and f. One solution of the plane wave equation (30) under initial conditions p(0,t) = p(t) is always — x since p(t + x/c) satisfies the wave equation and also the imposed initial conditions. Thus, if we restrict our attention to progressive waves traveling in a single direction, the solution of the wave equation with the given initial conditions is a cos anil + ; — .) P or x p = acos anil — ‘A _ .) (34) Clearly, the pressure changes represented by equa- tion (34) are at most a; for that reason, ais called the amplitude of the disturbance. Also, it is clear that at a fixed point of space, p goes through f periods in one second; and so f is called the frequency of the dis- 18 WAVE ACOUSTICS turbance. The quantity ¢ is called the phase constant because it fixes the position of the disturbance in time. Two vibrations of the same frequency and the same ¢ have their zeros simultaneously, also, their maxima. If they have different e’s, one has its zeros a fixed time interval ahead of the other, and we say that there is a phase difference between the vibrations. 2.3.3 Spherical Waves The sound at large distances from an actual source resembles the sound from a point source more closely than it does the sound from an infinite plane. Hence, for some purposes it is more realistic to abandon the assumption of plane waves, and assume instead a point source at the origin which causes the pressure in the surrounding medium to be a function only of the distance r from the origin and of the time ¢. That is, the pressure is given by some function p = p(r,t) (35) and is thus independent of the direction of the line joining the source to the point in question. We shall now show that the wave equation (27) re- duces, for the assumed case of spherical symmetry, to the simple form (37). From simple analytic geometry, Bape pede 2 oe aia ae (36) In order to transform the wave equation, the vari- ables x,y,z must be eliminated, and the variable r in- serted. In order to do this, we must use the relations (36) to calculate 0?p/dx’, 0°p/dy’, and 0*p/dz? in terms of r and the derivatives of p with respect to r. This is done as follows. Op _opodor odpx ox Ordx because spatially p depends only on r. By differen- tiating again, Op 4] 3 ae ® =| ne a) dx? dxLéorrt or rs rox\or orr 2 Op r? or 2p a — a] az2—soar rs a Addition of these expressions for 02p/dz, 02p/dy?, and 0?p/dz?, in order to obtain the right-hand side of equation (27), gives OD Co , GO Fp Ox? —s oy? a2 or? The latter expression is easily verified to be (1/r) (8?/dr?)(rp), so we finally obtain ap 18 ax? dy? act ran’?» 2 0p T Or + and the general wave equation (27) reduces to d?(rp) 3° ——— = © (79). a ari?) (37) This equation has a form similar to that of equa- tion (30) for plane waves with p replaced by rp, and x by r. By using an argument similar to that in Section 2.3.1, it can be shown that equation (37) is satisfied by rp = i. aE ") , c where f is an arbitrary function of one variable. By dividing out ther, we get the following expression for p(r,t) as the general solution of equation (37): e-Dvdes?) r eA (38) Assume that the following initial conditions are given. The initial disturbance is confined to a spheri- cal shell of infinitesimal thickness at a distance r = 7 from the origin. We suppose that the excess pressure in this spherical shell source is given by _ Bw) P(Toxt) = (39) To between the times ¢ = 0 and t = t. We also suppose that the excess pressure at points outside this shell is zero at the time ¢ = 0. The general solution of equation (37) with these initial conditions is aed + (1—«)p (+! — | , (40) because first, the right-hand expression is in the form of equation (38), and therefore satisfies equation (37) ; second, the right-hand expression satisfies the initial conditions imposed. In particular, if the spherical- shell source has a very small radius so that it approxi- PROPERTIES OF SOUND WAVES 19 mates a point source at the origin, the following solution is obtained. —, Tr t—- pit ‘) Pp al as eo a) Al) Physically, we can eliminate the solution (1/r) p(t-+ r/c). This solution corresponds to wave propagation in the negative r direction, with the speed c; in other words, to a wave which starts out at some negative time with a great radius and con- tracts into the point x = y = z = Oat the time? = 0. The first solution is physically valid since it resembles actual propagation from a point source. It implies that the spherical wave spreads out from the point source into ever-increasing spheres with the speed c. Therefore, an initial ping of duration 7 seconds will cause the resulting sound energy to be contained within an expanding spherical shell of thickness cr. If the source is harmonic (emits a pure tone of the frequency f), the initial conditions are of the form a cos 2rf(t — e) i ee Sea LT Smt? To and if ro is nearly zero, the pressure at the distance r from the source and time ¢ is given by @ COs anil oe .) c (42) iP Pp = The constants f and e have the same physical signifi- cance as for the plane wave case; f is the frequency of the vibration and e¢ is the phase constant which orients the vibration in time. There is a difference, however, in the interpretation of a. In the plane wave case, a represents the maximum pressure change in the wave at all distances from the source; since a is a constant, all these pressure changes are equal. For spherical waves described by equation (42), however, it is clear that the maximum pressure change at the distance r is given by a/r, decreasing as r increases. The constant a is no longer the amplitude at all ranges, but merely the amplitude at the particular range r = 1. 2.4 PROPERTIES OF SOUND WAVES 2.4.1 Pressure versus Fluid Velocity PLANE WAVES For a plane wave we have, from equation (4), Oo Ou Gi be? where u is the particle velocity in the positive x direction. Because of equation (18), this equation can be transformed into (43) From equations (21) and (18), there results the following expression for dp/dx: (44) Equations (48) and (44) will be used to derive the general relationship between the excess pressure and the particle velocity in a plane wave. Assume that as initial conditions the plane « =0 has its excess pressure given by p(0,t) = p(t) between ¢ = 0 and = t, and p(0,t) = 0 for all other values of t. Then the general solution of the plane wave equation, if we assume that the wave moves in the positive x direction, is given by p(a,t) = ol - z). By differentiating both sides of this equation with respect to ¢ and also with respect to x, we obtain op af a) op 13 _) ——— #—=}): —= —-j7lti——)- Ei) Pale c/’ ax cP c dp _ldop dz Sst ee) Combining equations (43) and (45) gives ; ; 2(u — a) = (i), (46a) Ox K and by combining equations (44) and (45) () =—(p — = 0. 46b ave pocu) (46b) Equation (46a) means that (wu — cp/x) is a func- tion of ¢ alone; and equation (46b) implies that (p — pocu) is a function only of x. But these two parentheses are proportional, differing by the factor (—poc) since pyc? = x. Therefore, each parenthesis must be identically equal to some constant. This constant turns out to be zero for both since at any given point both p and u vanish before and after the disturbance has passed. The following relation be- tween particle velocity and pressure results: 1 c “wS=) => 70: K Poc (47a) 20 WAVE ACOUSTICS Equation (47a) can also be shown to hold good for the wave moving in the positive direction if the initial conditions are for the space interval 0 < x < 2» at the time t = 0. If the wave is moving in the nega- tive x direction, it can easily be shown by an argu- ment similar to the above that the pressure and par- ticle velocity are related by (47b) For the particular case of a plane harmonic wave, we have from equations (47a) and (47b) a us + cos 2nf(t — e). Poe The following interesting result stems directly from equations (47). If the particles in a plane wave are moving in the direction of wave propagation, they are in a region of positive excess pressure; if they are moving in a direction opposite to the route of the wave, they are in a region of negative excess pres- sure; and if the particles are not moving, they are in a region of zero excess pressure. Also, we can argue from equations (47) that if the initial conditions ful- fill neither 1 u(z,0) = —p(z,0) poc nor 1 u(x,0) = == D(2,0)p Poe then waves are propagated in both a positive and a negative direction from the initial source of pressure disturbance. SPHERICAL WAVES At great distances from the source a small section of a spherical wave approximates a plane wave. For this reason, many of the foregoing results can be re- written in a form valid for spherical disturbances far from the source. Since the mathematical proofs, though straightforward, are rather cumbersome they will not be reproduced here. For a general spherical wave far from the source, the following relation exists between the excess pres- sure and particle velocity: c 1 WS sE=f) S sof K Poc in analogy with equations (47). For a spherical har- monic wave it will be remembered that the maximum pressure change at the distance r from the source is given by a/r. Thus, for the case of a spherical har- monic wave, 1G (48) Umax >= poe r The relations (47) are not necessarily true for the general solution of the wave equation (27). 2.4.2 Acoustic Energy and Sound Intensity The vibration of the particles of a fluid disturbed by wave propagation is a process which involves both kinetic and potential energy. The energy of vibration of the sound source is propagated through the fluid along with the sound wave. In a perfect fluid where frictional heat losses are zero, the energy content of the wave is unchanged as the wave travels. The en- ergy passes from one region to another, “activating”’ the region through which the wave is passing. Thus, there are two quantities of interest. One is the energy found at any location as a function of time; the other is the rate at which energy is transported from one region to another as a function of time. In the follow- ing sections, both of these quantities are expressed in terms of the wave parameters we have introduced. The kinetic energy possessed by a volume element v, whose volume was v at equilibrium, and whose speed is u, is given by (49) The potential energy possessed by the volume ele- ment v is the work which was done on it to change its volume from vp to v. This work can be calculated as follows: By equation (9), the relative change in volume produced by an infinitesimal alteration of condensation from ¢ to « + do is just —dc. The total volume change caused by this infinitesimal alteration of o is, to a first approximation, —vdo. The work done during this infinitesimal alteration is merely the pressure times the small volume change, that is, puods, which, because of equation (18), equals xovoda. The total amount of work done on the volume element as its volume changes from » to » can be obtained by integrating this infinitesimal amount of work between a condensation of zero and condensa- tion of o. Kinetic energy of » = spout. oe Potential energy of v = vox { ado = 4 9Ko” 0 Vop” = — 50 D: (50) PROPERTIES OF SOUND WAVES 21 because of equation (18). By adding equations (49) and (50), we obtain pool? —-Uo 2 2k By dividing equation (51) by the volume w: Total energy of v = p. (51) Energy density at (x,y,z) = Bad +E (62) K We are now in a position to give a general expres- sion for the intensity of a progressive sound wave, the characteristic which determines its loudness. Inten- sity is defined for a general progressive wave as the amount of energy which crosses a unit area normal to the direction of propagation in unit time. Since the energy travels at the same rate as the sound pulse, the instantaneous rate of energy flow will be equal to the energy density at the point in question times the sound velocity at this point. The intensity will be the time average of the instantaneous rate of energy flow, or Intensity at (x,y,z) O U2 2 = Time average of (oe + = 2 2k =) Cpou? cp 2 F 2k where the bar over a quantity denotes the time average of that quantity. We shall now attempt to calculate the intensities explicitly for various types of sound waves. We shall first consider plane waves and spherical waves, and then more general waves. (53) PLANE WAVES A plane progressive wave satisfies equation (47); and therefore equation (50) reduces, for that case, to (54) which is exactly equal to the expression (49) for the kinetic energy of v. We therefore get the result that for a plane progressive wave the kinetic and potential energies possessed by any small volume element at any time are equal. The kinetic and potential energies attain their greatest values at the spots where the particle velocity and excess pressure have their max- ima or minima; and they vanish at the spots where Potential energy of » = xpovou? the particle velocity and the excess pressure are both zero. Because of this equality of kinetic and potential energies for a progressive plane wave, equation (53) simplifies to 2 Intensity = ep = poe (55) by equations (47) and (26). In a plane progressive wave that is also harmonic the pressure is a sinusoidal function of the time with maximum value a. Since the average value of sin? @ over a complete period is 14, p? = a?/2. 2 : a Intensity = ae ‘poe (56) Also, from equation (55) and the fact that u is also a sinusoidal function of the time, Intensity = 4pocurrax- (57) SPHERICAL WAVES For a spherical wave far from the source the formulas derived for plane waves are approximately true. We must be careful in applying them, however, to remember that the amplitude of the pressure vibration is no longer constant, but diminishes in- versely with distance. Using equations (57) and (48), we obtain 2 ; la Intensity = —— — 2poc r? 8) for harmonic spherical waves where a is the maximum pressure change at a distance one unit from the source. Equation (58) is the familiar inverse square law of intensity loss for a spherical wave spreading out from a point source into an infinite homogeneous medium. Let F represent the amount of energy radiated by the source into a unit solid angle’ in one second. Then Total rate of emission = 47F. (59) 4Solid angle is the three-dimensional analogue to the ordinary, two-dimensional, plane angle. It measures the angu- lar spread of such three-dimensional objects as a cone, a light beam, or the beam of a radio transmitter. Its measure is de- fined as follows. Construct a sphere of arbitrary size with the apex of the solid angle as its center. The solid angle will then cut out a certain area of the surface of the sphere. This area, divided by the square of the radius of the sphere, is the meas- ure of the solid angle. It is dimensionless and does not depend on the sphere radius chosen. The unit solid angle is frequently called the steradian. The full solid angle, comprising all direc- tions pointing from the apex, has the value 47. 22 WAVE ACOUSTICS We can calculate F by means of equation (58). Since a sphere of radius r has the area 47r?, the total energy crossing such a spherical surface per unit of time is merely the intensity times this area: 4 27a" Rate at which energy crosses sphere = re - (60) Because of the assumption of conservation of sound energy, equation (60) must be equal to the amount of energy radiated by the source per unit of time. Dividing equation (60) by 47 gives a a (61) GENERAL SOUND WAVES We now examine the transport of acoustic energy for the case of a general solution of the wave equa- tion (27). In the general case, it is useful to start with an equation of continuity for energy flow analogous to the exact equation of continuity (2) for mass flow. Tt will be recalled that equation (2) followed directly from the law of conservation of mass. The law of con- tinuity for energy flow will follow from the law of con- servation of energy in exactly the same fashion. For the mass density p, the energy density which may be denoted by Z is substituted. Also, for the instanta- neous flow of matter with components w;,Uy,w2, the instantaneous flow of energy is substituted. The components of the instantaneous energy flow past normal unit area may be denoted by E.,E,,EH,. The equation of the continuity for energy flow becomes, in analogy with equation (2), OZ [= 0E, a ot ie oy oe Equation (62) is the mathematical expression of the assertion that the energy flow through a closed sur- face is equal to the decrease of energy inside this sur- face. A rather complicated argument must be used to calculate the components of energy flow E,,H,,H7.. Equations (21) are rewritten by using p = xo, as (62) OUz ay oy Ley an OUy -Op met = SSE 63 Poa By (63) du. _ _ 9p ucrena GY Also, from equations (4) and (18), we have the rela- tion 1 dp [& OUy a e ax oy 7 dz Multiplying the first equation of (63) by uz, the second by u,, the third by w., and equation (64) by p, and adding them all up, we obtain (64) afr 2 4 24 a2 Z| 2| + Uy + uz) + 2x _ [| epuz) , a(pwy) tee) = [ ap ar ay + ae (65) Because of equation (52), we see that the left-hand sides of equations (65) and (62) are equal. Hence the right-hand sides are also equal, and we must have (66) The instantaneous energy flow E is the resultant of its three components EH,,H,,E, and is numerically equal to V Ee + E; + E?. Thus, we have the general result that E, = puz; E, = pu,; E, = puz. (67) According to equation (66), this energy flow is always along the direction of the particle velocity. The intensity 7, which was defined as the time average of EH, is therefore always given by the fol- lowing formula: E = pu. = i (68) 2.4.3 Complex Representation of — Harmonic Vibrations The complex number e*” is defined by the equation e~ = cos w +7 sin w where 72 = —1. The one-dimensional harmonic vi- bration d = a cos 2rft can therefore be regarded as the real part of ae”™”. Similarly, the vibration d = a cos 2zf(t — €) can be rewritten as (69) d = real part of ae?“ . The latter relation can be expressed in the following less cumbersome form D= a e2ritt—9) (70) if the conventions are adopted that the actual phys- ical displacement is the real part of the complex dis- PROPERTIES OF SOUND WAVES 23 nnn ee EEE eI IEEUE EEEEEEIENE NEESER SEEDER placement D and that the numerical value of this actual displacement is the real part of the right-hand side of equation (70). With this understanding, equations (70) and (69) represent one and the same physical process. The complex form for a vibration simplifies some types of calculations and will be frequently used in the remainder of this chapter. We notice that equa- tion (70) can be rewritten in the form D = Ae?" (71) where A is the complex number ae""*. A is called the complex amplitude of the vibration described by equation (69). It is apparent that A =a [cos 2xfe — 7 sin 2zfe]. Thus, the complex amplitude has a cos 2rfe as its real component and —a sin 2zfe as its imaginary com- ponent. As an example of the convenience afforded by the complex representation of a vibration, we shall use it to find the harmonic solution of the plane wave equa- tion (30). We assume tentatively that p(2,t) = Aermi it mz) (72) and see if we can find a value of m which will make equation (72) a solution of equation (30). Substi- tuting equation (72) into equation (30), we have (2rif)2p = c?(2rim)?p. In other words, a value of m equal to f/c or —f/c makes the expression (72) a solution of equation (30). These two solutions are, explicitly, p= Aerils (z/e)1. A= ae, (72a) These two solutions, interpreted according to the convention of this section, are obviously identical with the “real” solutions (84). Similarly, a point harmonic source in an infinite homogeneous medium gives rise to spherical harmonic waves according to the equation 2miflt—(r/c)] . nae —2rife at rie). A = ae rife A p(7,t) a a (73) 2.4.4 Sound Sources The wave equation (27) governs the manner in which disturbances will be propagated in the interior of a fluid, but does not say anything about the initial disturbances themselves. In this section we shall con- sider the various types of initial disturbances which can be produced by sound sources. First we shall dis- cuss the quality of the sound put out by various sources, where quality refers to the frequency char- acteristics of the emitted sound. Next, since some sources radiate equally in all directions while others do not, we shall consider in a general way the directivity properties of sources. FREQUENCY CHARACTERISTICS Strictly speaking, the concept of frequency can be applied only to simple harmonic disturbances. A simple harmonic disturbance of the pressure in a fluid is described by an equation of the form p = acos 2nf(t — e) and gives rise to what is called a pure tone. Most of the echo-ranging transducers used at present produce sounds which are very nearly pure tones, but cannot be heard by the ear because the frequencies are too high. If two or more pure tones are put into the water at the same time, the resultant is known as a compound tone. Some transducers, used mainly in research work, can produce compound tones. Any sound of this nature can be expressed as the sum of a finite number of harmonic vibrations. Many sources, however, produce in their immedi- ate vicinity an irregular change in pressure which cannot be represented as the sum of a finite number of sinusoidal vibrations. Such sound outputs are called noises. Ship sounds and torpedo sounds are examples of noises, and the reader can doubtless supply other examples. According to a mathematical theorem called the Fourier theorem, it is often possi- ble to represent such an irregular sound output as an infinite sum of simple tones, whose intensities, fre- quencies, and phase relationships are such that they add up to the given noise. If most of the component frequencies lie in a narrow frequency range, the sound is called a narrow-band noise; otherwise it is called a wide-band noise. Some types of echo-ranging gear put out a fre- quency-modulated signal. In this type of output, the pressure is at every instant a sinusoidal function of time, but the frequency changes during the signal in some designed way. In one type of frequency-modu- lated signal, called a “chirp” signal, the frequency increases linearly with time for the duration of the pulse: p = acos 2n[(fo + at)t]. In a typical chirp, 100 msec long, the frequency may increase from 23.5 ke at the beginning of the pulse to 24.5 ke at the end of the pulse. 24 WAVE ACOUSTICS Directiviry CHARACTERISTICS Also, So far we have been mainly concerned with the Ales GE IN = yg ES (75) simple point source which gives rise to a spherically symmetric disturbance in the immediate vicinity of the source. It is called a point source because the re- sulting sound field is discontinuous at only one point of space, at the source itself. If the discontinuity is of a more complicated nature, as in the case of a line source, the sound field will not, in general, be spheri- cally symmetric in the neighborhood of the source; that is, the amounts of sound energy radiated into different directions will be different. In this subsec- tion, sources giving rise to sound fields that are not spherically symmetric are discussed. Double Sources. Suppose there are two point sources, So and Sy’, one at (Xo,yo,20) and the other at (x0' Yo’ ,20), 28 indicated in Figure 6. The resulting P=p,+ Po! \ a N Po! % So! So Figure 6. Resultant pressure produced by two sepa- rate sources. pressure at any one point P and time ¢ will be the algebraic sum of the pressures that would be pro- duced by each source separately. That is, if f and f’ are the two frequencies emitted, « and ¢’ are the two phase constants, and A and A’ are the two complex amplitudes, the resulting p(r,t) is given by A’ 2uif[t—(r/c)] ah — p2rif'Lt—(r'/c)] ey A Die (74) according to equation (71). We shall restrict our attention to the case of two sound sources situated on the x axis, one at the origin and the other a small distance s away. We assume that these two sources produce initial pressure dis- turbances of equal real amplitudeand equal frequency and that the initial disturbances are opposite in phase. y co) 0’ Figure 7. Resultant pressure produced by double source 00’. This case, pictured in Figure 7, is a fairly good ap- proximation to many sources occurring in practice, such as a vibrating diaphragm. Because of these as- sumptions, the following relationships exist among the quantities in equations (74) and (75): a=a; f=f} 6—0; «= (76) Also, A = aand A’ = ae = tion (75). Of particular interest is the extreme case where the distance between the two sources is very nearly zero, but where the real amplitude a of the individual dis- turbances is so large that the product as is an ap- preciable quantity. Such a combination of two single sources with very small separation and very large individual amplitudes is called a double source. A double source may be described by two quantities: the product as, and its axis, the direction of the line joining the two sources. —a because of equa- PROPERTIES OF SOUND WAVES 25 With these assumptions, equation (74) becomes a ey @ peri Aro (r/c)] mt eT (r'/c)} (77a) p= where Pa=eaety+2; r2= («¢—s)P?+y +2. (77b) If F(r) is an arbitrary function of 7, and if r and r’ are very nearly equal, we have from simple calculus dF F(r) — Fr’) = (¥ — r')—- dr The quantity r — r’ may be calculated as a function of x,s,r as follows: Pr One roe) @r which equals sz/r from equation (77b). tity dF /dr may also be calculated: aF _ OF ax , oF ay oy Or nh because r = 1’, The quan- aP a dr dx or dz Or As the origin changes from O to O’ on the z axis, thereby changing r to 7’, the coordinates y and z of all points in space are unchanged. Thus, for all changes in r defined in this manner, CY Ge or or so that af OF dr OF r dr ax dr axe because of equation (86). r — r’ and dF/dr, Using these values of oF F(r) — F(’) = s— Ox and equation (77) may be rewritten as 0} 1 : p= asd emer | * OxrLr By calculating out the derivative of the bracket with respect to x, and by remembering that dr/dx = x/r, this equation becomes p = aserrifte rifle) J ae ‘). r\ ¢ i If @ is the angle between the z axis and the radius vector OP, x/r = cos a, and the preceding equation becomes Sgt juvinet cos af2rft 1 p= ase™""e es a a r Cc Tr, If r is very large compared with c/f, the second term in the brackets may be neglected, and as a result 2rfasi por = Gales cos alt e/o)1, cr P Replacing the factor as by b, we obtain the final result ae2Tfilt—C/o)1 Qf bi p= cos (78) By comparing equation (78) with equation (73), we see that the pressure changes produced at great dis- tances by the double source are identical with the pressure changes produced by a single source, which is situated at the same place, vibrates with the same frequency, and has the following complex amplitude. Qrfbi A = ee (79) It is clear from equation (75) that the real ampli- tude of the vibration is equal to the absolute value of the complex amplitude. From algebra, we know that the absolute value of a complex number A is just VAA, where A is the conjugate complex of A. Let a be the real amplitude corresponding to equa- tion (79). Then a is given by 2rfb a= oer COS a. (80) With this definition of a, the actual pressure dis- tribution defined by equation (78) is p(r,a,t) = a cos anil — *). (81) r Since :p is harmonic, its square averaged over a com- plete period is one-half the square of its amplitude (80); from equation (58) we have for the intensity at the distance r and angle a: 1/27fb 2 I I(r,a) = a cos a) — PN G 2poc ph Q1°f?b? * pocr COS? a. (82) Thus, the sound intensity caused by a double sound source is directly proportional to the square of b and to the square of the cosine of the angle a of emission and is inversely proportional to the square of the distance from the sound source. Let F(a) denote the average rate at which energy is emitted in the direction a. It is clear from Figure 8 26 WAVE AGOUSTICS that this average emission per unit solid angle is given by T(r,a)r'dw — 2f?b? = c ite) = dw poc® 0s? a (83) where dw is an infinitesimal solid angle in the direc- tion a. The maximum value of F(a) occurs in the direction of the x axis, for which Qf 2b? FO) = 84 (0) ae (84) Thus equation (83) can be rewritten as F(a) = F(O) cos? a (85) (0) Karta) Figure 8. Rotation of wedge aa about axis. and equation (82) as F(O) cos? Gg) = mens, (86) In order to find the total energy emitted by the double source in one second, we calculate the total energy traversing the surface of a sphere of radius 7 in one second. This is clearly equal to the rate at which the source is putting out power. To get this total energy, it is necessary to integrate the average energy flow (86) over the whole sphere. Such an integral is in general multiple, but in this particular case it can be expressed as a single integral because the energy flow depends only on a. First consider the average rate at which energy is flowing through the small area element intercepted on the sphere by the two cones defined by the angles a and a+ da, as in Figure 8. This small element of the sphere has the area 2rr? sin ada; and, therefore, it intercepts a solid angle of 27(sin a)da units. By equation (86), the average rate of energy flow through this element is F(O) cos? a@- 27 sina: da. The total emission in one second is this average rate of energy flow integrated between the angles 0 and 7; that is, 1/2 Rate of emission = 2 f F(O) co? a- 27 sina- da 0 4rF (0) é 3 It will be remembered that F'(0) is the maximum rate of emission per unit solid angle, by the double source. All sound projectors have pattern functions which describe the distribution of sound energy emit- ted in different directions. A general direction in space can be defined by the two coordinates (6,¢), where 6 is the angle of elevation of the direction OP relative to the horizontal zy plane, and ¢ is the polar angle in the zy plane between the zx axis and the projection OP’, as in Figure 9. Let F(6,¢) be the (87) y Figure 9. Coordinates specifying direction OP. emission per unit solid angle in the direction OP, and let Fax be the emission per unit solid angle in the direction of maximum emission, called the acoustical axis of the projector. Then the pattern function b(6,¢) is defined by F (6,6) = Frax 6(8,9). If we take the acoustical axis in the direction (0,0), this becomes F(6,¢) = F(0,0)b(6,¢). (88) The pattern function b clearly depends only on the nature of the projector. SOUND WAVES IN A VISCOUS FLUID 27 In analogy with the result (86) for the simple case of axial symmetry, hong = ee ; (89) 72 at a distance r from the source much greater than a wavelength. The rate at which the projector emits energy in all directions must be exhibited as a double integral in this general case. The area element on the sphere of radius r, intercepted between (6,6) and (@ + dé, ¢ + d¢), is r? cos 6déd¢; and the solid angle inter- cepted by this area element is cos @déd¢. Thus, in view of equation (88), Emission through area element = F(0,0) 6(6,¢) cos 6déd¢ (90) and therefore Total rate of emission 7 a/2 = F(0,0) it do f (0(6,0) cos 6d6. (91) Equation (91) can be put in the following form, which is directly comparable to the law of emission (59) of a point source: Rate of emission = 47F'(0,0)6 (92) where x x/2 = ins af b(,¢) cos 6dé. (93) 4nd —« —1/2 The factor 6 is a constant depending on the nature of the source and may be called the directivity factor of the source. For the point source of equation (59), this directivity factor is 1; while for the double source of equation (87) it is 14. 2.5 SOUND WAVES IN A VISCOUS FLUID In a homogeneous perfect fluid, the decrease of sound intensity with increasing distance from the source is due only to spreading according to the inverse square law (58). However, sound intensity measurements show clearly that the intensity loss in the ocean tends to be much greater than the value pre- dicted by equation (58). These extra losses above the theoretical loss, (58), due to the fact that the ocean is not a homogeneous perfect fluid, are called trans- mission anomalies or, more loosely, attenuations. In this section, we shall derive some results for sound intensity in a viscous fluid and see how much of the observed attenuation can be ascribed to fluid viscosity. In the derivation of the wave equation (27) for a perfect fluid, we used the equation of continuity (4), the equations of motion (15), the equation of state (18), and the law of forces (16). The equation of continuity, the equations, of motion, and the equation of state, it will be recalled, apply to any fluid, whether it is perfect or viscous. The law of forces (16), how- ever, is valid only for a perfect fluid. The exact law of forces operating in a viscous fluid is quite difficult to derive since it depends on the theory of viscous fluid flow. It is sufficient to say that this complicated law of forces, combined with equations (4), (18), and (15), can be used to derive a general wave equation for viscous fluids, analogous to equation (27). Under the assumption that the resulting pressure distribu- tion in the viscous medium depends only on the co- ordinate x, this general equation reduces to* ep Kp 4u 0p Ot py 2? 3p Oxdt where p is the coefficient of shear viscosity. Equation (94) is the plane wave equation for a viscous fluid. In the absence of viscosity (u = 0), equation (94) re- duces to equation (30). Let us see whether we can find a solution to equa- tion (94) of the form p= Aer itm) (94) By substituting this expression for p into equation (94), we obtain i f 4 Ff ont ©) Po 3po If cand a are defined by c=", (96) Po 8 uf? Bas Se ee, a 37 mL (97) the relation (95) becomes 1 ps ele = =) (98) ¢ \/ an ie c 2m T ae according to the binomial theorem, if we assume that ais so small that a? and higher terms can be neglected. In order to get the case of waves propagated in the positive x direction, the negative sign of equation (98) must be chosen and 277mz becomes 2Qrimz = —2ri-x — ac. Cc 28 WAVE The corresponding solution of equation (94) is there- fore p= Aer e or sla/ec) Cm We can write this expression for p in the form p= Ae at p 2riflt—(x/c)] (99) For » = 0, a vanishes, and equation (99) reduces to 2niflt—(x/c)] (100) which is just the solution for plane waves propagated harmonically into a perfect fluid. By comparing equations (99) and (100), we see that the effect of viscosity is to cause the amplitude of the pressure vibration to decay exponentially with distance, by the factor e “*, where a is the positive real number defined by equation (97). A vibration of the type of (99) is referred to as a damped vibration, and e “” is called the damping factor. To see whether this energy loss due to shear viscosity is the cause of the attenuation observed in the sea, one can first calculate a for sea water, by using the known values of p,c,u for sea water, and the known frequency f of the sound source. The in- tensity loss is measured between two points so far from the sound source that the wave propagation be- tween those two points approximates plane wave propagation. Then this observed intensity loss is compared with the theoretical intensity loss calcu- lated from equations (97) and (99). It is found that only for very great frequencies (much higher than 100 ke) can an appreciable fraction of the observed attenuation be ascribed to shear viscosity; at lower frequencies, the theoretical loss from viscosity makes up only a very small part of the observed attenua- tion. Thus other causes must be sought for the extinc- tion of sound energy in the sea. The sound transmis- tion studies of Section 6.1 of NDRC have had as one of their primary objectives the discovery of the factors governing the intensity loss of sound in the sea. Although some progress has been made, the problems of attenuation in the sea have by no means been completely solved (see Chapters 5 to 10). Since the observed attenuation of sound in the sea is much greater than the value indicated by equation (99), it appears possible that there is another type of viscosity, in addition to the classical shear viscosity, which may be responsible for part or all of the re- maining attenuation. The classical theory of the flow of viscous fluids is based on Stokes’ hypothesis that frictional forces within a fluid arise only from a p = Ae ACOUSTICS change in the shape of a volume element; in other words, that a change in the size of a volume element, if its shape remained unaltered, would meet no re- sistance. The concept of a compression viscosity has been suggested to represent the resistance of the fluid to pure volume dilatation. Such a compression viscosity would not be discovered in a stationary flow of the type employed to measure shear viscosity be- cause in these experiments the fluid acts essentially as an incompressible fluid. But in the transmission of sound this conjectural compression viscosity would contribute a term to the expression for a which would also be proportional to the square of the frequency f. Actual determinations of the constant a at many different frequencies show that between 0 and 100 ke the attenuation increases less rapidly than the square of the frequency. There are no theoretical grounds for assuming any power law for the depend- ence of attenuation on frequency. If a power law is as- sumed, the empirical curve is best fitted by a 1.4th power dependence, but even this best fit is poor. It thus appears that factors other than viscosity must account for much of the attenuation of sound ob- served in the sea. 2.6 EFFECT OF A BOUNDARY Conditions of Transition and Boundary Conditions 2.6.1 We shall now return to the assumption of a perfect fluid and turn our attention to the effects of bounda- ries. Consequently, we now drop the assumption that waves are propagated in a single homogeneous infi- nite medium. Instead, we shall consider the case that all space is filled up by two different homogeneous media separated by a plane, which we choose as the plane y = 0. For the one medium (the sea), at y < 0, we denote the density, excess pressure, bulk modulus, and sound velocity by p, p, x, and c respectively; for the other medium, air, for example, at y > 0, we call these quantities p1, pi, «1, and cy. It is necessary, from a physical point of view, to assume that the pressure in both media is the same at the boundary. Otherwise, the force per unit mass at the interface would become infinite. We have, thus, p=piaty =0. (101) Also, if the two media are to remain in contact with each other at all times, the displacements normal to EFFECT OF A BOUNDARY 29 the boundary must have the same value in both media at the boundary. In symbols, if (S.,S,,S.) are the components of particle displacement in the first medium, and (Sj,,S1,,Si.) are the components of particle displacement in the second medium, S, = Sy at y = 0. (102) No restrictions of the form of (102) can be placed on the displacements S, and S, because displacements parallel to the boundary will not cause loss of con- tact. Since equation (102) holds for all time, the time derivatives of S, and S,, must also be equal at the boundary; in other words, OUy OUny = ing == = (0). 103 Uy = Uy; at at y (103) Because of equation (17), equation (103) implies 1d 10 = SE Nee aty = 0 a poy pidy 0 0 2 a BSE iy SO. (104) Oy pi OY We shall call equations (101) and (104) conditions of transition. In the general case, the propagation in one medium depends on the exact nature of the propagation in the other medium, because of the con- ditions of transition. In the case of the sea, however, conditions are often such that we can ignore the exact propagation in the surrounding medium; the transi- tion conditions of the type (104) may then be re- duced to boundary conditions for the sea itself. In the next section, the conclusion is reached that at the yielding boundary between sea and air the follow- ing condition holds: p=0 (105) and that at the solid boundary between sea and rock bottom we always have, approximately, Op | oy a Relations of the type of (105) and (106) will, in many cases, suffice for calculating the sound field in the medium of interest. By use of such boundary condi- tions explicit consideration of the sound field beyond the boundaries may be made unnecessary. (106) Reflection and Refraction of Plane Waves 2.6.2 Consider now what happens to a plane wave when it hits the plane boundary y = 0 between two dis- similar media, in one of which the sound velocity is c, and in the other of which it is c¢. For generality, we assume that the direction of propagation of the inci- dent wave is oblique to the boundary, making an angle 0; with the normal to the plane boundary. We can also assume, without losing generality, that the direction of propagation is parallel to the zy plane; y represents the vertical direction positive upward, x a horizontal direction, and z a front-back direction, as in Figure 10. y z SOUND VELOCITY c, DENSITY Py SOUND VELOCITY ¢ DENSITY Ficure 10. Splitting of plane wave at boundary be- tween two media. Since the incident wave is plane, it may be de- scribed by the equation (72a) with x replaced by x sin 6; + y cos 6:, in view of the oblique direction of propagation. That is, for the incident wave, 4 _ sind; +y cos 6; DUA Gea Woe RAAT) where p; represents the sound pressure of the inci- dent wave, and A; its complex amplitude. We can consider that the incident wave terminates its existence when it hits the boundary and expends its energy in producing a disturbance of the interface. Thus the boundary will act as a sound source, which vibrates with the frequency f of the incident wave. The vibration of the interface will send out sound waves of the frequency f into both media. We shall assume that these two waves are plane waves; this 30 WAVE ACOUSTICS result is intuitively apparent, but can be proved only by a long tedious argument. For brevity, the wave propagated by the boundary into the second medium will be called the transmitted wave, and the wave propagated back into the first medium is called the reflected wave. We shall now cal- culate the amplitudes and directions of propagation of the transmitted and reflected waves. The pressure and complex amplitude of the trans- mitted wave are denoted by p, and A,;; the same quantities for the reflected wave are denoted by p, and A,. Let the transmitted wave have the direc- tion 6;, relative to the normal, and the reflected wave have the direction 6,, as indicated in Figure 10. The angles 6, and 6, are usually called the angle of refrac- tion and angle of reflection, respectively, and the angle 0; is called the angle of incidence. Because the reflected and transmitted waves are plane, Qwifi (senor vicos on) p, = Ae c (108) 2mif (t— ae inle taleiGOSiet) p= Ae Me . (109) The sign of y is different in equations (108) and (109) because in equation (108) y decreases with the time on the wave front; in equation (109) it increases. Equation (109) gives the resultant total pressure in the second medium. The resultant pressure in the first medium is the sum of the pressures of the inci- dent and reflected waves, which is obtained by adding equations (107) and (108). Denoting the resultant pressure in the first medium by p, we obtain Qnif (1—zaindetucos?s) p=pitp, = Ae (110) 2nif (t= sin 6, — y cos bry + A,e ¢ The pressure must be the same on both sides of the boundary. Therefore, p: + p, = p: at y = 0; that is, Qrif (Ee DS) amis (¢_78 Aje c "+ A,e = Ae xsin rent) 2Qnt if (t— or e| Ae sr on if SiB Or omnis Sint Sey aa ae VAN nema: | =0 (111) for all values of t and x. Therefore, the bracket itself must be zero. Furthermore, the sum of three har- monic functions of z can vanish for all values of x only if their periods are the same. It follows that sin@; sin 6, = = . (112) Ci J Cc Cc The second equation of (112) implies that 6; = 6,; that is, the angle of incidence is equal to the angle of reflection. The first equation may be rewritten as sin 6; sin@; ¢ sin 6, aii a relation which is well known in optics as Snell’s law. Because of equation (112), the exponential factor is the same f for all three terms in the bracket of equation (111) and can be divided out, giving A, =A:i+A,. (114) The individual amplitudes A; and A, are calculated by making use of the transition conditions (104). By calculating dp/dy from equation (110), and dp,/dy from equation (109) and by substituting these values into equation (104), we obtain A; cos 6; 2nif (1-72) A, cos 6, aris (280) ASG == SAS c Cc A,cos 6; 2nf {—zsin Oe sO OSA EI (115) Pi C1 In view of equation (112), the exponential factor is the same for all three terms and may be divided out. Also, 0; = 6,. Thus, equation (115) becomes cos 6; cos 6; (Ag = A) = Sy (116) Pp Cy Equations (114) and (116) are two linear equations in A, and A,. By solving them in terms of Aj, the amplitude of the incident wave, and by replacing c:/c in the result with its equivalent from equation (118), pici COS 0; — pc cos A, = (117) “pic: COS 8; + pc cos 6; 2pici COs 6; A, =A (118) “pic: COS 0; + pc cos 0; To eliminate the angle 0, from equation (117), equa- tion (113) is used which can be transformed by trigonometric identities into 3 0 4 2 co Wi + tanta(1 - a) = B. cos 0 Cc Thus, equation (117) becomes Ar _ pia — pcB (119) A; pit: + pcB EFFECT OF A BOUNDARY 31 Equation (119) gives interesting results when it is applied to the case of a sound wave in water hitting the surface separating water from air. The numerical values are (subscripts for air; no subscripts for water): Sey We Peay 1/1) C1 PL By substituting these values into equation (119), A, 1—3,311V1 + 0.95 tan? 6; Ai 143,311V/1 + 0.95 tan? 6; For perpendicular incidence 0; vanishes, and A,/A; differs from —1 by less than one part in a thousand; for greater values of 6; the approximation to —1 is even better. A wave in water reflected by air thus preserves its real amplitude almost exactly, that is, almost all the energy in the incident wave remains in the water. But it reverses its phase; this means tuat pr = —p;at the boundary. This conclusion, that the resulting total pressure at this type of interface should be very nearly zero, was called a boundary condition in Section 2.6.1. The derivation of this section furnishes the justification for assuming this boundary condition, which was stated without rigor- ous proof in Section 2.6.1. Equation (119) provides an estimate of the error caused by replacing transi- tion conditions at a boundary with the more simple boundary conditions. In the case of the interface separating water and air this error is clearly very slight. Another interesting case is the incidence of under- water sound on a hard bottom like solid rock or tightly packed coarse sand. The treatment of sound waves in solids is rather more involved than the treatment of sound waves in fluids because a solid has two different kinds of elastic forces: those which resist changes in volume; and those which resist changes of shape (bulk modulus and shear modulus). Consequently, two different kinds of propagation of sound are possible in a solid. The two types are usually referred to as longitudinal waves and trans- verse waves. In the oblique incidence of underwater sound in a water-solid interface both types of waves are generated in the solid and the transition condi- tions are, therefore, more involved than those dis- cussed previously. If, however, the solid is quite rigid — that is, if both bulk modulus and shear modulus are appreciably greater than the bulk modulus of water — then it may be assumed, in good approximation, that the interface will not permit displacements perpendicular to itself. In other words, u, Will vanish approximately. If u, at the interface is zero, then its time derivative vanishes as well; and by reason of the equations of motion (17), op = Oat 7 = 0: oy This is the boundary condition which is often as- sumed in the treatment of reflection from a hard bottom. If this boundary condition is realized, it can be shown that the incident and reflected waves will have equal amplitude and the same phase. Thus, when sound is reflected from a rock bottom, almost all the energy of the incident wave will be found in the reflected wave. For a soft bottom like mud, this boundary condition will no longer be satisfied, even in approximation, and considerable sound energy may be lost by transmission through the interface. 2.6.3 Homogeneous Medium with Single Boundary Point Source NEAR SEA SURFACE We shall now solve the problem of finding the solu- tion of the wave equation which satisfies the bound- ary condition p = 0 at the interface y = 0, and cor- responds to a sound wave radiated by a point source at the depth h. This situation is illustrated in Figure 11. The depth of the ocean is assumed to be y ORIGIN aH — PZ) Se ome Figure 11. source O. Pressure produced at location P by sound infinite. The initial conditions are specified by the assumption that in the immediate vicinity of the source, that is, for points whose distance from the source 7, r =Vete+ (y + h)? is very small, the pressure satisfies the relationship rp(r,t) = Fd). (120) 32 WAVE ACOUSTICS If it were not for the boundary condition p = 0 at y = 0, the problem would be solved by means of the expression fs ") (121) c prt) = *r(e — r We shall have to modify this solution in order to satisfy the boundary condition as well. To this end, we resort to a trick. We solve a fictitious problem, one in which a source exactly like the first one is lo- cated at a distance h on the other side of the inter- face with the water extending through space and with the initial conditions r’p'(r’,t) = —F(d) (122) at points very close to the new source. This problem has the solution = Jl Y p (r’,t) = x(t = ") c r= NV x2 22 (yy —h)?. Clearly, sincer = r’ at y = 0, wehave p + p’ = 0 at x = 0. Thus, the wave given by the sum of the two disturbances described by equations (121) and (123), or (123) where (r/o)] _ Mt = (r'/c)] te th a EE ale) (124) satisfies the imposed boundary conditions. Also, equation (124) satisfies the initial conditions (120) because the expression (124) can be rewritten as rp = ri = 1) = rH = ) c r c which reduces to equation (120) in the vicinity of the actual source, where r ~ 0. Finally, equation (124) satisfies the wave equation itself since the difference of two solutions of that equation is itself a solution. If the source S executes a harmonic vibration, the solution (124) becomes Like i Qnflt — (r/o)] _ iP r cos 2rfLt — / w/o, (125) Formula (125) fully describes the effect of surface re- flection on harmonic waves emitted by a single source under the assumptions that air has negligible density and elasticity and that the sea surface is a perfect plane. From the method of construction of the solution (124), it is possible to deduce that there should be a zone of low intensity near the surface. The reason is that, at points near the surface, r and 7’ will be nearly identical; and the two resulting fictional pressures will almost balance each other. This type of destruc- tive interference near the surface is called the Lloyd mirror effect or image interference effect. The next few paragraphs will discuss the width of this low intensity zone and the intensity within this zone. Consider the intensity measured by a receiver at the depth h:, located at a horizontal range R from the source, as in Figure 12. We also assume that R Poe a aneee aiae TESS PUR-n,0) lal z Figure 12. Fictitious scheme for solving wave equa- tion and surface boundary conditions. is so large compared with h and h; (the depths of source and receiver) that second-order products of h/R and h,/R may be neglected. By applying the Pythagorean theorem to Figure 12, then (it) ae, (li + h)? | Sy eee ven) cee . Since h/R and hi/R are small, these equations may be rewritten as (hi — h) remit a | yo efi am fe ahs aT (126) 2 R? and as a result aa ve — =) L Angi Nut rT R ; any n+ ar (127) 1 ONE ro R because 1/(1 — e) = 1 + € if c€ is small. Putting these in equation (125), we obtain p= 2 cos 2uf (« = ") — cos 2rf' (« - 2) EFFECT OF A BOUNDARY 33 plus negligible terms, which may be rewritten as a f - r+r’ p= RU —2 sin [2ar( Vintage ) sola) by the trigonometric identity for the difference of two cosines. This equation reduces approximately, be- cause of equation (126), to p= 3 sin [2(2)™4 | sin Ec - Ry - (128) At the point P, equation (128) tells us that the amplitude of the pressure variation with time is ’ Amplitude = = sin ae where \ = c/f is the wavelength of the sound. The resulting sound intensity at P, which is proportional to the square of the maximum acoustic pressure, will be very small if the argument of the sine in equation (129) is small. That is, the intensity will be low if hyh/d is very small compared with R, or, in other words, if (129) AR h<< Tia Assuming that equation (1380) holds, the sine in equation (129) will be approximately equal to its argument, and we have (130) 4rahh, Rr In terms of the intensity, this means f 167°a7h?hi (1 Intensity « meme vee)” That is, in the layer of poor sound reception the sound intensity falls off inversely as the fourth power of the horizontal range at great ranges. For smaller values of horizontal range R, we find that the amplitude vanishes wherever the argument of the sine in equation (129) is an integral multiple of 7, or, in other words, where Amplitude = (131) while the amplitude will show greatest values in the neighborhood of those points where the argument is 1/2, 3x/2---; in other words, where Ahhy =e = 1,3, 5,-°° Rr This sequence of interference minima and maxima is called the image interference pattern. The image interference effect described here is only occasionally observed in the sea for reasons which are discussed in Section 5.2.1. Pornt Source Far FROM SHA SURFACE We assume now that the source is located so far from the surface that the sound waves near the sur- face can be regarded as plane waves. Only incident waves propagated purely in the y direction are con- sidered, that is, normal to the surface. Then equa- tion (125) has to be replaced by p= af cos mi ae ) — cos oni + ai (132) By applying to equation (132) the trigonometric formula for the difference of two cosines, we obtain p = —2asin on sin 2nft. (133) We notice a very curious thing about the disturb- ance described by equation (133). The acoustic pres- sure is zero over the entire fluid when ft is any in- tegral multiple of 44. Further, the acoustic pressure is zero for all time at points where y/) is an integral multiple of 14. Thus we see that the interference be- tween two plane waves of equal amplitude and of the same frequency traveling in opposite directions pro- duces, at least in this case, a disturbance of the medium for which at any instant all points have identical, or opposite phase. We no longer have pro- gressive waves, but a phenomenon which we call stationary or standing waves. The points where the amplitude is zero for all time are called nodes; the points where the amplitude term of equation (133) is a maximum are called loops or antinodes. This state of affairs is permanent as long as the source keeps vibrating. The nodes are permanent re- gions of silence; and the loops are permanent regions of maximum pressure amplitude. Such a state, in which all points of the medium perform vibrations of the form sin 2zft with an amplitude dependent on position is called a stationary state of the medium. REFLECTION FROM SEA BorroM If water is separated by the plane y = 0 from a medium with a density much greater than its own, the boundary condition which must be fulfilled at this plane is 34 WAVE ACOUSTICS It turns out that a solution synthesized as was equa- tion (124), but with a plus sign in equation (122) in- stead of a minus sign, will satisfy this boundary con- dition. This solution is = sa a oe) (134) ry’ We verify that equation (134) satisfies dp/dy = 0 by differentiating equation (134) with respect to y, and noting that dr/dy = —dr’/dy at y = 0. We now examine the possibility of stationary states for the case where the boundary is a hard sea bottom. Again, we assume a harmonic source so far from the bottom that waves reaching the bottom are plane and we assume perpendicular incidence. Then equation (134) must be replaced by p= of cos anil’ — 4) + cos anil’ =P ] (135) which, by trigonometry, reduces to p = 2a cos an sin 2zft. (136) We easily see that equation (136) also represents a stationary state of our fluid. The nodes of utter silence are situated where cos 27(y/d) disappears, that is, at y =/4, 3d/4, 5d/4,---; the loops of maximum sound intensity are located where cos 2r(y/d) equals +1, that is, at y = 0, d/2, d,---. 2.7 NORMAL MODE THEORY Plane Waves in a Medium with Parallel Plane Boundaries 2.7.1 The problem of sound propagation in a medium bounded on two sides is extremely complicated and cannot be solved in general. The difficulty lies in the fact that the solution must satisfy not only the wave equation, but also the initial conditions and the boundary conditions at each boundary. In Section 2.6 it was shown that certain definite and instructive results could be obtained for the case of a single boundary by considering the case of plane waves and assuming (1) perpendicular incidence and (2) an infinite change in density at the boundary. The result was a standing wave pattern whose geo- metrical properties depended on the wavelength and whose maximum amplitude depended on the energy in the incident wave. We shall keep these two assumptions in this sec- tion and shall first find out under what conditions a stationary wave pattern of any sort can be set up in our bounded medium. The general expression for a standing wave pattern is p = W(y) cos 2nf(t — ¢) (137) where y is any function of y. In other words, equa- tion (137) means that all points of the fluid perform vibrations with the same frequency f and phase con- stant «, but with amplitude ¥(y) depending on the position coordinate y. The immediate problem is to find out what sort of functions y(y) are necessary to make equation (137) a solution of the plane wave equation ep op ae ay? and also a solution of the boundary conditions. on the boundaries y = 0 and y = L. These boundary conditions are either equation (139a), (139b), or (139¢). (138) p = Oat bothy = Oandy =L (139a) 0 p=O0aty =0; ay 7 Oty = (139b) op eens 0 at both y = Oandy = L. (139¢) y It is immediately apparent that the condition for equation (137) to satisfy the plane wave equation (138) is : 2 Ar? dy any =, dy? ? First consider the case of the boundary conditions (139a). The boundary conditions (139a) can be re- stated as (140) ¥(0) = 0, ¥(L) = 0. (141) The problem is thus reduced to the case of finding the solution of an ordinary differential equation with boundary conditions on both ends of an interval OsySL. Equation (140) is a simple differential equation whose general solution is well known to be 2 2 vy) = A sin ¢; Wy) (158) = as our initial condition; and yt) = A; cos 2af,(t — €; ply,t) x ; C08 2af,(t — €;)W,(y) 00) A; cos 2nfje; = c; as the set of solutions to the total problem. While each term in the sum (159) represents a sta- tionary state of vibration, the infinite sum is not sta- tionary, in view of the fact that the terms have different frequencies f;. General Waves in a Medium with Parallel Plane Boundaries 2.70.2 Section 2.7.1 showed how the assumption of a stationary state led to a possible solution which satis- fied the wave equation, the boundary conditions, and the initial conditions. However, the treatment in Section 2.7.1 was restricted to the case of plane waves moving perpendicular to two enclosing plane bound- aries. This section explains how this method may be generalized for the case of general waves in a medium with two parallel plane boundaries. We assume again.a stationary state in the medium, of the form P(x,Y,2,t) = cos 2rft-p(x,y,z). (160) If this solution is substituted into the wave equation (27), we find that w satisfies a partial differential equation of the form Oy dp dy 4m? Ox? a oy? w 02? i ” which is time independent. In addition, y must satisfy the boundary conditions imposed at the bounding planes y = 0 and y = L. The boundary conditions may be of the form (139a), (139b), or (139¢). We shall treat only the case characterized by the conditions (139a). The treatment of the other cases is completely analogous. We attempt to find a solu- tion of equation (161) of the form ¥=0 (161) 2 V(e,yyz) = sin 5 (az) (162) U in which the constant \, must have one of the values 2L ly = ~3j=1,2,3, °°: (163) J to satisfy the boundary conditions. For the G(z,z) we have the equation eG AG 1 1 Ai + ai + =e 6 = 0. (164) Any solution of this equation when multiplied by cos 2nftsin 27y/\, is a solution of the wave equation (27) and also satisfies the boundary conditions (139a). Equation (164) will be satisfied by any plane wave solution in the zz plane with the wavelength \* given by Asay WZ ay * NUN AAG Such a solution will be of the form 2 2 F G= a, cos 3s + a? sin satis = xcosé+zsind- (164a) It is easily verified that this function G satisfies (164). If \, < X, A* is imaginary instead of being real. In that case, the solution should be written in the form @=s be 27) aL yg CI in which 0’ is the real constant given by NORMAL MODE THEORY 37 G is thus the sum of two terms, one increasing ex- ponentially with the distance s from the source, the other decreasing exponentially. The only solutions of this character which have physical significance are those for which the first (increasing) term is zero. If b; were not zero, the sound intensity would increase rapidly with the distance from the source and that is physically impossible. Since the greatest possible value of i, is 2L, or twice the depth, it follows that sound of wavelength greater than 20 will have an exponential rather than a trigonometric solution; because b, must be zero, such sound will suffer an exponential pressure decay with increasing range. It is clear that the longer the wavelength, the more rapid will be the decay. A more detailed discussion of this type of transmission is given in a report by the Naval Research Laboratory [NRL], where the detailed properties of the bottom are taken into account.* The angle @ in the solution (164a) may be chosen arbitrarily. Thus to any value of j in the equation (163) belongs an infinite set of characteristic func- tions. These characteristic functions can be com- bined to satisfy particular initial conditions; how- ever, the rules for their combination are too involved to be presented here. The derivation of the wave equation (27) was based partly on the assumption that the velocity of propa- gation was everywhere the same, in other words, that the medium was homogeneous. Let c be an arbitrary function of (2,y,z). In the ocean, the variation in sound velocity is due mainly to the variation in water temperature with depth. To assume that the velocity is variable, amounts for most practical purposes to assuming that c in equation (27) is now a function of position. The method of normal modes can be applied to find a solution in that case just as in the case of constant sound velocity. As before, we assume a stationary state of form (160). Substituting equation (160) into the wave equation, we get as the time-independent differential equation the following: Py Ob ay 4af? Ox? zs ay? iy ry a (xyz) y=0 (165) which differs from equation (161) only in that c is now variable. The solution of equation (165) satis- fying the imposed initial and boundary conditions can be found as before by the superposition of an infinite number of normal modes; in this case of vari- able c, however, the computation of the characteristic values and functions is more troublesome. An ap- plication of this type of analysis is discussed in Section 3.7. Intensity as a Function of Phase Distribution 2.7.3 Whenever the sound source is harmonic, the pres- sure distribution resulting from given initial and boundary conditions can be written in the form Dade a | aes) a and e being real functions of x,y, and z. For some purposes it is convenient to set V 97) e(X,Y,2) = ue 2 (167) so that equation (166) becomes p= dep oe e o (168) or, explicitly for the real pressure, V p = a(x,y,2) cos anil ¢ — ri (169) C Since we know from Section 2.4.2 that I = pu, we must derive an expression for pu. This is done by making use of equation (44), relating the derivatives of p and u,: = === 170 ot p Ox (wag) From equation (169), then QrfoV a TD Gin ery ee ee Os Cy Ox Ox where i = ani — Zl (171) At) Therefore, from equation (170), x 2 1 ) Gite pa ee wok s H— ot p Co OX p Ox Integrating this, we get a OV 1 da = — in H—- 172 U. ae cos H a Onfo sin aE (172) From equations (172) and (169), we obtain ae OV a da = — cos? H-— — ——~sinH cosH—-: (173 DUz oe cos? H a oer sin H cos ae (173) The average energy flow in the x direction [,, at the point 2,y,z, is just the time average of equation (173) over a complete period. The time average of the 38 WAVE ACOUSTICS square of the cosine is 14; the time average of the product of the sine and cosine is zero. Thus, | EN, ie 2pco) Ox 2\0 bea) 2pco oy n= (en) 2pcyJ dz Since I = (12 + I; + I2)3, we have the following ex- pression for the intensity fers) ray ele - i 2pcoL \Ox oy Oz The relations (174) and (175) will be found useful in Chapter 3 when the equivalence of wave acoustics and ray acoustics is investigated. Similarly, (174) (175) 2.8 PRINCIPLE OF RECIPROCITY The principle of reciprocity makes a statement con- cerning the interchangeability of source and receiver. Very crudely, the import of the statement is that if in a given situation the locations and orientations of source and receiver are interchanged, the sound pres- sure measured at the receiver will be the same as be- fore. To be true, under the most general conditions, this statement has to be qualified in detail. The fol- lowing is an attempt to formulate the General Reciprocity Principle. Assume that a source of a given directivity pattern b and a receiver of a different directivity pattern b’ are placed in a medium with a particular distribution of sound velocity c(z,y,z), enclosed by boundaries of any given shape with any particular boundary con- ditions; let the output of the source on its axis be given by an amplitude A at one yard. The receiver will then record some pressure amplitude, correspond- ing to the amplitude B on its axis. Now let the source be replaced by a receiver having the same orientation of its axis and having the directivity pattern 6’; assume also that the receiver is replaced by a source which has the same orientation of its axis, the direc- tivity pattern b, and the output A on its axis. Then the new receiver will again register a pressure equivalent to that of a sound wave incident on its axis with an amplitude B. The proof of this theorem is difficult in the general case, and will not be reproduced here. Instead, we shall give the exact proof for the simple case of a plane source emitting plane waves into a medium that satisfies boundary conditions of the type (139). We shall then indicate, roughly, how the proof can be generalized. Suppose the pressure is a function of y and ¢ only. The medium may be inhomogeneous, but both the density and bulk modulus are assumed to be a func- tion of y only. Then, the sound velocity will be a function of y. The wave equation for this case there- fore reduces to FD © (Ze a ee (176) and its solution, by equation (160), will be ply,t) = W(y) cos 2aft (177) where ¥(y) is obtained from dy Arf? 4? ke =0, R= = ==: ll get Oe AG ae We assume the medium is bounded by the planes y = 0 and y = L and satisfies boundary conditions of the type (139a), (139b), or (139c). We assume that the plane y = a is a source of sound. If by Wa(y) is meant the function defining the pressure amplitudes at every point of the medium, in- cluding near y = a, then y, satisfies equation (178) everywhere except near y = a. That is, it satisfies dba dy? + Py)pa = Aly) (179) where A(y) is very large in the immediate neighbor- hood of y = a, and zero everywhere else. Then the magnitude of the plane source Sz, located at y = a, will be defined by a+6 L s.= [ Awdy =f Away where it is understood that S, is the limit of the integral as 6 approaches zero. In the same way, we define a plane source at y =b by assuming an amplitude function ¥(y) which satisfies equation (178) everywhere except near y = b, that is, it satisfies t+ BU) = BY) 'Yy (180) (181) where B(y) is very large in the immediate neighbor- hood of y = b, and zero everywhere else. Then the magnitude of the source at y = b will be b+5 Biy)dy. b=5 Sp = (182) INADEQUACY OF WAVE ACOUSTICS 39 By multiplying equation (181) by Wa, and equation (179) by ys, and by subtracting the latter result from the former, we get which may be rewritten as d( dr ae —{ y.— — Ww] = u.B — pA. aC Fi Yo ti v Yo Equation (183) holds if y. and y are arbitrary functions of y, and A and B are defined by equations (179), (181). We can integrate (183) over the entire extension of the fluid between y = 0 and y = L, and get (183) dy, a i I: e—— — hw] = aB — dy. 184 (v ih Voy F B (v Me y. (184) Since ¥. and y each satisfy some combination of vy = 0 or dy/dy = 0 at y = 0, and y = L, the left- hand side vanishes identically, and LL f (YaB — YrA)dy = 0. (185) Equation (185) is valid for all functions of Ya and yy and satisfies equations (179), (181), and the boundary conditions (139). Since B = 0 except near y = b, by equation (181), +6 L J, vaBay = yet), Bay = yald)Ss 0 b—6 because of equation (182). Similarly, L a+6 it Ady = wsla) { Ady = W(a)S.. 0 a—6 Therefore, equation (185) becomes Spa(b) a Says(a) = 0. (186) If both sources are of equal magnitude, then S. = S,, and ¥a(b) = (a). (187) That is, if two plane sources of equal strength are emitting plane waves into a ‘stratified’ medium, where the sound velocity obeys an arbitrary law, and where boundary conditions are of the form of (189), then the first source (at y = a) produces at y = b the same acoustic pressure which the source at y = 6 would produce at y = a. It is interesting to note that we have proved equa- tion (187) without solving explicitly for the pressure amplitudes. We remember that even in this one- dimensional problem the equation (178) with bound- ary conditions usually cannot be solved for y in terms of elementary functions if the sound velocity is an arbitrary function of y. However, we found we could prove equation (187) merely by assuming that y. and ¥ were solutions of the wave equation with initial and boundary conditions, and by following up the consequences of that assumption. In the general case, in which we no longer assume perpendicular incidence on plane parallel boundaries, the proof is more complex. Instead of equation (178), we have the more complicated form of (165). Equa- tions (179) and (181) must be replaced by equations with the same left-hand sides as equation (165), but with right-hand sides which are different from zero only in the immediate vicinity of a particular point, (2a,Ya,2a) and (25,Yo,2s), respectively. The distribution of the functions A and B about these two points de- termines the directivity of the source considered. The integration (184) must be replaced by a volume integral, or rather, by an infinite series of volume integrals to account fully for the two directivity patterns, the left-hand sides of which can be shown to vanish. From there on, the proof runs similarly to the plane case. The foregoing remarks have applied only to the case of propagation in a perfect fluid. It can be shown that the reciprocity principle holds, with additional qualifications, for propagation in a viscous fluid also. 2.9 INADEQUACY OF WAVE ACOUSTICS In this chapter, we have set up a schematic picture of the transmission of sound in the ocean, and pro- ceeded to derive a rigorous mathematical description of our schematic picture. Unfortunately, the results obtained cannot be used directly as a basis for the prediction of the performance of sonar gear. The schematic picture is not nearly complete enough from a purely physical point of view; furthermore, even the simplified schematic picture can be solved rigor- ously only for simple cases; and in the cases where solutions are possible, the calculations are very difficult. The physical picture is inadequate on several counts. For one thing, boundary conditions like p = 0 of dp/dy = 0 at the boundaries are only a vague description of what actually happens at the bound- aries. The surface is not a perfect plane, but is usually disturbed and uneven, with the result that even plane waves are not reflected according to the law of reflec- 40 WAVE ACOUSTICS tion; they are partly reflected in a direction depend- ing on the direction of the surface and also partly scattered in all directions. Neither does the bottom obey the postulated conditions; it is never infinitely. dense; at best it is rocky; at worst it is so muddy that it can hardly be called a boundary. The medium itself, the sea water, is not completely described by its density and its bulk modulus. There are many inhomogeneities in the sea volume, such as bubbles, floating plant and animal life, fish, and others. For all we know, such inhomogeneities may produce a very important part of the observed transmission loss, perhaps as important a part as the variations in sound velocity. The mathematical difficulties should be apparent to anyone who has even glanced at the remainder of the chapter. Even when the boundary conditions can be formulated exactly, and initial conditions are simple, the exact solution of the problem usually can- not be presented. In the general case, it can be proved that a solution exists and is unique, but the solution cannot be written in a formula which would provide a practical basis for intensity calculations. The primary benefit of the rigorous approach is that one can derive certain very useful properties of the sound field, such as the principle of reciprocity and the de- pendence of intensity on the phase distribution, with- out going into the exact solution itself. How, then, are we going to predict the sound field intensity? We certainly cannot go out and measure the intensity in all cases; such measurements are time-consuming, and provide information only about that particular part of the ocean at that particular time. We should have some method for estimating the intensity field, at least qualitatively, so that the observed intensity data can at least be interpreted according to a frame of reference; mere data without some reference to a theoretical scheme are useless. In the theory of light, this problem was solved by using the methods of ray optics. The fundamental problems about optical instruments, like those for tele- scopes, can be solved by ray tracing methods without resorting to the exact solution of the wave equation. This ray theory is based on the assumption that light energy is transmitted along curved paths, called rays, which are straight lines in all parts of the medium where the velocity of light is constant, and which curve according to certain rules in parts where the velocity of light is changing. This light-ray theory is valid in all cases where obstacles and openings in the path of the radiation are much greater in size than the wavelength of light. In the next chapter, we shall describe the applica- tion of ray methods to underwater sound transmis- sion and shall also examine the validity of this type of approximation. Chapter 3 RAY ACOUSTICS ae 2 was devoted to the rigorous computa- tion of the acoustic pressure p as a function of position in the fluid and of time. In situations where the acoustic pressure could be determined the sound intensity at an arbitrary spot and at an arbitrary time could be calculated. However, it was noted that, in most situations involving initial and boundary conditions similar to those met with in actual sound transmission in the ocean, this computation was at best very laborious and at worst completely impos- sible to carry out. Ray acoustics provides a more con- venient though less rigorous approach. In the study of sound the ray concept has not played so great a role as in optics. The reason for this is that the wavelengths of most audible sounds are not small compared to the obstacles in the path of the sound. Consequently, sounds audible to the ear do not travel straight-line or nearly straight-line paths; they bend around corners and fill almost all of any space into which they are directed. However, for the short wavelengths used in supersonic sound, the ray methods have an important, if limited, application. This chapter elaborates the theory of sound rays, describes the computation of sound intensity from the tay pattern, and finally, examines the conditions under which ray intensities may be expected to ap- proximate the intensities calculated according to the rigorous methods of the second chapter. 3.1 WAVE FRONTS AND RAYS Bale Spherical Waves The wave equation was solved explicitly for p in one very important case: an impulse sent out by a point source into a homogeneous medium under the assumption of spherical symmetry. The pressure as a function of time and space was found to be SA ovate In this expression, r is the distance from the source, and the function f(t — r/c) is determined by the out- put of the source. Specifically, the source can be characterized by the statement that, while the pres- sure itself at the source is infinite, the product rp in the immediate vicinity of the source has a finite value at every instant, namely f(t). Obviously, this function f(t — r/c) determines when a pulse emitted by the source at a particular instant will arrive at a given point of space. If the pulse should, for instance, have started in time at some instant ¢ = e so that for all values of the argu- ment less than ¢ the function f(¢) vanishes, then the onset of the disturbance at a distance r from the source will be observed at the time r t=e+-- c Likewise, we find that the front of the pulse will have reached at the time ¢ a distance r, given by r= c(t — «). (2) What has just been stated about the front of the pulse might just as well have been said about any other specified part of the pulse; only, e would in that event characterize the time at which the specified part of the pulse was radiated by the source. What has been called, vaguely, part of the pulse, is often more concisely called phase, particularly in connec- tion with harmonic pulses. The term ¢ then char- acterizes the phase of the pulse considered and is ordinarily referred to as a phase constant. The surface defined by equation (2) is, of course, a sphere of radius c(t — e) at the time ¢. As the time increases, the radius of the sphere increases at the rate of c units per second. The surface of this sphere of constant phase is called the wave front. The energy represented by the disturbance of equilibrium condi- tions clearly spreads out radially from the source. We may focus attention on the direction of energy flow by mentally drawing an infinite number of radii from the source to the wave front. These radii may be re- garded as representing the paths of energy flow and 41 42 RAY ACOUSTICS may be called sownd rays, in analogy with light rays. Sound energy may be regarded as traveling out along these rays with the speed c. The wave fronts assume in this description the secondary role of surfaces everywhere normal to the rays. An individual sound ray cannot exist as a physical phenomenon. An isolated sound ray would mean a state of the fluid where the condensation was confined to the immediate neighborhood of a particular straight line. Beams of narrow cross section can be produced by directing a wave front onto a very nar- row slit; but if the size of the slit becomes comparable to the wavelength of the sound, the sound leaving the slit is not a narrow beam, but a cone. This phenome- non is called diffraction, and will be discussed in Section 3.7. It is mentioned here only to indicate that the concept of a sound ray refers not to the propagation of a narrow beam with sharp edges, but merely to the direction of propagation of actual wave fronts. Spherical wave fronts represent only one particular case of sound propagation, even in an infinite fluid. First, the wave front can be spherical only if the initial disturbance has no preferential direction (a vibrating bubble satisfies this condition). Second, the expanding surface remains a sphere concentric with the origin only if the sound velocity c is either con- stant throughout the fluid, or has spherical symmetry about the sound source. The wave factor f(t — r/c) in equation (1) is re- sponsible for the conclusion that the disturbance is propagated with the velocity c. The remaining factor, 1/r, is called the amplitude factor since it is responsi- ble for the decrease in sound intensity as the distance from the source increases. The rate at which sound intensity is weakened with distance can be easily computed by using the concept of rays as carriers of sound energy, provided we assume that energy is generated only at the source and then flows through space without gain or loss. For reasons of symmetry, the energy flow from the source must take place along the radial sound rays. There will be a definite number of rays inside a unit solid angle. These rays will intercept an area of 1 sq ft on a sphere of radius 1 ft whose center is at the source, an area of 4 sq ft on a sphere of radius 2 ft, and generally r? sq ft on a sphere of radius r. Since the total energy flow is the same for all these spherical surfaces, the energy flow per unit area, or sound intensity, must be inversely proportional to the square of the distance of the unit area from the source. 3.1.2 General Waves The frontal attack on the wave equation was the solution of the boundary problem by the method of normal modes. This method was found to be too com- plicated. It was shown in Section 3.1.1 that the method of sound rays gave a simple and plausible ac- count of sound propagation for the case of spherical symmetry. A natural approach to the general prob- lem would be to generalize the definition of sound rays, and see if light could thereby be thrown on the general case of variable sound velocity and arbitrary initial distributions of p. First we must generalize the definition of wave fronts. In what follows we shall restrict ourselves to harmonic sound waves, that is, sound waves which have been produced by a sound source which under- goes single-frequency harmonic vibrations. In ac- cordance with Section 2.4.3, the pressure at any point inside the fluid can be represented as the real part of an expression having the form 20 (x,y,2,t) p = A(z,y,z)e (3) in which the angle @ at each point in space increases linearly with time, 6= 2nflt nee e(x,y,2) ]. (4) We shall now call a wave front all those points at which the phase angle 6 has a specified value, say 4. At any time t, this surface is defined by the equation €(2,Y,2) St om (5) For later convenience, we shall replace e(x,y,z) by an expression W(z,y,z)/co, in which ¢ is the velocity of sound under certain designated standard conditions. Equation (8) then takes the form p = A(z,y,2) eerie Teu), (6) both A and W being real functions of the space co- ordinates. The defining equation (5) of an individual wave front assumes the form where = = 0 The term ¢ has different values for different wave fronts, but is constant both in space and in time for a given wave front. The function W clearly has the dimension of a length. In order to make use of the concept of sound rays to describe the energy propagated by such generalized FUNDAMENTAL EQUATIONS 43 wave fronts, we must also generalize the definition of a ray. We can no longer assume that the rays are straight lines since we concede the possibility of re- fraction and reflection. We shall, however, retain the property that the rays are everywhere perpendicular to the wave fronts. It is, of course, by no means obvious that the results of this new approach will agree with results from a direct solution of the wave equation plus initial and boundary conditions. A comparison between the results from the ray pattern approach and the results from a rigorous treatment. of the wave equation will be carried out in Section 3.6 once the ray method has been fully described. It will be found that in many practical situations these two approaches lead to similar results. Geometrically, the rays and successive wave fronts ean be constructed as in Figure 1. The wave front at w=c dt Figure 1. Huyghens’ method for constructing suc- cessive wave fronts. time ¢ = 0 (whose equation is given by W = — colo) is first drawn. In order to determine the wave front at the time dé, the small ray elements are drawn as straight-line segments perpendicular to the initial wave front, as at (21,y1,21). In the time dé, the end point of the ray starting at (71,y1,2:) will have pro- gressed to a point a distance c dé from the initial wave front, where c is the velocity at the point (z1,41,21). If this process is carried through for all the points on the initial wave surface, the end points of all the small ray elements will determine a second surface, which may be regarded as the wave front at the time dt. By performing this process many times, the wave front can be obtained at any time t. This method of determining wave fronts by gradually widening an initial wave front was first suggested by the Dutch physicist, Huyghens, in the seventeenth century, for the solution of problems in optics. ae FUNDAMENTAL EQUATIONS Differential Equation of the Wave Fronts 3.2.1 Since the construction of wave fronts described in the preceding section is purely geometrical, it must be reformulated in mathematical terms for use in an algebraic analysis of the sort we are carrying out. Figure 2. Differential ray path. Let P in Figure 2 be any point on the wave front at time ¢. The equation of the wave front is given by equation (7). Let the coordinates of P be (2,y,z); let PP’ be the ray element emanating from P at the end of a time interval dé; and let @,8,y be the direction cosines of PP’. Then the coordinates of P’ are (x + acdt, y + Bcdt, z + ycdt). Further, the wave front at the time t + dt is given by the equation W(x + acdt, y + Bcdt, 2 + yedt) = co(t — to + dt). (8) If cdt is assumed to. be very small, the left-hand side of equation (8) is very nearly ae to Gs) ow W(z,y,z) + (2¥ =F S + ot If we substitute this expression ae equation (8), and use equation gis te (8) reduces to ow 4 Bs tt "5 ==, (9) c The ates ele cosines se tte next be eliminated from equation (9). It is a well-known theorem! of analytical geometry that the direction cosines of the normal to the surface W = constant at the point (x,y,z) satisfy the proportion ow oW OW SIE aay lay Wor 44 RAY ACOUSTICS Because the sum of the squares of the direction cosines equals unity, ef +P +y= 1, the constant of proportionality in the multiple pro- portion above can be determined, and we obtain ow \2 aw \2 ow \2 |-20W «= ((2) +(%) +(2y] @z_ (10) 20W alee heey esd Nae © ~ LN\ az oy ama Woz By De By substituting these values of @,8,y into equation (9) and squaring both sides, Sk eae Ox oy dz C?(x,Y,2) If we define n, the index of refraction, by n(x,Y,2) = = ’ (12) c(x,Y,2) equation (11) becomes (2) +B) <() eran. Bus + oy =e dz =n (x,y,z). ( ) Equation (13), often called the eckonal equation, is the fundamental equation of ray acoustics. It is a partial differential equation satisfied by all functions W which can define wave fronts according to equa- tion (7). Initial conditions for equation (13) are usually of the form that W has the value zero for all points (x,y,z) on a particular surface. Once the solution W of equation (13) has been found, the ray pattern can easily be drawn. The direc- tion cosines of the rays at every point of space can be computed from equation (10); more simply, if equations (10) and (13) are combined, (14) Later we shall eliminate the function W from equa- tion (14), and derive a set of ordinary differential equations, which together determine the course of each individual ray. First, however, we shall give a simple example illustrating how the ray pattern may be calculated from the partial differential equation (18) for the wave fronts. Let us consider the special case where the sound velocity c depends only on the vertical depth co- ordinate y. Thus, the sound velocity is assumed constant everywhere on a particular horizontal plane. We shall examine only the ray pattern in one vertical plane, which we can take as the xy plane. Then equa- tion (13) reduces to (ie Ox a oy ae: To find a simple solution of equation (15), we assume that W (x,y) is the sum of a function of x and a function of y. W(a,y) = Wi(x) + Waly). Substituting this expression into equation (15), we obtain oe) ey es ( AW Tae (y). To obtain a family of solutions, we put dW,/dz = k, where & is an arbitrary constant. Then, the differ- ential equation will be satisfied if dW. dy Therefore, in view of the assumed nature of W, the equation (15) will be satisfied by all functions W de- fined by Wey) =ke + [ Ve@—Fdy (06) where k is any constant. A particular choice of k corresponds to a particular solution W(a,y) and therefore to a particular set of wave fronts (7). The direction cosines of the rays, corresponding to this choice of k, can be calculated from equations (14) and (16). (15) = Vii) —B; Wr = f Vit) — Fay. We use these expressions to obtain the equations y = y(z) of the sound rays. If we denote by dy/dz the slope of the direction of the ray at the point (x,y), then ou B P= |/e@ 4, This equation integrates immediately to eee (17) where 2 is an arbitrary constant. Regard the k as fixed, and the 2 as variable; then equation (17) gives an infinite set of curves which satisfy the definition of rays. FUNDAMENTAL EQUATIONS 45 If n is a constant, that is, if the sound velocity is independent of depth, the rays (17) are clearly straight lines. 3.2.2 Differential Equations of Rays It may be argued that the replacement of the wave equation by the ray treatment as represented by the differential equation (13) has little to recommend it- self. It appears that one difficult partial differential equation has merely been replaced by another, which might resist attempts at solution as effectively as the first one. Figure 3. Specification of direction of ray element ds by direction cosines. Further examination shows, however, that the new equatior (13) has two properties which tend to sim- plify its solution. First, equation (13) contains no time derivatives. This means that it describes the propagation of a disturbance in terms independent of the frequencies which make up this disturbance. Second, it is possible to set up ordinary differential equations that describe the path of individual rays; the latter equations will be derived in this section. We start with the equations (14), from which we proceed to eliminate W. This can easily be done by use of the formulas 0?W/dxdy = 0?W/dydz, etc. By differentiating the first equation of (14) with respect to y, the second with respect to z, and by equating the results, a relation between a, 8, and 7 is obtained. Proceeding similarly with the other equations, we obtain the following relationships which must hold at any point of the ray pattern: (na) ds d(nB) A(na) 4 d(ny) _ 9(nB) _ a(ny) oy GR Ge aeAlgg WT Liga (18) These equations can be developed further to yield the changes of a, 8, and y along the path of an indi- vidual ray. If the arc length along the ray path from a given starting point is denoted by s, we have d(na) O(na)dx | d(na)dy | A(na) dz i enn aah a aaa Fgh! We see from Figure 3 that dx dy dz aes eee aaa ts (20) Thus equation (19) turns into d(na) 0(na) 0(na) 0(na) Rds airvtdrvel ayw dz’ which, upon using the relations (18), becomes d(na) x 0(na) 0(na) O(na) ase Ri Ox Ox Ox to} = (@L eae Ox ae gt?) + (2% 40% 45% Hes ACY The first parenthesis equals unity, because it is the sum of squares of direction cosines; while the second parenthesis, which is equal to one-half times the derivative of the first one, vanishes. Thus equation (21) simplifies to d(na) dn ds Ox After similar calculations are carried out for d(nB)/ds and d(ny)/ds, we get the following set of three ordi- nary differential equations: d(na) _ dn _ d(nB) _ on _ d(ny) _ an (22) ds dz’ ds dy’ ds ~~ Oz It is understood that n, the index of refraction, is a given function of z,y,z. We now deduce an important result for the special case where the sound velocity is a function of the vertical depth coordinate y alone. We shall show that for this case the entire path of an individual ray lies in a plane determined by the vertical line through the projector and the initial direction of the ray. Let the origin of coordinates be taken at the pro- jector, and let the direction cosines of a ray leaving 46 RAY ACOUSTICS the projector be ao,80,70, as in Figure 4. Since nde- in detail. It is intuitively obvious that if we carry pends only on y, equations (22) simplify to through the solution for the zy plane, then the ray d(na) d(nB) dn d(ny) pattern in any other plane through the vertical (y) Tage ee = 0. (23) is will be identical in si a ah ay ade axis e identical in size and shape. Since the water depth increases in the downward direction, we shall take the y axis positive downward. We shall denote the angle which a direction in the zy en plane makes with the positive x direction by 6, as in a Figure 5. To avoid ambiguity, we must specify care- along the ray. Then, the initial direction of the ray 1s (a0,80,Ka). Thus, along any individual ray we have na = con- stant, ny = constant, which in turn implies B LG Ficure 5. Change in ray direction over ray element BRE fully the sign of the angle 9. We shall be concerned only with rays moving in the direction of increasing Figure 4. Change of ray direction between point oe a ° ine woes, te une mig a une guises Ht tne pa (G, 0, 0 ) and point (2, y, 2). ° is gaining depth with increasing range, we give the angle 6 a positive sign; while if the ray is losing depth The direction at a general point P along the ray With increasing range, we give # a negative sign. These will be characterized by the direction cosines a,8,«a conventions, illustrated in Figure 6, enable us to use because of the equations (23). It can easily be shown , by the methods of analytical geometry that the nor- mal to the plane determined by OY (direction cosines 0,1,0) and OA (direction cosines a,f,kao) has the direction cosines [Vie +1,0,-1/V"+1. The direction of the ray at P is characterized by the direc- tion cosines a,8,«a; thus the ray direction at P is perpendicular to the normal to the plane AOY ; hence the segment PB lies in that plane. Since P was any point on the ray, the entire ray must lie in the plane AOY. z CLIMBING RAY Oo NEGATIVE DESCENDING RAY 6 POSITIVE y ae RAY PATHS FOR VERTICAL Ficure 6. Conventions fixing sign of 0. VELOCITY GRADIENTS the following relations both for climbing and de- 4 i f scending rays: 3.3.1 Derivation of the Equations eo Ses0s @= ends 7 SO. (24) of Ray Paths Since the sound velocity is assumed to depend We now solve the equations (22) for the special only on y, we have case where the sound velocity depends only on the an _ on : 4 é : : ahi Needle") vertical depth coordinate y and discuss this solution Ox z : RAY PATHS FOR VERTICAL VELOCITY GRADIENTS 47 and by reason of relations (24) the equations (22) re- duce to d(n cos 0) _ Gi d(nsin@) dn ds et ai ds From the first equations it follows that n cos 6 has a constant value along a particular single ray. That is, if P and P’ are two points on the ray, then = (25) dy a Co — cos 0 = — cos 0. c c If, in particular, P is located at the depth where c(y) = co, and if @ is the direction of the ray at this point, this equation becomes = — = —- (26) Equation (26) is identical in form with Snell’s law in optics. The second equation in (24) is used to compute the curvature of the ray at any point. The curvature of a curve at a point on it is defined as d6/ds, the angle through which the tangent turns as one travels along the curve for unit distance. Because of our conven- tions for the sign of the direction angle 0, upward bending is always associated with negative curvature, and downward bending with positive curvature. From the relations (25), we have dn d(sin @) . dn dy x ds al eee d(sin 6) dé dn dy SS = (= 27 d@ ds Sika dy ds Co dé dn g— in2 6 = nN COS Fe + sin fi since dy/ds = sin 6, from Figure 5. The solution of equation (27) for d6/ds yields dé idn d(log n) = S=— COS0(= ———— Co ds ndy SO. (28) Since log n = log @ — logc, equation (28) can be rewritten as dé ___—— d(logc) ds dy i We can use equation (29) to describe, qualitatively, what happens when a ray travels to a layer just above it (dy < 0) of different sound velocity. If the new layer has higher sound velocity, the curvature d0/ds has a positive sign, and the ray is bent downward. If the layer just above has lower sound velocity, the curvature d@/ds is negative, and the ray is bent up- ward. We get the opposite result if the ray is traveling os 0. (29) to a layer just below it (dy > 0) of different sound velocity. Thus we can say, in general, that a ray en- tering a layer of higher sound velocity is bent away from the layer, and a ray entering a layer of lower sound velocity is bent into the layer. In the open ocean the vertical velocity gradient usually falls into one of two types, depending on the temperature-depth variation. If the temperature does not depend on the depth, the velocity is determined by the pressure, which increases with depth; there- fore, in such isothermal water the sound velocity in- creases gradually with depth, and sound rays should possess slight upward bending. Another common case has the temperature decreasing with depth. Since velocity is much more sensitive to changes in tem- perature than to changes in pressure, the velocity will also decrease with depth, and the sound rays will bend strongly downward. The water temperature rarely increases with depth; when it does, the sound rays are bent strongly upward. We shall now examine, quantitatively, the change of curvature along an individual ray, and derive cer- tain relationships between the range and depth reached at time ¢ by a ray leaving the projector at a certain angle. Assume that the projector is situated at the depth where c = qm; thus the ray may be characterized by its initial angle 4 at the projector. Because of equation (26), equation (29) becomes dc cos 4 (30) The advantage of the representation (30) is that it gives the curvature along a single ray as a function of. dc/dy only, since is constant for that particular ray. We consider, in particular, the case where the velocity gradient has the constant value a; that is, c=@+ ay, (31) if the origin of coordinates is taken at the projector. At all points on the ray, in view of equation (30), dé a COS % blr er ag (32) We see from equation (32) that the curvature is constant along the ray; this means that the ray must be an arc of a circle. As the radius of curvature is the reciprocal of the curvature d@/ds, the radius r of this circle must be given by Co a COS % (33) If a is positive, the curvature (32) is negative, and 48 RAY ACOUSTICS the circular arc bends upward; but if a is negative, the circular arc bends downward. We can determine the center of the circle defining the ray by a simple geometrical construction. Figure 7 shows the path of a ray leaving the projector at the a VELOCITY GRADIENT IN UNIT OF VELOCITY PER UNIT OF LENGTH Figure 7. Geometrical construction of ray path. angle @ into a medium of constant negative velocity gradient. The center of the circle is obtained by fol- lowing the perpendicular to PQ down through the medium a distance /a cos 6. It is a simple conse- quence of the geometry of the situation that this center will lie on the horizontal line a distance c/a below the projector. For, from the illustration, RO = (e@/a)(cos 0:/cos 6), and 4 clearly equals 4. Similarly, if the constant velocity gradient is positive so that the rays are bent upward, the centers of the defining circles lie on a horizontal line a distance o/a above the projector. It is easily shown that the dashed horizontal “‘line of centers’’ in Figure 7 is at the depth where the velocity c would equal zero if the assumed .linear gradient extended indefinitely. An approximate solution in the general case where cis an arbitrary function of y can be obtained by re- peated use of the solution for constant gradient. Even a complicated velocity-depth curve can be closely approximated, as in Figure 8, by dividing the depth interval into a relatively small number of segments in each of which the velocity is assumed to change linearly with depth. Within each layer the ray path is an arc of a circle; and the total ray path is a con- secutive series of such arcs. VELOCITY OEPTH —— TRUE CURVE ———APPROXIMATING CURVE Figure 8. Approximating velocity-depth curve by a succession of linear gradients. In practice, the path of the ray cannot be conven- iently plotted as a sum of circular arcs because the horizontal ranges are much greater than the depths of interest, and therefore their scale must be contracted. Instead, the ray is usually traced by calculating the angles 6; and 6, at which it enters and leaves a given layer, and the horizontal distance it travels in the layer. This calculation is illustrated in Figure 9, Figure 9. Ray path in layer of linear gradient. where the top of the layer is at depth y:, the bottom is at depth y2, and the thickness of the layer is h. The ray leaves the projector at an angle %, enters the layer at the angle ;, and leaves the layer at the angle 6. Then, by equation (26), E cos | 6, = are cos “Tage (34) EE cos =] 62= arc cos eeu where c(y:) and c(y2) are calculated from equation (31). RAY PATHS FOR VERTICAL VELOCITY GRADIENTS 49 Consider now the chord P;P2 converting the end points of the circular are. The direction @ of this chord is bysimple plane geometry 19(6; + 42); and its length is therefore given by h sin 3(0; + 62) The increase in horizontal range due to the passage of the ray through the layer is P,P. cos 6, or [22 = (35) (36) This result may be applied to the following prob- lem. Suppose we have a sum of layers of the sort shown in Figure 10; and we wish to find the horizontal — + Range in layer = h cot 3(6; + @). VELOCITY PROJECTOR DEPTH DEPTH Figure 10. Succession of linear gradients. range attained by the time the ray reaches the depth H below the projector. We let the bottom layer ex- tend just to the depth H; suppose this is the third layer below the projector. We know 6 and we cal- culate 61,42,0; by the relations (34). Then the hori- zontal range to the depth A will be the sum of terms of the form (86): Horizontal range to H = hi cot $(@) + 0:1) + he cot 3(01 + 0) + hs cot ¢(62 +63). (37) The inverse problem is a little more complicated. Suppose we wish to find the depth reached by a ray of initial direction @ by the time it has traveled a horizontal distance R in a stratified medium that consists of layers of thickness h1,h2,h3, etc. We cal- culate the range R; in the first layer, R2 in the second layer, and so on, until the sum of these partial ranges is greater than A: R, + R. + R3 < R Ri+h,+ Rh + Rs > R. Then the depth the ray reaches at range R& will be greater than hy + he + hg and less than hy + ho + hs + hy. Its value may be obtained with sufficient accuracy by interpolation. The ray-tracing methods described in this section are too cumbersome to use in practice. A number of devices have been developed to facilitate the plotting of rays bent by known velocity gradients; these de- vices will be discussed in Section 3.5.1. 3.3.2 Application to Depth Correction The ray-tracing methods described in Section 3.3.1 have had a valuable application in correcting the depths determined by the use of tilting beam sonar gear. These instruments are used on surface vessels to determine the true depths of submerged sub- marines. They employ a transducer with good verti- cal directivity and tiltable in the vertical plane, which sends out echo-ranging pings at various angles of de= pression. When velocity gradients are absent, the sound rays are straight lines, and the true depth of the target is just the slant range times the sine of the angle of depression at the orientation for which the target returns the loudest echo. The depth finder computes this latter product automatically. When velocity gradients are present, however, this simple method often leads to serious underestimation of the target depth. In this section, we shall describe a method for estimating the error produced. For simplicity, we assume that the projector is at the surface, as in Figure 11. Let the apparent target PROJECTOR Yo APPARENT TARGET POSITION TARGET Figure 11. Error in target position due to refraction. angle be 4, and the apparent target depth indicated by the depth finder be Yo; the true depth of the target is designated by Y. Our aim is to derive an expression for Y — Yo in terms of the way the sound velocity c varies with depth. Let y represent the actual depth attained by the sound ray at time f, and y represent the apparent depth reached by the ray. Then Yo = Co Sin Oot (38) 50 RAY ACOUSTICS where c is the velocity of sound at the projector. Since the actual ray path is curved, all we can say is that y is some function of the surface velocity, the apparent angle %, and the velocity-depth pattern. We can, however, give an exact expression for the increase in y during the time interval dé. If c is the sound velocity at the depth y, and @ is the inclination of the ray at the depth y, we have dy = csin 6dt. (39) We now take differentials of both sides of equation (38), obtaining (40) By dividing equation (40) by equation (39) to elimi- nate the time, dyo = c sin bode. dyyo sin % = = SS 41 dy c sin 0 . We eliminate @ from equation (41) by using Snell’s law (26): sin @ = V1 — co 6 =V 1 — [(c/co) cos 4) so that equation (41) becomes (42) The quantity c/c) represents the variation of velocity with depth. If ¢ is defined by the relationship Cc == lee Co then e represents the relative change in velocity as a function of depth. Rewriting equation (42) in terms of «, we obtain dyo _ sin 69 dy (1+6[1 — (1 +6)? cos? } = {(1 + )? ese? [1 — (1 + ©)? cos? ]} = [1+ 2(1 — cot?) e+ (1 — 5 cot?) & — 4 cot? & & — cot? & 4]? upon multiplying out and collecting terms. Since percentage changes of sound velocity are always small in the sea, the quantity eis a very small fraction, almost always less than 0.02. Consequently, the terms with é or higher powers of e« in equation (43) may safely be neglected, giving approximately (43) d 1 a2 = fil Sl = ea aT dy If we define w by w = 2(1 — cot? 6), (44) equation (44) may conveniently be rewritten as ee) 7 @4& we)? It may be noted that although ¢ is always much less than one, we is not necessarily so. We now integrate both sides of equation (45) between 0 and the true depth Y, obtaining Y to | o (1 + we)* The expression (46) provides a functional relation- ship between the true depth Y, the apparent depth Yo, and the velocity-depth variation e(y). In any practical situation it is possible to determine Yo and 4, and e(y) can be deduced from the temperature- depth variation indicated by the bathythermograph slide. Thus, all quantities in equation (46) are known except the true depth Y, which occurs only as the upper limit of integration. The value of Y may be estimated by using trial values for the upper limit of integration and by seeing which trial value yields a value for the definite integral closest to the known left-hand side Yo. If the velocity-depth variation is not simple, the integrals must be evaluated by numer- ical integrations; but if «(y) is a linear function of y, or a succession of linear functions of y, the integrals can easily be evaluated exactly. Tables have been developed by the use of such methods for the depth errors expected in the presence of various types of velocity gradients. In preparing these tables it was assumed that the sound velocity versus depth curve could be approximated by judi- ciously chosen straight-line segments without intro- ducing too much error in the calculated depth error. Though equation (46) is the relation used in the construction of depth correction tables, it is interest- ing to carry the approximation two steps further. If we expand the integrand in powers of we and neglect all but the first two terms, we get Y Yo= [ (duet ---)dy which becomes dyo (45) (46) “ Y—Yo= of e dy + terms in (we)?. To the same order of approximation, Yo may be sub- stituted in place of Yas the upper limit of integration, which gives Yo ) ae As f e dy + terms in (we)?. (47) CALCULATION OF SOUND INTENSITY FROM RAY PATTERN 51 When we is small the terms in (we)? may be neglected; under these same conditions Y — Yo is small com- pared with Yo. Equation (47) thus provides a useful approximation when the depth error is relatively small. By translating from ¢« back to c this equation becomes Yo eet yen heey 2coJ 0 The expression (48) has a simple interpretation in terms of the velocity-depth diagram. The integrand (co — c)dy is just the black area in Figure 12; thus (48) Co y Figure 12. Depth correction as area under bathyther- mograph trace. the integral from 0 to Yo is the shaded area between the velocity-depth curve, the vertical line c = c, and the horizontal line y = Yo. Qualitatively, we may conclude that the depth correction will be large for steep gradients and larger if these steep gradients are located at shallow depth. 3.4 CALCULATION OF SOUND INTENSITY FROM RAY PATTERN The foregoing sections were devoted exclusively to tracing the paths of individual rays and stated nothing about sound intensity in the ray pattern except for the special case of spherical waves. For that situation an assumption that the energy flows out radially along the sound rays led to the same inverse square law of intensity decay which was de- rived rigorously under “Spherical. Waves” in Section 2.4.2. It is a plausible generalization to assume that energy always travels out along the rays even when the sound velocity is not constant and the rays are curves. The assumption for the case of constant sound velocity and its generalization for the case of variable velocity are illustrated in Figure 13. In the left-hand VELOCITY DISTANCE x SOURCE a w a SOUND VELOCITY CONSTANT WITH DEPTH VELOCITY DISTANCE SOURCE x z= a 4 WwW a Wa SOUND VELOCITY CHANGING WITH DEPTH Figure 13. Effect of vertical velocity changes on ray paths. drawing, the rays are straight lines; and the energy radiated by the source into a small solid angle is con- fined inside the indicated cone. Because of this as- sumption, we get the exact inverse square law of in- tensity loss. In the right-hand drawing, the rays are curves; and the energy radiated by the source into the same small solid angle is confined inside the horn- shaped surface displayed. In this general situation the energy flow through normal unit area depends not only on the distance r from the source, but also on the total cross-sectional area of the horn which, in turn, depends on the way the rays are bent. Thus it is clear that the inverse square law will not, in general, be predicted even by this simplified ray treatment. General Formulas for Change of Intensity along a Ray 3.4.1 The prediction of shadow zones as described in the preceding section is only one part of the description of the expected intensity distribution. There remains the problem of calculating the intensity in regions traversed by the rays. We already know that this in- tensity loss will not exactly obey the inverse square law except in very special cases. 52 RAY ACOUSTICS We restrict ourselves to the case where the sound velocity is a function only of the depth coordinate y. The ray pattern in the zy plane can be computed according to the methods of Section 3.3. We can get the entire ray pattern in space by rotating the ray pattern of the zy plane about the vertical (y) axis; because the velocity depends only on y, the ray pat- terns in every plane through the y axis will be identi- cal in size and shape. We assume that the projector is a point source located on the y axis at the depth yo, which radiates energy at the rate of F energy units per unit solid angle per second. Then, energy will be projected into x S Y Figure 14. Specification of solid angle. a very small solid angle dQ at the rate of FdQ energy units per second. The rays bounding this solid angle will curve in some fashion depending on n(y) and the angle of emission; suppose at the point P some- where out along the ray bundle, the cross-sectional area of the bundle is dS. Then the intensity at P will be the energy crossing this area dS per second, which equals FdQ, divided by the cross-sectional area dS. \ dQ Intensity at P = Fs Because of the cylindrical symmetry of the rays with respect to the y axis, we shall find it convenient to define our small solid angle as indicated in Figure 14. It is the solid angle swept out in space by rotat- ing the portion of the zy plane between the angles 0) and 4 + d@ about the y axis. On a unit sphere the (49) solid angle so defined intercepts a spherical zone of radius cos 6) and width d@ which is therefore of area 27 cos 4d6. Thus our solid angle dQ is given by dQ = 27 cos Ood6o. (50) We wish to calculate the intensity for the ray of initial direction in the zy plane, at the horizontal range . By equations (49) and (50), this intensity is given by F. (2a cos 4 do) ds where dS, the cross-sectional area, is clearly the area swept out when the segment PP’ in Figure 15 is rotated about the y axis. We proceed to calculate dS. I(R,6) = (51) x y Figure 15. Diagram used in deriving intensity formulas. The horizontal range RF is clearly a function of the depth h and 6: R = R(h,6o). (52) Therefore, the horizontal separation dR at a fixed depth is given by dR = — one 06 The minus sign is inserted because FR at fixed depth decreases as 6 increases. Let 6, be the direction of the ray at the-point (R,h) asin Figure 15. Then PP’, the shortest (norma!) distance between these two rays near P is dK sin 6, = —(AR/06)d% sin 6,. By rotating PP’ about the y axis we get dS, the desired cross section of our bundle. This area is clearly that of a spherical zone with radius R and thickness—(dR/06) sin 6,d6); its area dS is therefore dS = (53) OR —2rR— sin 6, dA. (54) 06 CALCULATION OF SOUND INTENSITY FROM RAY PATTERN 53 Substituting this expression into equation (51), we obtain for the intensity J at the range & the expres- sion I(R,40) Ar cos % Re on ; R— sin 4, (55) 00 It is now necessary to obtain expressions for R and for the partial derivative of R with respect to 0. We begin by calculating the range R. We have hda h r= wa -f t ody, », dy” ae u which, upon substituting cos 0 = (c/co) cos 4 from equation (26), becomes (56) P iL cdy 7A = || [SS 5 "J nV & — & cos? Go We differentiate this expression for RF as a product of functions of 6, assuming that the usual formulas for differentiating under the integral sign are valid. When the two resulting integrals are put over one denomi- nator, the whole expression for the derivative simpli- fies to oR Ae it ‘ cdy —— — —Crsin Se 08 “ : yo (e = C cos? 6)? Substituting the expressions for R and for 0R/06p, equations (57) and (58), into equation (55), we find for the intensity J the expression (58) a ice PE eR 1a : f COND) (ia eR sin 4 sin 6;, = 5 y sin @ Jy, sin®*6 with 2 2 | Oo C and (60a) 2 2 sin 6 = 1 = (*) oasis Co For application, this formula suffers from two de- fects. First, it is not sufficiently simple; second, it is not sufficiently general. The second point will be taken up later, where it will be seen that equation (60) does not cover the important class of conditions where the sound ray becomes horizontal anywhere en route. As for the first point, we shall simplify equation (59) for application to these cases where it is valid. Under ordinary circumstances, c does not vary be- tween the sea surface and operational depths by more than 5 per cent of the surface velocity. As a result, those rays which leave the projector at a moderate angle will not become so steep that the sine of the angle cannot be replaced in good ap- proximation by the angle itself. Thus we may re- place the expression cj — c? cos? 4, which appears three times in equation (59), by the approximation ce — @ cos? ~ afi me i a é) | Co co (% ot 2e), in which e¢, as in Section 3.3.2, stands for the ex- pression (¢ — ¢)/c. As a result, we can replace equa- tion (60) by the approximate relation I 1 —-Dw fe TERN ET ORR Ve — 2e J =! v(6 — 2c)! while the range is given approximately by the expres- sion v (61) (62) R f pee 63 + Y% Ve Be ce) In most transmission work, the sound field inten- sity is reported either as transmission loss or as transmission anomaly. The transmission loss H is defined as the ratio of the source strength F and the sound field intensity in decibels, F H = 10 log —- oF The transmission anomaly A is defined as the ratio of the intensity predicted by the inverse square law and the sound field intensity J, also in decibels, F/R? F ‘ike 10 log per On the basis of this definition and equation (62) the transmission anomaly will be given by the approxi- mate relationship (64) A = 10log (65) he dy h dy A =~ —10lo i — + 10lo {i "Ju, 9(y) * J, 8) + 10 log % + 10 log 4, (66) where 6(y) = V6 — 26. Where it applies, this formula is simple enough to lead readily to results of practical significance, as under “Layer Effect”’ in Section 3.4.2. From its mode of derivation the expression (59) for 54 RAY ACOUSTICS the intensity at a point P on a ray is not valid if at any place between the projector and P the ray has become horizontal. For, at a point where the ray is horizontal, 6 = 0, which implies that c — c cos 4 = 0, by equation (26). This means that the inte- gral in equation (57) becomes infinite at points where the ray is horizontal and cannot be differentiated under the integral sign. We therefore conclude that the expression (59) is valid only for rays that are always climbing or always dropping, but cannot be used, for example, to examine the intensity near places where the ray diagram predicts shadow zones. We now derive an expression for J(6),R) which will be valid even at points on a ray beyond where the ray has become horizontal. This will be done by deriving an expression similar to equation (58) in which the variable of integration is @ instead of y. In all cases, we have, because of equation (56), h R= ii cot Oddy Yo nh dydc = t a— — dé. fi a dc d@ dc al cos Z| COunE i =— sin 6, do doe wane % COS 4 (i) : d 2 f cos a8. COS Ov 6, dc This expression for R has the advantage over equa- tion (56) in that the integrand does not become infinite for 6 = 0. This expression can, therefore, be differentiated with respect to 4 even when the ray passes through points at which it is horizontal. The variable 4 occurs explicitly in the factor in front of the integral and as the lower limit of inte- gration and implicitly in the terms 4, and dy/dc. Taking this into consideration, it follows that Since it follows that R = — (67) oR in 0 ("d Eh) 2k me 2 oath 00 cos? 4 6 dc 2s) In Co Sin 8 ad? == 2 f oe, cos? 6d@ cos? 0) J a dc? Co Sin a a cos? 2) | 68) cos? 0 Lsin 6; \dc/r — sin % \dc/o ( Though the expression (68) is much lengthier than the expression (58), it has the advantage of being valid at points on the ray where 6 = 0. The resulting intensity, calculated by using equation (55), will also be valid at all points on the ray. The quantities R and 0k/ 0% must be substituted from equations (67) and (68). These expressions can be simplified by means of the assumption that all angles are small. With this assumption all cosines of angles can be replaced by one; the sines may be replaced by the angles them- selves; and among a number of terms those multi- plied by higher powers of the angles may be disre- garded. The simplified expressions for R and for 0R/06 then take the form (69) aR a) m Ae) | By ol ( iy ON\Golis ) From equations (55) and (65) we have, as a general expression for the transmission anomaly A, and — sin On 0 R cos % 4 If we assume that 4) and 4, are both so small that cos 4 can be replaced by one, and sin 6, by 6, formula (71) becomes A = 10 log (71) A = 10 log (71a) By putting in the approximate values of R and 0R/0% from equations (69) and (70), an explicit ex- pression can be obtained for the transmission anomaly. : In the application of equations (67) to (70) one precaution must be taken. While the integrands of the integrals that occur remain finite when the angle of inclination becomes zero, these expressions ap- proach infinity as the gradient approaches zero. They are, therefore, not suitable for the treatment of propagation through isovelocity layers. Another method of computing the transmission anomaly that may be used whether or not a ray has become horizontal and is in a more convenient form for numerical computation is given under ‘“Combina- tion of Linear Gradients” in Section 3.4.2. 3.4.2 Applications Section 3.4.1 was devoted to deriving formulas for the intensity out along a ray as a function of the hori- zontal range and the velocity-depth variation. These formulas involve line integrals and are too compli- cated to use for practical intensity computations. CALCULATION OF SOUND INTENSITY FROM RAY PATTERN 55 The formulas are’ simplified in this section by using various simplifying assumptions concerning the velocity-depth variation. Drrect BEAM IN LINEAR GRADIENT Let us assume that the sound velocity increases or decreases linearly with depth, with the gradient a. Then, a (72) Since the velocity is never constant with increasing depth in this case, the approximate equations (69) and (70) are applicable. Using these equations, we find that the range F is given by the expression R= SQ] O) (73) a and that the derivative of the range with respect to 6 is given by dee 1 (1 *) a a om Substituting these expressions into equation (71a), we obtain for the transmission anomaly the expres- sion (74) A = 10log1 = 0. (75) The transmission anomaly vanishes, at least in this approximation. If we had used the rigorous expres- sions (67) and (68) for R and 0R/06, and the exact form (71) for the transmission anomaly, the following formula would have been obtained, which is rigor- ously correct, A = 20 log cos 4. (76) It may seem surprising that the transmission anomaly (76) does not depend on the sharpness of the velocity gradient. This seeming discrepancy results from the use of the horizontal range R in the defini- tion (65) of the transmission anomaly instead of the slant range r. The results (75) and (76) for this case of uniform downward refraction apply only to the sound field at points actually reached by the direct rays. If the water is very deep, there are portions of the ocean where no sound ray penetrates, as illustrated in Figure 24; in such regions the ray theory predicts a vanishing sound intensity and thus an infinite trans- mission anomaly. REFLECTED Bram IN LINEAR GRADIENT We now calculate the intensity along a ray which has suffered one or more bottom reflections, for the same linear velocity gradient assumed under “Direct Beam in Linear Gradient”’ in Section 3.4.2. First we shall assume one bottom reflection. This situation is pictured in Figure 16A where the ray hits the bottom OCEAN BOTTOM A ONE REFLECTION R a INVA Lo NAINA. OCEAN BOTTOM B SEVERAL REFLECTIONS Figure 16. Reflection of sound ray from sea bottom. A. One reflection. B. Several reflections. at an angle 6, (6, > 0) and is reflected at the angle —@,. The rays will be refracted downward, as indi- cated; and the incident and reflected rays will be circular ares with equal radii. We can compute the horizontal range R by equa- tion (69), which is valid for all cases where the ray path is made up of several arcs, provided care is taken in breaking up the interval of integration cor- rectly. 9b], ond Y] y — | de — if "8 oof, dc 2 dc =— a, = (i) = "6, + %) a a R= C =— nate + 0, — 4). To use equation (71a), we must also calculate 0R/ 06. Using equation (74), we obtain OR a » ai ae) pata a eee Uy ee Elba, ee ee 00 a Oy i a On, Substitution of these expressions for R and 0R/0% into equation (71a) gives om ’ 64 2 = — += ( a 26, + 6, — Oo For the case of. a ray suffering n + 1 reflections, pictured in Figure 16B, the procedure is similar ex- A = 10 log (77) 56 RAY ACOUSTICS cept that n complete journeys from the bottom back to the bottom must be added to the interval of integration. The calculated transmission anomaly for a ray which leaves the source at the angle , suffers n+ 1 bottom reflections, and strikes a receiving hydrophone at the inclination 6, turns out to be 6 6; afam+1 —24 b h 2(n + 1) 0 + A — 4% A = 10log (78) Layer EFrrect When sound originates in an isovelocity layer or in a layer with a weak velocity gradient and then passes into a layer with a sharp negative gradient, the sharp refraction results in an extra spreading of the sound rays and a consequent drop in intensity. This phe- nomenon is called layer effect, and is of operational importance. We shall consider only rays which leave the projector in a downward direction, so that the formula (66) for the transmission anomaly will apply. Two separate cases will be treated. First, we shall consider the velocity-depth pattern shown in Figure 17: an isovelocity layer, followed by a layer of VELOCITY, Go-ac HYDROPHONE Figure 17. Bending of ray by temperature discon- tinuity. negligible thickness with a very sharp gradient and a total drop of sound velocity of amount Ac, followed in turn by a second isovelocity layer with the velocity c — Ac. If the ray direction in the first isovelocity layer is 4, and in the second isovelocity layer 6,, we have by Snell’s law (26) cos, & — Ac _ Ac cos & Co Co If the angle @ is small, we may replace its cosine by its approximate equivalent 1 — 62/2. Using this ap- proximation, and dropping the negligible term (Ac/co)0, the preceding equation becomes a = Ve tgp, (79) (67) If h, is the height of the sound source above the abrupt velocity change, and h, is the depth of the hydrophone below the velocity change, we easily find from formula (66) that isin hy ) ie rs) A= 10 og (+ + 10 log Bt 9 + 10 log % + 10 log 4, = he 2) = 10 log} ~3(1+ 2 2 o5| RG a h, 07/1’ since Ih BS (80) Next we shall consider the velocity-depth pattern shown in Figure 18: an isovelocity layer extending to VELOCITY SSS SOURCE ° GRADIENT -a Figure 18. Bending of ray by deep thermocline. a depth h; below the sound source, followed by a layer of indefinite extent with the constant velocity gradi- ent —a. At a depth y’ below the top of the gradient layer the sound velocity will be c + ay’. We there- fore obtain the following expression analogous to expression (79) for the ray direction @(y’) at the depth y’. ay!) = V 65 — 2(a/a)y. (81) We shall use equation (66) to calculate the intensity of the sound received by a hydrophone at a depth hz below the top of the gradient layer. Since the ray direction in the isovelocity layer is constant at 6, the separate integrals in equation (66) have the values [2 — Ee ID Gua aeiaa o.v6) as 0 V8 — 2(a/e)y’ The last term may he integrated directly, and with use of equation (81) ee ces 6 0 a/c hy he a Gr 2@aSGoe CALCULATION OF VELOCITY TOR DEPTH GRADIENT a, GRADIENT gq, DEPTH HYDROPHONE DEPTH es SOUND INTENSITY FROM RAY PATTERN 57 SEA SURFACE Figure 19. Ray path in succession of linear gradients. since 6; — 6; has the value 2(a/cy)h by equation (81). Similarly, we have ie a hy ze iis dy’ = hy (4 = On) 08 6 Jo (0% — 2(a/c)y’)? 6% (a/c) 0% _ th 1 he ———— 8 09, 3 (Bo + Fn) Substituting these expressions into equation (66), we find for the transmission anomaly = ae en )| s = 10104 | @ + 6d, 3 + Os) Onha he 62 | = 101 Le = ——t __}) |, 08l Re! + hr, 46,60 + 0) since from equation (37), hy he 8 3(4 + %) If the gradient is sharp, and if the range is con- siderable, the angle 6) will generally be small com- pared with the angle at the hydrophone, @,. Also, if the hydrophone is not too far down, we may assume that the fraction h2/h; is not too large. In that case, the second term in the parentheses in both equations (80) and (82) is small compared with unity and may be omitted as negligible. In either case we have, then, as a rough estimate of layer effect, the simple rela- tionship =) A = 10 log{| —,]- oF (82) (83) CoMBINATION OF LINEAR GRADIENTS In this subsection we shall derive a formula for the intensity along a ray which has passed through a suc- cession of layers in each of which the sound velocity changes linearly with depth in the layer. This con- dition is of considerable practical importance since most velocity-depth curves can be replaced in good approximation by a number of linear gradients. The assumed velocity-depth pattern is shown in Figure 19. There are n + 1 layers, labeled 0, 1, 2, 3, --- n, in which the velocity gradients are do, @1,---Qn respectively; the term a; represents the velocity change in the 7th layer in velocity units per foot of depth increase in the layer labeled 7, where z takes any integral value from 1 to n. The velocity at pro- jector depth is c; at the top of layer 1, c; at the top of layer 2, @; and so on, as indicated in Figure 19. The ray direction is 6) at the projector, 6, at both the bottom of layer 0 and the top of layer 1, and so on; and, finally, 6, at the bottom of the (nm + 1) layer, which is assumed to be the depth of the receiving hydrophone. The total horizontal range covered by the ray is R; the component of horizontal range covered in each layer is designated by Ro, R1,---Rn, as indicated. We shall compute the intensity at the range R by means of the formula (71), which is generally ap- plicable. To use this formula, we must first derive an explicit expression for 0R/ 06 in terms of param- 58 RAY ACOUSTICS eters which may be calculated from the given veloc- ity-depth distribution. Theray path in the 7thlayer willbe an arc ofa circle whose radiusisc;,/(a;cos) according toequation (33). Let the small ray element ds be inclined at the angle @, as in Figure 20. In traversing this small distance, the VELOCITY Cj DEPTH ie Ficure 20. Ray path in 7th layer. ray travels horizontally a distance ds cos 6. By equa- tion (32), we see that ds is given by the expression —c,d6/(a;cos@;); and the element of horizontal range covered by the ray in a distance ds is CG: - — eos 6d. a; Cos 6; To get the horizontal range covered by the ray in its entire journey through the 7th layer, we must inte- grate this result between 6; and 6:41. Oix1 —¢; Ri = Ht cos 6d0 6: A; COS 0; Ci = (sin 6; — sin 6:41) a; Cos 0; (84) C sin 6; — sin 6:44 COS 6 Qi because of Snell’s law (26). The results of this para- graph apply without changes of sign both to layers where the velocity increases with depth and to layers where the velocity decreases with depth. The total horizontal range FR from the projector to the receiver is the sum of the range components in the n + 1 layers. = 2, R= “ COS 6p i=0 ai — sin 6; — sin 6:41 (85) where the symbol 2 indicates summation. By dif- ferentiating both sides of equation (85) with respect to 4 OR 30 Co Sin 6) sin 6; — sin Oi41 COS? 0) i=0 a; ye COS 4 7=0 a; 00; 06544 cos 6;— — COS 8:4:.—— “00 06 which may be written oR _ & sin 0) < » 005 Cos? 0) 7=04; (sin 6; — sin O44 cos 6; Cos 0 00; cos Oi+ cos 8, 0654 : ° ) . (86) 00 sin 4 sin 4 005 For equation (86) to be usable, we must calculate 00;/ 00. By Snell’s law, Ci cos 6; = — cos . (87) Co By differentiating both sides of this with respect to 4% 06; ci sin % 065 c sin 6; (88) sin 4 cos 6; cos % sin 0; By using 06;/00 from equation (88), and a corre- sponding expression for 06;:1/00, the expression in parentheses in equation (86) becomes . | cos? 6; cos? Oi44 sin 6;— sin 0:41 + — eis sin 6; sim O;41 gill 1 ~ sin@; sin 8:41 Thus equation (86) becomes OR asin 1 1 1 ) 0 cos? % i=0a;\sin 6; sin B44 sin 0 ~< R; COS 6) i=0 Sin 6; sin Os Putting this value of 0/06) into formula (71), nie noting that 6, is simply 6,41, we obtain the final result sin 4 SIN O41 < R; | A = 10lo | < = - (89 : R cos? % p> sin 6; sin 0541 ) The expression (89) is in a form well-suited for practical intensity calculations. The various angles 6; can be computed from the known velocity-depth pat- tern by equation (26), and the R; can be obtained either from equation (84) or equation (36). FoRMULAS FOR TRANSMISSION ANOMALIES In this section, the formulas obtained for the transmission anomaly resulting from refraction will be summarized. In the absence of a velocity gradient, or if the sound velocity increases or decreases linearly with depth below the projector, the transmission anomaly in the direct beam is negligible. If the sound velocity decreases linearly with depth RAY DIAGRAMS AND INTENSITY CONTOURS 59 below the projector, and if the ray has been reflected once by the ocean bottom, the transmission anomaly is given in good approximation by the equation 8 ) (7 (to) See ere ( rer where 0; is the angle of inclination at the bottom. In the case of multiple bottom reflections the transmis- sion anomaly is given by rey ee *) a 2041) 4, aF i 2(n + 1) + A — % where the number of bottom reflections is n + 1. The transmission anomaly for sound propagated through a thermocline (layer effect) is approximately given by A = 10 log where h; is the height of the projector above the top of the thermocline, is the inclination of the ray in the overlying isovelocity layer, and @, is the inclina- tion of the ray at the receiver. The transmission anomaly for sound propagated through a succession of layers, each of which possesses a constant gradient of velocity, is given by sin 0 sin 6,+1 < R; ) R cos? 6 A = 10lo ( : ; S 7=0 SIN 6; Sin O;+4 where the various terms are defined under “Combi- nation of Linear Gradients” in Section 3.4.2. 3.55 RAY DIAGRAMS AND INTENSITY CONTOURS Methods The differential equations (25) which govern the path of a ray in a medium where the sound velocity depends only on one coordinate cannot be easily integrated if the velocity depends on depth in a very complicated manner. We have seen, however, that the integration can be accomplished, and the path of a ray with a specified initial direction calculated if the depth interval is divided into layers in each of which the velocity gradient is constant. For this reason, rays are traced in practice by replacing the actual velocity-depth curve with a series of straight- line segments, as in Figure 8. 3.95.1 The bending of sound rays in the ocean is too slight to be evident in a drawing that uses the same scale for range and depth. The deviation from straight-line propagation is never more than a few hundred feet in a mile; although such deviations are extremely important to a surface vessel seeking a submarine with echo-ranging gear, they do not show up well on paper unless the depth scale is expanded. For this reason, ray diagrams cannot be constructed geometrically in practice as the sum of circular arcs through the various layers. Instead, the change in range as the depth increases must be computed alge- braically as described under “Combination of Lin- ear Gradients” in Section 3.4.2, and the results plotted on a graph with suitably chosen scales. A special circular‘slide rule has been invented to simplify this calculation.” This instrument, developed by WHOL early in the war, gives the horizontal range covered by a ray in its passage through a layer with a constant gradient. It does this by using several scales arranged for convenience as concentric circles. The thickness of the layer, the temperature at the beginning and the end of the layer, and the direction of the ray at the projector are given to start with. By use of the slide rule one calculates directly the direc- tion of the ray when it enters and leaves the layer; from the average of the two directions the horizontal range covered in the layer can then be computed by use of equation (36). The instrument exactly dupli- cates the calculations described in Section 3.3 and avoids the necessity of consulting trigonometric tables. Since the direction of the ray as it enters the following layer is given as an intermediate step in calculating the range in the layer, the process may be duplicated until the ray has been traced through all the layers. This slide rule may also be used to com- pute intensities by integration along each ray since the scales provided may be used for evaluating equa- tion (90), which appears later. Similar mathematical aids were developed by UCDWR, an@ by other re- search groups doing a large amount of ray tracing.® Another instrument developed by NDRC for the facilitation of ray tracing is the sonic ray plotter,* pictured in Figure 21. The ray plotter is a device which integrates the differential equations (25) me- chanically and exhibits the solution not as an alge- braic function but as a curve denoting the ray path, drawn, of course, with a much expanded depth scale. The ray plotter has one advantage over the slide rule — it can plot the ray paths for any type of velocity- depth variation, no matter how complicated. 60 RAY ACOUSTICS The accurate plotting of many rays is facilitated by use of a method called the method of proportions (see reference 2). When this method is used, a few rays are drawn with the aid of a slide rule; the posi- tions of intervening rays can be estimated rapidly by an interpolating process. A full description of the method of proportions is given in reference 2. If the ray plotter or the method of proportions is used, it is not difficult to obtain a ray diagram with many rays drawn for closely adjacent values of 4, the ray inclination at the projector. From such a diagram Figure 21. The sonic ray plotter. the intensity at any point can be determined graphi- cally by measuring the vertical separation between rays at that point. In most situations this is the simplest method for computing approximately the theoretical intensities in the sound field. The basic equation used in this graphical procedure for determining the sound intensity may readily be derived from the analysis in Section 3.4.1. By com- bining equations (49) and (50) the following results for the intensity J dA I = 2xrf cos O78 The area dS is given by dS = 2xR cos 6dh, where dh is the vertical distance between the two rays at the point where the intensity is measured, and R is the horizontal range. By combining these formulas, and by substituting into equation (65) for the trans- mission anomaly the following equation results 36 dh A = 10 log (= aa — 10 log R. COS 65 dé For most cases of practical importance, cos @ and Cos 8 may be replaced by one; dh and dé may be re- placed by finite increments Ah and Aé. Thus, we have, finally, the simple result A = 10 log =) — 10 log R. (90) A@ In equation (90), Ah and R must be expressed in yards; while A@ must be given in degrees. Although this equation usually gives sufficiently accurate re- sults, it is difficult to apply practically in regions where the intensity is changing rapidly, such as near the shadow boundary below an isothermal layer. The practical application of equation (90) is given in reference 2, which includes a graph giving the theoretical intensity J in terms of the measured ray separation in feet at the range R and the initial angular separation of the rays in degrees. 3.5.2 Ray Diagrams for Various Temperature-Depth Patterns In practice, the ray paths are usually computed not from the velocity-depth curve, but from the temperature-depth curve obtained with a bathy- thermograph. This is done because the sound velocity is very sensitive to changes in temperature of the magnitude usually encountered in the ocean and rela- tively insensitive to changes in pressure and salinity. The effect of pressure, although small, is usually al- lowed for in the drawing of rays because it is con- stant, causing an increase of 0.0182 ft per sec in sound velocity per foot increase of depth. The effect of salinity on the ray paths is usually ignored, except near regions where fresh water is continually mixing with ocean water; in such cases, the velocity-depth pattern must be calculated explicitly by use of both the bathythermograph record and the salinity-depth variation. The following paragraphs describe ray diagrams for various commonly observed temperature-depth patterns. A more detailed explanation of ray dia- grams along with explicit diagrams for some 380 temperature-depth patterns of the sort found in the ocean is given in a report by WHOI.® Very Derr IsoTHERMAL WATER In deep isothermal water all the rays show slight upward bending because of the constant effect of pressure. This bending, for a ray leaving the pro- jector in a horizontal direction, amounts to about RAY DIAGRAMS AND 50 ft in a distance of one mile. Figure 22 is the ray diagram for this case. IsOTHERMAL LAYER ABOVE 'T'HERMOCLINE The most common temperature-depth distribution observed in the ocean possesses a surface layer of reasonably constant temperature, which overlies a TEMPERATURE-F RANGE IN YARDS 1000 OEPTH IN FEET 1500 Figure 22. Ray diagram for deep isothermal water. layer where the temperature decreases rapidly with increasing depth, called a thermocline. A ray diagram for such a temperature-depth pattern, with the sur- face mixed layer extending down to a depth of 100 ft, and an underlying thermocline is shown in Figure 23. Tt will be noted that all the rays which issue from the projector at higher angles than 1.44 degrees remain entirely within the top layer; the rays become hori- zontal at some depth less than 100 ft and bend back to the surface. All the rays leaving the projector at RANGE IN YARDS 2000 TEMPERATURE-F 7O_ -6°_® +1000 aes 3000 4000 100) DEPTH IN FEET Figure 23. Ray diagram for isothermal layer above thermocline. angles lower than 1.44 degrees reach the thermocline while still inclined downward; the thermocline pro- gressively increases this downward bending. For theoretical reasons, then, the beam should split into two parts; one heads back toward the surface and the other heads down into the thermocline. Between these two beams the sound intensity should be very low according to the ray theory. All velocity-depth patterns for which the ray theory predicts such a splitting of the beam have been called split-beam patterns. The existence of the INTENSITY CONTOURS 61 predicted low-intensity zone which lies between two zones of higher intensity has been verified in experi- ments with explosive sound. The experiments are described in Chapters 8 and 9. However, experiments with single-frequency sound, which are designed to test whether or not the beam splits when predicted, have frequently failed to indicate any splitting of the beam at all. The reasons for this discrepancy are not completely understood. Diffraction of sound into the low-intensity zone, although predicted to a limited extent by wave acoustics, is not sufficient to explain why the beam does not split. Possibly the sound in the predicted low-intensity zone may be largely due to scattering of sound either by in- homogeneities in the predicted path of the rays, or by roughness of the sea surface, or by irregularities of the temperature distribution in the ocean. Strone NeGative GRADIENT Negative temperature gradients are a frequent oc- currence near the sea surface, especially when the sur- face is receiving more heat than it is losing. Under such temperature conditions the rays are bent strongly downward. Figure 24 gives a ray diagram TEMPERATURE-F RANGE IN YARDS a 79 -6 a1 2000 3000 7 cs GH; Ww Ww & 100) z OUTER SHADOW ZONE x Ye 200) -4.87 Ww a 300'S5 Ficure 24. Ray diagram for strong negative tempera- ture gradient. drawn for a case where the negative temperature gradient amounted to about 8 F per 100 ft of depth. It is clear from the figure that the ray which left the projector horizontally has been bent down 400 ft by the time it has covered a horizontal range of 1,000 yd. The most important quality of the ray diagram for this case is the indicated shadow cast by the surface. It is clear from the figure that no ray leaving the pro- jector can possibly penetrate into the zone marked shadow zone if the water is deep. All rays lower than the ray leaving the projector at a climbing angle of 4.8 degrees stay within that ray and are bent down- ward. The 4.8-degree ray itself becomes tangent to the surface and then bends downward. All rays higher than this “limiting ray” are reflected by the surface 62 RAY ACOUSTICS TEMPERATURE-F (o) 1000 RANGE IN YARDS OEPTH IN FEET A COMPLETELY ISOTHERMAL OCEAN RANGE IN YARDS Sas TEMPERATURE-F 4000 SY D 150-FOOT ISOTHERMAL LAYER OVER SHARP THERMOCLINE a pal 508 NCEE wo o - DEPTH IN FEET B 50-FOOT ISOTHERMAL LAYER OVER SHARP THERMOCLINE OEPTH IN FEET G SO-FOOT ISOTHERMAL LAYER OVER WEAK THERMOCLINE FIGURE 25. inside the place where the limiting ray hits it; and so all the surface-reflected rays remain inside the 4.8- degree ray, also. If the actual sound intensity obeyed the predic- tions of the ray diagram, no sound at all would pene- trate out to horizontal ranges of more than a couple of thousand yards in the top 500 ft of the ocean. Thus, a submarine further from the projector than 2,000 yd could be almost certain of escaping detection by sonar gear. In practice, as with split-beam patterns, the shadow zone in the strict mathematical sense of a region of zero intensity does not exist. However, un- like split-beam patterns, the transmission anomaly invariably increases sharply at or near the indicated separation of sound from shadow when the down- ward refraction is strong. As discussed in Section 5.4, such zones of weak sound are observed whenever the temperature gradient is strong enough to cause the predicted shadow zone to start nearer the projector than about 1,000 yd; the sound in them is about 30-40 db weaker than would be predicted by the in- verse square law. If the negative gradient is weak, however, so that the predicted shadow zone does not begin until a range of about 1,500 yd or more, sound usually decreases gradually and at a more uniform rate; the shadow zone in such a case can scarcely be said to have any real existence. Possible mechanisms for the penetration of sound into predicted shadow zones are discussed in Section 3.7. SG UM 22s Ne ae E 150-FOOT ISOTHERMAL LAYER OVER WEAK THERMOCLINE F STRONG NEGATIVE GRADIENT FROM SURFACE DOWN Sample intensity contour diagrams. 3.5.3 Intensity Contours Not all the characteristics of the sound field become apparent from a glance at the ray diagram. Although the distribution of intensity is governed by the spreading of the rays, the degree of spreading cannot be accurately estimated visually, and it is even diffi- cult to judge qualitatively. If a ray diagram is avail- able, the intensity at a pomt may.be quickly esti- mated by measuring the vertical separation of the rays nearest that point, in accordance with equation (90). Since it is assumed that the ray bending is in a vertical direction only, the predicted sound intensity will be directly proportional to the measured separa- tion of the rays. Intensity contours provide a very graphic method for displaying the results of such computations. In practice, the intensity loss is usually reported in decibels below the sound level on the axis at a dis- tance of 1 yd from the projector. The exact value of the spreading loss at maximum echo range depends on many factors, such as the strength and direction- ality of the projector, the efficiency and operating condition of the gear, the intensity of background noise, and the amount of intensity loss due to absorp- tion and scattering. In many cases, it is useful to know at what range the intensity loss due to spread- ing will be 55 db, at what range it will be 60 db, ete. It is clear that the range at which the spreading loss has a specified value will depend on the depth of VALIDITY OF RAY ACOUSTICS 63 the point to which the range is measured, and on re- fraction conditions, or more specifically on the tem- perature-depth variation indicated by the bathy- thermograph. The intensity contour diagram is a set of lines drawn on a ray diagram indicating the intensity loss. On each contour the intensity loss has a constant value, in a fashion similar to the curves of constant barometric pressure on a weather map. The contours are obtained from a ray diagram by using one of the methods discussed in Sections 3.4 and 3.5. On each ray, or for each pair of adjacent rays, the intensity, or transmission anomaly, is computed at suitably chosen intervals. Then one finds, by interpolation, the points where the intensity loss is 55 db, 60 db, 65 db, and soon. After this process is carried through for all the rays, intensity contours can be drawn by joining the points of equal transmission loss on all the Trays. Sample intensity contour diagrams for the oceano- graphic situations treated in Section 3.5.2 are given in Figure 25. The contour diagram for isothermal water is shown for comparison since it indicates optimum sound-ranging conditions, that is, the intensity losses which would be observed if the water had no tem- perature gradients, and if there were no attenuation losses; for this situation, the intensity loss out to the range R is given by the inverse square law and amounts to 20 log R. The contour diagrams for the split-beam cases are identical with that for the iso- thermal case at depths near the sea surface and at short to moderate ranges; at depths below the ther- mocline, however, the predicted spreading loss is much increased; the amount of increase depends on the depth to the thermocline and the sharpness of the thermocline gradient. In the case of downward refrac- tion, the intensity contours which denote large values of the intensity loss are piled together in the vicinity of the predicted shadow boundary. A more detailed discussion of intensity contours with a derivation of some of the basic equations de- rived at the beginning of this chapter is given in a report by UCDWR.*® Sample theoretical intensity contours for different temperature patterns are also discussed in this reference. A comparison of these pre- dicted intensities with sound intensities found from explosive pulses is given in Chapter 9. The encyclo- pedia of ray diagrams in reference 5 includes intensity contours on most of the diagrams and thus may be used to find the type of predicted sound field for many different varieties of temperature-depth patterns. It will be seen in Chapter 5 that the intensity pre- dictions of the contour diagram are not, in general, sufficiently accurate to be trusted for the prediction of maximum echo ranges. However, they are useful for various special purposes, such as indicating howsound intensities should vary with depth at a fixed range. 3.6 VALIDITY OF RAY ACOUSTICS In Sections 3.1 to 3.5 of this chapter the method of ray acoustics has been presented as an independent theory without much connection with the rigorous treatment of wave propagation presented in Chap- ter 2. We first noted in Section 3.1.1 that the im- portant features of the propagation of spherical waves could be derived equally well by using the concept of wave fronts connecting points which have equal phase of condensation, and the concept of energy transported by rays perpendicular to these wave fronts. Then we generalized the definition of wave fronts and rays, derived differential equations for the ray paths from these definitions, and solved these differential equations for the ray paths and the re- sulting sound intensity. It is important to remember, however, that the method of wave fronts for the general case placed no requirement on the wave front, except for stipulating that it be of the form (7) for some function W(z,y,z). To make the idea of wave fronts intuitively signifi- cant, it was implied that the wave front should always join points of constant phase of condensation; but this implication was never used. The ray paths depended only on the form of the function W and the variation of c; the intensity calculations depended, in addition, on the assumption that energy is transported out along the rays. In this section, where we try to find a connection between ray acoustics and wave acous- tics, we must assume a physical significance for the wave fronts. Accordingly, we shall make the explicit assumption that the wave fronts join points of equal phase of condensation since we already know that the assumption brings ray acoustics and wave acous- tics into agreement for the case of spherical waves. In this section, we shall examine whether wave acoustics and ray acoustics with this definition of wave fronts are equivalent in general or only under some special conditions. Since sound field calcula- tions are much simpler by the ray method than by a rigorous solution of the wave equation, it will be ex- tremely valuable to know when the ray theory can be applied without much error and when it will lead to definitely wrong results. 64 RAY ACOUSTICS Eikonal Wave Fronts versus General Wave Fronts 3.6.1 It will be remembered that the entire method of rays was based on the eikonal equation (13), which in turn was based on the assumption that the wave fronts (7) “grow” perpendicularly to themselves. That is, the eikonal equation was derived by assum- ing that the wave front at time ¢ + dt is found from the wave front at time ¢ by moving each point on the latter a distance cdt along the outward normal. We shall now show that wave fronts ordinarily do not obey this law of propagation rigorously, but that the assumption often provides a good approximation. It is intuitively apparent that wave fronts, defined purely as surfaces of constant phase without refer- ence to the way they grow, exist in the exact case, at least when the dependence on time is harmonic. We shall define these wave fronts in the rigorous case by V(a,y,2) = a(t — to) (91) reserving the expressions W for those cases where the wave fronts grow perpendicularly to themselves, and where W therefore satisfies the eikonal equation. We shall call surfaces (91) general wave fronts, and sur- faces defined by similar equations, with V replaced by W, eikonal wave fronts. We know that in instances where the sound source vibrates harmonically with a single frequency f the solution of the wave equation can be expressed as the real part of the complex expression p = A(xyyyz)eerf ULV ane) eo} (92) This expression is identical with equation (6), except that we assume that the expression (92) with the function V(z,y,z) is a rigorous solution of the wave equation, while the expression (6) with the function W (a,y,z) was obtained by means of a Huyghens con- struction so that W(z,y,z) would satisfy the eikonal equation. We now shall see under what conditions the ex- pression (92) can satisfy the wave equation and, simultaneously, V (x,y,z) satisfy the eikonal equation. Suppose p satisfies equation (27) of Chapter 2, and simultaneously V satisfies the eikonal equation (13). The latter condition is dG (OD (ODE (2) + (ZY +(2) Si a The former condition may be simply calculated by noting that equation (92) may be written as =, ” } % p= log A 2nif (V co) (2aift (93) (94) Substitution of the expression (94) into the wave equation, performance of the indicated differentia- tions, and collection of terms is a straightforward calculation which will not be reproduced here. The real and imaginary parts must vanish separately; these parts are ae pica Ga lees Sere (2 i oy v 0z ia Ox? d(log A) 82(log A) | =i at oy” 02? vr Ox O(log A) |? O(log A) |? +| (log | al (log Th <0. (95) oy 0z and CAN, CY GH [Ee Ox? oy? 02? Ox Ox OV d(log A) dV d(log 1] ie F oy oy name fi Clearly, V will satisfy condition (93) only if d?(log A d?(log A d*(log A Eee), oie, ee Ox? oy” 02" O(log A) |2 O(log A) |? O(log A) |? +/ (log | +| (log | ral (log Pao. Ox oy Oz (97) This can happen if )y is zero, or if 1/PA AA A 133 = | = op op Se) (= a oy? v 4) % Ce) since the expression in braces in (97) easily reduces to the above. This condition (98) is usually not satisfied. While it happens to be satisfied by the pres- sure wave of a point source in a homogeneous medium, it does not hold, for instance, for the radia- tion of a double source. In general, equations (93) and (95) will be rigorously equivalent only if the wavelength Xo vanishes. Conditions for Nearly Eikonal Wave Fronts 3.6.2 We derived in Section 3.6.1 the conditions under which wave fronts, defined as expanding surfaces of constant phase of condensation, expand perpendicu- larly to themselves. It is more useful to know how large the frequency must be, relative to the other parameters of the problem, before the function V(a,y,z) of equation (92) very nearly satisfies the eikonal equation; we will then know under what con- ditions the wave fronts are very nearly perpendicu- larly expanding. SHADOW ZONE AND DIFFRACTION 65 Clearly the expression B of equation (98), the re- mainder term will be negligible compared with the other terms if No(log A)!’ « V’ (99) Ne(log A)” « (V’)?, (100) where the prime denotes any spatial derivative, and < means “‘is negligible compared with.’”’ If V even approximately satisfies the eikonal equation (13), then V'~wn, (101) where the symbol ~ signifies “‘is of the same order of magnitude as.” Another useful relation is obtained from equation (96). The functions A and V must satisfy equation (96) as long as the surface (91) has the significance of a general wave front. But equation (96) implies that V" ~ V'(log A)’, (102) which in turn implies that NV” ~ V’ro(log A)’ «K V’V’ (103) because of equation (99). Combining equations (103) and (101), No V" Kn’. (104) In the ocean the index of refraction n is of the order of magnitude of unity. Then, the relation (104) may be stated in the following words. The first spatial de- rivative of V must not change much over a spatial dis- tance of one wavelength. The first spatial derivatives of V give the direction of the rays; while the second derivatives, yielding the rate of change of ray direc- tion, give the curvature of the rays. Therefore, the condition (104) becomes the following. The direction of the ray must not change much over a distance of one wavelength. In regions where the ray curves very strongly, ray acoustics cannot be applied safely. Differentiating the eikonal equation (13), we get V'V" ~ nn’ or V" ~ n’ because of equation (101). In view of equation (104), this means that hon’ K ww 1. (105) In other words, the index of refraction must not change much over a distance of one wavelength. We derive one more restriction — this time on the amplitude function A. From equations (102) and (104), we also have Ao(log A)’ <1. (106) The relation (106) means that log A must not change much over a distance of one wavelength. Since this change is very nearly \)A’/A, this means that the percentage change in A over one wavelength must be very small. We can summarize our conclusions as follows. The eikonal equation usually will not lead to a good ap- proximation (1) if the radius of curvature of the rays is anywhere of the order of, or smaller than, one wave- length, or (2) if the velocity of sound changes ap- preciably over the distance of one wavelength, or (3) if the percentage change in the amplitude function A is not small over the distance of one wavelength. 3.6.3 Comparison of Ray Intensities and Rigorous Intensities It follows from the results of Section 2.7.3 that if the general wave fronts are defined by equation (91), and the instantaneous acoustic pressure by equation (92), then the rigorous intensity is given by a2 oV \2 AV \2 oV \2 roe () a () + () (ta) and, further, that the direction of energy flow is char- acterized by the direction numbers 0V/dx : 0V/dy : 0V/dz. The latter direction is perpendicular to the general wave front; thus, if the wave fronts are eikonal wave fronts, the energy flows along the rays in the rigorous case. If the wave fronts are approximately eikonal wave fronts, then the directions perpendicular to these wave fronts represent very nearly the true direction of energy flow. Thus, if the conditions for eikonal wave fronts de- rived in Section 3.6.2 are satisfied, the energy ema- nating from the source into all solid angles will re- main within the tubular confines assumed in deriving the ray intensity. We can therefore say, intuitively, that if the wave fronts are very nearly eikonal wave fronts, the ray intensity will be very close to the rigorous intensity. Further, we can say that in both cases the intensity will be given by a? iss Qpc’ m2) +8) +E =o Seno Bin ay ae a hess 3.7 SHADOW ZONE AND DIFFRACTION When the velocity decreases from the surface down- wards, the ray theory predicts a sharp shadow boundary across which no sound ray penetrates; a typical ray diagram for such an instance is shown in Figure 24. At the shadow boundary the ray theory (108) 66 RAY ACOUSTICS predicts a discontinuous drop of intensity from a finite value on one side to a zero value on the other. It was shown in Section 3.6 that the ray theory can- not be trusted whenever it predicts such a rapid change of intensity in a distance of only a few wave- lengths. Thus, it is necessary to use the wave equa- tion directly to compute the intensity of sound which penetrates the so-called shadow zone. The simplest case of a shadow zone is that pro- duced by a screen in front of a light source. As shown in Figure 26, the ray theory predicts that no light SOURCE oH I \ Ficure 26. Optical shadow zone produced by screen. can reach the shadow zone behind the screen. When the rays carrying the energy are curved, as in Figure 24, it is the surface of the ocean that intercepts the curved rays and ‘‘casts a shadow.’’ In either case, however, some energy actually appears inside the predicted shadow zone, and the wave is said to be “diffracted.” The computation of diffracted sound in the shadow zone is a rather complicated problem in the general case. To indicate the type of analysis required, and to show the general nature of the results, a simplified problem will be considered here. As shown in Figure 27, a sound projector is assumed to be placed against a vertical wall, which extends down to great depths. The introduction of the wall simplifies the problem without changing the final results essentially. The water is assumed to be so deep that bottom-reflected sound may be neglected. The projector face is as- sumed to be so wide that the horizontal spreading of the sound beam may be neglected; thus, only the two-dimensional problems need be considered. The sound velocity c is assumed to vary according to the law (109) where B is a constant, and y represents depth below the surface. Since B is in practice very small, this gradient is indistinguishable from a linear gradient at depths of interest. The exact velocity gradient at the depth y is given by ie & B dy —- 2c (1 + By)? Thus, at the surface, where y = 0, the velocity gradient —b is given by (110) (111) y=0 The gradient (109) is chosen instead of a simple linear gradient not for physical reasons, but because it simplifies the following computations. y PROJECTOR SURFACE x VELOCITY > SHADOW ZON Ficure 27. Sound shadow cast by sea surface. To solve the wave equation under these conditions, it is necessary to use the method of normal modes developed in Chapter 2. In particular, we must find a solution to the wave equation (27) in Chapter 2 which satisfies the boundary conditions we shall impose. As in Section 2.7.2, we look for a solution which is the product of three functions, one dependent only on the time ¢, another dependent on the depth y, and the third, a function of the horizontal distance x. The coordinate z need not be considered in the two- dimensional case under discussion. Following the analysis of Section 2.7.2, we there- fore write P(x,y,2,t) = er F(y)G(a). (112) By substitution of equation (112) into the wave equa- tion (27) of Chapter 2, and by dividing through by C?, it is found that F and G satisfy an equation of the form 2 PG == F ay Cp vr dx? 2 Sol Gl = e ath (113) SHADOW ZONE AND DIFFRACTION 67 If equation (118) is divided through by FG, and equa- tion (109) used for c, 1 ) [ 1@F 4rf | pees = (1+ By)|=0. (114 Gs 1G Te a ee | a Since the first bracket depends only on x and the second only on y, equation (115) can be satisfied only if each bracket is constant. If we denote the first bracket by — 12, the second bracket must be +y?, and we have 4a? f? oF Fa + By — 2] =0 (115) dy? Co @G = 2G = 0. 1 wt HG =0 (116) The basic problem is to find solutions of equations (115) and (116) which satisfy the boundary condi- tions. First, we have the boundary conditions for equation (115). In the analysis in Section 2.7.2, these boundary conditions were that the pressure vanished both at the surface and at the bottom. Here, also, the pressure must vanish at the surface. However, the water is so deep that the condition at the bottom disappears. Instead, there is simply the condition that at some distance below the projector no ‘sound is coming upwards; that is, any sound present at these depths is coming down from shallower depths. Al- though this boundary condition is somewhat compli- cated to formulate exactly, the general result is the same as that found in the solution of equation (161) of Chapter 2. In this earlier instance it was found that sin 27y/\,, corresponding to F(y) in equation (112), when B is zero, satisfied the two boundary con- ditions only if \, had one of a number of fixed values. Similarly, the function f(y) can satisfy the two bound- ary conditions only if u has one of a certain number of values. These values, which are called characteris- tic values of », may be denoted by si, ue, us, and so on, or more generally by y;, where j can be any integral number. For each of the characteristic values p,;, equation (115) has a particular solution F';(y) which satisfies the boundary conditions. Once a value of 1; bas been chosen, the solution of equation (116) is very simple. For each value of p,, G = Ae” (117) where A, is an arbitrary constant.2 Thus the wave = The negative sign must be taken in the exponent so that p; in equation (118) will correspond to a wave moving away from the projector; that is, p; must be a function of 27ft — jz, where y; is positive. equation is satisfied by any product of the type Pp; = Aje™" F (yes. (118) Equation (118) satisfies the boundary conditions at the surface and at great depth since F’,(y) satisfies these conditions. However, the boundary conditions at the vertical plane x = 0, the assumed vertical wall, must also be satisfied. These conditions are that the particle velocity at the sound projector must be % cos 2rft, and that the particle velocity at all other points in the plane z = 0 must be zero. To satisfy this boundary condition at the plane x = Orequires a combination of an infinite number of possible solutions of the form (118). Hach A; must be chosen in such a way that the sum has the re- quired properties. Methods for doing this have been developed, but are beyond the scope of this discus- sion. However, the final result is that the pressure p is the sum of many terms of the type (118) with ef the only common factor. Within the direct sound field a large number of these terms are important, and an exact computation is necessary to find p. In the shadow zone, on the other hand, one term dominates, and the other terms may be neglected. This is because all the u; are partly real, partly imaginary, with the result that the abso- lute value of exp (iu;7) decreases exponentially for sufficiently great values of x. It can be shown that the range at which only one term dominates is ap- proximately the range to the shadow boundary com- puted from the ray theory. This dominant term is the one for which yu; has the smallest imaginary part. Thus, the theory predicts that in the shadow zone the sound intensity falls off exponentially with increasing range, or, in other words, that the predicted trans- mission anomaly in the shadow zone increases line- arly with increasing range. Although the exact determination of the different characteristic values »; is somewhat involved, it is relatively simple to show how these values depend on the frequency f, the velocity gradient, and the sound velocity ¢ at the surface. This is useful since it indi- cates how the attenuation into the shadow zone may be expected to vary under different conditions. In order to investigate this dependence of y; on the other variables, we rewrite equation (115) in a simpli- fied dimensionless form. Let 2F2 a (119) and “o (120) 68 RAY ACOUSTICS where D is an arbitrary constant to be determined later. Then equation (115) becomes, on dividing through by D?, PF n So mat (E+ c D3 age If D is chosen so that D3 (121) _ ips hg then equation (115) becomes 2 22 (RL NPS du? ? (122) where nee (1°f?Beo)* Equation (122) has solutions of the type desired only for certain characteristic values of K, denoted by the symbol K,. The different values of K; are determined only by the nature of the differential equation (122) and by the two boundary conditions, namely that the sound pressure is zero at the surface and that no sound is coming up from below the projector. Thus the values of K; are independent of the frequency, sound velocity, and velocity gradient. Once these characteristic values of K have been found, the corresponding values of » to be used in equations (115) and (116) can be found directly. By substitution in equations (123) and (119), we find 22th _ ePBa)s (123) cs & The second term in equation (124) is always very much less than the first in cases of practical im- portance. Even for a temperature gradient as large as 1 F per ft of depth increase, and for a frequency of only 100 c, the second term is less than 1 per cent of the first for K; less than 10, the region of practical interest. Thus we may take the square root of equa- tion (124), expand in a series, and retain only the first two terms. This process gives Pf (58) Hao 2 \8af KGS (124) - (125) Let K, be the characteristic value of K with the smallest imaginary part, and let this imaginary part be denoted by 7K;. Let the theoretical sound pressure associated with the characteristic value K, be p;. In the shadow zone the intensity is proportional to the square of 7p; since the sound pressures associated with the other characteristic values K; may be neglected. The intensity level found from equation (119) is Ki (afB\? L = 20 log p: = C — 20(logw e) =e) z (126) where C includes A; and the other variables taken over from equation (118). While C changes gradually with position, it is nearly constant along the shadow boundary. Multiplying out terms in equation (126), and using equation (111) for B, we get, finally, 5.05K;f?(—de/dy)?x to) It should be emphasized that equations (126) and (127) apply only in the shadow zone. In the main beam other terms corresponding to other values of K; must be considered. The analysis in a report by Columbia University Division of War Research’ considers the radiation in three dimensions sent out by a point source and is thus more general than the simple analysis presented here. However, the final result for the sound in the shadow zone is nearly identical with equation (127); the only difference is that the term 5.05K, becomes 25.7 in the exact computation of reference 7. With this substitution, we have the following formula for a, the attenuation coefficient beyond the shadow bound- ary in decibels per unit distance. ga otf (= de/ dy) Co In this equation f is the sound frequency in cycles per second, and de/dy is the velocity gradient in feet per second per foot. If co is in feet per second, formula (128) gives the attenuation in decibels per foot; if co is in yards per second, the result is the attenuation in decibels per yard. Since inverse-square spreading is quite negligible compared to the intensity drop at the shadow boundary, equation (128) gives the slope of the transmission anomaly at points beyond the shadow boundary. However, this equation cannot be used at shorter ranges and must therefore be regarded as an expression for the local attenuation coefficient in the shadow zone. Equation (128) is compared with observational data under Attenuation Coefficient at Shadow Bound- ary in Section 5.4.1, where it is shown that the ob- served local attenuation coefficients beyond the shadow boundary are not more than about half the predicted values. In other words, in practice much more sound appears in the shadow zone than is pre- dicted by equation (128). VS (Oh (127) (128) Chapter 4 EXPERIMENTAL PROCEDURES | es PRECEDING chapter was concerned chiefly with the development of the ray-tracing technique, the earliest theoretical approach which led to practical results in the prediction of maximum ranges. This method was, however, only partially successful. Its chief accomplishment was the prediction of the shadow zone boundary in the presence of pronounced negative gradients at the surface. Predicted maximum echo ranges computed by ray- tracing methods agreed with the available observed range data to a fair degree of accuracy, but it was clear that these prediction methods were too simple. The evidence relating maximum observed ranges to temperature conditions was too incomplete to be analyzed with a view to improving range-prediction methods. Navy vessels could not often be made avail- able for range determinations under carefully con- trolled conditions, and the scattered observations made in the course of routine operations were incon- clusive. It was decided, therefore, to initiate a pro- gram in which the sound field produced with standard Navy echo-ranging gear would be measured in much greater detail than before. It was contemplated that this study would place the prediction of sound ranges on a firmer basis and in general would lead to a better understanding of the basic factors important in trans- mission of sound through the ocean. Subsequently, this program was broadened to include sound of fre- quencies between 100 and 60,000 c, and to cover situations somewhat different from those encountered in routine operation of standard gear. Only such a broad experimental investigation of the propagation of sound under various conditions can possibly fur- nish an adequate insight into the mechanisms deter- mining the sound field in the sea. This chapter deals with the experimental methods which have been developed in connection with the sound field program. The results obtained will be discussed in Chapters 5 and 6. 4.1 QUANTITIES CHARACTERIZING TRANSMISSION Before launching into a detailed discussion of these experimental methods, it will be necessary to review briefly the principal quantities which characterize the transmission of sound energy in the sea. In general, sound power is transmitted at a particular frequency or ina specified frequency band; all statements in this section concerning power, intensity, and sound level refer to the frequency or frequency band once speci- fied. Let F denote the power output per unit solid angle on the axis of symmetry of the sound source; at a moderate distance r from the source, the sound in- tensity on the axis therefore equals F'/r?. The power output per unit solid angle in any other direction will be given by bF, where b, the pattern function defined in Section 2.4.4, is a function of the direction; by definition, b equals unity for the direction of the projector axis. Since decibels are commonly used in sound field measurements, we shall transform F into a more con- venient quantity, the source level S. At a point on the axis at a distance of 1 yd from a point source, the sound intensity I) will be proportional to F. The source level S is defined as this sound intensity at 1 yd in decibels above a suitably chosen reference in- tensity I,: Iy S = 10 log ay (1) The reference intensity J, is usually chosen as that corresponding to an rms pressure of 1 dyne per sq cm. Actual sound sources, such as a battleship gen- erating propeller and machinery noise, frequently have large spatial extensions, and the sound level 1 yd from the sour¢ée is not well defined. However, at dis- tances large compared with the linear dimensions of 69 70 EXPERIMENTAL PROCEDURES RECEIVING SHIP SENDING SHIP ° HYDROPHONE DEPTH i= w w “200 = x = a w a « 400 Ww e 6 = -20 > a = on te) ra) <3 58 a: == 20 Qa aq c = 40 (0) 2000 4000 6000 RANGE IN YARDS Figure 1. such a source, the sound field intensity drops off like that of a point source producing similar sounds; in other words, for intensity calculations at long ranges, the actual source may be replaced by an equivalent point source. The reported source level of an ex- tended source is nothing but the sound level of the equivalent point source at a range of 1 yd. For echo- ranging transducers, the sound level is often meas- ured at a distance of a few yards and then extra- polated to a distance of 1 yd by means of the inverse square law. Consider now the sound field intensity J at some specified location, presumably at a fair distance from the sound source. This intensity is commonly ex- pressed as a sound pressure level LZ in decibels above an rms pressure of 1 dyne per sq cm (decibels of a pressure level are defined as twenty times the loga- rithm of the ratio between the rms acoustic pressure and 1 dyne per sq em). If the sound source is highly directional, like an echo-ranging projector, it is usually understood that the projector is trained, that is, rotated about its vertical axis, toward the point at which the transmission loss is to be determined. But in the absence of a tilting device, the axis ray 150 200 VELOCITY ANOMALY IN FT PER SEC TRANSMISSION LOSS HIN DECIBELS Transmission loss (H) and transmission anomaly (A). leaves the projector in a horizontal direction and may then be refracted to a depth different from that of the recording hydrophone. The difference S — LZ will therefore. depend, in general, on the directivity pat- tern of the transmitter. As long as the distribution of acoustic pressure does not approach the conditions of explosive sound, S — L will be independent of the absolute power output of the transducer. The dif- ference S — LZ in decibels is called the transmission loss and is denoted by the symbol H. Frequently, the transmission loss is represented in terms of its deviation from the law of inverse square spreading. If this law vere valid, the transmission loss should amount to 20 log R, where £ is the hori- zontal range from the transmitter to the chosen point. The expression H — 20 log R is called the transmission anomaly and is denoted by A. Figure 1 shows the experimentally determined values of H and A in a particular run with plots of the sound velocity against depth and of the computed ray diagrams. H and A are quantities depending on the trans- mission characteristics of the path under considera- tion and on the directivity pattern of the transmitter. They are independent of the power output of the TRANSMISSION RUNS 71 SHAPE OF PING SHAPE OF PING SHAPE OF SIGNAL GOOD COHERENCE Figure 2. transmitter; moreover, A is an absolute quantity which is independent of the system of units chosen.® Transmission loss and transmission anomaly are the principal quantities which characterize the propa- gation of sound from the source to any point of in- terest. For some purposes, it is also desired to obtain information on the steadiness of the transmitted sig- nal and on its “coherence.”’ Slow changes in signal strength that occur in the course of several minutes are called variation. Changes that take place in the course of seconds are called fluctuation. The coherence of a signal may be loosely defined as the degree of fidelity with which the envelope of the transmitted signal is duplicated by the envelope of the received signal. If transmission conditions in the ocean did not change rapidly, one would be a perfect copy of the other, except for a negligible transient. Actually, conditions sometimes change so rapidly that the shapes of the transmitted and received signal re- semble each other only slightly. Figure 2 shows (A) a case of good coherence, and (B) a case of poor coher- ence. Both of these figures show oscillograms of re- ceived supersonic signals recorded on the same equip- ment. A detailed discussion of variation, fluctuation, and coherence will be given in Chapter 7. 4.2 DETERMINATION OF TRANSMISSION LOSS Information on transmission loss has been ob- tained by three distinct methods: first, transmission runs; second, echo-ranging runs; and third, the sta- 2 A is ten times the logarithm of the ratio between the power flow per unit solid angle close to the source and the power flow per unit solid angle at the specified location; both of these quantities should be expressed in the same units. The apex of the solid angle is in both cases formed by the sound source. SHAPE OF SIGNAL POOR COHERENCE Examples of good and poor coherence. tistical analysis of observed echo and listening ranges. Sound transmission runs include all investigations in which the source of sound is separate from the receiv- ing instrument and in which the sound travels from source to receiver without suffering reflection from a target; slanting reflection from the surface or the bottom of the sea is, however, not excluded. While various setups have been used for transmission runs, the most common one involves the use of two ships. One ship carries the sound source, whereas the other ship is equipped with hydrophones whose outputs are recorded. In echo-ranging runs, the same transducer is used as both source and receiver. The sound is propagated toa target and then reflected back toward the point of origin. The target may be a vessel, but is more frequently an artificial target, that is, a device used exclusively for research and training purposes. The observed range information has been furnished to the research groups in the form of log books and patrol reports by naval craft on active duty. Of the three methods of investigation mentioned, transmission runs have proved by far the most power- ful and reliable tool. The other two methods, analysis of observed ranges and echo runs, are now merely subsidiary. 4.3 TRANSMISSION RUNS The characteristic feature of the transmission run is the employment of separate devices for transmit- ting and receiving the sound. It is, therefore, possible to measure the transmission over any type of path by varying the depth of the projector, the depth of the hydrophone, and the horizontal distance between them. Depending on the instrumentation, it is further possible to vary other important acoustic parameters, such as signal frequency and signal length, or to em- ploy signals composed of several frequencies or a continuous range of frequencies. The temperature 72 EXPERIMENTAL PROCEDURES distribution in the ocean during transmission, the depth and nature of the sea bottom, and in many in- stances factors such as condition of the sea surface, wind velocity, the presence of ocean currents, and the presence of salinity gradients, will affect the trans- mission characteristics of the ocean; these must be recorded along with the geometry of the transmission path itself. In recent experiments, not only the level but also the coberence and the degree of fluctuation of the received signal have been studied. All in all, the number of variables determining a signal is almost overwhelming; also, the characteristics of the result- ing signal are quite complex. In any given investiga- tion, both field procedure and the analysis of data are necessarily concerned with only part of the complete picture. Ordinarily, the sound source in transmission runs is a transmitter driven by a harmonic oscillator through suitable amplifying stages, so that single- frequency sound is put into the water. Frequencies used range from 200 c up to 100 ke and more, but more runs have been carried out at 24 ke than at any other frequency. A second ship carries the receiving gear, hydrophone, amplifiers, and recorders. The hydrophones are usually cable-mounted bydrophones, which can be lowered to any desired depth from a few feet to several hundred feet below the surface. In the most common form of run, the depth of the hydrophone or hydrophones is kept constant during one run. The range, however, is varied during the run from 100 or 200 yd to several thousand yd, by having the sending ship either approach or recede from the receiving vessel. The run is usually completed in less than half an hour. It is hoped no major changes in temperature distribution or other oceanographic variables will have taken place during that time. A more detailed description of the field procedure will be given in Section 4.3.2. First, however, a brief description of sound-transmitting and sound-receiv- ing equipment will be given. 4.3.1 Sound Sources and Receivers A sound source suitable for transmission runs should satisfy several requirements. It should be easily controlled. It should be capable of being mounted on a ship or towed by a ship. Its output should be stable. Its frequency characteristics should be simple, that is, it should produce either single- frequency sound or wide-band noise with a smooth spectrum; and the acoustic power output should be high so that even at long ranges the received signal will usually be above the background. In practice, most results have been achieved with the use of single- frequency sources, such as echo-ranging projectors. Some work has also been done with noise makers of the type used for acoustic mine sweeping. Single-frequency sources have been of three kinds, electromagnetic or dynamic speakers for sonic fre- quencies, and magnetostrictive and piezoelectric pro- jJectors for supersonic frequencies. Work has been re- ported by UCDWR at 200, 600, and 1,800 ¢, and 14, 16, 20, 24, 40, 45, and 60 ke, and by WHOI at 12 and 24 ke. More transmission runs have been carried out at 24 ke than at all the other frequencies combined because echo-ranging gear used by the Navy was de- signed for use at approximately that frequency. Oc- casionally, transmission runs have been made with “chirp” signals; these are frequency-modulated sig- nals in which the frequency rises linearly from 23.5 to 24.5 ke or some similar frequency range during a pulse. Other important parameters of the sound source are its directivity and its power output. The direc- tivity may be reported in the form of pattern plots in the horizontal plane and in the vertical plane. Ten times thelogarithm of 6, the pattern function of the pro- jector, is plotted on a circular graph against the angle from the axis. These plots are incomplete since no in- formation is given concerning the value of 6 off the two planes plotted. Most echo-ranging projectors, however, approach rotational symmetry with respect to the axis so that a single plot including the axis gives adequate information on the pattern in all directions. Figure 3 shows the directivity pattern of the JK crystal projector which has been used by UCDWRE for many transmission runs at 24 ke. Frequently, the directivity of a projector is re- ported by means of a single quantity, the directivity index D. The directivity index is defined (see Section 2.4.4) by means of the equation D = 10 log Z A baa) (2) in which © denotes the full solid angle. The units are decibels. The directivity index so defined has the value of zero decibel for a spherically symmetric sound source. Since the axis for echo ranging is invar- iably the direction of greatest power output, b no- where exceeds unity, and D is a negative quantity. For the standard Navy sound heads QC (magneto- TRANSMISSION RUNS 73 > SW HORIZONTAL PLANE VERTICAL PLANE Figure 3. Directivity patterns of the JK SK4926 at 24 ke. strictive) and JK (X-cut Rochelle salt), the direc- tivity index is approximately —23 db at 24 ke. The total power output of a projector is usually of less interest than the power output per unit solid angle on the axis. This quantity is customarily re- ported in terms of the source level S, which has al- ready been defined in Section 4.1. The source level of the gear used in the UCDWR transmission studies is about 107 db above 1 dyne per sq cm 1 yd from the projector face. The receiving instruments in transmission runs are usually cable hydrophones. The sound head consists of the electroacoustical element itself and sometimes contains a preamplifier which boosts the output voltage before it passes through the cable to the main sound stack. The electroacoustical element itself may be either a crystal element (as in the CN8 series used for a long time at UCDWR), or it may be a magnetostrictive device (similar to the Harvard Underwater Sound Laboratory [HUSL] B19-H). The receiving response of a hydrophone is defined as the ratio between the rms voltage across the out- put terminals of the hydrophone or the preamplifier and the sound pressure of a plane wave incident on the axis of the hydrophone. It is ordinarily reported in decibels above 1 volt per unit sound pressure and then denoted by s. Waves incident in directions not Y NS Wil! Ll|\ IIS SX ZG, NS SU Ll Figure 4. Response pattern of the CN-8-2 No. 597 hydrophone at 24 ke in horizontal plane. parallel to the axis will produce lower voltages than sound waves of the same amplitude incident on the axis of the hydrophone; in other words, most hydro- phones discriminate against off-axis sound inputs. 74 EXPERIMENTAL PROCEDURES \ : = i W Figure 5. Response pattern of the CN-8-2 No. 597 hydrophone at 60 ke in horizontal plane. The degree of discrimination is reported in a manner analogous to the statement of the directivity pattern of a projector. The ratio of sensitivity in a given direction to the sensitivity on the axis is denoted by b’, which is often plotted in decibels relative to axis sensitivity. Tbe degree of discrimination of the hydrophone may also be reported asa single quantity, its directivity index, defined by the equation 1 D’ = 10 log ic { vao), (3) TT which is completely analogous to equation (2). Cable hydrophones cannot be trained since they are freely suspended from their cables. It is, there- fore, extremely important for a cable hydrophone to be nondirectional in the horizontal plane; otherwise appreciable unknown errors in the received intensity result. Several models, which have been used in re- search, come fairly close to nondirectionality. Figure 4 shows the horizontal directivity pattern of the CN-8 crystal hydrophone, used extensively for trans- mission runs at both UCDWR and WHOI. This pat- tern was determined at a frequency of 24 ke. Figure 5 shows the horizontal directivity pattern of the same hydrophone at 60 ke. It will be noted that at this fre- quency the CN-8 is quite noticeably directional in the horizontal plane. More recently the HUSL Figure 6. Horizontal directivity pattern of the Har- vard B19-H hydrophone at 20, 60, and 100 ke. B19-H magnetostrictive hydrophone has found favor because of its great stability and high degree of nondirectionality in the horizontal plane at a wide range of frequencies. Figure 6 shows the horizontal directivity of the B19-H at 20 ke, at 60 ke, and at 100 ke. The response of a hydrophone is a measure of the strength of the electrical signal which will be passed into the cable with a given intensity of incident sound. Since the cable and subsequent amplifying stages-will produce a certain amount of instrumental back- ground noise, the response alone may, under exceed- ingly favorable external conditions, determine the level of the minimum detectable signal. The thermal noise in a I-c band is determined by the receiving response and by the effective resistance G of the hydrophone according to the following equation.! N = 10 log G — s — 195. (4) G is measured in ohms while N represents decibels above 1 dyne per sq cm. The output of the hydrophone, or preamplifier, is transmitted through the hydrophone cable into the receiving sound stack. There the signal is filtered, amplified, possibly rectified or heterodyned, and then fed into the recorder. Most commonly used for re- cording are cathode-ray or galvanometer oscillo- graphs with a very nearly linear> response, which re- > The circuit is said to respond linearly if the amplitude of the recorded signal is proportional to the amplitude of the incident sound field. TRANSMISSION RUNS 75 Figure 7. Oscillograph record of received 50-msec single-frequency signals; F is the radio signal, T is the 60-c timing trace, I and III are rectified traces, and II is the heterodyned trace. The range is approximately 80 yd. cord the received signal on film or sensitized paper moving past the oscillograph at a constant speed. A timing trace, usually provided, permits accurate measurements of time intervals on the film or paper strip. The signal is transmitted not only as an under- water sound signal, but also as an airborne radio signal over a radio link, usually FM, between the two ships. At the ranges involved, the radio signal arrives practically without time delay and without distortion. Tn addition to providing a convenient monitoring de- vice, the radio signal serves as a means of accurately determining the range between the two ships. Since it is reproduced as a separate trace on the oscillograph record, it is an easy matter, with the help of the tim- ing trace, to measure the time interval between the arrivals of the radio signal and the sound signal, and thus determine the distance traveled by the sound for each separate transmitted pulse. The resulting record will be similar to the strip in Figure 7. The top trace is the radio trace, the bottom trace the timing trace, and the three traces labeled I, II, and III, are the outputs of three different hydrophones, recorded simultaneously. The outputs of I and III were recti- fied before being recorded, and the output of II was heterodyned down to 800 ¢ before recording, but not rectified. The range can be read off the record with an error of less than 15 yd. In the example shown in Figure 7 the range is approximately 80 yd. In the past, transmission runs have also been re- corded by means of power level recorders. These re- corders are electromechanical recording instruments with a logarithmic response. A stylus records on a moving paper strip the received signal level in decibels above the reference level. Although these records are much easier to read than oscillograph rec- ords, they suffer from a certain unreliability of the instrument. Frequently, the stylus “‘sticks’”’; that is, it follows a change in signal level only when this change exceeds an appreciable threshold value. Fur- thermore, the stylus travels only a certain number of decibels per see (50 to 500 db per sec, depending on the model); the instrument, therefore, cannot record correctly the level of very short signals. For this reason, power level recorders have not been used in recent transmission work. WHOT has developed an electronic device designed to combine the advantages of both oscillograph and power level recorder. It consists essentially of a rectifier, an amplifier with logarithmic response over a range of approximately 80 db, and a galvanometer oscillograph. This device has a time constant of roughly 0.5 msec.” Up to the present, it has been used only for reverberation studies; whether it will prove: useful in transmission work remains to be seen. The amplifier used in this device has been improved since reference 2 was published. The output of the hydrophone is passed through filters at some stage before it reaches the recording instrument. The purpose of the filter is to improve the signal-to-background ratio. All the unwanted background (see Division 6, Volume 7) contains energy in a very broad frequency band. A band-pass filter centered at the signal frequency will discrimi- nate against the broad-band background in favor of sound at the signal frequency. Most of the filters used are approximately 44 ke wide. Such a width leaves an adequate margin for possible drift of the driving oscillator in the sending stack and for dop- pler. Both the amplifiers and the recording instruments will be linear and otherwise satisfactory only in a limited range of signal amplitude. On the other hand, actual signal levels are likely to change by as much as 80 db between short and long ranges of transmis- sion. For this reason, step attenuators are provided. These attenuators are usually operated by hand; however, in the most recent installation at UCDWR 76 EXPERIMENTAL PROCEDURES the attenuator is automatically actuated when the received signal level rises above or drops below certain limits for several successive signals. All changes in attenuator setting are recorded, either in a separate log book, or automatically on the oscillograph record.* 4.3.2 Field Procedures ! In this section the field procedures used in trans- mission runs will be described. First, a number of oceanographic facts are ascertained and recorded, either immediately preceding or immediately follow- ing each transmission run. These include the depth of the ocean, the type of bottom (in shallow water), the state of the sea, the swell, the wind strength, and, most important, the vertical temperature distribu- tion in the ocean. The bathythermograph, an instru- ment which measures vertical temperature gradients, is in general use in the Navy wherever echo ranging is involved. It is a recording device which can be lowered into the water down to considerable depth (as much as 450 ft for the “‘deep”’ model) and which, upon being returned to shipboard, indicates the temperature versus depth distribution as a trace marked on a smoked slide. Ordinarily, a bathyther- mograph is lowered on each of the two vessels partici- pating in a transmission run; the source vessel makes its lowering at the point of greatest distance from the receiving vessel and frequently one or two lowerings at intermediate points. Figure 8 shows a blank which contains the oceanographic information belonging to a simple transmission run. This blank has been used at UCDWR. Another subsidiary step is the calibration of equip- ment. In this chapter the term calibration will be used with a definite meaning. Calibration is a pro- cedure which translates sound field data taken off the oscillograph trace into the transmission loss. The transmission loss was defined in Section 4.1 as the difference in decibels between the source level S of the vrojector and the sound level Z at the hydrophone. Snicze the source level of the projector is defined in turn as the sound pressure level at a range of 1 yd, the transmission loss is then the difference in decibels between the sound levels at a range of 1 yd, and the range r of the hydrophone. In theory, then, one would obtain the transmission loss according to the follow- ing formula. , \2 H = 10 log (2) 5 OAT where a is the sound pressure amplitude in the water at the hydrophone, and a; is the sound pressure amplitude at 1 yd. If it were possible to bring the receiving hydro- phone up to a distance of 1 yd from the projector, the absolute transmission loss could thus be readily de- termined without knowing either the projector source strength or the hydrophone response. The squared ratio between the signal amplitude (on the oscillo- gram or on the tube screen) at 1 yd and the signal amplitude at R yards would give the transmission loss, provided the design of the receiving stack guar- antees proportionality between received pressure amplitude and recorded trace amplitude. Actually, it is next to impossible to bring the projector of the sending vessel and the hydrophone of the receiving vessel closer together than about 30 to 50 yd without inviting a maritime catastrophe. Correction of the observed signal level at 30 or 50 yd back to the pre- sumed level at 1 yd has at times been done by straightforward application of the inverse square law. However, this method is probably too simple. There is some evidence that, even at ranges of 50 yd, the transmission loss cannot always be expected to follow the inverse square law of spreading.* Several more complicated methods have been em- ployed, which in theory should enable determination of the absolute transmission loss. Although none of these suggested calibration procedures have proved completely satisfactory, some may be preferable to the simple correction by means of the inverse square law. The following paragraphs are devoted to a description of some of these more refined calibration procedures. During a substantial part of its supersonic trans- mission program, UCDWR carried out runs called calibration runs at very short range, approximately 100 yd. During these runs, both the sending vessel and the receiving vessel were permitted to drift. The signal level at 100 yd, obtained from this run, was arbitrarily assigned a transmission anomaly value of zero; and all other transmission data obtained on the same day were referred to the 100-yd level obtained in the calibration run. Somewhat later, these special runs were discontinued. Instead, an average was taken of all the short-range data ac- cumulated during the day, and a value of zero for the transmission anomaly was assigned to this average. In these two methods no attempt is made to calibrate in terms of a distance of the order of 1 yd; that is, no test is made which would relate 77 TRANSMISSION RUNS DEPTH IN FEET (UMGON) vorysutojur o1ydeisouvad0 10; yuR[G “8 TUNSIT GE de "ON LO3f°0ud Va HAN GNIM SIL T1a4S vas ONO SS ae ae ae SdAL WOLLO8 NOY a ea re ed —______ Bid Yava VIVO SIHdVHSONWS90 NI SuNIWeadW3L INVSNOHL NI SONVISIG NOISSINSNVYL QOHLAW '03ud SONVY ‘SVAZ == or See ee ala uGd ae ae NYSLIVd —— Le 40 SIL aaa ‘ON 3010S Tas YAGWNN 18 GNSS —<“<—~C~;~*C«SS SSAA 78 EXPERIMENTAL PROCEDURES the sound level at 100 yd to the source level as defined in Section 4.1. WHOL used a related method of calibration until the summer of 1945. For each individual run, the ob- served sound field levels were each increased by 20 log R and plotted against range. The resulting points between a range of 100 yd and the range where the observed intensity was 40 db less than the in- tensity at 100 yd were then fitted by inspection with a straight line. This line was extrapolated to a range of 1 yd to give the presumable sound level (in decibels above 1 pv) at that range. More recently, both institutions have put into use new methods, which are designed to obtain a calibra- tion directly in terms of the short-range (1 to 10 yd) sound level. These methods are of two kinds, which may be described as unaided calibrations and as calibrations with the help of standards. Asan example of unaided calibration, WHOI floats one of the re- ceiving hydrophones out to a distance where it can be picked up safely by the crew of the sending ship. The hydrophone remains connected by cable with the re- ceiving ship, and its output is recorded by the same equipment used in the transmission runs. The hydro- phone is then secured at a measured distance of a few yards from the face of the projector. The projector is trained on the hydrophone, and a signal is put into the water and received by the hydrophone. At this short range, the effect of the surface is minimized by the directivity of the projector, and tests have shown that the sound field obeys the inverse square law within the limits of observational accuracy. This method of calibration would be expected to yield the most accurate and most trustworthy results. A sub- stantially identical method is occasionally employed by UCDWR for checking the results of other calibra- tion methods. The unaided methods are not always practical in the field since in a heavy sea the transfer of a hydrophone from one ship to the other may not be possible. At best, the hydrophone transfer is a cumbersome and time-consuming maneuver. Never- theless, unaided calibration is standard procedure at WHOI. Most of the field calibrations at UCDWR are now made with the help of calibration standards, sound units which are designed primarily for this purpose and which are more stable than other units. Typical is a UCDWR procedure which involves the use of two OAX transducers; these transducers are HUSL de- signs. One of these two transducers is kept aboard the receiving vessel, the other aboard the sending ship. Aboard the sending ship, where the projector source strength S is to be determined, the OAX is used as a hydrophone. It is attached to a boom which can be swung over the side and which insures that the OAX is always at the same distance (13 ft) from the face of the keel-mounted JK projector. The projector is trained on the OAX, and the voltage generated across the terminals of the OAX unit by the sound field of the JK projector is measured. Aboard the recelving ship, where the hydrophone response S is sought, the second OAX unit is hung over the side amidships to a depth where it clears the keel. The receiving hydrophone is also hung over the side, on the other side of the ship, and is lowered to the same depth as the OAX. The distance between the two units is, therefore, with fair accuracy, the beam width of the receiving ship. The OAX is then energized as a projector with a standard power input, and the gen- erated voltage across the terminals of the receiving hydrophone measured. If these two tests lead to the same results or very nearly the same results day after day, it is assumed that all four units are constant. Large jumps (several decibels) are presumably indi- cations that either one of the four sound heads or the electrical follow-up (amplifiers and associated equip- ment) has changed its characteristics. If the change cannot be assigned to the electrical follow-up, it is assumed that the standard OAX units have remained unchanged and that either the JK power output or the receiving hydrophone response has changed. In a word, the characteristics of the projector and hydro- phone used in the transmission runs are measured in terms of auxiliary standards, which are presumed to be stable. The standards themselves are thoroughly tested every few months at a special calibration station. For a time, WHOI also used a similar method of calibration that depended upon the use of HUSL monitor standards. This method of calibration was later abandoned by that group in favor of unaided calibration. Clearly, not one of the calibration methods which have been described is both convenient and wholly satisfactory as a method for translating observed hydrophone voltages into accurate estimates of the absolute transmission loss. Yet, until electroacoustical equipment is developed which can be relied on to re- main stable, field calibration remains a necessity. It is to be hoped that rapid and adequate procedures of calibration will be developed in the future. Once the equipment is calibrated, the transmission TRANSMISSION RUNS 79 loss can be determined by measuring the received signal level at the range and depth of the hydro- phone. There are several types of runs. It is possible, for instance, to make a vertical transmission run, in which the range between the two ships is kept very nearly constant and in which the hydrophcne is slowly raised or lowered, so that the transmissic n loss is determined as a function of depth at a fixed range. In horizontal runs, the depth of the hydrophone is kept fixed, while the range is changed. Horizontal RECEIVER STACK RECEIVING SHIP SUPPORTING CABLE OO- LB WEIGHT fq-HY DROPHONE 25-LB WEIGHT Figure 9. Method of suspension of deep hydrophones. runs are much easier to carry out than vertical runs. In a vertical transmission run, the depth of the hydrophone can be changed only slightly each time. One or several pings are transmitted while the hydro- phone is kept at-a constant depth; and then the hydrophone depth is changed again. Also, whenever the hydrophone is moving through water, the flow of the water past the hydrophone gives rise to noise, which may effectively mask the signal. In a horizontal run, the receiving ship is permitted to drift, or, in shallow water, anchored. The noise due to water current thus is minimized and the hydrophone cable tends to hang straight down. Proper training and control of depth is thus facilitated. The sending ves- sel then runs either toward or away from the receiv- ing ship. In this manner the range can be varied con- tinuously by an amount of several thousand yards without ever interrupting the transmission of signals. A supersonic transmission run of the horizontal type takes, on the average, about 20 or 30 minutes. If it is desired to obtain the transmission loss at several depths, two or three hydrophones can be suspended at various depths from the receiving vessel. The over- whelming majority of transmission runs made up to the present have been horizontal runs. In all transmission runs, elaborate precautions have always been taken to keep hydrophones at their nominal depth. Because of the wind drift of the re- ceiving vessel, and because of ocean currents going in different directions at different depths, deep hydrophones, which are lowered occasionally as far as 450 ft below the surface, will rise to a much shallower depth unless special care is taken to make them hang straight. To this end, a 300-lb weight is suspended from a strong steel cable; the hydrophone hangs down from this weight and is held down by an addi- tional 25-lb weight as shown in Figure 9. The hydro- phone cable carries relatively little weight in this arrangement. Horizontal transmission runs can be either ap- proaching runs or receding runs, that is, the sending ship can either close or open the range. In the reced- ing run, the wake of the sending vessel is located be- tween the two ships. Since it has been found that wakes are capable of absorbing sound, the sending ship usually changes its course from time to time in a RECEIVING SHIP 6 TO 12 KNOTS | WIND Figure 10. Course followed during a receding run. manner illustrated in Figure 10 in order that the direct sound path between the two ships may never pass through the wake laid by the sending ship. How- ever, over shallow bottoms where change of course would result in a changing bottom type, this pro- cedure is sometimes not followed. During an approach run, the sending ship remains between its wake and the receiving ship, and it may, therefore, run along a straight course and pass the receiving vessel at a 80 EXPERIMENTAL PROCEDURES range of approximately 100 yd. During a run, the sending vessel keeps its projector trained at all times on the receiving vessel. This aiming is done by means of a pelorus, mounted either on the flying bridge or vertically above the sound projector to eliminate parallax. In a recently developed installation, selsyn repeaters cause the sound projector to follow auto- matically the changes in bearing of the pelorus. In former installations, an operator in the ward room of the sending ship had to match the two bearings by hand. At WHOI, projector training is now completely automatic, with the help of a radio compass. In transmission work at supersonic frequencies, ping lengths are usually either about 50 msec or about 10 msec. It is believed that as the signal length decreases below 50 msec aural perception of the re- sulting echoes becomes more and more unsatisfactory (see Volume 9 of Division 6). Very short pings have an important use, however, in transmission studies. Tf the water is fairly shallow and the ping length is sufficiently short, the directly transmitted and the bottom-reflected signals can be examined separately. This is possible when the time resolution of the fol- low-up circuit is sufficient to resolve the time differ- ence of arrival between direct signal and bottom-re- flected signal. The minimum requirements of resolu- tion in a specific case will depend on the geometry of the paths, depth, range, and refraction pattern of the ocean. For very long ranges, the signals often arrive with badly distorted envelopes and with tails known as forward reverberation. When such tails are present, no resolution of the electrical circuit will result in satis- factory separation of the two sound paths. In the absence of such tails, resolution is frequently possible even with fairly long, square-topped signals; in other words, it is possible to distinguish three portions of the received signal trace, the direct signal alone, the composite signal, and the bottom-reflected signal alone. Signals are emitted at a rate of about one signal per second. Once a minute pinging is interrupted, and a long signal of 10 seconds’ duration is sent out. These long signals serve two purposes. First, a received long signal often provides a very instructive, graphic il- lustration of the degree of coherence of the transmis- sion. Furthermore, this long signal makes it possible to correlate the received short sound signals with the radio signal, and thus to determine the range at which the signal was received. In a transmission run carried out at 5,000 yd, by the time a sound signal arrives at the receiving ship, three additional signals have al- ready been put into the water. The once-a-minute breaks facilitate the identification of particular sig- nals. The overall accuracy of the determination of the transmission loss of an individual signal has been im- proved steadily in the course of the transmission pro- gram. A distinction must be made between the deter- mination of the absolute transmission loss, and the determination of the difference in transmission loss between two signals received at the same range, or one signal received at different ranges. The deter- mination of relative loss is not affected by errors of calibration, while the determination of the absolute loss is affected by calibration errors. The uncertainty of calibration in the earlier data taken both by UCDWR and WHOL is very large and probably ex- ceeds 10 db in many instances. Improvement in pro- cedure has cut this uncertainty down to approxi- mately 1 db. Both absolute and relative determina- tions are affected by training errors of the projector and by the horizontal directivity of the hydrophone. Training errors are small at long range where the bearing is changing slowly. At ranges of the order of 100 yd, where the bearing changes rapidly, training errors can be significant even when great care is used in following the target. The uncertainty of training has been almost eliminated by improved instrumenta- tion and is probably well within 1 db at the present time. Even in earlier work, training errors probably never caused an error in received sound level much in excess of 1 db. The most recent hydrophone models in use are practically nondirectional at 24 ke, but the directivity of the CN-8 model used in earlier studies introduced errors of about 2 db. Thus, the experi- mental error of most of the transmission loss deter- minations at UCDWR is probably about 2 db, while for the most recent data the experimental error is probably considerably smaller. 4.3.3 This section will be concerned with the analysis of data in which single-frequency supersonic sound is received by one of the recording systems with a linear response. The procedure used in the analysis of runs carried out with sonic sound will also be sketched. Figure 11 shows that the received sound intensity is subject to rapid changes in intensity, which obvi- ously cannot be related to observed changes in range or temperature distribution. Figure 12 shows the re- ceived amplitude of a continuous, 10-sec, 24-ke signal. Analysis of Data 81 TRANSMISSION RUNS “SUIBIBO][IOSO [BUBIS 99S-NT OMT, “ZI WAND “UOI}eNyJONY SULMOYS s[VUSIS PaAtedaI Jo sp1000qy TI wang 82 EXPERIMENTAL PROCEDURES The upper strip was obtained at a range of 110 yd in the direct sound field, while the lower strip is a typical record of sound received at a long range, 1,700 yd, in the shadow zone. This fluctuation of received sound field intensity has become the subject of special in- vestigations, which are summarized in Chapter 7. The principal purpose of transmission runs, however, is to obtain the average transmission properties of the ocean with a given set of oceanographic conditions. To obtain a representative average, it is necessary to select a sample of signals, assign to each signal an individual sound field amplitude, and then to strike an average. The final result of these steps, the average sound field intensity or sound field level, will depend not only on the record obtained, but also on the details of the sampling and averaging procedure employed. UCDWR has standardized these pro- cedures to insure intercomparability of results ob- tained at different times and by different research groups. The procedure is described in a report by UCDWR: and will be briefly recapitulated in the following paragraphs. In the selection of a sample several requirements must be satisfied. The sample of individual signals must be large enough so that the standard deviation of the average is not much larger than of the order of 1 db. Moreover, the benefit of averaging will be obtained only if the sample covers a period of time in which the transmission passes through a number of maxima and minima, for otherwise the average would be an average of individual signals most of which may be relatively high or relatively low. On the other hand, the period of time covered by the sample must be short enough so that it corresponds to a negligible change of range between the two ships and a negligible change in the large-scale temperature structure. The standard procedure for supersonic work, de- signed to strike a compromise between these require- ments, has been to select five signals, equally spaced during a period of 20 sec. Since the standard devia- tion of an individual signal from average intensity is between 2 and 4 db in most samples, the standard deviation of the arithmetical average of five signals from the average of a very large number of signals is between 1 and 2 db, (1/ Vn — 2 times the standard deviation of the individual signals). At WHOI, the rule has been to use as a sample ten consecutive signals. Since signals are transmitted about 1.2 sec apart, a sample extends over a period of 12 sec. This method, although slightly different from that employed at UCDWR, leads to averaged ampli- Figure 13. Signal with high noise background. TRANSMISSION RUNS 83 Figure 14. tudes which differ from those obtained by the other method, but probably by no more than the internal spread of either method. The next step consists of the assignment of an amplitude to each member of the selected sample. If the received signal were square-topped, like the emitted signal, this step would raise no questions. However, received signals are often far from square- topped. It was decided at UCDWR to assign to each signal the peak value of the amplitude registered any- where during one signal, with two qualifications. One concerns noise received simultaneously with the signal. At low signal levels, the noise which is re- ceived continuously shows up as a very striking “spiny” record (shown in Figure 13). Such spines superimposed on the signal are disregarded. This rule presupposes that noise spikes can be distin- guished from rapid signal fluctuations. It has been found that all persons competent to evaluate record film are able, with but little practice, to make that distinction. The other qualification concerns ‘‘end spikes.” Frequently, there is interference between sound traveling via two different routes, for example, direct and surface-reflected sound. As the two paths do not have exactly the same length, the intensity at the beginning and end of the signal may be markedly different from the intensity during the signal. In the case of destructive interference, the signal then as- sumes the shape shown in Figure 14. The end spikes appearing in such signals are also disregarded. The rules just outlined have certain advantages and certain drawbacks. The principal advantage is that the peak amplitude of a signal can be read much more rapidly than such quantities as mid-signal amplitude; also, it is unambiguous. The drawbacks appear when the signal envelope is not smooth. In id, vane Sr eck erat . 1 / SAE Rao nN PA ANT WANNA MWY WW VA WN, vi Ww Signal with end spikes. that case the sound emitted during a short signal interval arrives at the receiver during a much longer period of time, with the result that the energy re- ceived during an interval equal to the signal length is substantially less than all the energy received. This effect will be quite conspicuous for very short signals, but negligible for continuous transmission (10-sec signals). As a result, the average amplitude is very definitely a function of ping length, when peak amplitudes are used; it very likely would not be a function of ping length if the amplitudes of individual signals were defined in a different manner. One possible solutiga has been suggested by the group which is carrying out transmission experiments at WHOI. They have constructed an integrating cir- cuit. If the received signal is squared and fed directly into this integrating circuit, the recording instru- ment shows the total energy received. This would be strictly proportional to the signal length and would thus provide a measure typical for the ocean and its overall transmission properties. Any deviation from strict proportionality would be indicative of non- linear transmission and would, therefore, be of the greatest importance. At the time of this writing, no such experiments had been carried out. Once individual amplitudes have been assigned to the five signals that comprise one sample, the average amplitude is found by taking the arithmetical mean of the five individual amplitudes. This procedure has the advantage of simplicity. Alternatively, one could compute the mean level or the mean intensity (squared amplitude). A very rough estimate shows that in a typical record the averaging of amplitudes and of intensities will lead to results which are dif- ferent by about 1 db. While this difference depends on the assumed distribution function of amplitudes, 84 EXPERIMENTAL PROCEDURES it is not likely to be a dominant cause of error in a determination of the transmission loss. Transmission work at single sonic frequencies began only recently, and the analysis procedure has not yet been very well standardized. Records ob- tained up to the present appear to indicate that fluctuation of signal intensity is much less severe at sonic frequencies than at supersonic frequencies. On, the other hand, because of image interference, sys- tematic changes in signal level are observed at short ranges which vary so rapidly with range that any averaging procedure would obscure them. For this reason, in transmission work at frequencies from 200 to 1,800 ¢c individual signal levels rather than sample averages are reported. In the records obtained at sonic frequencies the en- velope of the signal trace, as a rule, is badly serrated. The fuzziness of the envelope is probably caused by the unfavorable signal-to-noise ratio, which is about 1 db for 1.8 ke and lower, somewhat higher for 22.5 ke, and by the relative narrowness of the filters used in the recording channels. If random noise is received through a wide filter, the oscillograph trace has a typical “spiked” appearance, that is, the noise is characterized by sudden high peaks of short dura- tion. If the filter is narrow, as it must be in sonic transmission work, the individual peaks are lowered and broadened, and their separation from the single-frequency signal is more difficult. For this reason, the person reading the film record does not attempt to measure the “peak”’ level, which would be fictitious, but estimates and reports the average amplitude. It has been found that the uncertainty introduced by this estimate is less than 1 db, on the average. The final step in the processing of a transmission run consists of the recording of the computed signal intensity. Since in most transmission runs the range is altered by a factor of 10 to 100, the signal levels change in the course of a run by a large number of decibels. It has, therefore, been found useful not to plot signal level in decibels below transducer output directly, but to take out the bulk of the variability by plotting the transmission anomaly. Usually, the transmission anomaly is plotted as the ordinate down- ward, with range as the abscissa. Theoretically, this curve should pass through zero for zero range. In view of the great experimental difficulties involved in the determination of the signal level at very short ranges, the curves usually stop at a range of 109 yd or more. 4.4 ECHO RUNS As mentioned before, transmission runs are by far the most important useful method of obtaining in- formation on the propagation of sound in the ocean. The other two methods, which are of secondary im- portance, will be discussed in the next two sections: for the sake of completeness. Kicho runs have been carried out both on specially designed standard bodies and on chance targets, such as wrecks, in order to study the dependence of echo level on range and in order to study fluctuation and coherence. The principal purpose of echo runs has usually been to study not the propagation of sound between echo-ranging transducer and target, but rather the effect of certain targets on the received echo. (See Chapters 18 to 26 of this volume.) The equipment used in echo runs differs from that used in transmission runs in that sending and re- ceiving stacks are aboard the same ship and the same sound head is used both for sending and receiving. A change-over relay connects the sound head first with the sending stack and then, immediately following the emission of the signal, with the receiving stack. Artificial targets have been developed for research and training purposes. Natural targets usually re- flect very differently at different aspects; most arti- ficial targets are designed to minimize this kind of directionality without sacrificing too much overall reflecting power. There is one geometrical shape which remains the same regardless of any twisting of the cable from which the target is suspended. That is the sphere. From the point of view of constant re- flecting power, spheres constitute ideal artificial targets. Unfortunately, the reflecting power of a sphere, while constant, is fairly small. To obtain useful echoes from spheres at distances similar to the ranges commonly encountered in practical echo ranging, one would have to use spheres with a diameter greater than 30 ft. It was found, however, that a 10-ft sphere was almost unmanageable at sea. The only spheres which could be handled with ease were spheres with a diameter of 2 or 3 ft. In the search for an artificial target with a large target strength, the best solution found so far has been the triplane®” shown in Figure 15, which com- bines ease of handling with a reflecting power com- parable to that of a submarine. It is a well-known fact, sometimes used in optical signaling, that a ray which has been reflected from three planes which are INFORMATION OBTAINABLE FROM REPORTED RANGES 85 mutually perpendicular leaves in a direction exactly opposite to the incident direction. The action of a tri- plane can, therefore, be compared with that of a single plane perpendicular to the incident rays. That is why a triplane reflects a larger percentage of the incident energy back into the transducer than any other body of equal size. Fictre 15. Triplane. Another type of artificial target is the so-called echo repeater. This is a device which acts essentially as a relay. It consists of a transducer with power out- put proportional to the incident sound energy re- ceived by a hydrophone. Echo repeaters have been used only for training purposes. A full description of the echo repeater can be found in two UCDWR re- ports.®? 4.5 INFORMATION OBTAINABLE FROM REPORTED RANGES The research methods described in Sections 4.3 and 4.4 of this chapter are of comparatively recent origin. During the first year of the war, the only information available consisted of observed maximum echo and listening ranges obtained by surface ships on escort or patrol duty and by research vessels on ocean cruises. In testing the performance of echo-ranging gear, several workers recognized the strong variability of achieved maximum echo ranges.’°!4 Vessels at- tached to the West Coast Sound School at San Diego found in practice maneuvers that ranges in the after- noon compared unfavorably with ranges in the morn- ing of the same day. These observed maximum ranges gave the first clue to the existence of shadow zones in the presence of sharp downward refraction. Maxi- mum ranges obtained by echo ranging on submarines above and below the depth of the layer revealed the existence of a layer effect (see Section 5.3.3). Also, the tabulation of observed ranges over various types of ocean bottom in shallow water gave clues as to the effect of the bottom on sound transmission. Because of the unexplained variability of observed maximum ranges, it was decided to set up the investi- gation of the underwater sound field as a research program, and the quasi-laboratory methods of trans- mission runs and echo runs were developed. Since the inception of the transmission program, observed echo ranges have rarely served as scientific evidence; they have continued to serve as a stimulus for the investi- gation of new problems and as signposts on the road to solutions. The SS Nourmahal, a converted yacht with a deep projector and with unusually quiet machinery, has reported extreme echo ranges in the presence of very deep isothermal layers; as a result, the sound field in deep mixed layers was investigated. To give another example, earlier experience had shown that the sound field in shallow water over MUD bottoms is very nearly the same as the sound field in deep water, because MUD reflects sound rather poorly. Unexpectedly long echo ranges in certain areas in which the bottom was classified as MUD led to a new program aimed at a differentia- tion of the various bottom sediments now called MUD. Also, data obtained at WHOI seem to show that attenuation increases at very high wind forces. These observed ranges have been obtained both by naval vessels in regular operations and by research vessels. A number of naval vessels have sent to WHOI records of observed maximum echo ranges along with bathythermograph slides. Additional observed ranges were obtained by research vessels on extended cruises in various parts of the world. The range data thus obtained do not permit any detailed conclusions con- cerning the transmission loss as a function of range but serve to indicate whether sound transmission was good, fair, or poor. A summary of the theory of maximum echo ranges is presented in Volume 7 of Division 6. Chapter 5 DEEP-WATER eines 1N DEEP WATER, where bottom- reflected sound is unimportant, is somewhat simpler to study than transmission in shallow water. Even when the effects of the bottom have been elimi- nated, however, sound transmission in the ocean re- mains an exceedingly complex phenomenon. The theoretical results based on the elementary ray theory and on an idealized ocean stratified in uni- form horizontal layers are seldom realized exactly in the sea. Sometimes this simple picture leads to er- roneous results, even qualitatively. Moreover, the only constant element in underwater sound trans- mission is change. No one ping resembles the preced- ing. In this chapter, the rapid fluctuation of trans- mitted sound from one second to the next is ignored and reference is made throughout to averages based on many consecutive pings. However, even these averages sometimes vary considerably. Although the theory developed in the previous sec- lions is admittedly imperfect and may be incorrect in principle, this theory is nevertheless retained as the framework on which to hang the discussions of the observational material. The theory is believed valu- able, partly in indicating which results may be ex- pected to have general validity beyond the particular conditions under which the results were obtained. Even more important, a discussion of the interrela- tion between facts and theories is essential for an in- telligent formulation of research programs. In the long run, progress in any scientific problem can be achieved most efficiently by formulating hypotheses and then testing them in critical experiments. To lay the groundwork for such future research is, in large part, the objective of the present chapter. FACTORS AFFECTING DEEP-WATER TRANSMISSION In principle, the propagation of sound can be com- pletely determined if the nature of the medium through which the sound passes is known. In the present section a description is given of the known properties of the sea which are believed to influence underwater sound transmission. Dell 86 TRANSMISSION 5.1.1 Meaning of “Deep Water” For the purposes of this chapter, water is deep when the bottom has a negligible effect on under- water sound propagation. From a theoretical stand- point this has the very simple meaning that the bot- tom is ignored; the ocean is thought of as extending to infinite depths. From the observational stand- point, this means that only those observations will be considered here on which the bottom is believed to have no effect. In general, the bottom can have several effects on underwater sound. Sound energy reaching the bottom may be partly reflected back at various angles into the body of the sea and partly transmitted into or absorbed by the bottom. The relative amounts re- flected and absorbed depend on the depth and the nature of the bottom, prevailing refraction condi- tions, and sea state. This dependence and, generally, the effect of the bottom on sound transmission will be discussed in detail in Chapter 6. Furthermore, the presence of the bottom affects the background. Some of the sound reflected backward by the bottom reaches the receiver and gives rise to a ringing sound known as reverberation. For most of the observations discussed in this chapter, short pulses of sound are used. With this technique, sound that has traveled to the bottom and has been reflected toward the hydrophone can readily be distinguished from sound that has traveled directly from projector to hydrophone. Thus, for most obser- vations the bottom-reflected sound can readily be distinguished. If the bottom is rough, an appreciable amount of sound may reach the hydrophone after having been scattered from various portions of the sea bottom so that the signal is followed by a long single-frequency train of forward reverberation. Usually, the direct signal is so far above this back- ground of scattered sound that forward reverbera- tion is negligible in the evaluation of transmission observations. It is of importance to know when the ocean is ef- fectively deep in practical applications of underwater sound. With present echo-ranging gear, an echo from FACTORS AFFECTING DEEP-WATER TRANSMISSION 87 PERCENTAGE OF TOTAL OCEAN AREA IN EACH 200-METER DEPTH ZONE Figure 1. a submarine or typical surface vessel cannot ordi- narily be detected more than 3,000 yd from the pro- jJector in water of any depth, except under unusually favorable conditions. All supersonic projectors are highly directional with not much energy radiated at angles more than 6 degrees from the axis. If the bot- tom is more than 150 fathoms below the projector, and the water is isothermal, very little of the energy in an echo-ranging pulse will reach the bottom at ranges less than 3,000 yd or return to the surface at ranges less than 6,000 yd. Thus, the echo from targets near maximum range will contain very little bottom- reflected sound; however, the background for such echoes may contain some bottom reverberation. If sharp temperature gradients are present in the upper layer of the ocean, the sound beam will be bent down more sharply, and a considerable amount of bottom- reflected sound could reach a target 3,000 yd away in water 150 fathoms deep. The bottom reverberation in suck conditions may be quite intense even at 1,500 yd. To insure that bottom-reflected sound cannot re- turn an echo in practical echo ranging, a depth of more than 200 fathoms is required, while twice this depth is required to eliminate bottom reverberation. For various types of tilting beam equipment, sound scattered from the bottom can be important even in 00 DEPTH IN METERS Distribution of depths in the sea. somewhat deeper water. For most echo-ranging situa- tions, however, 100 or 150 fathoms is a more repre- sentative dividing line between deep and shallow water. It makes very little difference whether the point of division is taken as 150 fathoms, or 100 fathoms, as has been done in the manuals of echo-ranging predic- tion issued by the Navy,!" or 200 fathoms, as has been suggested. Water depths between 100 and 1,500 fathoms are quite uncommon. Figure 1 shows the distribution of depths in the sea.* It is evident from Figure | that almost all of the ocean bottom is either less than 100 fathoms, about 200 meters, below the surface, or more than 1,500 fathoms, about 3,000 meters, below the surface. Water which is deep for echo ranging may be shallow for sonic listening, since average listening ranges are so much longer than average echo ranges, and since sonic listening gear is nondirectional. Lis- tening ranges are often greater than 10,000 yd. Ex- cept in the deepest parts of the ocean, sound arriving from such long ranges will contain bottom-reflected sound. Sonic gear is usually nondirectional in a verti- cal plane, at least at low frequencies, and bottom-re- flected sound in 2,000 fathoms may contribute ap- preciably to the received signal. Thus, for sonic lis- 88 DEEP-WATER TRANSMISSION SURFACE LAYER DEPTH IN METERS THERMOCLINE LAYER DEEP WATER LAYER TEMPERATURE IN DEGREES F Figure 2. Typical temperature-depth curve. tening at long ranges the ocean is rarely if ever deep. for supersonic echo ranging will also be deep for Supersonic listening gear, on the other hand, is supersonic listening, even though supersonic listening usually sharply directional in the vertical plane and ranges may exceed sonic ranges. will discriminate against bottom-reflected sound from Evidently, at supersonic frequencies most of: the ships in the same way that it will discriminate against ocean is effectively deep for practical purposes. Even bottom-reflected echoes. Thus water which is deep at low sonic frequencies, transmission in water deeper FACTORS AFFECTING DEEP-WATER TRANSMISSION 89 than 1,500 fathoms is practically deep-water trans- mission under many conditions. Thus the study of sound transmission in deep water is of considerable practical importance. 5.1.2 Vertical Temperature Structure and Computed Ray Diagrams The temperature distribution in the ocean largely determines the sound velocity distribution, which we have seen is an important factor in sound intensity. For this reason, measurement of ocean temperatures at various depths has been an integral part of the re- search on sound transmission and is also important in the tactical use of sonar equipment. The temperature in the ocean is affected by the absorption of radiation from the sun and sky, by the cooling of the surface layer by evaporation, by dis- placements due to currents and upwelling, and by the addition of fresh water near shore. Usually a water column in the deep sea can be divided into three principal layers, shown by the sample temperature- depth plots in Figure 2: (1) a relatively warm surface layer, which is subject to daily and seasonal changes in thickness and vertical temperature gradients, (2) a layer of transition at mid-depths called the thermo- cline, in which the temperature decreases rapidly with depth, and (3) the cold deep-water layer, in which the temperature decreases only gradually with depth. A detailed discussion of the temperature dis- tribution in the ocean is given in Volume 6 of Divi- sion 6. Here, only a few of the basic temperature- depth patterns are described and their expected in- fluence on underwater sound transmission briefly dis- cussed. It will be pointed out in later sections that the transmission loss is least and sound ranges are longest when the surface layer is reasonably isothermal and deeper than about 100 ft. Such deep isothermal layers tend to occur when the water at the surface is losing more heat than it is gaining, as in midwinter in the high latitudes. The colder surface water will be heavier than the water just beneath and will mix with it. As a result, a surface layer of more or less constant temperature will be formed. In midwinter the isothermal surface layer is usually several hun- dred feet deep, except in tropical waters, where this depth varies from 50 to 500 ft depending on ocean currents and other factors. In very high latitudes the isothermal layer may extend down to the ocean bot- tom in February or March. The ray diagram for sound transmission in the case of an isothermal surface layer above a thermocline has approximately the characteristic shape shown in Figure 3. According to the simple ray theory, the sound beam should split at the bottom of the isother- mal layer, with the upper portion bending gradually TEMPERATURE RANGE ——s EROUECTCR | —aeanns(c inane DEPTH Figure 3. Ray diagram for isothermal water above thermocline. back to the surface because of the effect of pressure and the lower portion bending sharply down into the thermocline. Temperature-depth patterns resulting in such a predicted ray diagram are called “split- beam’’ patterns. If the intensity of the sound field were measured along the vertical line SS’ in Figure 3, the intensity immediately below the isothermal layer should decrease sharply with increasing depth. After having reached a minimum, the sound field intensity should begin to increase slowly with increasing depth until finally the edge of the main sound beam is reached. At longer ranges with split-beam patterns a meas- urement of intensity, as along the vertical line RR’ in the diagram, should indicate a substantial amount of sound in the isothermal layer, but in or below the thermocline very little sound should appear. As shown in Section 5.3.2, these predictions of theory for long range are not confirmed by the observations, which show no clear trace of the predicted shadow boundary below the layer. It may be pointed out that the shaded area in Figure 3 is usually not a true shadow zone, even in theory. The temperature-depth graph usually curves continuously from the isothermal layer into the thermocline, rather than breaking at the sharp angle shown in Figure 3. Asa result of this curvature, some direct sound always theoretically penetrates the “shadow zone”’ in this case, but this theoretical sound is much weaker than the sound actually observed. Heating of the surface layer of the ocean some- times produces a temperature gradient which extends all the way to the sea surface. Such conditions are most common at high latitudes during the summer months, when the surface water is gaining more heat 90 DEEP-WATER TRANSMISSION than it is losing. Surface heating and marked temper- ature gradients in the top 30 ft of the ocean are par- ticularly marked on summer afternoons with calm seas and cloudless skies. At night, or during periods of high winds, gradients near the surface tend to dis- appear. The ray diagram computed for a temperature gradient extending up to the surface is shown in Figure 4. The decrease of sound velocity with in- TEMPERATURE RANGE —=— PROJECTOR DEPTH Ficure 4. Ray diagram for negative gradient extend- ing to surface. creasing depth bends the entire sound beam down- ward, as discussed in Chapter 3. Beyond a certain limiting range, which increases with increasing depth, no sound ray can penetrate, and a shadow zone of complete silence should result. Observations of under- water sound transmission confirm the presence of this shadow zone when the temperature gradient: is suf- ficiently strong, about 1 degree or more in the top 30 ft. The only sound reaching such a shadow zone is the scattered sound, which also produces reverbera- tion back at the echo-ranging projector. When the surface gradients are weak, however, other effects become important, and shadow zones do not appear. In addition to these two basic but simplified tem- perature-depth patterns, innumerable varieties of intermediate situations occur. Complicated tempera- ture structure is especially likely in the surface layer; and the accurate computation of a ray diagram from a temperature-depth record can be very laborious. Since the observations usually do not confirm the detailed predictions of the simple theory, which is based on small details of the temperature structure, the computing of ray diagrams for these intermediate cases is of limited usefulness. Sharp positive temperature gradients are extremely rare in deep water. Such gradients are stable only when accompanied by positive salinity gradients. Salinity gradients may also affect sound velocity, but their effect is usually negligible compared to that of temperature gradients. Salinity gradients may be appreciable in some near-shore areas, where large rivers drain into the sea, and at the coastwise margins of the permanent ocean currents, such as the Gulf Stream. In such regions, sharp positive temperature gradients may occur. In the open ocean, however, they are usually less than a few tenths of a degree in 30 ft. Because of the rarity of sharp positive gradients, there is a complete absence of data on sound trans- mission in deep water through regions of strong up- ward refraction. Sales Variability of Vertical Temperature Gradients The way in which ocean temperature changes with depth is variable from time to time and from place to place. Gradual changes from day to day and from one geographical region to another have an important effect on the performance of sonar gear. These changes are discussed in detail in Volume 6 of Division 6, and form the basis for the Sound-Ranging Charts‘ and the Submarine Supplements.® In some areas these changes are so rapid that they greatly complicate the study of underwater sound transmission. In the coastal waters off San Diego, a bathythermograph lowered at one end of a transmis- sion run frequently showed marked differences from the bathythermogram obtained at the other end, with wholly different ray diagrams resulting. Two samples of such records are presented in Figure 5. Some of this variation represents a change with time, while much of it arises from changes with location. In early com- parisons between transmission data and the com- puted range to the shadow boundary, an average was taken of the ranges computed from several bathy- thermograph records. More recently, a single bathy- thermograph record taken on the receiving vessel has been used at UCDWR in studying the relation be- tween the transmitted sound intensity and the tem- perature-depth record. TEMPERATURE MICROSTRUCTURE AND EFFECTS In addition to these large temperature changes over several thousand yards, smaller changes take place over much smaller distances. These changes may affect the way in which the sound beam travels through the water. In Chapter 3 the predictions of the ray theory were discussed for a sound beam passing through an ocean in which the sound velocity depends only on depth, but decreases gradually with depth. In such an ideal ocean an exact temperature-depth record would be similar to that shown in Figure 6. A plot of temperature against range at any depth would FACTORS AFFECTING give the horizontal lines shown in the figure. Under such temperature conditions a shadow zone is pre- dicted at a certain limiting range. In practice, the ocean is never stratified in plane parallel layers, each of uniform temperature. In- stead, an exact temperature-depth record might be | Be ey, |= TEMPERATURE TRACE AT 810 le | == TEMPERATURE TRACE AT 1831 5O 60 70 ° DEPTH IN FEET fo} ° 200 TEMPERATURE IN DEGREES F Ficure 5. Temperature conditions at beginning and end of transmission run. similar to that shown in Figure 7. Plots of tempera- ture against range at different depths would be curves similar to the wavy curves also shown. This ‘‘tem- perature microstructure” must be taken into account in any explanation of observed underwater sound transmission. The evidence available on thermal microstructure is very limited. Measurements on a surface ship are difficult to interpret because of the rise and fall of the measuring instrument through the water. Some of this vertical motion arises from the roll and pitch of the measuring ship, and some from the distortion of the temperature-depth pattern by the surface waves. For this reason, a very small-scale microstructure is difficult to measure from a surface ship, although changes over a hundred yards or so can usually be disentangled from the more rapid changes resulting from roll and pitch. Measurements from a submarine show conclusively the presence of complicated ther- mal microstructure. In Figure 11 of reference 6, fluctuations of the vertical gradient are shown which amount to about 0.020 degree per ft over patches about 100 yd long. This result was obtained with DEEP-WATER TRANSMISSION 91 large temperature gradients present near the surface. When the bathythermograph shows mixed water to more than 100 ft, the microstructure is much less marked. DEPTH IN FEET VERTICAL SECTION OF OCEAN TEMPERATURE F Ficure 6. Temperature distribution in ideal ocean. A general theory of underwater sound transmission which takes microstructure into account has not yet been formulated. However, certain general results seem apparent. These temperature fluctuations are usually fairly small compared to the smoothed gradi- ent, and on the whole the actual temperature-depth DEPTH IN FEET VERTICAL SECTION OF OCEAN TEMPERATURE F Figure 7. Temperature distribution in actual ocean. SOURCE MICROSTRUCTURE ABSENT MICROSTRUCTURE PRESENT Fictre 8. Distortion of sound beam by microstruc- ture. pattern portrayed in Figure 7 does correspond to the ideal pattern shown in Figure 6. Thus some corre- spondence may be expected between observed sound transmission data and predictions based on the smoothed temperature-depth pattern. 92 DEEP-WATER TRANSMISSION The chief effect of temperature microstructure is to introduce irregularities into the path of the indi- vidual sound rays. They will be slightly bent away from the average ray path in random fashion, as in- dicated in Figure 8. Considering a sound beam as a whole, we may expect that microstructure will very slightly broaden the beam pattern, although such broadening effects have never been determined with assurance. Within the sound field, local intensities may show deviations from the average values which would be observed in the absence of microstructure; these local deviations will be discussed in Chapter 7. Another effect of these irregularities is that sound may penetrate with a small but observable intensity into regions which are shadow zones according to the large-scale ray pattern. It is possible to estimate the effect which micro- structure will have on the ray trajectories of indi- vidual sound rays. Theoretical analysis shows that with certain simplifying assumptions the rms lateral displacement Ay of a sound ray because of micro- structure is given by the formula’ Ay = ere (1) In this equation G is the rms value of the fractional velocity gradient caused by microstructure, R is the range, and 6 is a quantity having the dimension of a length, which may be called the patch size of the microstructure. Roughly speaking, b is the average distance over which the vertical velocity gradient caused by microstructure retains the same sign. To derive this formula, an expression was first obtained for the lateral displacement of a ray passing through a given microstructure. This expression was then squared and averaged. The square root of the final result gave expression (1). It has already been noted that fluctuations of the vertical temperature gradient, amounting to 0.02 F per ft over patches about 100 yd in length, have been reported. If these values for G and for 6 are substi- tuted into equation (1), it is found that at a range of 1,000 yd the rms lateral spreading of the sound beam amounts to about 20 ft; while at 2,500 yd it amounts to 70 ft and at 4,000 yd at 150 ft. These figures indi- cate that at these ranges random spreading of the transmitted sound beam, because of microstructure, will obscure bending of the sound rays due to large- scale vertical temperature structure if the vertical gradient is of the order of 0.1 F in 30 ft. Actual obser- vation shows that even negative gradients of four times this magnitude often fail to produce clearly recognizable shadow zones, although the sound does weaken gradually with increasing range. It is not known at present whether microstructure will fre- quently have a magnitude appreciably in excess of that assumed for the estimate of lateral beam spread. If not, some other cause must be invoked for an ex- planation of why weak negative gradients do not produce shadow zones. Since no complete theory exists at the present time capable of explaining in detail the results obtained in transmission runs, much of the discussion of under- water sound transmission must be empirical in char- acter. It is possible, for example, that some of the empirical relationships found between the smoothed temperature-depth curves and the measured trans- mission anomalies result primarily from an oceano- graphic correlation between the temperature micro- structure and the smoothed distribution of tempera- ture with depth. Such observed empirical relation- ships are valuable, but until their basic physical cause is explained they should be used with caution since they may be valid only for the particular time and place in which the observations were made. 5.1.4 Classification of Bathyther- mograms For practical use of temperature-depth informa- tion some simple method of classifying bathythermo- graph records is essential. Even if the predictions of ray theory were exactly fulfilled, practical require- ments would probably rule out the time and effort required to construct ray diagrams and to compute theoretical intensities. Thus, a set of rules has been devised to classify temperature-depth records by the properties which are acoustically significant. Such classifications have also proved useful in transmission research. Since the simple ray theory was clearly inadequate, some other basis was required for comparing measured anomaly curves with the corresponding bathythermograms. In view of the complexity of possible temperature-depth curves, no classification can be entirely satisfactory. All such classifications must be regarded as preliminary until sufficient acoustic information is available to indicate exactly what features of the temperature-depth pat- tern are significant in any situation. Present systems of classification are primarily de- signed to correspond to different types of transmis- sion loss for a shallow projector, about 15 ft. When FACTORS AFFECTING the temperature increases with depth sufficiently for the temperature at some depth below the projector to be greater than the projector temperature, rays leaving the projector at slight downward inclina- tions will in theory be bent back up to the surface again; the transmission anomaly for a shallow hydro- phone should therefore be low, although experimental data on this point are lacking. When such positive gradients are present, the temperature pattern is called positive, sometimes denoted by PETER. Such patterns may be more completely character- ized by the depth of the layer of maximum tempera- ture and the difference between maximum tempera- ture and the temperature at projector depth. The sharpness of the underlying thermocline may also be acoustically significant. Other temperature-depth records are classified by the temperature difference in the top 30 ft. If this difference is 0.3 F or less, the water is said to be isothermal, and the temperature pattern is called mixed, sometimes denoted by the word MIKE. When this difference is greater than 1/100 of the surface temperature the computed range to the shadow boundary is less than 1,000 yd, for projector at 15 ft, hydrophone at 30 ft. For this temperature condition, the predicted shadow zone is commonly observed, and transmission to a shallow hydrophone becomes poor for ranges greater than 1,000 yd. Such a tem- perature distribution is called a sharp negative pat- tern, sometimes denoted by NAN. Temperature dif- ferences intermediate between MIKE and NAN tend to be somewhat variable and are classified as weak negative or changing patterns, denoted by CHARLIE. One exception is included in this relatively simple scheme. When the temperature difference from 0 to 30 ft is large enough to give a NAN pattern, but the temperature difference from 15 to 50 ft is 0.2 F or less, the pattern is classified as CHARLIB. With such an extremely shallow and negative gradient and with the projector in isothermal or nearly isothermal water, good but variable sound conditions may be expected. This type of pattern is the most favorable for the formation of a sound channel. With MIKE and CHARLIE patterns the depth and sharpness of the thermocline would be expected to affect the transmission of sound to a deep hydro- phone. Appropriate methods for characterizing these quantities are discussed in Section 5.8, where the acoustic measurements made with a hydrophone in or below the thermocline are summarized. DEEP-WATER TRANSMISSION 93 Another more detailed system of classification, which supplements the classification of negative gradients into MIKE, CHARLIE, and NAN pat- terns, has been devised at UCDWR. This system utilizes the depths at which the temperature is 0.1, 0.3, 1.0, 5.0, and 10 F below the surface temperature. These depths are called, respectively, D1, Ds, Ds, Da, and D;. For statistical analysis, these depths are given code numbers between 0 and 9 by the following numerical scale. Code digit Depth D in feet = 80 80 < D < 160 160 = D < 320 320 < D D greater than greatest depth reached by bathy- thermograph ONBHNIPRwWNE OS _ oO IA | Any bathythermogram may then be coded by giv- ing the code digits corresponding to D,, D2, D3, Da, D;. The surface temperature T is also coded by giving T/10 to the nearest whole number and by placing this digit after the other five and separating it by a deci- mal point. The code numbers for D, through D; and also T'/10 are denoted by dh, de, dz, ds, ds, and dg, re- spectively. The series of numbers is then written as ddzd3dids.ds, aS for example 23 457.6. The accurate determination of d; is very difficult because of the wide trace made by the bathythermograph near the surface; since the variability of this small tempera- ture difference will usually be high, there is some question whether this quantity is usually significant. Examples of bathythermograms classified by the two methods are given in Figure 9. These two systems of classification supplement each other and should probably be used together. The code system is probably most useful for surface gradients, where considerable detail is provided. For example, it is shown in subsequent sections that the transmission of sound to a shallow hydrophone depends markedly on d» for different NAN patterns. On the other hand, for a fixed d2, transmission to a shallow hydrophone differs markedly between NAN and MIKE patterns. For deep gradients the code system is somewhat less useful, owing to the very expanded depth scale. For example it is frequently not clear from the present code whether a deep hydrophone is above or below the thermocline. It seems likely that when more com- 94 DEPTH IN FEET DEPTH IN FEET DEPTH IN FEET DEEP-WATER TRANSMISSION DEGREES FAHRENHEIT 60 es 4s PRATT Tero 2590 Ey 2 =e 4-4 a SORE EEE 100 Gobryseageerscstcseeel: SgSgSeEeS Estes: ISO sasscsncaat reg essedseet esre 200 : bgsezcersspirss 250 a FH 300 aapesegee f 350 szaatl atc F H 7 Nolo) sezzeaezer)aeisussezercs sue 4504 MIKE CODE SYMBOL 66 679.7 DEGREES FAHRENHEIT 530.25 40 45 50 eoneones 70 75 80 85 99 ees bese Fett : H Stee a 1H ay aaam goeeca, Br sgtesseosetissstecasiactate DEEp 450 MIKE CODE SYMBOL 44 555.7 DEGREES FAHRENHEIT 55 60 65 70 75 5 40 45 SO 80 8 0s a ES sBEaRESTE 80 5 ites 10 pessoas: = 150 Heath HH 200 sosaeeees tees 250 eed ieee 300 tt : 3504 _ D 400 iE aetetatas EEP CHARLIE CODE SYMBOL O02 799.8 DEPTH IN FEET DEPTH IN FEET DEPTH IN FEET DEGREES FAHRENHEIT 45 50 55 60 65 Neat = 4-14 0 35 40 022 sacar 24 50 100} 150 200}= 250-5 300 3505 4008 450 CHARLIE CODE SYMBOL 22 569.6 DEGREES FAHRENHEIT 55 60 65 35 40 45 Om 0 75 80 85 99 fo) o2 gadeB! FEE FRE eee om y Eee SOR Hates SAH # 100 ge eeeeeeeeeae Seresee Age 150 peace Bp) aoa 2.00 EEE eet sageeaee 250 aH suaasaie 300 EE HH 350 E 400+ DEEP 450 NAN CODE SYMBOL OO 156.7 DEGREES FAHRENHEIT 45 50 55 60 65 70 75 go 9 35 40 85 9 (oheeeessecseziee soebeeaaeeTectece Bp aasstetat > Efe) sezestasesssesbssetetesssariet/*- 2 10 = fo} 2 .-¢ z 9 7) ) 3 BD z Fe SLANT RANGE IN YARDS Figure 11. 200 CYCLES BT CODE: MIKE 66 679.6 600 CYCLES BT CODE: MIKE 66 679.6 1800 CYCLES BT CODE: MIKE 35 579.5 Typical transmission anomalies at sonic frequencies, source at 14 feet, hydrophone at 50 feet. IN DOB TRANSMISSION ANOMALY TRANSMISSION IN ISOTHERMAL WATER 200 CYCLES (28 RUNS) SOURCE AT 14 FEET HYDROPHONE AT 50 FEET ISOTHERMAL LAYER AT LEAST 80 FEET DEEP e@ INDIVIDUAL POINT Oo MEDIAN —— QUARTILE 600 CYCLES (36 RUNS) CYCLES RUNS) 2000 4000 SLANT RANGE IN YARDS Figure 12. Average transmission anomalies at sonic frequencies. 99 100 law of simple geometrical spreading. A detailed analysis was given in Section 2.5 for the absorption resulting from viscosity. However, it was pointed out in that section that the observed absorption of sound in the ocean is far greater than can be accounted for by viscosity. The effect of absorption on underwater sound trans- mission is shown most simply by considering the propagation of sound in an unbounded homogeneous ocean. Let J be the total energy proceeding outward from a sound source in each second. If the ocean were not at all absorbing, this same amount of energy would spread to all ranges. If the source is nondirec- tional, this energy would be spread over an area of 47? at a range R, and the sound intensity I would be given by the equation == (5) where F, defined in Chapter 2, is given by 1 F = —J- re In the presence of absorption, a constant fraction n of the sound energy is absorbed in each yard of sound travel. Thus in a distance dr, an amount of energy 4rnF dr will be absorbed per second, and con- verted into heat energy. The constant n is called the absorption coefficient of the water. The decrease 4a dF of the sound energy over this distance will equal the energy absorbed, or 4anF' dr. Thus we have the equation dF dha which has the familiar exponential solution ie nF, (6) iPS [gee (7) The sound intensity is then given by the equation Fye-n® Lee ®) In terms of decibels, this equation may be written 10 log J = 10 log Fy) — 20 log R — mls (9) 1000 where a/1000 = 10n logy e = 4.34n; a expresses the absorption in decibels per kiloyard and is called the coefficient of absorption. The transmission loss H, as defined in Chapter 4, is 10 log Fo — 10 log I. The transmission anomaly A is simply H — 20 log R. DEEP-WATER TRANSMISSION Thus we have the simple equation aR ~ 1000. my) Hence, in an unbounded medium, absorption pro- duces a transmission anomaly which increases lin- early with the distance covered by the sound beam. Scattering of sound is more complicated than ab- sorption. When scattering rather than absorption is present, equation (10) may still be used to describe the decay of the unscattered sound. However, the sound that has been scattered must also be considered in computing the expected sound intensity. It is shown in Section 5.4.1 that scattering is probably not very important in isothermal water. However, since the exact role of scattering in isothermal water is not certain, and since some forms of scattering may be very important when temperature gradients are present near the surface, it is customary to refer to the combined effects of absorption, scattering, and similar phenomena as attenuation. The quantity a, determined by direct measurement of A and use of equation (10), is then called the coefficient of attenua- tion. Attenuation, as so defined, includes all effects which may produce a transmission anomaly. Extensive observations at a number of laboratories indicate that in isothermal water the transmission anomaly does, in fact, increase linearly with increas- ing range, in accordance with equation (10). Thus, the attenuation coefficient a for each frequency is a con- stant for any one run. The data at 24 ke, in a report on attenuation issued by UCDWR, provide a check of this point.!2 Of the many runs available in deep water off the coast of southern California and Lower California, at the time reference 13 was written, 65 were made when the temperature difference from the surface to a depth of 30 ft was 0.1 F or 0.0 F. For all these runs the graphs of transmission anomaly against range could “reasonably be approximated by straight lines beyond a range of 1,000 yards.’’ Two sample plots of transmission anomaly, with the corresponding temperature-depth records, are shown in Figure 13. Each point represents the average amplitude of five different pings. The linearity of the observed points is evident in Figure 13. On the average, about half of the plotted points lay within 2 db of the straight-line curve drawn for each run. Thus it is reasonable to conclude that in water which is isothermal from the surface to 30 ft, the transmission anomaly increases linearly with range from 1,000 to more than 6,000 yd. Since TRANSMISSION IN RAY DIAGRAM DEPTH IN FEET TRANSMISSION ANOMALY IN DB RANGE IN YARDS ISOTHERMAL WATER 101 BT INFORMATION 4840 4890 4940 SOUND VELOCITY IN FT PER SEG DATE TIME 2-28-1944 1705 BT CLASS__ MIKE WATER DEPTH 2200FM SEA ae ee Eee SWELL 3 4 WIND FORCE Figure 13A. Sample transmission anomaly in isothermal water. only about half the runs were made with shallow hydrophones, between 16 and 30 ft, and the other half with deeper hydrophones, usually below the thermocline, this result apparently applies for sound transmitted below the thermocline as well as for sound in the isothermal layer. This linearity of the transmission anomaly for deep hydrophones is dis- cussed again in Section 5.3.2. it is perhaps surprising that the transmission anomalies should be straight lines out to long ranges when the isothermal layer is at most a few hundred feet thick. In a completely isothermal layer the up- ward bending of the sound, caused by the increase of pressure with depth, should give rise to a shadow zone near the surface at 3,000 yd for a thermocline starting at 150 ft below the projector and at 6,000 yd for one starting at a depth of 600 ft below. While sound reflected from the surface would penetrate this shadow zone computed for the direct sound, some drop in transmission might nevertheless be expected at the shadow boundary. It is possible that slight negative gradients of about 0.1 F in 30 ft were present in most of these measurements since slight gradients are common off San Diego and are very difficult to measure exactly with the bathythermograph. Such a slight gradient would offset the effect of pressure on sound velocity and give nearly straight-line propagation out to considerable range. The observed results could also be qualitatively explained on the assumption that the temperature in the isothermal layer is not completely constant, but varies irregularly from point to point. It was shown in Section 5.1.3 that the microstructure ob- served in regions of sharp temperature gradient can broaden the sound beam in the vertical direction by a hundred feet in several thousand yards. If micro- structure of similar effectiveness were present in the isothermal layer, this alternate up-and-down bending from microstructure would wash out the pressure effect entirely and would enable some direct sound to travel to an indefinite range in the isothermal layer. Since a fraction of this sound would be bent down into the thermocline at all ranges, the linearity of the transmission anomaly curve at depths below the thermocline might also be explained on this basis. 102 DEEP-WATER TRANSMISSION RAY DIAGRAM BT INFORMATION f ——SENDING SHIP ' I—— RECEIVING SHIP i 0 TL HYDROPHONE DEPTH A t 100 Ww wu = =x e a — —— — w OPHONE DEPTHO 200 SENDING 300 fo) 2000 4000 6000 8000 SOUND FIELD DATA 4890 4940 4990 SOUND VELOCITY IN FT PER SEC 10,000 12,000 14,000 DATE I1- 27-1943 TIME 1830 BT CLASS MIKE WATER DEPTH _2000 FM SEA 2 TRANSMISSION ANOMALY IN DECIBELS 8000 IN YARDS 4000 6000 RANGE (9) 2000 SWELL. 3 WIND FORCE 3 10,000 12,000 14.000 Ficure 13B. Sample transmission anomaly in isothermal water. However, since there are no extensive measurements on small-scale thermal structure in the isothermal layer, no conclusions about these problems can be reached at the present time. The straight line of best fit drawn on a plot of measured transmission anomaly against range gives, with moderate precision, the attenuation coefficient for the time and place of the measurements. Because the transmission anomaly usually approximates a straight line for measurements at fixed depth in isothermal water, the probable error of the attenua- tion coefficient found in this way is only about 4% db per kyd, for runs extending out to about 6,000 yd. The detailed variation of the attenuation coefficient in isothermal water from place to place and from time to time has not been thoroughly explored; and so it is most useful to deal with an average attenua- tion coefficient. The evidence on variation of trans- mission loss in isothermal water will be given in Sec- tion 5.2.3. The most complete investigation of the average attenuation coefficient at 24 ke in isothermal water is apparently that presented in a report by UCDWR." o a z ca eee ee i : a NCS 10° Ee ae SS | on i ie A @ FRESH WATER @ FRESH WATER 4 FRESH WATER & WHO! - 10° FREQUENCY IN CYCLES Figure 17. Dependence of attenuation coefficient on frequency. The points on the curve are from the following refer- ences: [|] NRL, 13 and 16; V UCDWR, 14; O UCDWR, A Fresh Water, 23; A WHOI, Chapter 9 of this book. coefficients of 18, 26, and 32 db per kyd, respectively, for an assumed amplitude reflection coefficient of the bottom of 0.5 (energy loss of 6 db per reflection), cor- responding to the SAND-AND-MUD bottoms over which the measurements were made. A change of the reflection coefficient by 0.2 in either direction changes the attenuation coefficient by about 2.5 db per kyd in the same direction; this variation may be taken as a rough estimate of the probable error of the re- sults. Owing to this high probable error, the values of 2.0 and 7.0 db per kyd, found at 24 and 40 ke re- spectively, are of relatively low weight and may be disregarded. At frequencies between 500 and 2,500 ke, extensive measurements of attenuation have been made by the Canadian National Research Council. A projector was mounted on a dock in Vancouver Harbor in 13 to 25 ft of water. The receiver was also mounted on the same dock at distances varying up to 100 ft. As a result of the high directivity of the projector, surface and bottom-reflected sound were largely eliminated. The slope of the transmission anomaly was measured 18; e CNRC, 19; @ Fresh Water, 20; [J Fresh Water, 22; to give an attenuation coefficient at each frequency. Relative probable errors of these coefficients, esti- mated from the reproducibility of the results, averaged about 7 per cent. No temperature measure- ments were made. Over such short ranges any gradi- ents would have had a negligible effect. No measurements at frequencies above 3 me are available for sound in the ocean. However, such measurements have been made in the laboratory.” * Those of reference 20 extend down to 2.8 mc, where the values found are of the same order of magnitude as those determined in the ocean. Other determina- tions of absorption in fresh water in the frequency range between 200 and 4,000 ke are about ten times as high as those found in the sea.**~° These fresh- water measurements are not in good agreement with each other and may be affected by systematic errors. Since the sea-water values taken from reference 16 were made over a much greater sound path, these should be much more reliable, and in any case, consti- tute better evidence for the attenuation of sound in the sea; the fresh-water measurements in references 106 DEEP-WATER TRANSMISSION > (o) to) N @ o oe in) & ahs O Pale ATTENUATION COEFFICIENT IN DB PER KILOYARD TEMPERATURE CG Z a a eae cole ae LTS ie a cE 1800 2000 2200 FEB 14 1600 TIME OF DAY 0400 0800 1000 FEB IS 0600 Fieure 18. Variability of attenuation coefficient during one day. 24, 25, and 26 may therefore be ignored in the present discussion. At frequencies below 14 ke no data on deep-water attenuation in the surface layers of the sea are avail- able. The long-range measurements of explosive sound propagation, discussed in Chapter 9, give an upper limit on the attenuation coefficient deep in the ocean. Although the data are uncertain, the results quoted in Chapter 9 indicate that the attenuation at 2,000 ¢ is probably less than 5 X 10-? db per kyd. These different determinations of the attenuation coefficient are combined in the plot of a against fre- quency shown in Figure 17. The dashed line gives a curve of best fit drawn through the plotted points. The solid line in this figure gives the value of a to be expected from viscous damping, taken from Section 2.5. Evidently, at frequencies above 1 me the at- tenuation coefficient is three to four times the classi- eal value. In a UCDWR memorandum,” this dis- crepancy is attributed to an additional viscous force proportional to the rate of compression of the water. Such a force has usually been neglected in hydro- dynamics, since ordinarily water flows like an in- compressible fluid. Although no tests of such an hypothesis have been suggested, it is entirely possible that this ‘compressional viscosity’? may be respon- sible for the observed values of a at frequencies above 1 me. No explanation has yet been advanced for the ob- served values of the attenuation at somewhat lower frequencies. Since all the measured values between 10 and 100 ke were obtained in temperate latitudes in water well above the freezing temperature, it is possi- ble that these values are not applicable for all oceano- graphic conditions. It is still not wholly certain that the attenuation observed for supersonic frequencies in isothermal water is entirely the result of absorp- tion rather than scattering; however, the weakness of scattered sound observed for backward scattering (reverberation) and for forward scattering (incoher- ISOTHERMAL WATER TRANSMISSION IN ent sound measured in shadow zones) makes this highly probable. At frequencies below 15 ke the attenuation of sound is largely conjectural. The shallow water meas- urements discussed in Chapter 6 show that a is not greater than about 1 db per kyd at frequencies below 2 ke. On the other hand, if the attenuation in this frequency range were as low as 0.1 db per kyd, high- speed warships could be heard consistently many hundreds of miles away. Since such listening ranges are apparently not obtainable, it may be inferred that the attenuation coefficient in the surface layers of the ocean is greater than 0.1 db per kyd at all sonic frequencies. Such a high value is not necessarily in- consistent with the much lower value observed in the deep sound channel since the attenuation at low frequencies near the ocean surface may result pri- marily from scattering of sound out of the isothermal layer into the thermocline, where it is bent sharply downward, and is lost. 5.2.3 Variation of Transmission Loss The previous section has discussed average values of the attenuation coefficient at each frequency, but has ignored changes in this coefficient. It has already been noted that from a single run out to 6,000 yd the attenuation coefficient can be determined with a probable discrepancy of only about 14 db per kyd from its true value at that particular time and place. Since the scatter of the observed values exceeds this, it may be inferred that the attenuation coefficient in sea water is probably not constant. In the measurements reported in reference 16, for example, half the attenuation coefficients at 24 ke differed by more than 1 db from the average value of 4.4 db per kyd. Corresponding variations also ap- peared at the other two frequencies (17.6 and 30 kc). However, those runs in which more than one fre- quency was used show a good correlation (correlation coefficient 7 between 80 and 85 per cent) in the varia- tion of the coefficients for the three frequencies. A good illustration of this correlation is provided by the runs made during one 24-hr period (February 14-15, 1944), analyzed in reference 13. Seven meas- urements of the vertical temperature structure were made during this period. In each case, no variation of temperature of more than 0.2 F was noted down to depths of 120 ft, but this was also the limit of ac- curacy of the thermometer on these days. The surface temperature changed appreciably during the period, 107 however, probably as a result of the changing position of the vessels. The variation of the attenuation coefficients for sound at 17.6, 23.6, and 30 ke is plotted in Figure 18 with the measured surface tem- perature. Evidently the attenuation coefficients at the three frequencies changed very substantially; however, the difference in the attenuation coefficients between the different frequencies was more nearly constant. Under some conditions, bowever, the attenuation coefficient during a 48-hr period is less variable. A series of transmission measurements was made by UCDWR in the deep water off Point Conception, California, where a persistent well-mixed layer was to be expected. Measurements were carried out dur- ing and after a storm, with winds of force 3 to 6 (Beaufort scale). The surface layers of the sea were probably better mixed during these transmission runs than for any other reported transmission experiments. A typical temperature-depth record taken during these measurements is shown in Figure 19. DEPTH IN FEEF WATER TEMPERATURE .IN DEGREES F Figure 19. Depth record for Point Conception runs. The cumulative distribution of attenuation coeffi- cients for the data taken with the shallow and deep hydrophone is shown in Figure 20. These are plotted on probability paper, so that a normal, or Gaussian, distribution of plotted points would lie on a straight 108 DEEP-WATER TRANSMISSION line. For the data shown in this figure, half of the ob- served values of a fall within about 0.6 db per kyd of the average values. This is so close to the probable error of 0.5 db per kyd for a single determination of a that the result may be entirely due to observational errors. Certainly the reduced scatter can be attrib- uted to the relatively uniform conditions prevailing during these tests. The observed attenuation coef- ficients for the deep hydrophone will be discussed in Section 5.3. In addition, the relatively small scatter shown in Figure 15, about the same as the observa- tional error in the measurement of transmission loss, = esape | ests a i mis Sess SINC SUS Cia ae TTT Repeats ! 5 20 40 60 80 95 99 99.9 PER CENT OF ALL ATTENUATIONS GREATER THAN THE VALUE PLOTTED ATTENUATION COEFFICIENT IN OB PER KILOYARD ow > Figure 20. Distribution of attenuation coefficients for Point Conception runs. suggests that in isothermal water the attenuation may be relatively constant. Further information is required, however, before any very definitive conclu- sions can be drawn about the variability of the at- tenuation coefficient when the temperature of the water in the upper 100 ft of the sea is approximately constant. Most of the UCDWR data have not been analyzed with this purpose in mind. In reference 13, where a high variability of a is found, the runs in isothermal water are not treated separately. Also, some of the runs included do not extend out very far; when marked peaks in the anomaly curve are present, the attenuation coefficient for such runs may be as low as 1 db per kyd. An examination of a sample run of this type, shown in Figure 21, indicates that such values are not necessarily indicative of the attenua- tion over longer ranges. Thus, these data do not cast much light on the variability of the attenuation coefficient in isothermal water. It is tempting to assume that the values shown in Figure 17 represent pure absorption, and that any variations from these values found in approximately isothermal water represent distortion of wave front by temperature gradients too small to be detected reliably on the bathythermograph. More accurate data on transmission in mixed water and more ac- curate thermal measurements would be required to test such an hypothesis. 5.2.4 Short-Range Transmission The goal of transmission studies is to relate the sound intensity at any range to the sound output of the source. Most transmission measurements at sea, however, do not measure the sound level closer than about 100 yd from the source. In principle it should be possible to measure the absolute sound level in the water, and also to measure the absolute level 1 yd from the projector; in practice, the measuring equip- ment has apparently not been sufficiently stable to make these absolute measurements possible. Thus transmission measurements give only relative sound levels and may be used to give the sound level at long range relative to the level at several hundred yards. The methods used in computing transmission anom- alies are discussed in some detail in Chapter 4. For most of the data discussed here, the transmission anomaly at short range, usually about 100 or 200 yd, has been taken equal to zero. Thus, to find true transmission anomalies at long range requires infor- mation on the true value of the transmission anomaly at ranges around 100 yd. Since refraction can be ignored at such close ranges, the transmission anomaly for the sound passing di- rectly from projector to hydrophone must be very close to zero. If the surface were perfectly flat, sur- face-reflected sound would, on the average, double the sound intensity at ranges of several hundred yards, provided that the intensity is averaged over the interference pattern discussed in Section 2.6.3; the transmission anomaly A at these ranges would then be —3 db. Sound intensity measurements between 1 yd and several hundred yards would then show a gradually decreasing transmission anomaly as the range increased and as surface-reflected sound ap- proached the same average strength as the direct sound. However, the sea surface is never perfectly flat, and this fact may be expected to alter the simple relationships to be expected for a flat surface. Al- TRANSMISSION THROUGH A THERMOCLINE RAY DIAGRAM DEPTH IN FEET SOUND FIELD DATA o [=] z > x @ = fo} z 4 a = ° =z less than 10 ft). We shall call this slope a’ and SEA SWELL WIND FORCE | NAN pattern with deep isothermal layer below. may regard it as a sort of “local attenuation coef- ficient,”’ that is, GY = = (15) The local attenuation coefficient defined by equation (15) is not to be confused with the actual attenuation coefficient characterizing the transmission from the sound source to the range R, defined as 1,000A/R. The observed slope beyond the break is to be com- pared with the local attenuation coefficient at the shadow boundary which would result from diffrac- tion. From Section 3.7, we have the following formula for a’ in the case of a linear velocity gradient. (16) In formula (16) c is the sound velocity in yards per second; dc/dy is the velocity gradient in feet per second per foot; and a’ is in units of decibels per yard. TRANSMISSION WITH NEGATIVE GRADIENTS NEAR SURFACE If the temperature difference between 0 and 30 ft is about 1 F, and the surface temperature is about 70 F, then for sound of 24,000 cycles equation (16) gives an attenuation of about 0.1 db per yd, or about 100 db per kyd. This is at least twice as great as the values usually obtained. The discrepancy seems somewhat too great to be explained as observational error, although the fact that observed and predicted attenuations are of the same order of magnitude is suggestive. It is possible that the presence of thermal microstructure may explain this difference. No at- tempt has been made to correlate the observed at- tenuation across the shadow boundary with either the frequency f or the velocity gradient dc/dy at the surface. SCATTERED SOUND IN THE SHADOW ZONE Most transmission anomaly plots for NAN pat- terns are characterized by a nearly constant trans- mission anomaly well beyond the computed boundary of the shadow zone, amounting to between 40 and 60 db and extending out to the limit of measurement at about 5,000 yd. An examination of the oscillograph record shows that the signal received at ranges between 1,500 and about 3,000 yd bears very little relation to the shape or length of the original pulse. Figure 44 shows the signals received at different ranges when a marked negative gradient was present at the surface. At moderately short ranges, less than 1,000 yd, the re- ceived signal reproduces the outgoing pulse rather faithfully. At moderate ranges into the shadow zone, however, the signal has an appearance similar to that of reverberation, and is much prolonged, as shown in trace C made at 1,840 yd. At these intermediate ranges, the use of peak amplitudes in reading each signal gives a value about 7 db higher than the use of average intensities. For coherent 100-msec signals, the difference between peak amplitude and rms am- plitude is negligible. At slightly longer ranges even the few traces of the direct pulse, visible for some of the signals in trace C, completely disappear. At the long- est ranges the signal begins again to resemble the emitted pulse, as shown in trace D. - The intensity of the sound received in the shadow zone increases with increasing pulse length. The observed difference in intensity for 100-msec and 10-sec pulses is shown in Figure 45. In the shadow zone at intermediate range, where the received signal is prolonged and incoherent, the effect is large, amounting to between 4 and 8 db. At very long 125 RANGE IN YARDS to} 1000 2000 TRANSMISSION ANOMALY IN DB NO DATA FOR III D2 BETWEEN O AND 10 FEET I 50=h al ¢ = fo} za qt 2 Q w @ = on Zz 9 c = Dz BETWEEN 10 AND 20 FEET 60 Figure 43. Average transmission anomalies for NAN patterns with hydrophone deeper than 50 feet. ranges and within the direct sound field at shorter ranges the signal is coherent and the difference is small. This small difference results from the fact that the peak amplitude of a long signal, which fluctuates from minimum to maximum values many times, tends to be greater than the peak amplitude of a short signal, which may be altered by fluctuation to a low value. 126 DEEP-WATER TRANSMISSION RADIO SOUND SIGNAL SIGNAL 830 YDSK— ATTENUATION i aalemeaaties SIGNAL 54 gp RADIO SIGNAL DIRECT BOTTOM OUND SIGNAL REFLECTION _ HYDROPHONE DEPTH = 75 FEET OCEAN DEPTH = 6I5 FATHOMS ne A RECORDS AT 450 YARDS ue B RECORDS AT 830 YARDS ~ 9 db ATTENU. Sage G RECORDS AT 1340 YARDS _ D RECORDS AT 4350 YARDS - Ficure 44. Records of received signals at various ranges under downward refraction. n a WwW o oO Ww [=) = Ww Oo z iy c Ww iS = © 100 200 500 1000 2000 5000 10,000 RANGE IN YARDS Ficure 45. Increase of average peak amplitudes for long pulses. The sound received in the shadow zone, at least out until about 2,500 or 3,000 yd, is probably sound scattered from the main beam at considerable depth up to the hydrophone near the surface. The scatter- ing coefficient required to explain the observations can be readily estimated from the transmission anomalies of the long pulses. The transmission anom- aly for 100-msec pulses in the shadow zone is usually between 40 and 60db. Figure 45 shows that about4to 8 db must be subtracted from these values to find the transmission anomaly of the 10-sec pulse, which may be regarded as essentially a continuous tone in these observations. On the other hand, 7 db must be added to find the anomaly in terms of average intensity instead of average peak amplitude. Since these two corrections about cancel out, 40 to 60 db is the trans- mission anomaly for the intensity of long pulses in the shadow zone. A theoretical value for this transmission anomaly may be computed on a somewhat simplified picture. Although the scattered sound is itself refracted by the prevailing vertical velocity gradient, most of the scattered rays are inclined so steeply that they may be regarded as straight lines. With this assumption, TRANSMISSION WITH NEGATIVE GRADIENTS NEAR SURFACE s SURFACE Figure 46. the scattered sound is transmitted to the entire ocean as if refraction were not operating. Also, for the pur- pose of calculating the sound scattered into the shadow zone, the actual direct beam may be replaced by a tilted beam traveling in a straight line, as in Figure 46. To calculate the scattered sound received when a long (10-sec) pulse is sent out, it may be assumed that sound scattered from the entire length of the beam is received at the hydrophone. Let the total initial power in the beam be denoted by J. If attenua- tion is neglected, this energy will remain constant as the sound travels outward. If the scattering coef- ficient is m per yard, a fraction m of the sound energy will be scattered per yard of travel of the beam (see Chapter 2 of Part II). This energy will be scattered in all directions; and the intensity of the sound scat- tered from this cross section of the beam 1 yd thick and reaching the hydrophone at a distance r’ yd away will be mJ/4zr”. While in actual fact the sound scattered from the lower side of the beam will be more weakened than that scattered from the upper side, the distance r’ from the hydrophone to a point on the axis of the beam should be a reasonable approxima- tion for each separate cross section of the beam. Let / represent the distance from the projector to the point where scattering is taking place. The sound scattered LIMITS OF ACTUAL SOUND BEAM ee LIMITS OF ASSUMED TILTED BEAM A HY DROPHONE Diagram used in calculating scattered sound intensity. between / and / + dl is thus (mJ/4zr’)dl; and the total scattered sound received at the hydrophone is “mJ _ md s dl _— = = — ——_—_ 0 4nr2 4nJo (L—1/P?+@ (17) where the quantities L and d have the meanings shown in Figure 46. The integration yields approxi- mately not ae _ md "ape \al BG While in the general case m will be a complicated function both of position in the ocean and of the direction in which the scattered sound is measured, here m is assumed to be constant. Equation (18) thus refers in practice to an average value of m. The expression (17) does not take into account the transmission anomaly resulting from absorption or refraction. When a sound beam is refracted sharply downward, the intensity in the direct sound field is not reduced much below the inverse square value. The scattered sound, which reaches the hydrophone at steep angles, is also relatively unaffected by re- fraction. The absorption must be considered, how- ever. For points in the sound beam to the left of the point B in Figure 46, the sum of the absorption loss for direct and scattered sound will not depend much (18) 128 DEEP-WATER on whether the sound was scattered close to the pro- jector or some distance out. Sound scattered from points to the right of the point B will suffer a two- way absorption loss, and may be neglected. There- fore, to calculate the sound intensity at the hydro- phone, taking absorption into account, we must first integrate equation (17) with the infinite upper limit replaced by L, obtaining approximately mJ/8d; then we must multiply this value by some factor to take the absorption into account. Because we are neglect- ing the sound scattered to the right of B, we may con- sider that the sound reaching the hydrophone has traveled a total path length, on the average, equal to R, the range from the projector to the hydrophone. lf ais the attenuation coefficient in decibels per yard, the intensity I, is therefore given by mJ. y-aR/i0 8d Equation (19) was derived without considering the possibility that sound could be scattered up to the surface by points to the left of B and reflected back to the hydrophone. We can allow for this extra intensity due to surface reflection by multiplying the expression (19) by 2. Our final result is 1, = M2 19-err0, Tag) It is convenient to restate equation (20) in decibels: 10 log 7, = 10 log m + 10 log J — 10 log (4d) — ak. (21) The total emitted power J is related to F, the power output per unit solid angle in the direction of the projector axis, by the formula 10 log J = 10 log (47F) + D, (22) where D is the directivity index of the projector. Combining equations (22) and (21) gives 10 log J, = 10 log m + 10 log F + D — 10 logd — ak + 10 log zx. Ile (19) (20) (23) The transmission anomaly A of the scattered radia- tion is given by A = 10log F — 20 log R — 10 log J;. If R sin 6 is substituted for d, from Figure 46, and J, is taken from equation (23), the transmission anom- aly A becomes A = — 10log R — 10 logm — D+ aR — 10 log + 10 log (sin @). (24) TRANSMISSION For a total temperature decrease of about 20 de- grees in the thermocline, the limiting ray PSC in Figure 46 bends downward below the thermocline at an angle of 12 degrees. Thus, a typical value of @ is 12 degrees. For a directional transducer of the type normally used in echo ranging, the directivity index Dis —23 db. If an absorption coefficient a of 4 db per kyd is used in equation (24), and D and 6 are set equal to —23 db and 12 degrees, respectively, then A is nearly constant from 1,000 to 3,000 yd, and equals —10 log m — 14. Since A is observed to lie between 40 and 60 for transmission to points well in- side the shadow zone, 10 log m is between —54 and —74. Because the receiver is directional in a vertical plane, these values for 10 log m must be increased somewhat to take account of the directivity pattern of the receiver. An examination of the receiver pat- terns in reference 34 indicates that this correction should be about 6 db. Thus, we finally have for 10 log m a value between —48 and —68 db. This result is in general agreement with the value of —60 + 10 db for the scattering coefficient of volume reverberation given in Chapter 4 of Part IT. A value greater than —40 db seems definitely ruled out by the observations. Thus one may conclude that the scattering coefficient for sound at angles between roughly 10 and 120 degrees is not more than about 10 db greater than for the backward scattering which gives rise to reverberation. It is possible that the scattering of sound by the volume of the sea is the same in all directions. More exact conclusions would require simultaneous determinations of reverberation and sound scattered into the shadow zone. In addi- tion, the change of scattering coefficient with depth, frequently observed in the deep scattering layers dis- cussed in Chapter 14, would demand consideration. The present very rough analysis is adequate, how- ever, to indicate that the attenuation observed in deep isothermal water is not the result of scattering, unless one makes the improbable hypothesis that scattering in the isothermal layer is very much greater than the scattering in the thermocline. If an attenua- tion coefficient of 4 db per kyd or 4 x 10-*db peryd is attributed entirely to scattering, the scattering coef- ficient m would be 10 log e times a or 1.7 X 107°, giving more than —20 db for 10 log m. This is 20 db greater than the maximum possible value of m con- sistent with the low intensity of sound observed in the shadow zone. If not all the sound in the shadow zone is due to scattering, the disparity becomes even greater. TRANSMISSION WITH NEGATIVE GRADIENTS NEAR SURFACE co 20! 202 © OBSERVATIONS e MEAN VALUE IN 0.005 INTERVAL OF + LEAST SQUARES FIT TO OBSERVATIONS ATTENUATION COEFFICIENT IN DB PER KILOYARD DEPTH D> TO THERMOCLINE IN FEET Figure 47. Attenuation coefficient above the thermocline. Although the scattered sound observed at ranges between 1,000 and 2,500 yd is readily explained, the coherent signals received in the shadow zone at 4,000 yd are less simply explained. These coherent signals could be produced by a suitable variation of the scattering coefficient m with depth similar to those found in deep scattering layers. A preliminary UCDWR analysis of reverberation measurements made with a vertical projector at the same time as transmission measurements indicates that this hy- pothesis is correct; the scattering coefficients found by these two methods agree to within a few decibels. 5.4.2 Weak Temperature Gradients in Top 30 ft When the temperature gradients in the top 30 ft are intermediate— CHARLIE patterns—it has been clearly demonstrated in Table 1, that at least half of the transmission anomaly curves are approximately straight lines while the others have more complicated shapes. Thus the type of transmission likely to be en- countered is highly unpredictable. This observational result may be in part caused by the rapid variability of temperature conditions for this type of pattern; small changes of temperature, of the sort very com- mon near the surface, can change the theoretical ray diagram completely in a matter of minutes. Various methcds have been developed for analyzing the trans- mission conditions to be expected with these patterns. Since the average transmission loss observed with shallow MIKE patterns is very similar to that ob- served for CHARLIE patterns, these two tempera- ture types are combined in the present discussion. Most of this section refers to average results obtained at 24 ke. Some special temperature distributions are discussed at the end of this section. ATTENUATION COEFFICIENTS In reference 13, an attenuation coefficient was found from the slope of each straight-line transmis- sion anomaly graph. Attempts to correlate these coefficients with various temperature differences either in the surface layers or in the thermocline were not very successful. However, a significant correla- tion was found with the depth Dr to the thermocline. The plots showing this correlation are reproduced in Figures 47 and 48. A least-squares solution gave the following equations of best fit: 170 Above the thermocline a = 3.5 + ce (25) T F 260 Below the thermocline a = 4.5 + D, (26) T Although these mean curves are unquestionably sig- nificant, only half of the individual points lie within 2 db of the values predicted from equations (25) and (26). 130 DEEP-WATER TRANSMISSION o OBSERVATIONS @ MEAN VALUE IN 00S INTERVAL — LEAST SQUARES FIT TO O'S a=4.5 + 260 OT “ATTENUATION COEFFICIENT IN DB PER KILOYARD O;= DEPTH TO THERMOCLINE IN FEET Ficure 48. Attenuation coefficient below the thermocline. Figure 48 shows an increase of attenuation below the thermocline with decreasing thermocline depth. This effect has already been noted in Section 5.3.4 for thermocline depths greater than 40 ft; the change of Ra with layer depth (Figure 29) was shown to be consistent with the theoretical curve based on equa- tion (18). It isapparent from Figure 47 that this same effect persists to much shallower layers. A comparison of Figures 47 and 48 indicates that layer effect in- creases steadily with decreasing depth of the ther- mocline. This result is also consistent with expecta- tions based on equation (13). The increase of the attenuation coefficient in the isothermal layer as the depth of the layer decreases is quite marked in Figure 47. It may be noted that the values shown for Dr between 20 and 30 ft are not inconsistent with the attenuation coefficient of about 13 db per kyd found from the upper curve in Figure 40, drawn for D, between 20 and 30 ft. The origin of this high attenuation is hard to explain. As sound travels along through the layer of nearly isothermal water, with the sound rays continually reflected from the surface and distorted by temperature microstruc- ture, it may be expected that a certain fraction of the sound would be bent out of the isothermal layer in each yard of sound travel. Any such sound reaching the thermocline will be bent down so sharply that it is unlikely to return to the isothermal layer. It is possi- ble that a quantitative theory along these lines, based on more accurate information on the properties of the isothermal layer, may explain the observed decrease of attenuation with increasing thickness of the layer. In any case there is little question as to the reality of the effect noted in Figures 47 and 48. RANGE IN YARDS (0) 1000 2000 3000 20 —-—MIKE, 80=D, <320 ----MIKE,40= 0D, < 80 MIKE, 30=D, =<40 — — CHARLIE, 20 =D,<30 TRANSMISSION ANOMALY IN DECIBELS 40 Figure 49. Average transmission anomalies for MIKE and CHARLIE patterns (hydrophone shallow). Whether the scatter evident in these figures is greater than can be explained by the observational scatter of all observed transmission anomalies is not evident from an examination of reference 13. It has already been noted, in Chapter 4, that for most of the UCDWR data transmission anomalies determined by averaging 5 successive received pings have a probable error of about 2 db as a result of hydrophone direc- tivity, training errors, and sampling errors; the errors TRANSMISSION WITH NEGATIVE GRADIENTS NEAR SURFACE 131 RANGE IW YAROS : be) tooo *20 TRANSMISSION ANOMALY IN DB Nn ° 40 © MIKE @ CHARLIE 60 Figure 50. due to calibration do not affect the accuracy of the attenuation coefficient determined by fitting a straight line to the observed anomalies. With such a scatter, the uncertainties in the attenuation coef- ficient a can be very substantial for those runs in which the length of run was short. However, it does not seem likely that observational error can account for all of the scatter shown in Figures 47 and 48. The attenuation coefficients shown in Figures 47 and 48 should probably not be used for estimating transmitted sound intensities when temperature gra- dients are present in the top 30 ft. The results are valid on the average when the transmission anomaly graph is a straight line, but unfortunately there is no 2000 3000 4000 ° say >! UPPER QUARTILE @0e e ‘| MEDIAN CURVE LOWER QUARTILE Individual anomalies for D, between 20 and 40 feet. way of predicting whether or not the measured sound intensities will yield a straight line, except when the top 30 ft are isothermal. Exclusion of those situations where the transmission anomaly curve is not a straight line may be expected to give results system- atically different from those obtained when runs are classified only by the temperature distribution. Therefore the average anomaly curves given below are preferable as a tool for estimating the sound in- tensities to be expected in any situation. AVERAGE TRANSMISSION ANOMALIES The average transmission anomalies obtained with MIKE and CHARLIE patterns have been combined 132 in reference 14 to give average curves. The curves for a shallow hydrophone (16 to 30 ft) are reproduced in Figure 49. The curves are again plotted for different values of Do. To illustrate the scatter of the individual observa- tions, all individual anomalies averaged to give the curve for D. between 20 and 40 ft are shown in Fig- ure 50. The open circles represent the anomalies for MIKE patterns, the solid circles those for CHARLIE patterns; no systematic difference is apparent be- tween these two sets of points. The upper and lower quartiles of the distribution are shown by dashed lines. The increase of spread with increasing range is very marked and is an evidence of the unpredicta- bility of transmission conditions for such shallow isothermal layers. The quartile spread at short range is much smaller and represents the more normal scatter, apparent also in Figures 15 and 41. ao a z > =) 4 = [e} 2 4 FS I HYDROPHONE DEPTH h=30F 8 50=h =100FT = 100=h < 200 FT FS 200=h <~300FT f= 300=h = 400FT RANGE IN YARDS Figure 51. Average transmission anomalies for D, between 20 and 40 feet. The change of transmission anomaly with changing hydrophone depth has not been analyzed separately for MIKE, NAN, and CHARLIE patterns. The average results for D, between 20 and 40 ft, combin- ing results for all three patterns, is shown in Figure 51. The change with depth down to 200 ft is negligi- ble, but at greater depths an appreciable increase in sound intensity is noted. This change is greater than the probable error of each curve resulting from the internal scatter of the points and is therefore probably significant, even though different temperature pat- terns were present when different hydrophone depths were used. The corresponding plot for D2 between 40 and 80 ft has already been given in Figure 30. For such tem- perature structure, the intensity first decreases with increasing depth as the hydrophone goes below the DEEP-WATER TRANSMISSION thermocline, and then increases. As pointed out in Section 5.3.4, the transmission anomaly below a sharp thermocline is likely to show better correlation with the depth and sharpness of the thermocline than with the temperature code used in Figures 49 and 51. In fact the limited results available are consistent with the belief that for MIKE and CHARLIE patterns in general the average transmission anomaly below a thermocline is approximately given by equation (13) in Section 5.3.4. a ra) = > F4 = ro) z < z ° a a) = 7) z z id S RANGE IN YARDS Ficure 52. Average transmission anomalies for hydrophone 50 to 100 feet deep. An example of the way in which transmission anomalies change with changing temperature struc- ture is shown by the set of average curves for hydro- phones between 50 and 100 ft deep shown in Figure 52, again taken from reference 14. Most of the data used in these curves were actually obtained with hydrophones at 50 ft. The curves are in terms of the digits in the temperature-depth code explained in Section 5.1.4. The successive curves may be inter- preted as follows: dod 13 Temperature decreases to 0.3 F below surface tem- perature between 5 and 10 ft; between 20 and 40 ft it has decreased to 1 F below surface temperature. This is a moderately sharp NAN pattern with the sharp gradient extending practically up to the surface, and TRANSMISSION TEMPERATURE -F 200 DEPTH IN FEET 300 WITH NEGATIVE GRADIENTS NEAR SURFACE 133 RANGE IN YARDS 2000 3000 Ficure 53. Sound channel ray diagram, extreme case. shows the corresponding rapid rise in the transmission anomaly. 23 This isamuchthe sameas above except that the gradient in the upper 10 ft is somewhat weaker. This curve and the preceding one are closely similar. 33 This is a NAN, MIKE, or CHARLIE pattern, but in any case the thermocline is shallow and the attenua- tion high. 34 The main thermocline is deeper here since the tem- perature is within 1 F of the surface temperature down to at least 40 ft. The hydrophone may be below or above the thermocline. 35 The top of the main thermocline is not much above 80 ft, and the hydrophone is either close to the top or above it. However, there are gentle gradients above the hydrophone, and these act to reduce the sound intensity. The reduction in sound intensity produced by the weak gradient between the projector and the hydrophone may be regarded as an example of layer effect. 45 The gradients above the hydrophone are weaker and transmission is improved. 55 The water is virtually isothermal down to 80 ft, and the results discussed in Section 5.2 are applicable. The deviation of this curve from a straight line is probably not significant. Sounp CHANNELS When the sound velocity at the projector is less than the velocities above and below, rays leaving the projector at sufficiently small angles will, in theory, curve back and forth within two fixed depths of equal sound velocity, giving rise to the curious ray diagram shown for an extreme case in Figure 53. This situa- tion is called a sownd channel, and should in theory give rise to high sound intensity at long ranges. When sharp negative temperature gradients are present over sharp positive gradients, such sound channels should be persistent and very marked. However, very few measurements have been made with positive temperature gradients present in the water. In the absence of positive temperature gradients, the effect of pressure on sound velocity can produce a positive velocity gradient below the projector. How- ever, this gradient is very small, and a temperature decrease of only 0.3 F (at 60 F) between the projector and the isothermal layer will bend the sound rays down so sharply that an isothermal layer 100 ft thick is required to bend the rays back up again. More- over, as a result of this small gradient, rays bent down into the isothermal layer would return to the surface only at ranges of many thousands of yards. This bending is so gradual that the presence of thermal microstructure might be expected to mask com- pletely any sound channel effects resulting from up- ward bending in nearly isothermal water. However, since some striking acoustic effects are observed with shallow gradients overlying isothermal layers, and since thermal microstructure has never been measured under such conditions, it is instructive to examine what the sound field would be like in truly isothermal water underlying slight gradients at the surface. If the projector were in such a hypothetical layer of completely isothermal water, the effects of the sound channel would not be particularly noticeable since the sound that has curved first up into the negative gradient and then down into the isothermal layer would be indistinguishable from the rays that have traveled through the isothermal layer for their entire path. In fact downward bending by a very shallow surface gradient above the projector is proba- bly very similar to reflection by the surface. To produce marked effects the negative tempera- ture gradient at the surface must extend below the projector depth, so that the entire sound beam is bent downward, resulting in low sound intensities meas- ured by a shallow hydrophone at short range. Then when the rays are curved back to the surface thou- sands of yards out, the sound intensity should show a marked increase. On the basis of the simple ray theory, which neglects thermal microstructure and diffraction, the theoretical intensity at the projector depth is infinite at the range where the axial ray from the projector becomes horizontal again; this singu- larity results from the crossing of many adjacent rays at this point. Although of course diffraction and ther- 134 OEPTH IN FEET TRANSMISSION ANOMALY IN DECIBELS RANGE IN YARDS Fiaure 54. mal irregularities will reduce this theoretical inten- sity, sound intensities considerably above normal would be expected at certain ranges if a sound chan- nel were produced by a slight temperature gradient lying above rigorously isothermal water. The conditions for a sound channel, when no posi- tive temperature or salinity gradients are present, are thus rather critical. There must be a temperature gradient at the surface which extends somewhat be- low the projector depth. Below this must be a layer of completely isothermal water a hundred feet or more in depth. The temperature difference between the isothermal layer and projector depth must be not less than about 0.1 F but not greater than about 0.3 F, the exact limits depending on the surface tem- perature as well as on the depth of the layer. Thus, even if no thermal microstructure were ever present, sound channels of this type would normally be quite transitory, appearing during the development of sur- face heating in deep isothermal water; they would be expected to become prominent when the temperature difference between the projector and the isothermal layer increased to 0.1 F, and to disappear as the gradient extended downward and the temperature DEEP-WATER TRANSMISSION BT INFORMATION ——-—SENDING SHIP ——RECEIVING SHIP DATE TIME W-29-1943 1400 BT CLASS CHARLIE WATER DEPTH 880 FM SEA ! SWELL 2 WIND FORCE 1 Peaked transmission anomaly possibly resulting from sound channel ray diagram. at projector depth gradually increased by another one or two tenths of a degree. Sound transmission measurements at the UCDWR laboratory in San Diego show transitory effects simi- lar to those which may be expected to result from sound channels. Because of the difficulty in reading the bathythermograph slide accurately to 0.1 F, and because of the high variability of thermal conditions in space as well as in time, it is not possible to predict from the bathythermograms exactly when a sound channel may be present. However, marked peaks in the measured transmission anomalies are occasionally found when thermal conditions are appropriate. A good example of the type of effect that can be observed is shown in Figure 54. As shown in the ac- companying temperature-depth record, a sharp nega- tive gradient extends from about 15 ft to the surface while below this a nearly isothermal layer extends down to 100 ft. A careful reading of the trace indi- cated a slight negative gradient in this layer (about 0.3 F in 100 ft) giving a constant sound velocity be- low the projector. Moreover, the sharp surface gradi- ent did not extend below the projector depth. Thus, the ray diagram in Figure 54 does not show a sound TRANSMISSION WITH RAY DIAGRAM DEPTH IN FEET TRANSMISSION ANOMALY IN DECIBELS RANGE IN YARDS NEGATIVE GRADIENTS NEAR SURFACE BT INFORMATION ==—SENDING SHIP J —— RECEIVING silly 4890 SOUND VELOCITY IN FT PER SECOND. DATE TIME 3-22-1944 1600 BT CLASS__NAN WATER DEPTH_650FM SEA | SWELL Ly WIND FORCE_1 Figure 55. Peaked transmission anomaly with sound channel unlikely. channel although barely perceptible changes in the temperature-depth record would make it so. Never- theless, the anomalously high intensity at long range in the shallow hydrophone is very striking. If an absorption coefficient of 4 db per kyd is assumed, the increase of intensity at 6,000 yd shown in Figure 54 is about 25 db. Until a more accurate means for de- termining ocean temperatures is available, it cannot be decided whether peaks of this type are actually the result of the focusing action predicted in sound channel theory. However, it is suggestive that an examination of all cases in which the anomaly curves show peaks of 10 db or more shows that the temperature-depth curves for most of these are very similar to the curves which would, on the simple theory, give rise to sound channels. Out of the many hundreds of runs made off San Diego, only about 25 show these peaks. In almost all of these the thermocline is below 100 ft with nearly isothermal water above, and the receiving hydro- phone is at shallow depth. Moreover, in most cases slight negative gradients are present close to the sur- face. A few significant exceptions are present, as for example the run shown in Figure 55 where the deep hydrophone shows a peak of 20 db at 4,500 yd. As an example of the transitory nature of most such peaks, Figure 55 may be compared with Figure 42, which plots the run immediately preceding and shows no trace of any peaks. Also on one day (March 15, 1944) the shallow hydrophone showed a marked peak. throughout the day while the temperature records only occasionally showed the deep layer of constant temperature required for a sound channel. Thus, while the evidence suggests that sound channels may in fact occur, there may quite possibly be other factors, still unexplored, that play a part in producing anomalously high intensities at certain ranges. 5.4.3 Transmission at 60 ke The effects produced by temperature gradients on the transmission of underwater sound have been thoroughly explored only at 24 ke. A few measure- ments are available, however, at 60 ke. 136 DEEP-WATER TRANSMISSION RANGE (IN YARDS oO 1000 2000 3000 4000 ——02 BETWEEN 5 AND 20 FEET =—-—02 BETWEEN 20 AND IGOFEET TRANSMISSION ANOMALY IN DECIBELS Figure 56. Average transmission anomalies at 60 ke. Average transmission anomalies for a shallow hydrophone at 60 ke are shown in Figure 56 for dif- ferent values of D2. All runs for D2 between 20 and 160 ft are combined in the upper curve, since no systematic variation of transmission anomaly with changing D» was noted for these data. The agreement between the curves for D2 between 20 and 40 ft and for greater D, is in marked contrast to the differences shown at 24 ke (see Figure 49). Another, more conclusive indication of the compli- cated differences between the two frequencies is shown by simultaneous measurements at both frequencies on a single shallow hydrophone. Measurements were made with a CHARLIE pattern (temperature dif- ference about 0.5 F in the top 30 ft) and a thermocline RANGE IN YARDS 2000 fo) 1000 3000 TRANSMISSION ANOMALY IN DECIBELS Fieure 57. Simultaneous transmission at 24 and 60 ke. at 50 ft. The resulting curves are shown in Figure 57. The difference of anomalies between the two fre- quencies increases by 15 to 20 db per kyd, as com- pared to a corresponding difference of not more than about 10 db per kyd in isothermal water; see Figure 17. If data from many more runs under somewhat similar conditions are used, however, the average dif- ference of transmission anomaly between 60 and 24 ke has a slope of about 9 db per kyd. This is in close agreement with the difference of 8.5 db per kyd found in Section 5.2.2 for isothermal water. Further data on the difference of transmission loss between dif- ferent frequencies are required to show how much and in what way this difference varies. Chapter 6 SHALLOW-WATER TRANSMISSION | CHAPTER 5, it was shown how the propagation of sound in deep water is affected by temperature gradients in the sea and by the sound frequency. In shallow water, these factors continue to operate; added to them is the effect of the bottom. The bottom affects the sound field in two different ways. Some of the sound incident on the bottom will be reflected and may penetrate into shadow zones. Also, some of the sound incident on the bottom will be scattered back- ward and will form part of the reverberation back- ground against which an echo must be recognized in echo ranging. This latter effect of the bottom will be considered in Chapters 11 to 17 of this volume. In this chapter, only the transmitted sound reaching a receiving hydrophone will be considered. 6.1 PRELIMINARY CONSIDERATIONS In deep water, it was found that the most impor- tant single factor determining the transmission of sound of a given frequency is the vertical temperature structure of the ocean. The roughness of the surface of the sea plays a poor second, and nothing is known concerning the effects of other oceanographic varia- bles on sound transmission. In shallow water, the number of factors which may conceivably affect sound transmission is greater; it would be impractical to make a large number of sound transmission runs and then obtain rules of sound propagation empiri- cally merely by subjecting the data amassed to an unprejudiced statistical analysis. Rather, it was found necessary to assess beforehand the possible effects of bottom character, roughness of the sea surface, and refraction conditions, and then to analyze the trans- mission run data purposefully. This procedure proved successful in bringing order into a mass of data, and it will also be followed in this discussion. 6.1.1 Effects of Sea Bottom If bottom-reflected sound is added to the sound field which reaches the receiving hydrophone (or the target in echo ranging), interference between the signals transmitted via the different possible paths may be either constructive or destructive, depending on the geometry of the paths. However, if the sound field intensity is averaged over a volume of the ocean sufficiently large to include several maxima and minima of the interference pattern, the averaged sound field intensity will be the algebraic sum of the intensities of sound resulting from each path by itself. In this sense, averaged sound field intensities in shallow water are always higher than sound field intensities in deep water under otherwise identical conditions. The extent to which bottom-reflected sound will increase the ‘‘deep water’’ sound field in- tensity and to which it will eliminate shadow zones depends on a number of factors, which will be treated in this chapter. One of these factors is the reflectivity of the sea bottom. A theoretical treatment of bounding surfaces indi- cates that the reflectivity of a surface is determined by two factors: the degree of roughness of the surface itself, and the density and elastic moduli of the two adjoining media, such as sea water and granite. For the special case of two fluid media, it was shown in Section 2.6.2 that the percentage y. of reflected energy depends on the ratio of the densities as well as the angles of incidence and refraction, according to the formula i E — peV 1 + tan @ = 2/e) (1) Ve = pit + peV 1 + tan (1 — 2/c?) in which pand p, are the densities of the two adjoining media, c and ¢ are the sound velocities, and @ is the angle of incidence. This quantity y,. is called the coef- ficient of reflection of the separating surface. The coefficient of reflection equals unity when the angle of incidence exceeds the critical angle for total re- flection. Equation (1) is based on the assumption that a smooth plane interface separates two perfect fluids. This assumption is not entirely correct for either the surface or the bottom of the ocean. The surface of the ocean is not smooth. With high winds, it may contain a large number of air bubbles (whitecaps), which absorb and scatter sound. The bottom often consists 137 138 of material capable of shear stress, like rock, and is frequently rough. It is, therefore, simpler to deter- mine the coefficient of reflection experimentally than to attempt to obtain it from the mechanical param- eters of the two substances separated by the inter- face. A rough bottom will not only give rise to a trans- mitted sound wave, which disappears from the ocean, and a specularly? reflected wave, but will also scatter sound in a random fashion. The sound beam resulting from the reflection of a plane wave incident on a bottom of moderate roughness has a certain direc- tivity pattern. If the roughness is not excessive, this directivity pattern will show an intensity maximum in the direction which corresponds to specular re- flection from the bottom. The smoother the bottom, the more highly directive or collimated is the re- flected sound beam. Any nonspecularity of the reflection at the bottom has essentially the same effect as would a broadening of the directivity pattern of the projected beam. This increased divergence will not, however, always cause a decrease in the peak level of the received signal. If the bottom is smooth or only moderately rough, and if projector and receiver are not too highly directional, there should be little or no decrease in the peak signal level. This is because, on the average, as much energy will be deflected toward the hydrophone by non- specular reflection as will be lost out of the main beam through the same mechanism. Excessive roughness of the bottom, however, should cause a decrease in the peak signal level unless the projector is nondirec- tional and a long pulse is used. The reason for speci- fying a long pulse is that some of the sound scattered toward the hydrophone by a very rough bottom will travel paths much longer than the path correspond- ing to specular reflection. Thus, when short pulses of sound are projected over a rough bottom, the re- ceived signal will last longer than the transmitted signal. Although the total energy received at the hydrophone may be the same as if the bottom were smooth, the peak signal level will be lower. 6.1.2 Velocity Gradients and Wind Force The magnitude of the contribution of bottom-re- flected sound to the total sound field will depend not only on the acoustic properties of the bottom, but 2 Specular reflection is reflection for which angles of inci- dence and reflection are equal. SHALLOW-WATER TRANSMISSION also on refraction conditions in the body of the sea and on the reflectivity of the sea surface. The bottom should probably be of little importance if upward refraction in the sea volume prevented most of the sound energy from ever reaching the bottom. We may therefore tentatively predict that in the presence of positive gradients there will be no difference between deep-water transmission and shal- low-water transmission. On the other hand, in the presence of downward refraction the bottom should usually play a role of some importance. If the bottom is a very poor reflector of sound, then the sound field should not differ significantly from the sound field in deep water under similar refraction conditions, since bottom-reflected sound will make only a slight con- tribution to the total sound field. But if the ocean bottom is a good reflector, then the contribution of bottom-reflected sound will be significant. This con- tribution will increase in importance as the down- ward refraction becomes sharper and removes more and more energy from the direct sound field at long range. In addition, if the ocean bottom reflects sound fairly well, the sound field intensity at long range will probably be increased appreciably by sound which has been reflected several times between the ocean surface and the ocean bottom. We should, therefore, look for a dependence of the shallow-water sound field on the roughness of the sea surface. The quantitative prediction of sound field levels in shallow water, by combining the information on bot- tom and surface reflectivity with that on refraction conditions, would be very difficult. The bulk of the reliable information on shallow-water transmission has been obtained directly by means of transmission runs. The qualitative considerations of this section, however, have been valuable in planning these trans- mission runs and in interpreting the resulting sound field data. 6.1.3 Effects of Frequency on Spreading Factor It was shown in Section 5.2.2 that the attenuation of sound in deep water depends strongly on the frequency. It has been tentatively suggested that the observed dependence of attenuation on frequency might be fitted by a 1.4th power law.! It might be expected that a formula of the form H =ak + 20 log Rk (2) would not be applicable for transmission in shallow SUPERSONIC TRANSMISSION 139 ee SSSSSSSSSSSSSSSSSSSSSSSSSsSSS water since this formula was derived in Section 5.2.2 without taking into account the contribution of sound reflected from bottom and surface. For the higher supersonic frequencies, this fear is frequently un- justified. As a matter of fact, in the presence of a well-reflecting bottom, equation (2) provides a better fit to the observations, considering all types of re- fraction conditions, than in deep water. This appar- ent paradox can be explained easily. For 24-ke sound, for instance, it is known that the attenuation in the direct beam amounts to 4 or 5 db per kyd. Beyond a range of 2,000 yd, the second term in the expression (2) increases less rapidly than the first term. As a result, modifications in the second term of equation (2) due to changes in the geometry of spreading are insignificant compared with the first, or absorption, term —at least at ranges where the effect of the bottom might be expected to become noticeable. On the other hand, deviations from equation (2) in deep water are common in the presence of downward re- fraction in the shadow zone. These deviations are mitigated by the appearance of bottom-reflected sound in the shallow-water sound field at long range. It is, therefore, convenient to plot and to analyze transmission anomaly in supersonic shallow-water transmission, since at short range the sharp inverse square drop is taken out of the transmission loss, while at long range the variations of transmission loss and transmission anomaly are not very different (see Figure 1 of Chapter 4). At sonic frequencies the situation is different, since the attenuation as determined in deep-water trans- mission experiments is very small, certainly less than 1 db per kyd. As a result, the first term of expres- sion (2) does not overpower the second term even at ranges of the order of 10,000 yd. Moreover, sonic sources and receivers tend to be nondirectional, and bottom-reflected sonic sound tends to become im- portant at shorter ranges than does bottom-reflected supersonic sound. It may, therefore, be expected that the contribution of bottom-reflected sound will sig- nificantly affect sonic transmission at all ranges of operational importance for all refraction conditions except sharp upward refraction. At sonic frequencies, a modification of the inverse square law to take bot- tom-reflected sound into account thus is more neces- sary than at frequencies above 10 ke. If both the sur- face and the bottom were perfectly reflecting, sound energy would spread only in two dimensions, and as a result, the sound field decay at long range should be approximated by an inverse first power law. Actual interfaces permit sound to “leak” across, and the power law of sound field decay must be obtained by fitting a curve to the observations. Even under these circumstances, however, the consideration of trans- mission anomalies based on the inverse square law should reveal the essential features of sound trans- mission and may be preferred on the grounds of uni- formity of approach. In addition, a plot of transmis- sion anomaly has the practical advantage that a more open decibel scale is possible than for trans- mission loss. 6.2 SUPERSONIC TRANSMISSION To study the acoustic properties of various sea bottoms, both UCDWR and WHOI have carried out transmission and reverberation runs in shallow water. The purpose of these experiments has been both to measure specific parameters characterizing the sea bottom and to obtain information on the overall properties of the sound field encountered in shallow water. Reverberation experiments are discussed in detail in Chapters 11 to 17 of this volume; but it is necessary to refer to them in this chapter, because they have incidentally furnished tentative values for the reflection coefficients of sea bottoms for slant. incidence.” 6.2.1 Acoustic Properties of Sea Bottoms Tyres or Sea Borroms Analysis of observed echo and listening ranges, which began in 1941, indicated that ocean bottoms could be roughly subdivided into a few geological types with fairly consistent reflection characteristics for each type. The classification of bottoms for sound ranging purposes has been standardized and includes the following: SAND, SAND-AND-MUD, MUD, ROCK, STONY, and CORAL. These bottom types are described as follows.’ SAND Firm, relatively smooth bottom. SAND-AND-MUD Relatively firm, smooth bottom. MUD Soft, smooth bottom. ROCK Rough, broken bottom. Includes bedrock, outcrops, and areas covered by boulders. STONY Hard bottom, commonly rough. Pre- dominantly cobbles, gravel, and shells. Varying amounts of sand and mud commonly present. CORAL Hard bottom, with sandy patches, irregular to smooth. Includes var- ious marine forms which secrete masses of lime covering the bottom. 140 In the actual classification of sea bottoms, the criterion established for estimating the relative firm- ness or softness of the bottom was grain size, as de- termined by mechanical analysis. The size limits were set as follows (Division 6, Volume 6). MUD SAND-AND-MUD 90% by weight smallerthan0.062mm. Between 10% and 90% smaller than 0.062 mm. Less than 10% smaller than 0.062 mm and 90% smaller than 2.0 mm. Rounded or angular pieces of rock more than 2.0 mm and less than 10 ecm, which appear to represent glacial drift or other transported material. Rocks of a size greater than 10 cm or pieces broken from rock ledges or where bottom photographs show projecting rocks or rock ledges. Samples containing calcareous masses of coral, algae, or other lime secreting organisms, or bottom photographs showing their existence. SAND STONY ROCK CORAL This classification has been reasonably satisfactory from the acoustical standpoint except in the case of mud, for which it is now likely that texture alone is not an adequate criterion. Although the bulk of experimental work on sound transmission in shallow water has consisted of trans- mission runs, some special experiments have been made to determine numerical reflection coefficients of sea bottoms. REFLECTION COEFFICIENTS It may be gathered from the discussion in Section 6.1.1 that different values of the reflection coefficient are to be expected for sound incident vertically on the bottom and for slant rays. Although the deter- mination of reflection coefficients for vertical inci- dence has some interest at sonic frequencies, the most important situations at all frequencies, from an opera- tional point of view, involve slant rays. To obtain the reflectivity of the sea bottom for slant incidence, three different experimental methods have been con- sidered. The most direct method uses transmission runs with very short signals (10 msec), which often permit the separate reception of the direct signal and the bottom-reflected signal. (The surface-reflected signal cannot be resolved for the usual projector depth of 16 ft, but would be resolvable if the projector could be lowered to several hundred feet.) No coefficients SHALLOW-WATER TRANSMISSION of reflection resulting from these experiments have been reported. It is possible to make a very crude estimate of the reflection coefficient from published standard transmission runs in those cases where the curve showing the transmission anomaly plotted against range indicates at least two well-marked peaks corresponding to single and double bottom re- flections. Reading the level difference between con- secutive peaks and correcting for transmission loss due to absorption between reflection (say 4 or 5 db per kyd) leads to the following estimates: (1) for SAND, the loss through reflection amounts to be- tween 0 and 6 db per reflection, corresponding to in- tensity reflection coefficients between 1 and 0.25; (2) for MUD, the loss is between 10 and 30 db per reflec- tion, corresponding to coefficients of intensity reflec- tion between 10-* and 10-!. The wide spread in each of these estimates indicates both the uncertainty of the estimate in a given case and the wide varia- bility among ocean bottoms falling within one clas- sification. The second method involves the measurement of bottom reverberation. If an echo-ranging transducer is tilted downward about 30 degrees, a peak of the reverberation is received at the range at which the sound beam strikes the bottom. Sometimes, over well-reflecting bottoms, a secondary peak is observed, at a range at which the sound beam, specularly re- flected first by the bottom and then by the surface, strikes the bottom a second time. This secondary reverberation peak has been observed over a coarse sand bottom, and the average amplitudes of princi- pal and secondary peaks have been determined by UCDWR.*! If reasonable assumptions are made con- cerning the transmission loss between the primary and the secondary reverberation peak, and if the re- flection at the sea surface is assumed to be perfect (a calm sea and a low wind), then the reflection coeffi- cient of the sea bottom can be estimated. It was found be somewhere between 0.25 and 1.0, with the most probable value 0.5. These values were obtained for coarse sand, probably the bottom with the highest reflectivity. Reflection coefficients have been estimated as 0.031 for foraminiferal SAND, between 0.005 and 0.025 for SAND-AND-MUD, and 0.0017 for MUD.* These determinations are not very reliable, because they involve unrealistic assumptions concerning the trans- mission loss between consecutive reflections. In these computations, it was assumed that the transmission anomaly amounted to 1.6 db per kyd of vertical path SUPERSONIC TRANSMISSION RAY Bek Us AND BOTTOM PROFILE —~y* AMY. — Nie 141 BT INFORMATION 4800 4900 SOUND VELOCITY IN FEET PER SECOND 5000 DATE _8-6-1943 TIME 1330 BT CLASS NAN WATER ‘DEPTH_80 EM SEA SWELL WIND wu El AM z x = iQ FORAMINIFERAL SAND 600 SOUND FIELD DATA -20 a t=) = C > a ao a4 = fe) 2 qa 2 = 2) 2 = 2) 2 a9 [+4 e 4000 RANGE IN YARDS [o) 2000 FIGure 2. the presence of negative gradients have furnished the most valuable information on the reflectivity of sea bottoms, because of the large contribution of bottom- reflected sound under these conditions, even at relatively short ranges. Fortunately, the Pacific Ocean off southern California has sharp thermoclines most of the year, and the bulk of shallow-water trans- mission runs by UCD WR off San Diego were made in the presence of downward refraction. Figure 1 is a data sheet from a typical transmission run in shallow water over a SAND bottom. On the sheet, the sound data are plotted as transmission anomaly against range.” The transmission anomaly vs range di- agram would be a horizontal straight line if the sound field intensity obeyed the inverse square law. If the transmission obeyed a law of the form of equation (2), the transmission anomaly would be represented by > It will be recalled that transmission anomaly is defined as the excess of the transmission loss in decibels over the value computed in accordance with the inverse square law of spreading. SEA SWELL WIND 6000 Transmission run over sand showing linear transmission anomaly. a slanting straight line, whose slope would be a, the coefficient of attenuation. It has already been men- tioned that in many cases the transmission anomaly can be approximated reasonably well by a straight line. Such a straight line, fitted by inspection, is drawn as the dashed line of Figure 2. The slope of this line is approximately 4.5 db per kyd. Experience has shown that reasonably linear trans- mission anomalies are typical of well- and fairly well- reflecting bottoms. The slope of the transmission anomaly curve depends markedly on the degree of reflectivity of the sea bottom, at least for supersonic sound, but much less on the exact shape of the tem- perature distribution, as long as the downward re- fraction is strong enough to force the direct sound field out of the depth of the receiving hydrophone. EFrect or VELOCITY GRADIENTS This section deals with an analysis of several hundred shallow-water transmission runs, which were obtained by the UCDWR and by the WHOI SUPERSONIC TRANSMISSION laboratory groups. Plots of these runs were analyzed with respect to three factors: refraction pattern, depth of the receiving hydrophone and bottom character. Two other parameters which are also significant, the water depth and wind force, were not taken into account in order not to split the se mple into too many small divisions. No analysis is at present available concerning the effect of water depth on transmission. It is known that in very shallow water (5 fathoms) transmission is poorer in the presence of downward refraction than it is over the same type of bottom in deeper water.’ For the purposes of the analysis reported below, runs in water of less than 10 fathoms and runs in water of more than 200 fathoms have been omitted. As for wind force, a separate analysis has been made at UCDWR, which will be reported at the end of this section. The following types of bottoms have been treated separately: SAND, SAND-AND-MUD (including SAND-MUD and MUD-SAND), MUD (including only the soft muds), CLAY (the plastic muds), ROCK (including ROCK and CORAL), and STON Y (including gravel, cobbles, and similar notations on the original sheets). Sand-and-shells was treated as SAND. The depths of the receiving hydrophone were sub- divided into three classes: shallow (0 to 16 ft), inter- mediate (17 to 100 ft), and deep (more than 100 ft). These classes were chosen for convenience and uni- formity. A division of hydrophone depths into depths above and below the thermocline might have been preferable from a theoretical point of view; but on many bathythermograph traces the location of the thermocline is not uniquely determined. Therefore, a more mechanical division on the basis of hydrophone depth in feet was decided on. The bathythermograph patterns were divided into the usual classes, described in Section 5.1.4as NAN, CHARLIE, MIKE, and PETER. All patterns were classified as in deep water, that is, the classification BAKER (used for most conditions in shallow water) was never used. The MIKE patterns were subdivided into two classes, DEEP MIKE, consisting of all pat- terns in which the water was isothermal to at least 100 ft below the surface, and SHALLOW MIKH, in- cluding all other MIKE cases.¢ In the case of certain ¢ This division of the MIKE patterns was made for this analysis only. The designations DEEP MIKE and SHALLOW MIKE have no official standing in Navy doctrine. 143 well-reflecting bottoms, NAN and CHARLIE were combined into one class. As a preliminary step, median and quartile 4p ranges* were determined for all combinations of the three parameters considered in this analysis; quartiles were omitted wherever the number of runs was 7 or less. Table 1 lists the results obtained for the UCDWR runs in shallow water. Table 2 lists the results obtained for the WHOI runs available at the time of the analysis. For each class of runs, two figures are supplied in the upper right-hand corner of the box for median values of R4 in order to indicate the size and extent of the sample. The first of these two figures is the total number of runs making up the sample. The second number, which shall be called the “adjusted number of days” and is separated from the total number of runs by a slant line, indicates how widely distributed the sample is in time. The latter figure is supplied because it has been found that the acoustic data obtained on a particular day and at a particular location resemble each other more closely than data which have been obtained on different days, even though the oceanographic conditions are closely similar. Instead of simply noting the number of dif- ferent days on which the various runs making up the sample were obtained, it was decided to give an “‘ad- justed”’ number, computed as follows. If the number of runs made on & different days are denoted by m1, M2,°-+,Nx, then the adjusted number of days K is defined as the expression : IG) w=1 R= (3) K equals the number of days & if all the n; are very nearly equal; in other words, if the sample is evenly distributed over the various days on which runs in this classification were obtained. But if some of the days furnish only one or two runs with other days contributing large numbers of runs, the days with very few runs will not be counted fully. To give a 4 These ranges represent that range at which the trans- mitted sound level is 40 db below the level at 100 yd. In some cases, the level at 100 yd was ascertained by extrapolating in from several hundred yards. In the case of WHOI data, Ru is determined with reference to the sound level at 100 yd at the depth of the hydrophone in question, and R« is thus nothing but a measure of the slope of the transmission anomaly vs range; while for UCDWR data, reference is made to the sound level at 100 yd at a depth of 16 ft below the sea surface. SHALLOW-WATER TRANSMISSION 144 daaq 1/1009" ayeIpeulie}Uy AMIN O18'E | ce/s008'E | G¢9'2 MOTTBYS daad S81'% | «z/6000'% | 008‘T sr1/206‘T O8F'S | o-2/6061'S| OS0'S 0r2/2089'S deaq OSI'E | s:z/n066'S | OFL‘T s:z/9000°S GLS'Z | o-e/e008'S | GBS'S | O6ZE | re/ecOS'S| SEL‘T | oyeIpeutezUT AMIN 06S" | 2002'S | 009‘ | SFZ'E | e/ec00F'S| OF8'T G99'°S | ee2O1h'S | SHL‘'S | OSF'S | sz/e200L'S | OS0'S MOTTBYS | MOTIVHS 29'S | ee/s@OST'S | OBL'T | O8L'S | veieGLF'2| OOT'Z deeq | AITUVHO 61/9008‘ OOL‘S | r-e/e2018°S | 000'S | GO%‘E | s-1/190Z8'% | O08‘Z | eyerpouIZezUT aNv GSL'€ | saxS96'°% | SOT'S | STS‘S | o-e/80%B'S| OI6'T | OOT'S | 8/s.00L°S | OOT‘S MOTIBYS NVN 91/tG8G 1 82/9099 'T daaq v20S6 T 2-e/L008 ‘1 97YBIPsUeyUy 006‘ | «e/n00F'T | 008 se/L001‘S Moeys | AITUVHO MOTEq 99g OI6'T | vreO002‘T | OLF'T | $98°S | en089'T] SEE'T daeq OOL‘T | v2/01006 | O94 | 28'S | sv/OP8‘'T | SLS‘T ayeIpauria}Uy GLL'T | v/ee0ZO'T | OS9 | SESS | e-+/010Z0'% | OS8‘T moyeys NVN a9 uvIpay a9 aT uvIpeT a9 a9 uvIpey a9 Fe) 1a} uUBIpe|L ary a9 uvIpsyy Ee) 10% yj3dep U19998q -renb -ienb | -1enb -enb | -1renb -renb | -1enb -renb | -1enb -renb || suoydoipAyy La seddy) qaMOT | reddy aMoT | raddq raMoT | seddy qaMmo7Ty | roddq IOMOT qonw aqnw ANOLS MOOU GNVS -dNV -aNVS “IBYEM MOT[VYS Ul OF FZ 7B SUN UOISSTUISUBIY YM AGO 10j %Y Jo.son[va a[iqzenb pus uvipey~y “|[ T1aV 145 TRANSMISSION SUPERSONIC o/00F'E dag u720SL' 1/2888 1/8089°% 9yPIPaulIeyUT AMIN V2GLL'% 1/200F' MOT[eYS dadud 1/20S6' o1/1008'S dsoq 1/1008'T 1/200‘ o1/1069'G oPeIpoulozUy AMIN o2/rL89'T «1/9008'S z/O0L ‘8 oz/2GL G MOTeYS | MOTIVHS 9-1/400L'S deaq | AITUVHO 21/2006 T GLSS | oveGZE'S | 006'T || e781pewtEqUT aANV 1/2G26°S 12/sSS9 G 00S°S | ro/zeG22'S | SL2‘T MOTTBYS NVN «1/1006 T daa a} 8IPoUIazUT SE8'T | s/1SZ6 218 OSF'T | zz/2008'T | OSTT MOTTeYS | AITUVHO 1/20S6'T daa a7 BIPOUIO}UT EIT | ez/nSZ8 009 | OSS‘ |1-e/n001'T | $26 MOTTBYS NVN 319 uvIpeyy 319 319 uBIpay, Eyal e)ie) URIPI, 319 Eyal uvIpey, E) 1a} ctu UvIpIy 3119 y3dep ui9}48q -1enb -1enb | -1enb -1enb | -renb -renb | -renb -renb | -1enb -renb || euoydoupAyy La saddy JaMoT | reddy qaMo7T | reddy IaMoT | 1oddy qaMoTT | 1add() IaMOT qow qow ANOLS MOOU ANYS -ANV -GNVS "10}8M MOT[BYS Ul DF FZ YB SUNI UOISSTUISUBI} [OHM 10J Yay Jo sonyea o[j2enb pue ueipe~] °Z ATA, 146 numerical example, assume that a total number of 26 runs were obtained on 4 days. If the numbers of runs on these four days were 7, 7, 6, and 6, then the value of K for this case would be 4.0. If, on the other hand, two of the four days contributed 12 runs each, and the other two days only one run apiece, K would ke found to equal 2.3. The results in Tables 1 and 2 indicate that SAND, ROCK, and STONY are well-reflecting bottom types; they lead to values of R4o in excess of 2,000 yd in the great majority of cases, regardless of refraction con- ditions. CLAY also appears to be a well-reflecting bottom, although it should be emphasized that all the CLAY runs by UCDWR were made at a single loca- tion off San Francisco, and that a generalization of the results obtained should be based on a more ade- quate sample. In this method of analysis, no sys- tematic dependence on hydrophone depth can be dis- covered, although the samples for deep hydrophones are mostly too small for the results to be considered conclusive. Transmission over STONY bottoms ap- pears somewhat better than over any other type of bottom. For the classification MUD, it is apparent that the dependence of Ruy on the conditions of refraction is similar to the situation in deep water. The WHOI SAND-AND-MUD data resemble the MUD data in this respect. The Ray ranges are long when the water is isothermal, and short when the sound beam is bent downward by negative temperature gradients. In the classification SHALLOW MIKE, there is some evi- dence of layer effect over the poorly reflecting bot- toms. Layer effect is much weaker, if present at all, for SAND, ROCK, and STONY bottoms. There is also some evidence that in the case of MUD bottoms the transmission for the deep hydrophone is better than for the intermediate and shallow hydrophones. The transmission results obtained by UCDWR and by WHOI appear to be in fair agreement with each other except for the SAND-AND-MUD bottoms. In this classification, the transmission observed by WHOL is significantly poorer than the transmission observed by UCDWR. This disparity is not too sur- prising in view of the fact that the SAND-AND-— MUD classification covers a wide variety of bottoms, namely, all those bottoms in which very fine particles are mixed with sand grains and in which the per- centage of sand grains lies between 10 and 90 per cent. It appears reasonable to assume that the SAN D- AND-MUD bottoms investigated by UCDWR con- tained a larger percentage of sand grains and were SHALLOW-WATER TRANSMISSION TRANSMISSION ANOMALY IN DB TRANSMISSION ANOMALY IN 08 CG MEDIAN CURVES SHIFTED AND SUPERIMPOSED TO COINCIDE AT 1500 YARDS TRANSMISSION ANOMALY IN DB RANGE IN YARDS Fiaure 3. Comparison of WHOI and UCDWR trans- mission data over sand with downward refraction. harder, on the average, than the SAND-AND-MUD bottoms investigated by WHOI. The value of Ru is a useful parameter for the description of transmission; but in view of the fre- quent nonlinearity of the transmission anomaly range curve, no single parameter can be relied on to adequately characterize the transmission from very short to very long ranges. A more adequate method for describing a sample of transmission curves is the SUPERSONIC TRANSMISSION computation of a curve of median transmission anomaly. To obtain such a curve, values of the trans- mission anomaly are read off at a number of prede- termined ranges, such as every 500 yd. At each range, the median transmission anomaly is noted. If these median values are plotted against the range and the resulting points connected, the resulting curve will have the property that at each of the ranges at which values were read it separates the actual curves into two equally numerous portions. In a similar fashion, upper and lower quartile curves of transmission anomaly may be obtained. In general, the median anomaly curve will look smoother than the individual curves making up the sample, and “bumps” in the individual curves will not show up if they do not all appear at the same range. However, the median curve will be repre- sentative of average transmission conditions, and the quartile curves will provide a graphical measure of the spread of the sample. Obviously, median curves of transmission anomaly are valuable only if they are based on a fair-sized sample. For this reason, separate median curves were not constructed for all the classifications which were established in the analysis reported here. Not all individual transmission curves extend out to the same range. When transmission conditions are relatively poor, the reading of the traces must be stopped at a rather short range. When an appreciable number of curves in the sample cannot be read at the longer ranges, the median curve for the remainder apparently turns upward; since this upward turn has no physical significance, the median curve is stopped short in such cases. In the computations summarized in this chapter, median and quartile transmission anomalies were determined at 1,000 yd and every 500 yd from there on; the number of curves in the sample was noted at each of these ranges. As a first problem, the degree of consistency be- tween the UCDWR and the WHOI runs was investi- gated. For this purpose, all runs over SAND with downward refraction (NAN and CHARLIE), re- gardless of hydrophone depth, were collected for each institution separately, and median and quartile curves of transmission anomaly plotted. These curves are shown in parts (A) and (B) of Figure 3. There is some evidence that the discrepancy of about 5 db be- tween the curves is due, in part, to a different method of calibration. While UCDWR has usually referred transmission anomalies to the transmission level re- TRANSMISSION ANOMALY IN OB TRANSMISSION ANOMALY IN DB B WHO! MEDIAN AND QUARTILE CURVES RANGE IN YARDS Figure 4. Comparison of UCDWR and WHOI trans- mission data over sand-and-mud bottoms with strong downward refraction. corded at about 100 yd, WHOI has frequently ob- tained the reference level by extrapolating the meas- ured relative transmission anomalies backward. To illustrate the relatively good fit which results from vertical shifting of the curve, part (C) of Figure 3 shows the two median curves shifted so that they co- incide at a range of 1,500 yd. 148 SHALLOW-WATER TRANSMISSION TRANSMISSION ANOMALY IN DB RANGE IN YARDS Figure 5. Transmission over SAND for different hydrophone depths. We have already noted, from consideration of the Ro values, that for SAND-AND-MUD bottoms the agreement between WHOI and UCDWR is very poor. This discrepancy is confirmed by the median and quartile transmission curves of those runs over SAND-AND-MUD which were carried out with a shallow hydrophone in the presence of NAN pattern (strong downward refraction), shown in Figure 4. In this case, the discrepancy is undoubtedly real and not caused by different calibration methods; for not only are the transmission anomalies at a given range different, but the WHOI median curve has a much steeper slope. The slope of the median UCDWR is roughly between 8 and 10 db per 1,000 yd, while the slope of tke WHOI median curve is about 18 db per 1,000 yd. No other comparisons were made between WHOI and UCDWR transmission data because most of the WHOI samples were too small for such comparisons. All the median curves to be discussed later are based exclusively on UCDWR runs. To examine the effect of hydrophone depth over a well-reflecting bottom, median curves over SAND were determined for the three classes of hydrophone depth without regard to refraction pattern. In Fig- ure 5, the three resulting curves are superimposed on each other, identified as s (shallow), 7 (intermediate), and d (deep). Table 1 shows that the bulk of these runs were carried out in the presence of downward refraction, with about one-fourth of the BT patterns showing a shallow mixed layer above the thermocline. In Figure 5 there are no significant differences between the three curves. The quartile curves bave not been reproduced, but they are all fairly similar, deviating from the median curve by about 5 db at 3,000 yd. The transmission anomaly over SAND can be represented fairly well by a straight line passing through zero at zero range and having a slope of 5 +2 db per kyd. This numerical estimate is also good for the median and quartile curves shown in Figure 3, which do not in- clude the SHALLOW MIKE cases forming part of the sample used in constructing Figure 5. Figure 6 shows median curves, for shallow and for deep hydrophones, of all runs obtained over ROCK bottoms. It will be noted that the transmission over ROCK is not quite so good as over SAND, the aver- age slope for ROCK being 6 db per kyd. The quartiles deviate from the median, in this case also, by roughly 2 db per kyd. Figure 7 shows the median and quartile curves for all runs obtained by UCDWR over STONY bottoms. SUPERSONIC TRANSMISSION 149 TRANSMISSION ANOMALY IN DB RANGE IN YARDS Figur& 6. Transmission over ROCK for shallow and deep hydrophones. Transmission appears to be better out to 2,000 yd for STONY bottoms than for any other type of bottom, but deteriorates rapidly from 2,000 yd on out. The wide spread between the median and quartile curves is an indication that these bottoms are acoustically less uniform than SAND or ROCK bottoms. Figures 8 and 9 show two typical transmission anomaly plots, which were obtained over a ROCK and over a STONY bottom respectively. These runs were carried out in the presence of pronounced negative gradients from the surface of the sea down to well below the depth of the projector. The ray diagrams, which are shown in the upper parts of the figures, indicate that in deep-water transmission con- ditions would be confidently predicted to be poor. Because of the well-reflecting bottom, however, the observed transmission is comparable to that in a deep mixed layer in deep water. MUD bottoms were originally defined as bottoms in which the average particle was too small in size to be classified as SAND. However, evidence accumu- lated indicating that from an acoustic point of view there are two different types of bottoms which are composed of very small particles. These two types of bottom can be characterized by their consistency as soft and as plastic, and they have been designated in this analysis as MUD and as CLAY. Some evidence concerning the difference in acoustic properties of these two types of bottoms has been collected and published by WHOI.® This evidence for separating MUD bottoms into MUD and CLAY was apparently borne out by the analysis of the transmission data obtained by UCDWR, but doubts as to the correct classification of the CLAY samples involved detract from the value of this evidence. Figure 10 shows median and quartile transmission curves over MUD in the presence of negative gradi- ents, separated according to hydrophone depth. It appears that the quartile spread is appreciably re- duced for the shallow and deep hydrophone depths by this separation. Regardless of hydrophone depth, the transmission anomaly at 3,000 yd is approxi- 150 TRANSMISSION ANOMALY IN OB 305 1000 SHALLOW-WATER TRANSMISSION RANGE IN YARDS Figure 7. Transmission over STONY bottoms. mately 30 db. For shorter ranges, there is a significant difference between the anomalies at different hydro- phone depths. With the hydrophone deeper than 100 ft, the transmission anomaly is almost linear and increases at the rate of about 10 db per kyd. For the more shallow hydrophone depths, there is a much more precipitate drop at short range. With the hydro- phone at 16 ft, the median transmission anomaly at 1,000 yd is 26 db. From 1,000 to 3,000 yd, it drops only another 8 db, resembling in this respect the trans- mission of sound in the shadow zone in deep water. Figure 11 shows a typical run over a soft MUD bottom in the presence of a pronounced negative temperature gradient. Just as in deep water, the sound level at shallow depth begins to drop rapidly at a shorter range than does the level at considerable depth. At all hydrophone depths, the transmission anomaly increases sharply at the approximate range of the predicted shadow zone boundary. A slight recovery of the sound level recorded by the two shallow hydrophones is noted at almost exactly the range at which the axis of the reflected beam rises to the depth of the hydrophone. This recovery is, how- ever, not very pronounced. While it increases the sound level at 2,400 yd to approximately 10 db above the level which would have been recorded in deep water under similar circumstances, the transmission anomaly still amounts to about 25 db. EFrect oF WIND ForcE UCDWE has carried out an analysis of the effect which the roughness of the surface has on sound transmission in shallow water over well-reflecting bottoms such as ROCK and SAND. SUPERSONIC TRANSMISSION RAY DIAGRAM AND BOTTOM PROFILE BT INFORMATION ———-HYOROPHONE DEPTH O—— — —-—-— HYDROPHONE DEPTH 4—-—— DEPTH IN FEET RECEIVING VESSEL ——— SENDING VESSEL 50 60 70 TEMPERATURE F DATE 5-27-1944 TIME 1430 AT CLASS___NAN_ WATER DEPTH_42 FM SEA SWELL WIND TRANSMISSION ANOMALY IN DB ° 2000 4000 6000 RANGE IN YARDS Figure 8. Transmission run over ROCK. RAY DIAGRAM AND BOTTOM PROFILE BT INFORMATION oO fe) —— RECEIVING VESSEL RECEIVING END ——— SENDING VESSEL 200 ea 4890 4940 4990 DEPTH IN FEET IN FEET SOUND FIELD DATA DEPTH 4946 4951 4956 SOUND VELOCITY IN FEET PER SECOND DATE 1-9-1943 WS eco) BT CLASS___NAN__ WATER DEPTH_24 EM _ SEA SWELL WIND. TRANSMISSION ANOMALY IN DB RANGE IN YARDS Fieure 9. Transmission run over a STONY bottom. 151 152 \ \ SHALLOW-WATER TRANSMISSION ~A-A-A- SHALLOW HYDROPHONE \\ \ TRANSMISSION ANOMALY IN 0B TABLE 3. Rg versus wind force over ROCK. Figure 10. 2000 3000 RANGE IN YARDS Transmission over MUD with NAN pattern. Wind force (Beaufort) 1 2 3 4 Number of runs 29 54 31 21 Lower quartile Rao 2,050 2,150 1,550 1,500 Median Rao 2,950 2,400 2,100 1,750 Upper quartile Ryo 3,150 2,650 2,300 2,100 One hundred thirty-five runs were carried out over ROCK bottom at fairly constant depth. In Table 3 are listed the number of runs made at each wind force and the median and quartile values of Ro. Figure 12 shows the complete distribution. One hundred seventy-four runs were carried out over SAND bottoms in water of more than 6 fathoms. Table 4 shows the same data for these runs as Table 3 does for runs over ROCK. SUPERSONIC TRANSMISSION 153 RAY OIAGRAM AND BOTTOM PROFILE ——-—HYDROPHONE DEPTH O-—— — —-—-—HYDROPHONE DEPTH A—-—— DEPTH IN FEET TRANSMISSION ANOMALY IN DB RANGE IN YARDS Figure 11. Forty-six runs were made in water of 6 fathoms or less and over SAND bottoms. These runs were analyzed as a separate group. Table 5 summarizes the results. Figure 13 shows the complete distribu- tion of SAND runs. The majority of the runs used in this analysis were carried out in the presence of downward refraction, but no attempt was made to separate the runs with downward refraction from those with the projector located in a mixed layer. This method of analysis may account for the wide quartile spread and more par- ticularly for the great upper quartile spread for wind force 4. The reduction of Ray between wind force 0 or 1 and 3 amounts to an increase in the slope of the trans- mission anomaly curve of roughly 1 db per kyd. 6.2.3 Summary Transmission experiments at 24 ke indicate that the sea bottoms can be roughly divided into well- reflecting bottoms comprising ROCK, CORAL, STONY, SAND, and CLAY bottoms, and poorly reflecting bottoms, mostly MUD and some of the SAND-AND-MUD. Most of the SAND-AND- MUD bottoms are intermediate between well and BT INFORMATION —— RECEIVING VESSEL —-— SENDING VESSEL 50 60 70 TEMPERATURE F DATE 6-1-1944 TIME __J1415 BT CLASS__NAN ___ WATER DEPTH-47_EM _ Transmission run over MUD with NAN pattern. Tasie 4. Ray versus wind force over SAND in water depth greater than 6 fathoms. Wind force (Beaufort) 0 1 2 3 4 Number of runs 18 33 51 60 12 Lower quartile Ri 2,850 3,100 2,900 2,100 1,400 Median Rao 3,450 3,250 3,500 2,450 1,800 Upper quartile Rao 4,000 3,500 3,700 2,950 2,500 TaBLE 5. Rg versus wind force over SAND in water depth 6 fathoms or less. Wind force (Beaufort) 2 3 4 Number of runs 16 13 17 Lower quartile Rao 1,400 1,250 950 Median Rao 1,550 1,400 1,100 Upper quartile Rao 1,700 1,750 1,650 poorly reflecting bottoms. Present evidence indicates that in shallow water at least 10 fathoms deep and in the presence of downward refraction, transmission anomalies over SAND and STONY bottoms increase with the range by 5 + 2 db per kyd, and over ROCK bottoms by 6 + 2 db per kyd. The trans- mission is not significantly affected by hydrophone 154 6000; 4000) 3000 Rao 'N YARDS 2000 1000, WIND FORCE (BEAUFORT) Figure 12. 4 versus wind force over ROCK. depth or bottom depth. (In very shallow water, less than 10 fathoms deep, transmission is inferior to that found in moderately shallow water.) Transmission over MUD differs but little from transmission in deep water; secondary peaks due to bottom-re- flected sound are not likely to raise the level more than 10 db above the level that would be observed in a deep-water shadow zone. In isothermal water or with upward refraction, transmission over all bot- toms is about as good and sometimes slightly better than deép water. Transmission anomalies with negative gradients over the well-reflecting bottom types are affected ad- versely by heavy seas. For sea state 3, transmission anomalies are likely to be at least 1 db per kyd higher than in calmer seas. 6.3 SONIC TRANSMISSION Sonic transmission differs from supersonic trans- mission primarily in that dissipative processes within the water are much less important. The probable value of the absorption or attenuation coefficient at SHALLOW-WATER TRANSMISSION 4000 3000 UPPER QUARTILE Ryo IN (ARDS 2000 1000 x = 6 FM WATER e = ALL OTHER DEPTHS ° ' 2 3 4 5 WIND FORCE (BEAUFORT) Figure 13. Ry versus wind force over SAND. sonic frequencies has been discussed in Chapter 5. It has been estimated® that at the lower sonic frequencies (2,000 c and less) the attenuation of sound in sea water at a depth of several hundred fathoms is less than 1 db in 20,000 yd. While there is reason to believe that close to the surface, absorption at these low frequencies is appreciably higher,° it is probably no more than about 0.5 db per 1,000 yd. As a result, sonic sound in shallow water may show evidence of a spread less than that predicted by the inverse square law. This section summarizes the re- sults which were obtained by UCDWR,}°-” and by CUDWR-NLL. In these experiments, CUDWR- NLL used a single-frequency source, with higher e If dissipative processes near the surface at low sonic fre- quencies were as low as they were estimated at great depths in reference 10, then listening ranges on noisy surface targets should be of the order of 100 miles in the presence of deep mixed layers; actual listening ranges rarely exceed 20 miles even with the best sonic listening gear available. SONIC TRANSMISSION 155 harmonies present because of overloading, while UCDWR used a noise source of the type employed for acoustic minesweeping. The receiving equipment consisted of hydrophones, whose output was ampli- fied and sometimes put through band filters to be recorded by means of power level recorders. Trans- mission was always continuous throughout the run. 6.3.1 Long Island Area Survey Long Island Sound is mostly shallow, less than 15 fathoms deep, and the bottom is predominantly sandy, although some runs were made over MUD, SAND-AND-MUD, and STONY bottoms. All runs were made with a single-frequency source. Fre- quencies used were 0.6, 2, 8, and 20 ke. Geographi- cally, the survey was divided into three areas: the Fisher’s Island area, the New York Harbor ap- proaches, and Block Island Sound. In all three areas, hard bottoms were predominant. Depths varied from about 50 ft to 200 ft. During the New York Harbor and Fisher’s Island area surveys, re- fraction was mostly upward, owing in part to salinity gradients. Off Block Island, some negative gradients were found. Sea states were low with the exception of the New York Harbor runs where sea states up to 4 were encountered. Table 6 summarizes the results obtained. To obtain this table, the investigators at- tempted to fit each run by a formula of the form H =7n-10logr + ar (4) in which n is the power of spreading and a represents the attenuation in decibels per kiloyard. Since it is difficult to determine both n and a simultaneously by a best fit calculation, n was chosen arbitrarily to as- sume the values 1, 1.5, and 2. The best value of a was then determined by inspection for each of the three assumed values of n. The three fits for n = 1, 1.5, and 2 were classified in order of decreasing preference as I, II, III. In addi- tion, the individual fits were graded on an absolute standard as ‘‘good”’ (g), “fair” (f), “poor” (p). Table 7 is a Summary of the goodness of the fit obtained for these runs. Despite the equal standard deviation values of Table 6, the value 1 for n seems to be most frequently the best fit to the data, although 1.5 is probably the best average, especially at the higher frequencies. This survey resulted in the following general con- clusions. Higher frequencies were attenuated more than the lower frequencies. At high frequencies the transmission loss increased with increased disturbance TABLE 6. Statistical analysis of empirically determined values of a and n, poor fits omitted. oS =e SS b oH Ep OT bes iS es BA HAND SO AW — Rs 3 3 PS) s 3 3 & QA A aS qa CCS) loath Oa) 2 o N AND COIONWH N —< a) x2) I i & = ¢ B 2B 3 & = Bes fa Si COLO WA IN iO s NOmMS6 SO 15 Bg ee | MOK ATR OND eS Se i ee oe oe | Su Zo EG Gay | CSoSee 8#NO .DOnha Alas! Bett ANM ONAN a=) (0) Ns oO 0 Eg | BONA MOM AHH xe 2 SOnH NOK COA Or 4 < ] = 5 mao | HAAG oes OAS S SCOnt NNO TATE Bg as CWOOHM O1IDR ARON 3) Se ee | ao Su Zo as) {=| =e) ei Be, | Ceseo 8hem 2 OW Ss! dads ANH FBNANDD > oO ns o a0 By | OHARW BANS HAND o BAND OID D N19 od “A > =< ] = g 3 3% QIQICIN) Wego wpe te =* BANS HOHE ANDO ® & 22 lonom ann DON £3 Bc oe ce Be Sees mere Sw Zo asa SEEl|aHMN NDA KROOD og & ae NAN AA i) GH qo ia) =| 9900 2° 2°00 ro) od MM oO 4 © MY 3 |/enwmeo OMS CONDO 2 | S ao as a o|s Ze) Re) By o3 > oO talallalca g et et 524) SRS QAM Z244aa mM * Values of a and a are given in decibels per kiloyard. 156 SHALLOW-WATER TRANSMISSION of the sea for upward refraction, but there was no such effect at the lower frequencies. With downward re- fraction the state of the sea did not influence the transmission. Except at the lowest frequencies and over the softest bottoms, the type of bottom did not appreciably affect the transmission loss. Bottom types were MUD, SAND, and GRAVEL. Shoal areas and areas over sea valleys showed high transmission losses. The attenuation was virtually independent of depth for flat bottoms. Some correlation was found between the empirical value of n and refraction con- ditions; the power of spreading tended to assume large values in isothermal water. TasLe 7. Summary of shallow-water results (BI,FI) number of fits in indicated classification. n Classified | 0.6ke | 2.0ke | 8.0 ke 20 ke 1.0 I 18 U (9) 18 II 5 3 9 4 III 8 1 16 13 g 12 2 11 20 if 17 9 9 10 p 2 0 11 5 1.5 I 9 4 15 9 II 22 Ui 16 26 Ill 0 0 0 0 q 10 1 15 18 f 16 2 11 14 Pp 5 3 5 3 2.0 I 4 0 10 8 II 3 1, 6 5 III 24 10 15 22 9 4 0 7 6 ft 11 2 18 22 | p 16 9 5 7 6.3.2 Pacific Ocean Measurements Twenty transmission runs were made in the coastal waters off volcanic islands in the Pacific (see refer- ence 11). Measurements were made of overall trans- mission in the 1- to 3-ke band over SAND and CORAL bottoms. These measurements permitted the following conclusions. In the 1- to 3-ke band, the transmission loss from 100 to 3,000 yd could best be fitted by n equal to 1.5 and a equal to 2.5 db per kyd, under most hydrographic conditions. These condi- tions included slight upward refraction in water of depth about 200 ft, slight upward refraction over sloping bottoms, and downward refraction in water of depth about 100 ft. For downward refraction over a sloping bottom, however, the transmission loss at ranges above 1,000 yd was much greater. For this case, the best fit above 1,000 yd was estimated to be 10 db per kyd for the attenuation with the spreading factor n uncertain. This result is in agreement with ray theory, which predicts that sound multiply re- flected from the bottom under these conditions should run downhill, following the bottom slope and leaving a shadow zone near the surface. Also, some runs were made in the Thirteenth Naval District.!! These runs were made in water less than 300 ft deep over coarse gravel or rocky bottoms. Velocity gradients were slight and the sea calm. No correlation with computed limiting ranges was ob- served. The majority of the runs were best approxi- mated by zero attenuation and n equal to the values given in Table 8. One run through a tide rip was best approximated with n equal to 1 and the values of a given on the right-hand side of Table 8. Tas_e 8. Summary of shallow-water results (Thirteenth Naval District). F Special run through Ordinary Runs tide rip Frequency Average n a n a 0.1 1.4 0 1 3.0 0.6 1.3 0 1 1.5 2.0 1.5 0 1 3.5 8.0 2.5 0 1 8.0 20.0 3.2 0 1 8.0 6.3.3 Summary Recent experiments carried out by UCDWR with pulses of sonic single-frequency sound have not yet been reported; they are, therefore, not included in this summary. This summary lists conclusions which were reached in the spring of 1945 on the basis of data available then.'*!” It should be pointed out, however, that none of the conclusions reached at that time have been invalidated by later information. In shallow water, a distinction must be made be- tween transmission over MUD bottoms (which re- sembles deep-water transmission) and transmission over all other bottom types. No significant differ- ences were discovered in sonic experiments between any of the other bottom types including MUD- AND-SAND. Over sloping bottoms, a significant dependence on refraction pattern has been observed: with downward refraction transmission tends to be poor, while in isothermal water it is as good as in deep water. Over level bottoms, with isothermal water or in the presence of downward refraction, the transmission SONIC TRANSMISSION loss can be most adequately represented by an equa- tion having the form, H = l5logr+a, (5) where a, the coefficient of attenuation in decibels per kiloyard, depends on f, the frequency in kilocyeles, according to a = 0.25(f — 2) (6) above 2 ke. Below 2 ke the attenuation is very small. Equation (6) is believed to be adequate up to about 20 ke. Equation (6) represents merely the average de- pendence of the attenuation coefficient on frequency. TaBLeE 9. Variation of attenuation with frequency. a at a at Source a at aat a at 8.0 less | 20.0 less 2.0 ke | 8.0 ke | 20.0 ke | a at 2.0 | a at 2.0 San Diego 0.0 2.5 4.2 2.5 4.2 Fisher’s Island 0.6 ioe 4.1 1.1 3.5 Block Island 2.3 3.6 7.1 1.3 4.8 New York Har- bor 1.4 3.1 7.9 1.7 6.5 Average 1.1 2.7 5.8 1.6 4.7 In the portion of the sea fairly near to the surface, which is the only region of interest in sonic listening, the absorption coefficient probably depends on highly 157 variable factors, such as bubble content; thus large deviations from equation (6) may be expected to occur quite frequently. There appears to be little correlation at sonic fre- quencies between transmission loss and refraction conditions, depth of the water, and surface rough- ness. With strong upward refraction, an increase of attenuation with increasing sea state has been ob- served, undoubtedly caused by the poor reflectivity of a rough and aerated surface. At short ranges, out to approximately the range equal to the depth of the water, image interference maxima and minima have frequently been measured. However, except possibly at very low frequencies, the inverse fourth power decay has not been observed because of the disruptive effect of bottom-reflected sound at the ranges where the fourth power decay might be expected. In general, reliable information on sonic transmis- sion is scanty and is less consistent than the informa- tion on the transmission of 24-ke sound. In the future an increasing amount of stress is likely to be laid on the investigation of sonic transmission. However, sonic transmission will probably remain a more difficult field for investigation than supersonic trans- mission, because of the low directivity of most sources of sonic sound. Chapter 7 INTENSITY FLUCTUATIONS i IS CLEAR from the preceding chapters that the sonar officer or the research worker cannot pre- dict with precision the sound field intensity in the vicinity of a calibrated sound source, no matter how complete his information on oceanographic condi- tions. Chapter 5, in particular, mentions the wide range of sound field levels which are recorded under identical or nearly identical oceanographic condi- tions. This chapter will be concerned with the variability of the sound field which is found when a succession of single-frequency signals are transmitted over the same path and received and recorded through the same receiving sound head and stack. This variability within a single sequence of sound signals has been subdivided into fluctuation, changes in intensity ob- served to occur during seconds or fractions of a second; and variation, a slow drift of the average in- tensity, which becomes noticeable in the course of minutes. This division between short-term and long- term variability can be justified on practical grounds. Variation may well be correlated with those large- scale changes in the thermal structure of the ocean which would be revealed by a continuously recording bathythermograph. Fluctuation is caused by mecha- nisms which cannot be observed by means of any oceanographic instrument in current use. This chap- ter will be concerned, exclusively, with the short-term variability of the sound field. The longer-term varia- bility has already been discussed in Chapters 5 and 6. The first section of this chapter will set forth the mathematical concepts commonly used in the de- scription of fluctuation and will report the results of fluctuation experiments. In the second section, the significance of these experimental results will be as- sessed, and the contribution of various mechanisms to the observed fluctuation will be estimated tenta- tively. 7.1 OBSERVED FLUCTUATION 7.1.1 Magnitude of Fluctuation In describing fluctuation quantitatively, we need expressions which characterize both the magnitude 158 of fluctuation — roughly the amount by which an individual signal deviates from the mean for the run —and the time rate at which the sound field in- tensity changes. This subsection will be concerned with the magnitude of fluctuation. Three different quantities are commonly used to express the magnitude of a received signal: the pres- sure amplitude (in dynes per square centimeter), the intensity (in watts per square centimeter), and the level (in decibels above some standard). When we consider a sequence of N signals received under ap- parently identical conditions, we can characterize this sequence by three sets of figures: amplitudes, in- tensities, and levels of all the individual members of the sample. Each of these three sets of figures de- scribes the sample. Depending on our particular viewpoint, we may prefer one or another. These three sets of figures can be converted one into another by means of the two equations a? = 2Qpc’ (1) a ae). (2) in which a stands for the pressure amplitude, J for the intensity, and L for the level in decibels. To each of these three sets we may assign as an average quantity the arithmetical mean, such as I L = 10 log — = 20 log Iy ao (3) and refer to these quantities as the mean amplitude, the mean intensity, and the mean level of the sample. These average quantities are no longer related by the equations (1) and (2). Individual amplitudes will, of course, deviate from the mean amplitude. But some of these deviations will be positive, others negative, and it can be shown very easily that their sum vanishes. To express the spread of the amplitudes of the sample about the mean amplitude, a common procedure is to square the deviation of each individual amplitude from the mean amplitude and to average these squared devia- tions. The square root of the mean of the squared 1 GS AGiae Gear 2 © sp Gay) OBSERVED FLUCTUATION 159 deviations has the same dimension as an amplitude. It is called the root-mean-square (rms) deviation of the amplitude or, more briefly, the standard devia- tion of the amplitude. If it is divided by the mean amplitude, the resulting dimensionless quantity is called the relative standard deviation of the ampli- tude; this quantity is often expressed in per cent. The analogous quantities formed with intensities and levels bear analogous names. These names are, in fact, common in all fields of statistics. If the rela- tive standard deviation of the amplitude is very small compared with unity, the relative standard deviation of the intensity is about twice the relative standard deviation of the amplitude, while the abso- lute standard deviation of the level is approximately 4.34 times the relative standard deviation of the in- tensity. The fluctuation of underwater sound is usually so large that these relationships between the standard deviations do not hold. Relative standard deviations of the amplitude have been determined for transmitted signals of underwater sound under various conditions.‘ Most of the available data were taken at 24 ke. From the data at that frequency, an analysis was made of the dependence of the relative standard deviation on refraction conditions.? It was found that in the presence of strong downward refraction the median of the relative standard deviation, for the 29 samples collected, was 38 per cent." For eleven samples, in which the receiving hydrophone as well as the pro- jector were located within a mixed layer above a thermocline, the median of the relative standard deviation was 47 per cent. Seventeen samples, in which the hydrophone was in the thermocline be- neath a mixed layer, showed a median relative stand- ard deviation of 41 per cent, not much higher than the fluctuation in the presence of strong gradients from the surface down. Although these differences are probably significant, they should not be overesti- mated, in view of the wide spread within each of the a 4.3429 is 10 log e where the log is to the base 10 and e is the base of the natural logarithms. > In the discussion of the spread of a given set of data, it is very convenient to use the terms “median” and “quartile.” Their meaning is as follows. If all the determinations of a cer- tain quantity are arranged in the order of increasing magni- tude, the value corresponding to the midpoint of the array is called the median value of the spread. The point separating the lowest quarter of determinations from the rest is called the lower quartile, and the point which separates the highest quarter of all determinations from the rest is called the upper quartile. These terms will be used occasionally in the re- mainder of the chapter. groups of samples discussed previously. The lower and upper quartiles in the group of strong downward refractions are 46 per cent and 36 per cent, respec- tively, while the quartilesforthe isothermal group are 61 per cent and 44 per cent. It is probably justifiable to say that, on the average, the amplitude fluctuation in isothermal water is significantly higher than the amplitude fluctuation in the presence of strong down- ward refraction. The width of the quartile spread shows that even under similar conditions the magni- tude of the fluctuation itself fluctuates from sample to sample. In view of the large number of signals making up a sample, usually between 50 and 200, this variability is not to be explained as sampling error but represents an actual change in the transmission conditions as they affect signal fluctuation. The high degree of variability of fluctuation is an indication of the complexity of the underlying mechanism or mechanisms as well. Some information is available concerning the de- pendence of the relative standard deviation of the amplitude on frequency. One set of experiments, carried out at UCDWR, involved the simultaneous transmission of signals at two supersonic frequencies.? The frequency pairs used were 14 and 24 ke, 16 and 24 ke, 24 and 56 ke, and 24 and 60 ke. In 17 runs one frequency was either 14 or 16 ke, while the other was 24 ke. It was found that the mean of the relative standard deviations at the lower frequency (14 or 16 ke) was 38.8 per cent and at 24 ke 37.7 per cent. The difference is well within the root-mean-square spread, and is thus not significant. For the individual samples themselves, the difference between the fluc- tuations at the two frequencies is considerable for some runs, amounting to 19.2 per cent in one case. The root mean square difference is 8.5 per cent. Asa result, it may be concluded that the average fluctua- tion is the same at 15 and at 24 ke, but that for any individual run the fluctuation may be considerably different at these two frequencies. In the majority of cases, however, high fluctuation at one frequency is associated with high fluctuation at the other, and unusually small fluctuation at one frequency tends to be associated with small fluctuation at the other. An analysis of the runs carried out with the frequency pairs 24 and 56 ke, and 24 and 60 ke, leads to similar conclusions for these frequencies.° ¢ The correlation coefficient between the magnitude of the fluctuation for the frequency pair 14, 16 to 24 ke was found to be 0.65 and for the frequency pair 24 and 56 or 60 ke, 0.68. For a definition of the coefficient of correlation, see Section 7.2.3. 160 INTENSITY FLUCTUATIONS At frequencies below 10 ke, it was found both at San Diego and at the New London laboratory of CUDWR that the magnitude of the fluctuation de- creases with frequency. No quantitative data are available. A particularly interesting result was ob- tained in a single transmission run at 5 ke at San Diego. It was found that at moderately short range the relative standard deviation of the amplitude was 47 per cent, while beyond the computed last maxi- mum of the image interference pattern (see Chapter 5) it dropped to 10 per cent. 1.25 (o) 0.5 1.0 SIGNAL AMPLITUDE IN ARBITRARY UNITS Cumulative distribution function of four FRACTION OF SIGNALS WITH AMPLITUDE LESS THAN a fo) fy a fo) Ficure 1. signals. Complete evidence is not available concerning the dependence of the magnitude of fluctuation on range. It is known that at distances of a few feet fluctuation of transmitted sound is negligible. From 100 yd out to very long ranges the average magnitude of the fluctuation appears to be the same at all ranges. No analyses have been made comparing the magnitude of fluctuation at different ranges under identical thermal conditions. There have been recent experiments at UCDWR designed to determine the possible dependence of fluctuation magnitudes on the depths of the projector and receiver. In these experiments, a cable transducer was used as a projector which could be lowered to various depths up to 300 ft. When the projector depth was kept constant at 16 ft, the magnitude of the fluctuation was found to be independent of hydro- phone depth (except for MIKE patterns).2 When both the projector and receiver are deep, it is possible to distinguish between the direct and the surface-re- flected signal. Two runs were carried out with the projector at a depth of 140 ft and the hydrophone at a depth of 300 ft, and the direct and surface-reflected signals were analyzed separately. For the direct signal, the relative standard deviation of the ampli- tude was 9.8 per cent for the first run and 6.8 per cent for the second run; while for the surface-reflected signal the two fluctuations were 57 per cent and 51 per cent respectively. With both projector and hydro- phone at a depth of 300 ft, the fluctuation of the direct signal amplitude was 6.0 per cent and of the surface-reflected signal 50.5 per cent. These results indicate that much if not most of the observed fluc- tuation is caused by mechanisms operating at or near the sea surface. The remaining fluctuation is proba- bly caused at least in part by the slight directivities of the cable-mounted projector and receiver used in these experiments. 7.1.2 Probability Distributions The probability distribution of a set is that func- tion which tells how many members of the set lie between two specified values. Suppose, for instance, that we consider a sample of signals transmitted consecutively over the same transmission path. After these samples have been rearranged in order of in- creasing amplitude, it is then easy, by mere counting, to say how many of those signals have amplitudes less than a, how many have amplitudes less than a, and so on. If we divide these numbers by the total number of members of the sample, we obtain the fraction of signals with amplitudes less than a as a function of a, say P(a). P(a) vanishes for a = 0, equals unity for a = ©, and increases steadily be- tween these limits. This function is called the cumu- lative or integrated distribution function. As a very simple case, Figure 1 shows the integrated distribu- tion of four signals with amplitudes of 0.2, 0.4, 0.5, and 0.7. In the theory of statistics, it is usually as- sumed that if the number of members of the set is increased without limit, the shape of the function P(a) approaches a limiting shape in better and better approximation. It is this limiting shape to which a fundamental physical significance is ascribed. If we assume that the limiting function can be differenti- ated, then OBSERVED FLUCTUATION dP(a) da = (2) = p(a) (4) is the fractional density of members of the set at a. In other words, p(a‘da is the fraction of signals with amplitudes between a and a + da. The function p(a) is often called the differential distribution function. Analogous concepts can be formed for intensity and level distributions which have here been sketched for amplitude distributions. If we call the intensity dis- tributions Q(J) and q(J) (the capital denoting again Ss ees) 2G | aa ea SN REI RET ee PERCENTAGE OF SIGNALS WITH AMPLITUDES LESS THAN @ ($)° Ficure 2. Cumulative distribution of the amplitudes of 50 signals. the integrated and the lower-case symbol the differ- ential distribution), and likewise the level distribu- tions W(L) and w(L), then the integrated distribu- tion functions are, of course, related to each other very simply by the equation 2 WO) = W(2010g) = P@) = a0) = 0, ©) ao 2pc since the fraction of signals with an intensity less than J is identical with the fraction of signals having an amplitude less than a if a and J are related to each other by means of equation (1). For the differential distributions it follows that dW (L) _ @P@) da _ dL da dL since the amplitude and level are related fe equation (2); thus, w(L) = pla) w(L) = pla): (6) 8.69 161 Similarly dQ(I) dP(a)da pe T= al r da me OO} (7) because the amplitude and intensity are related by equation (1). Distribution functions can be determined experi- mentally, and the limiting distribution function will be approximated by the experimentally found distri- bution more and more closely as the size of the sample PERCENTAGE OF SIGNALS WITH AMPLITUDES LESS THAN @ 4 'a\2 () Fictre 3. Cumulative distribution of the amplitudes of 287 signals. is increased. On the other hand, distribution func- tions can also be predicted theoretically by assuming that fluctuation is caused by certain assumed mech- anisms. Figures 2 and 3 show two integrated distri- bution functions which were obtained from actual samples. One of these samples is plotted on proba- bility paper, on which any Gaussian distribution be- comes a straight line.4 Two theoretically predicted distribution functions will be discussed here. The first of these is the so- called Rayleigh distribution. Let us consider, as an 4 A Gaussian distribution is one in which the density p(a) is given by the function 7 (=40)2/26° a => — p(a) VOn8 A Gaussian distribution will usually result if a large number of random processes affect the value of the argument a. The two parameters @ and 6 are the average value and the standard deviation of a respectively. 162 example of Rayleigh distribution, the intensity which will result if a very large number of randomly located scatterers return a single-frequency signal to an echo- ranging transducer. This situation is significant, be- cause it is probably the most realistic model of volume reverberation. Each of these scatterers will return a weak echo, with definite amplitude but random phase. The resultant of all these individual echoes interfering with each other in a random manner will be the reverberation recorded. We shall not give a rigorous derivation of the resulting distribution func- tion but shall sketch the argument leading to it. IMAGINARY AXIS COMPLEX A PLANE REAL AXIS Ficure 4. many individual echoes. Reverberation amplitude produced by In Chapter 2, it was explained how the amplitude plus phase may be combined into a single quantity, the “complex amplitude,” which is designated by A to distinguish it from the real amplitude a. Obviously, a is the absolute value of A. In an interference prob- lem the complex amplitude of the resultant is the sum of the complex amplitudes of the interfering com- ponents. If we have a large number of interfering components with random phases, we may plot the individual complex amplitudes A;, A», ---, A, and the resultant complex amplitude A, in the complex A plane as illustrated in Figure 4. The direction of each individual component is completely random, while its magnitude is fixed. Obviously, the direction (phase) of the resultant A, will be random. As for its magnitude, it is well to consider at first only its component in one direction, say the x axis. This INTENSITY FLUCTUATIONS component of A, will be the algebraic sum of the x components of the individual complex amplitudes Aj, As, ---. In the mathematical theory of proba- bility it is shown that the distribution function for the sum of a large number of random terms is usually a Gaussian distribution, centered in this case about the zero point. In other words, the probability of the x component of A, having a value between x and x + dris (1/+/278)e~**dz, and the combined prob- ability of having the x component and the y compo- nent of A,in specified brackets of infinitesimal width is p(x,y)dxdy = 1 Ae drdy, (8) 276? It is convenient to introduce the polar coordinates a and 6 in the complex A plane; @=r+y, 9 tom Ge 2 9) x Equation (8) then assumes the form 1 2 2 p(a,0)déda = Sra /2®) adoda- TT If we wish to disregard the dependence on the phase angle 6, we may integrate over @ from zero to 27, with the result eee a p(a)da = mere Pedy = = (ocl/®) gy (10) (11) This last expression can be simplified by the intro- duction of the average intensity J. By definition, this average intensity is given by the formula i=f Iq(1)dl, (12) T=0 in which q(J) is, according to equation (11), IO) ae (13) Carrying out the integration, we find for J = (ya T=—) (14) pc which means that we have for q(J) and for Q(/) 1 i HOD) es a (15) and QD) =1— es, (16) respectively. Whenever a signal is the resultant of a large number of components with random phase re- OBSERVED FLUCTUATION lations, then the distribution function can be pre- dicted except for one single parameter, and this parameter is the average intensity. The Rayleigh distribution differs in this respect from a Gaussian distribution, which contains two adjustable param- eters, the average value and the standard deviation. The other distribution function to be discussed here is the zmage interference distribution. It is caleu- lated on the assumption that all the fluctuation of transmitted signal intensity is caused by the random interference of the sound transmitted directly and the sound reflected from the surface of the sea. The ex- tent to which this assumption is justified will be dis- cussed in Section 7.2.2. Let us assume that the ampli- tude of the direct signal alone is a, while the ampli- tude of the surface-reflected signal by itself is a2. If the phase angle between these two components is de- noted by 6, the resultant amplitude a will be given by the expression a = a? + az + 2a» cos 8. (17) With the large-scale geometry given, the values of a1 and a will not vary significantly; but the phase dif- ference between the two paths, 6, will change ran- domly because of the action of waves and because of the minute changes in position of both vessels. In other words, while a; and a, will be treated as fixed parameters (the values of which can, however, be specified only if the depths of source and receiver and their distance are known), 6 will be assumed to take all values between —z and z with equal probability. Since the value of a does not depend on the sign of 0, we shall restrict ourselves to values of @ between 0 and +7. The probability that @ exceeds a certain value 6* equals 1 — 6*/z, that is, the cumulative distribution function for A satisfies the equation P(a) =1— : (18) where a and 6 are related to each other by means of equation (17). In other words, the fraction of signals for which the phase angle exceeds the value @ is identical with the fraction of signals with an ampli- tude less than a. By differentiating both sides of equa- tion (18) with respect to a, we obtain an equation for the differential distribution, p(a), 1 dé pa) = --— (19) a with do —2a Be da Vo@-dp+2a@raa—a & 163 from equation (17). We find, then, for p(a) the ex- pression a 2 p(a) = - wV/ — (a? — a2)? + 2(a2 + a2)a? — a! la —a| SaSa+am. (21) Outside the limits indicated, p(a) vanishes since the amplitude cannot be greater than a; + a nor less than |a; — a.|. For P(a) we find, by means of the relationship Poa) = J plada, (22) the expression Tom 2 (+a P(a) ==+-—are Eee Cie a) ZT 20102 mg are sin | Cees (a= oF Tv 4a\d2 la —a|SaSut+aq (23) by trigonometric transformations. Both expressions (21) and (23) become much simpler if it is assumed that the reflection from the sea surface is perfect, that is, if a; = ae. We have, then Died a p(a) = EW ie = yee 0 < a < 2a, and 2 P(a) = —are sin on 0S aK 2a. (25) T 2a 1 At UCDWR, some experimentally obtained cumu- lative distribution functions of transmitted signals have very nearly the form of equation (25), while others are approximated by a Rayleigh distribution. All the distribution functions published at UCDWR are plotted as integrated distributions. This has been done because with a limited size of the sample the differential distributions would be very difficult to compute with any degree of reliability. Integrated distributions are reasonably accurate in the central part of the curve, but the ‘“‘tails’’ at both ends are necessarily based on very few experimental data. This is unfortunate, because the gross features of integrated distributions, and particularly the central portions, are not very sensitive to changes in the character of the distribution. By definition, all inte- grated distributions are functions which increase steadily from zero at —© to 1 at +o. The central portions of two different distribution functions will be determined in their gross appearance by the location 164 INTENSITY FLUCTUATIONS of the median point and by the slope with which the curve passes through the median point. The central portion of an integrated distribution function gives, de facto, no more information than is contained in the statement of two parameters of the distribution, such as the mean and the standard deviation.. The addi- tional information which is represented by the shapes of the two “‘tails,’’ must frequently be discounted because of the small number of signals which de- termine these shapes. It is true that the mere existence of a tail at the high amplitude end permits certain conclusions al- though these conclusions are mostly negative. If fluctuation were brought about exclusively by the interference of two signals, each having a fixed ampli- tude, then there should be a cutoff at an amplitude equal to the algebraic sum of the amplitudes of the components, corresponding to constructive inter- ference. According to equation (25), for instance, P(a) should reach its maximum of 1 at an amplitude twice a. The fact that there is a percentage of ampli- tudes, however small, exceeding that value proves that interference between two signals of equal ampli- tude cannot be the only cause of fluctuation. The variability of fluctuation magnitudes, which was touched on in Section 7.1.1, is reflected in the variability of the observed amplitude distributions. Even if large samples were processed consisting of thousands of signals for each sample, there is every reason to believe that their distribution functions would differ appreciably. At the present stage of the theory, the details of observed distribution functions do not lend themselves readily to theoretical inter- pretation. Additional plots of observed distributions can be found in references 1 and 2, while additional theoreti- cal distributions are discussed in a memorandum from HUSL.* celles Rapidity of Fluctuation So far, we have discussed only the typical devia- tions which individual signals show from the average. In this section, we shall be concerned with the time pattern of the fluctuation. Two sequences of signals could have the same relative standard deviation of amplitude, but could differ utterly in the nature of their fluctuations. For example, in one sequence the signal amplitudes might be distributed throughout the sequence in random fashion, so that a small ampli- tude signal is as likely to be followed by another small amplitude signal as by a large amplitude signal; while in the other sequence, each signal amplitude might be only slightly different from the amplitude of the preceding or following signal. In the second sequence, the total spread of amplitudes can be just as large as in the first one, if a rising or falling tendency is maintained through a number of consecutive signals. The self-correlation coefficient is the mathematical tool by means of which the time pattern of fluctua- tion can be expressed in quantitative form. THE COEFFICIENT OF SELF-CORRELATION Let us consider a sequence of signals which are re- ceived under apparently identical conditions. It is, of course, conceivable that each signal is completely unaffected by the strength of the preceding signal; this would mean that the distribution function of all those signals which follow immediately after signals of intensity J, are identical with the distribution func- tion of all signals (without restriction). On the other hand, it may be found that the signals immediately following signals with the intensity J, have a distri- bution function which depends on the choice of J:. Both of these situations seem to occur in practice. If the signals directly following those with intensity J; tend to have intensities not too much different from I,, it is said that, in the sequence considered, con- secutive signals have a positive correlation. In order to obtain some numerical measure for the degree of correlation in a given sequence, we shall compare the difference between two consecutive sig- nals with the difference between two signals picked at random. Focusing our attention on intensities, for instance (we might as well consider amplitudes or levels without changing the mathematics), we shall compare the mean squared intensity difference be- tween two signals chosen at random with the mean squared intensity difference between a signal and its immediate predecessor. We are then concerned with the expression he (be = (a= ina! = Wn = 2), (26) in which n and m are to be varied independently of each other. The expression on the right-hand side can be obtained as follows. We have ce a Ih)P ae In a een N?nm= 1 which in turn can be written as the product of two single sums, If there is no correlation between consecutive signals, then the two terms (J, — J,)? and (/, — I,-1)? in equation (26) are equal, and S,; vanishes. If the cor- relation between consecutive signals is positive, then the rms difference between two signals picked at ran- dom will be greater than the rms difference between two consecutive signals, and S, will be positive. If Si should turn out to be negative, that would mean that the average difference square between consecutive signals exceeds the random value; the correlation between consecutive signals would then be said to be negative. The quantity S, has the dimension of an intensity squared. If it is desired to obtain a measure of correla- tion which is dimensionless, it appears reasonable to divide S, through by the mean squared random dif- ference, (I, — In)? = 2(P — 1’). (27) For if the correlation were perfect (that is, if each signal pulse had the same intensity as its predecessor, a situation which can obviously not be realized ex- actly), this ratio would equal unity, while for nega- tive self-correlation, the ratio can be shown never to drop below the value —1. Hence, it is customary to measure the self-correlation of consecutive signals by means of the quantity 2 Nello = IE aS Sa IP = Il which is called the coefficient of self-correlation for unit step interval. In close analogy to this quantity, we may define the self-correlation coefficient for an interval of s steps, p., by means of the expression . bile =P ee i The averaging in the first term of the numerator is to be carried out by averaging over all values of the index n while keeping the step interval s fixed. For s=0, the self-correlation coefficient equals unity, by definition. It can be shown that for all values of s, pz lies between —1 (complete anticorrelation) and 1 , (28) De (29) (complete correlation). Furthermore, p, is an even function of its argument s, that is: Ps = p—s- A MEAN RANGE 115 YARDS “oOo 2 4 6 8 10 12 14 16 CORRELATION INTERVAL IN SECONDS 0 8 6 24 +32 40 48 #£=56 = 64 CORRELATION INTERVAL IN SIGNALS Figure 5. Self-correlation coefficients of two se- quences of supersonic signals. Figure 5 shows two self-correlation coefficients which were obtained at UCDWR and which were computed from samples at different ranges. In both cases, the receiving hydrophone was in the direct sound field. The abscissa represents the step interval, marked both in terms of the number of pulses s and in terms of the time in seconds. These two plots, which are typical of the others obtained, show that there is a marked positive self-correlation for step intervals of a few seconds. It appears that the longer the range, the longer is the step interval of positive correlation (that is, the slower is the fluctuation), al- though the evidence on that point is too scanty to be considered conclusive. For many of these plots, the self-correlation be- comes negative for some step interval before it drops down to zero. This anticorrelation has not yet been explained, although it is observed more often than not. 166 INTENSITY FLUCTUATIONS When the sound intensity is measured well inside a so-called shadow zone, there is usually no self- correlation for step intervals even as short as one second. As illustrated in Chapter 4, Figure 2B, the amplitude, or intensity, varies so rapidly that no correlation ean be expected between consecutive sig- nals with the usual keying intervals. However, the same figure illustrates another possibility for treating coherence in a quantitative manner. If we consider, instead of a sequence of signal pulses, the amplitude fluctuation in a continuous signal, we may define the self-correlation coefficient of the amplitude as a func- tion of the continuously variable interval 7 as follows: T =f A()A(t + 7)dt — A” 31 p(t) = =o) —9 ( ) where the interval of integration 7 must be large compared with the step interval 7. Figure 6 shows the self-correlation coefficient which was found during T IN SECONDS Ficure 6. Self-correlation coefficient of a 10-sec sig- nal received in the shadow zone. one run for sound transmitted into the shadow zone. The appearance of this function is similar to those in Figure 5, except for the enormous change in the time scale. Figure 7 shows the self-correlation coefficient which has been predicted theoretically for the in- tensity of reverberation produced by a square-topped single-frequency signal of length t&. The expression obtained by Eckart® for this coefficient is as follows: p(t) = (: ie rely for |7| S to 0 for |7| = to. (32) HippDEN PERIODICITIES In the preceding section, the coefficient of self- correlation was introduced primarily as a mathe- matical measure of the coherence of the transmitted signal or, in other words, as a measure of the rapidity of fluctuation. In addition, the self-correlation coef- Ficure 7. Theoretical self-correlation coefficient of the intensity of reverberation from a square-topped signal; ; ficient provides a powerful tool for discovering “‘hid- den periodicities.’”’ A hidden periodicity is essentially nothing but a tendency of the fluctuation pattern to repeat itself with a fixed period, a tendency which is modified by nonperiodic disturbances. Consider, for instance, an ordinary pendulum which is subject to random forces. This pendulum will be moved to carry out periodic swings, but the periodicity will not be strict since both amplitude and phase of its vibration are subject to random changes. But if we were to plot the motion of the pendulum for a long time (large compared with its period), we should find that the self-correlation coefficient will have a maximum (al- though not quite +1) for an interval equal to the period of the pendulum and a minimum (although not quite —1) for an interval equal to one-half the period of the pendulum. Extending the self-correla- tion analysis to longer intervals, we should find another minimum at 3/2 the period, again a max- CAUSES OF FLUCTUATION 167 imum at twice the period, and so on, these con- secutive minima and maxima becoming gradually more shallow until the self-correlation coefficient effectively approaches zero. In other words, hidden periodicities are revealed by the location of maxima and minima of the self-correlation vs interval curve.° It was believed, at one time, that part of the ob- served signal fluctuation could be explained as train- ing errors due to the roll and pitch of the transmitting vessel. Self-correlation coefficients were studied pri- marily with a view toward finding the fluctuation periodicities which would coincide with the known periodicity of the vessel. Although the results of these studies were at first disappointing, it is possible that future work will lead to more positive results. 7.1.4 Space Pattern of Fluctuation In order to discover to what extent the observed fluctuation varies in space, an analysis was made of the fluctuations observed in the simultaneous out- puts of two hydrophones.? The two hydrophones were kept either at the same or at two different re- corded depths. No determination of their horizontal separation was made, but they are believed to have had a horizontal separation of between 5 and 25 ft. Thus, both hydrophones were at about the same distance from the projector, which emitted 24-ke signals. The number of samples analyzed is too small to establish any quantitative law, but indications are that the correlation between the simultaneous out- puts tends to become weaker as the distance of the two hydrophones from each other or as their joint distance from the sound source is increased. How- ever, in the majority of samples analyzed, the correla- tion remains significant, even at the maximum verti- cal separation of the two hydrophones, which was 300 ft.! © This property of the self-correlation coefficient is incor- porated in a mathematical theorem frequently quoted as Khintchine’s theorem, which states that the coefficient of self- correlation is the Fourier transform of the (normalized) squared frequency spectrum of the time sequence considered. If, as in the case of the pendulum, the time sequence has a tendency to repeat its functional pattern, its spectrum will have a maximum at that frequency. This peak in the spectrum of the time sequence may remain undiscovered if the time sequence is inspected directly because of the changing phase relations. The squared spectrum, however, contains the abso- lute values of the squared frequency amplitudes, without regard for phase relationships. Consequently, its Fourier transform, the self-correlation coefficient, reveals the “hidden periodicities” more clearly than,the original time sequence. f With one hydrophone at 16 and the other at 300-ft depth, the correlation coefficient was 0.34 at a range of 950 yd and 7.2 CAUSES OF FLUCTUATION It has not yet been possible to develop a theory of fluctuation which would permit the prediction of its magnitude and time rate as functions of oceano- graphic or other parameters. Nevertheless, it is of interest to consider the various mechanisms which have been considered responsible for fluctuation. These mechanisms may be described under three headings: roll and pitch of the vessel, interference mechanisms, and thermal microstructure of the ocean. 7.2.1 Roll and Pitch of Transmitting Vessel Except in very calm weather, the transmitting vessel is subject to considerable roll and pitch, with the result that the bearing of the transmitter relative to the target and relative to the surface of the sea is not constant. Because of the directivity of the trans- mitted sound beam, a change in bearing will bring about a change in received signal intensity if the change exceeds a few degrees. This change may come about merely because the principal beam may miss the target during one phase of the roll and hit it during another phase. A more involved hypothesis considers the interference between direct and surface- reflected sound. In the presence of a slight upward refraction, the direct and surface-reflected rays to the target leave the projector at appreciably different angles. The sound received at the hydrophone is the result of interference between these two rays. If the training of the projector is changed slightly, the rela- tive intensity of the two rays will also change, as the projector will discriminate first against one and then against the other. If the two rays are out of phase by nearly 180 degrees, the resultant change in the in- tensity distribution of the interference pattern may become very appreciable, even with comparatively minor changes in the relative intensity of the two component rays. The chief argument in favor of roll and pitch as a cause of signal fluctuation was that in the earlier studies the self-correlation coefficient seemed to indi- cate a period of fluctuation similar to the known period of roll of the transmitting ship. Subsequent 0.38 at a range of 1,750 yd. The number of signals in each sample was 40. For a definition of correlation coefficient, see Section 7.2.3. 168 INTENSITY FLUCTUATIONS work has indicated, though, that the time rate of fluctuation is range-dependent, that is, the time rate decreases as the range increases, at least in the direct sound field. In addition, observations made when the 99 PERCENTAGE OF AMPLITUDES LESS THAN INDICATED VALUE Ficure 8. Observed cumulative distribution suggest- ing image interference fluctuation. roll of the transmitting ship was less than 2 degrees show about the same fluctuation as other data. Thus, while no definitive conclusions can be drawn at the present time, it seems unlikely that roll and pitch are dominant causes of the observed fluctuation in under- water sound transmission. 7.2.2 Interference If the sound signal received at the hydrophone were the resultant of several individual signals trans- mitted over two or more paths, any change in the relative phases and amplitudes would cause a change in received signal strength. If the properties of the transmission paths were subject to random variations, the resulting variability of the received signal would depend on the characteristics of these variations. Several different models of underwater sound trans- mission have been studied which involve multiple paths of transmission. Two Patus We shall consider, first, interference between the direct and the surface-reflected signal. If the geome- try of one or the other path could be changed ran- domly, the result should be a fluctuation in the rel- ative amplitudes or, at least, in the relative phases of the two interfering signals. Some evidence has been accumulated which indi- cates that interference between the direct signal and the surface-reflected signal is at least a major con- tributing cause for the observed fluctuation. Figure 8 shows a distribution of amplitudes similar to sev- eral which were observed at UCDWR. The theoreti- cal curve corresponding to the expression (25) (with a; equal to 7/4 times the mean amplitude @) is super- imposed on the observed points. The moderate agree- ment indicates that during the run from which this distribution of amplitudes was obtained, random in- terference between two equally strong signals could have been the principal cause of fluctuation. On the other hand, a large number of observed amplitude distributions fail to conform to the expression (25), suggesting that the assumed mechanism is not al- ways the principal cause of fluctuation or, at least, that it is frequently modified by other mechanisms. Another argument in support of the hypothesis that fluctuation is caused, in part, by interference be- tween the direct and the surface-reflected signal is provided by the absence of regular image interference patterns for most transmission runs at supersonic frequencies. While traces of the pattern are regularly observed for transmission at low sonic frequencies (see Section 5.2.1), they are almost never found be- yond 100 yd at frequencies exceeding 20 ke. This absence has usually been explained by the size of the irregularities of the sea surface. While at very low frequencies the wavelength of underwater sound is large compared with most of the water waves, this is not true for supersonic sound. Irregularities of the sea surface may well replace the theoretical image inter- ference pattern by an image fluctuation; this conjec- ture is supported by the fact that fluctuation at sonic frequencies is often markedly lower in magnitude than it is at supersonic frequencies. A very striking plot of a transmission run at 20 ke showing both image effect and image fluctuation has been pub- CAUSES OF FLUCTUATION a "RELATIVE INTENSITY IN DB —— 90 80 70 60 50 40 20 10 ° RANGE IN FEET — Ficure 9. Transmission run showing image interference effect. lished by UCDWR‘ and is reproduced in Figure 9. The signal level was recorded by a sound-level re- corder. It will be noted that the ranges are very short, extending to not more than 90 ft. The recorder trace shows clearly that the amplitude of the signal fluctuation is greatest near the minima of the regular interference pattern (drawn in asa theoretical curve) where the magnitude of the resultant would be most sensitive to phase shifts of the components. This rec- ord was taken in shallow harbor water, and the sur- face was undoubtedly quite smooth. Otherwise, the interference pattern might not have been so notice- able. Finally, attention is called to the experiments carried out with a deep transducer, which were men- tioned in Section 7.1.1. These experiments indicate that the fluctuation is often reduced to a fraction of its usual magnitude when both the sound source and receiving hydrophone are so deep that the direct signal can be separated from the surface-reflected signal. It is true that some fluctuation remains, even when interference with the surface-reflected sound is eliminated ; this small fluctuation may be the result of imperfect equipment. In all cases, however, the fluctuation of the direct signal is reduced drastically when it can be separated from the surface-reflected signal. The fluctuation of the surface-reflected signal by itself is somewhat higher than the fluctuation of the combined signal usually observed with a shallow projector. SEVERAL PATHS In shallow water, or even in fairly deep water with a transmitter of low directivity, sound will reach the receiving hydrophone not only over the direct path and through one surface reflection, but also through one bottom reflection, one bottom and one surface reflection, etc. The number of possible paths is, strictly speaking, infinite, and small changes in geometry may bring about random phase shifts be- tween the different arrivals. Nevertheless, the distri- bution cannot be expected to approach the Rayleigh case, because the intensity for the paths drops rapidly as the number of reflections is increased, both be- cause there are recurring energy losses on reflection and because the high-order paths are steep-angle paths and therefore discriminated against by the transducer. Only very few of the theoretically possi- ble paths of transmission will, therefore, be effective in contributing to the resultant signal. It has been found that in the presence of bottom-reflected sound the rapidity of fluctuation increases, as shown by the oscillograph trace reproduced in Figure 10. Unfortu- nately, no quantitative information is available con- cerning the decrease in the self-correlation coefficient due to the contribution of bottom-reflected sound. Many PAtus. Figure 2 shows a distribution function obtained at UCDWR, and superimposed on the experimental points is a curve representing the Rayleigh distribu- tion. The fit is good. A model of sound transmission was set up in an attempt to explain this observed approximation to Rayleigh distribution. The model is based on the thermal microstructure which has been found to exist in the ocean’ and which is de- scribed in Chapter 5. On the basis of ray acoustics, it was suggested that the irregular thermal structure of the ocean may give rise, simultaneously, to more than one ray path connecting the transmitter with the re- ceiving hydrophone. It seems reasonable to assume that these paths will have different travel times and that the signals transmitted along them are, there- fore, not in phase with each other. If the phase dif- 170 INTENSITY FLUCTUATIONS Ficvre 10. Oscillograph trace showing the effect of bottom-reflected sound. ferences were random and if the average number of paths were sufficiently great (at least five or six), the resulting distribution of intensities should approach the Rayleigh distribution very closely. Against this proposed mechanism two principal ob- jections have been raised: one, experimental, the other, theoretical. The experimental objection is simply that later research has revealed that the Ray- leigh distribution is only occasionally a very good fit to observed transmitted intensities. The theoretical objection concerns the phase shifts expected from the observed microstructure. It is pos- sible to compute the root-mean-square difference in acoustical path length between two alternative paths through the interior of the ocean on the basis of the average parameters of the observed microstructure.® It turns out that the magnitude of this variation in path length is too small to produce the random phase shifts as required for Rayleigh distribution. This argument is not entirely conclusive, because the microstructure parameters reported in Section 5.1.3 were obtained on a single run and have not been confirmed by a repetition of the experiment. No critical evaluation has as yet been made of the multiple path hypothesis on the basis of wave acoustics. The multiple path hypothesis is based, conceptually, on ray acoustics, and it may be that the ray concept has been stretched in this case be- yond the limits of its validity. A similar analysis for a different problem has been made by CUDWER.? 7.2.3 Lens Action of Microstructure If light passes through a medium with variable index of refraction, such as the turbulent heated air above a tarred road on a hot summer day, objects seen through this medium are often blurred. If sun- light falls on a white screen after having passed through such a medium, say the hot gases surround- ing an open flame, the surface of the screen is mottled, with bright and dark patches alternating and chang- ing rapidly as the thermal microstructure of the trans- mitting medium is varied. This random fluctuation in the brightness of the illuminated screen can be ex- plained by means of the lens action of patches of above-average and below-average velocity of propa- gation of light. A similar explanation has been sug- gested to account for part of the fluctuation of trans- mitted sound intensity in the sea.’ This role of the thermal microstructure in fluctuation is quite dif- ferent from the hypothetical interference effect dis- cussed in Section 7.2.2. While the interference effect is based on the coexistence of several distinct paths through the interior of the ocean, fluctuation because of refraction will be produced even over a single path. The theoretical treatment, not reproduced here, CAUSES OF FLUCTUATION leads to the result that if the refracting properties of the microstructure were alone responsible for fluctua- tion, the magnitude of fluctuation should increase with range. At moderate ranges the magnitude of the fluctuation should be proportional to the 1.5th power of the range. Since this hypothesis is based on ray acoustics, the fluctuation should be independent of the frequency, as long as the wavelength is short enough for ray acoustics to be applicable. The dependence of fluctuation on range predicted by this hypothesis has not been confirmed by obser- vations, although the variability of the magnitude of the fluctuation is so great that a small effect might not have been discovered. For that reason, some de- pendence of fluctuation on range cannot be definitely ruled out. The theoretical formula connecting the magnitude of the predicted fluctuation with the parameters of the microstructure appears to lead to a fluctuation of a magnitude much smaller than ob- served. There is, however, one feature which appears to suggest that refraction by microstructure is at least a contributing cause of the observed fluctuation. It was pointed out that fluctuation caused by micro- structure should be frequency-independent for a wide range of frequencies. In this respect it differs from hypotheses based on interference, since interference leads to fluctuation which is critically dependent on frequency. It has been possible to check the depend- ence of fluctuation on frequency by transmitting signals simultaneously at two widely separated frequencies and by noting the correlation between their instantaneous amplitudes.* These trials indi- eated a partial but significant correlation between the fluctuations at two widely separated frequencies. To understand the significance of this result, it is necessary to explain in a few words the mathematical meaning of the term correlation coefficient. If there are two time series, say Ki, Ko, --- , and Ly, Ls, --- , then the correlation coefficient between them is de- fined (in close analogy to the self-correlation coeffi- cient of one time series, introduced earlier in this chapter) as the expression K,L, — KL OKOL PKL = ox = K? — K’. (33) This expression equals unity if there exists a rela- tionship Dak Bo> 0 n = 1,2)3)-2. 64) or, in other words, if Z is a linear function of K with 171 a positive slope. If a < 0, px, will equal —1. If there is some tendency of large values of LZ to be coupled with large values of K, and small values of Z to be coupled with small values of K without the existence of a rigorous linear relationship (34), then px, will have a positive value less than 1; conversely, a negative value of px, (greater than —1) will signify a coupling of large values of LZ with small values of K and vice versa. If px, vanishes, then the deviations of individual K values from K are statistically inde- pendent of (uncorrelated with) the deviations of the corresponding LZ values from J. It was found that the correlation coefficient be- tween simultaneous signals at two different fre- quencies varied from 0 to 0.75 with an average of 0.3. In other words, while there was some tendency for strong 24-ke signals to be coupled with strong 56-ke signals, the simultaneous signal amplitudes at these two frequencies were far from proportional to each other. The same statement holds for each of the three other frequency pairs at which experiments were performed. It must be concluded that the observed fluctuation is caused by a combination of mecha- nisms, of which some operate independently of the signal frequency (refraction and roll and pitch), while others depend on the transmitted frequency (interference). Tone Summary The experiments carried out with a deep sound source and a deep hydrophone indicate that most of the observed fluctuation disappears if the whole transmission path is more than 100 ft below the sur- face. They also show that the surface-reflected signal by itself (without interference from another path) fluctuates more strongly than the composite signal observed in shallow transmission, at least when the incidence at the surface is not glancing. Unfortu- nately, these findings are not helpful in a choice be- tween the various mechanisms which have been considered. If roll and pitch contributed significantly to fluc- tuation, its effect on a cable-supported transducer would be very much less noticeable than the effect on a transducer rigidly connected with the hull of the ship; but there are not yet enough data with a shal- low-cable transducer to permit any conclusions. Image interference fluctuation will cease to operate when the surface-reflected sound can be separated from the direct signal. Microstructure will probably 172 INTENSITY FLUCTUATIONS be very appreciably reduced below the region of strong thermal gradients. Nevertheless, the composite evidence indicates that image interference is probably the most im- portant single factor contributing to the observed fluctuation. There may also be interference between the beamlets into which the irregular surface of the sea breaks up the incident coherent beam. In the deep transducer experiments, the surface-reflected signal showed a very high degree of fluctuation, but at glancing incidence the path differences may not be large enough to bring about random interference. If all interference effects could be eliminated, total fluctuation would probably be cut in half.# The remaining fluctuation is essentially frequency- independent. It may be due in part to pitch and roll and in part to the lens effect of microstructure. Elimi- nation of either of these effects is possible in principle, but would require very elaborate additions to present sound gear. £ Fluctuation by interference can be effectively eliminated by transmitting supersonic sound in a broad frequency band. The width of that band should probably exceed 5 ke in order to obtain maximum benefits. Chapter 8 EXPLOSIONS AS SOURCES OF SOUND 8.1 INTRODUCTION XPLOSIVE SOUND differs from sinusoidal sound both in the intensity which can be achieved with it and in the fact that it consists of one or more pulses of extremely short duration. These characteristics have prompted many suggestions for the employ- ment of explosive sound in communication and echo ranging, few of which, however, have so far been utilized in practice. The survey given in this chapter and the next of what is known about explosive sound is partly designed to facilitate an understanding of the possibilities and limitations of explosive sound in such applications. The study of explosive sound can be useful in another way, however, in that it can supply valuable additions to our information about the nature of the sea and its bottom, and about the causes of many of the phenomena observed in sound transmission. The possibilities of explosive sound as a research tool have accordingly been kept in mind in the selection of material for these chapters. To understand the complex phenomena which ac- company the propagation of explosive sound in the sea one ought to begin by finding out as precisely as possible just what the explosive disturbance is like, originally, before it has been propagated to any ap- preciable distance. Fortunately, much has been learned about explosions and the pressure disturb- ances which they create in the water near them. A detailed survey of what is known about underwater explosions would require a volume in itself; however, an effort will be made in this chapter to summarize briefly those parts of our knowledge of underwater explosions which have a bearing on the use of ex- plosions as sources of sound. In this chapter, there- fore, we shall be concerned with the disturbance at comparatively short ranges from the explosion, where its characteristics are presumably little affected by the departures of sea water from the concept of a pure homogeneous fluid. Most of the information in this field has been obtained in the course of experi- ments directed toward the elucidation of the damag- ing effects due to explosions. With this information as background, we shall be able, in Chapter 9, to dis- cuss the propagation of explosive sound through siz- able distances in the sea where departure of the medium from homogeneity, effects of the bottom, and other factors are important. 8.2 SEQUENCE OF EVENTS IN UNDERWATER EXPLOSIONS An explosion is a process by which, in an extremely short space of time, a quantity of “explosive” ma- terial is converted into gas at very high temperature and pressure. This conversion is due to a chemical reaction which converts the explosive material from a thermodynamically unstable state to a more stable one with the evolution of a great amount of heat. This reaction, when initiated at one point of a mass of explosive, propagates itself rapidly until all the mass is involved. The propagation may take place in either of two ways, called respectively burning and detonation. In burning, the contact of the hot gaseous products of the reaction with the untransformed por- tion causes a reaction to take place at the surface of the latter, the rate of transformation being slow enough so that the boundary between transformed and untransformed material advances with a speed slower than the speed with which the pressure gener- ated by the reaction is propagated through the mass. In detonation, on the other hand, the reaction takes place so rapidly that it can keep up with the pressure wave, which in this case is known as a detonation wave. These two processes, which will be discussed more fully later, permit explosive materials to be divided into two rather well-defined classes: ex- plosives which detonate, commonly called high ex- plosives, and explosives which merely burn, for which we shall use the term propellants since the most im- portant explosives of this type are used to propel projectiles from guns, or as rocket fuels. A given quantity of high explosive will radiate considerably more acoustic energy when it is set off than will a like 173 174 amount of a propellant; for this reason nearly all the material to be presented in these chapters concerns sound generated by high explosives. Let us therefore consider what happens when a quantity of high explosive is set off under water. First, detonation is initiated at some point of the explosive; this may be done, for example, by using (Bounpary OF GAS BUBBLE EARLY STAGE SHOCK FRONT PRESSURE DISTANCE FROM CENTER OF EXPLOSION [pee OF GAS BUBBLE | RESIDUAL FLOW WATER LATER STAGE AROUND BUBBLE | GAS PRESSURE SHOCK WAVE — SHOCK FRONT DISTANCE FROM CENTER OF EXPLOSION Pressure distribution in the water at two FicureE 1. instants of time following detonation of a charge of high explosive. a detonating cap containing a small quantity (about a gram) of an especially sensitive explosive traversed by a fine wire which can be suddenly heated to incandescence by a current of electricity. From the point of initiation a detonation wave spreads out in all directions through the explosive with a velocity of several thousand meters a second. In front of the detonation wave the material is in exactly the same state as before the explosion, while behind the wave front it is gas at a pressure of ten to a hundred thou- sand atmospheres and a temperature of several thou- sand degrees centigrade. When the detonation front reaches the boundary between the explosive and the water, this pressure is transmitted to the water, and a wave of intense compression starts outward through the water. If the pressure were not so enormous, this wave would be an ordinary sound wave. However, because of its great amplitude, the wave differs in a number of ways from ordinary sound waves and is called instead a shock wave; it bears somewhat the same relation to sound waves that a large breaker on the beach bears to an ordinary water wave. A shock wave is characterized by an almost discontinuous rise of pressure to a high value at the front of the ad- EXPLOSIONS AS SOURCES OF SOUND vancing wave and travels with a speed greater than the normal velocity of sound. The reasons for these characteristics will be discussed in the following sections. The pressure in the shock wave from an explosion dies off fairly rapidly behind the shock front, and by the time the shock wave has advanced to a dis- tance of the order of ten times the radius of the origi- nal mass of explosive it has become a fairly well localized disturbance, advancing outward and prac- tically independent of the motion of the water and gas in regions nearer to the center. Figure 1 shows schematically how the pressure may be expected to vary with distance from the original site of the ex- plosion at two successive instants of time as this state of affairs is becoming established. Although in these later stages it no longer affects the main part of the shock wave, the motion of the gas-filled cavity and the water immediately around it is by no means unimportant. At times such as those shown in Figure 1 the pressure in the gas cavity, hereafter called the bubble, is still quite high, and the water around it is rushing outward with a very high velocity. Because of the inertia of the water, this out- ward motion continues long after the force of gas pressure, which initiated it, has become negligible. As the gas bubble expands, the pressure in it drops, and eventually becomes far less than the normal hydrostatic pressure of the surrounding water. This excess of pressure on the outside finally brings the expansion to a halt, but not until the bubble has reached a radius which may be several dozen times the initial radius of the explosive. A contraction now sets in, and again, because of the inertia of the water, the bubble overshoots its equilibrium radius and the contraction does not stop until a very high pressure has been built up in the gas bubble. Several cycles of this expansion and contraction may take place before the oscillation dies out. The period of these bubble oscillations is of the order of a thousand times the duration of the pressure which the shock wave exerts as it passes a particular point in the water; it is usu- ally of the order of 1/30 to 1 sec, depending on the size of the charge and its depth. At each contraction a new pressure wave is sent out into the water; these so-called ‘‘secondary pulses” are many times less in- tense than the shock wave, but as they have a duration many times longer, they may contain a greater amount of impulse, and a comparable though smaller amount of energy. A quantitative theory o1 this phenomenon will be sketched in Section 8.6. SHOCK FRONTS 175 Most of the commonly used high explosives are re- markably similar to one another in the amount of energy they release per unit mass, and in the relative amounts of energy which go into the shock wave and the oscillations of the bubble. Of the total work done by the gas on the water in its initial expansion, about 40 to 50 per cent remains as kinetic and potcntial energy in the oscillations of the gas bubble and sur- rounding water, part of this energy being ultimately converted into heat by dissipative actions in the neighborhood of the bubble and part being radiated as acoustic energy in the secondary pulses. The re- maining 50 or 60 per cent of the original energy is at any stage divided between energy present in the shock wave and energy which has been converted into heat by dissipative processes occurring at the shock front. Dissipation of the latter kind is espe- cially rapid in the early stages so that by the time the shock wave has advanced a distance of the order of ten or twenty times the original radius of the charge, about a quarter of the original energy has been dissipated into heat, and the other quarter continues to be radiated outward in the shock wave. From this time on, the dissipation is much slower, although not negligible. In the preceding discussion, the phenomena have been described without reference to the size of the charge of explosive which is used. This is possible because explosions of all sizes are similar. If the range is not too great, the intensity and form of the shock wave, and many of the features of the bubble oscilla- tion, can be predicted exactly for one quantity of explosive if they are known for another quantity of the same explosive substance. To give a precise state- ment of the rule by which this prediction can be made: suppose two experiments are carried out with the same explosive material, the shape of the charge and the position of the detonation being the same in both cases, but the linear dimensions of the second charge being 8 times as great as those of the first. Then the rule states that if the pressure is p and the velocity of the water is u at a distance r from the first charge, at time ¢ after the detonation starts, the same pressure p and velocity wu will obtain at a distance Br in the corresponding direction from the second charge, at a time @¢t after the detonation starts. This rule can be applied to the shock wave provided that the range from the explosion is sufficiently short so that the dissipative or dispersive effects responsible for slowing the time of rise to maximum pressure (see Section 9.2.1) have not had an appreciable effect on the pressure-time curve. The applicability of the rule to the later oscillation of the bubble is more limited and will be discussed in Section 8.6. The physical basis of the rule will be taken up in Section 8.4.3. The phenomena which occur when a propellant charge is set off under water are similar to those just described for high explosives, with the important ex- ception that because of the comparatively slow burn- ing of the explosive, the pressure transmitted to the water builds up gradually over a period of time, and does not usually create a steep-fronted shock wave. Thus instead of the sort of pressure-distance graph shown in Figure 1, a propellant would give a graph more like Figure 2. The division of the disturbance | sounpary OF GAS BUBBLE PRESSURE (2) > yn RESIDUAL FLOW AROUND BUBBLE al OUTGOING PRESSURE WAVE | | WATER | | a DISTANCE FROM CENTER OF EXPLOSION Ficure 2. Pressure distribution in the water a short time after ignition of a propellant charge. into an outgoing pressure pulse and a residual bubble oscillation can usually still be made, but the propor- tion of the total energy which appears in the pressure pulse from a propellant is much smaller than that which appears in the shock wave from a high ex- plosive, and the maximum of the pressure is very much smaller.! The exact characteristics of the pres- sure pulse depend upon the rate of burning of the charge, which varies greatly depending on the type of propellant and the grain size. The following sections discuss in greater detail the previously mentioned features of the disturbance due to a high explosive. 8.3 SHOCK FRONTS The steep-fronted shock waves mentioned in Sec- tion 8.2 represent a form toward which all very in- tense pressure disturbances tend to develop. In this section and the following section we shall show why this is true and shall show that many of the charac- teristics of shock waves, such as the velocity of propagation of the shock front and the rate of dissi- pation of energy into heat, can be expressed as func- tions of the pressure jump, that is, the amount by which the pressure immediately behind the shock front exceeds the pressure in the undisturbed water in front of it. Characteristics of shock waves from 176 explosions which depend on other factors beside the pressure jump will be treated in Section 8.5. It isa familiar fact that the laws of acoustics are a limiting case of the laws of hydrodynamics for a com- pressible fluid and are valid only in the limit of very small amplitudes of pressure and velocity. According to these laws of acoustics, all pressure disturbances are transmitted as waves with velocity c = (dp/dp)', the derivative being understood, in the case of all ordinary fluids, to relate to the change of pressure p with density p in an adiabatic change. For the special case of a plane wave traveling in the positive x direc- tion, the pressure in the acoustic approximation is simply a function of (« — ct), where ¢ is the time; any such wave is thus propagated forward with velocity c without change of shape. For a disturbance of large amplitude this is no longer true. The shape of the wave will, in general, change as it progresses. The way in which the changes of shape take place can be calculated by a method of reasoning due to Riemann, the mathematical form of which will be given later in Section 8.4.1. Here we shall be content to give a simple qualitative explanation of Riemann’s ideas. P VELOCITY (c, + u,) S< VELOCITY (c.+u,) E x Figure 3. Development of a shock wave in one dimen- sion. Suppose we have a plane wave of the form shown in Figure 3A advancing in the positive z direction. Let the particle velocity at x = a be wm, and let that at x = 2 be ue. If we use acoustic theory as a first EXPLOSIONS as SOURCES OF SOUND approximation, we have m ~ p:/c, Ue ~ p2/c, where pi and p, are the pressures at x, and 2x2 respectively.* Now imagine an observer moving with thevelocity wu, so that to this observer the fluid at the point x is instantaneously at rest. Any small additional dis- turbance, such as the bump A, will seem to this ob- server to be propagated forward with the velocity c: = (dp/dp);_p,. Relative to the original system of reference, therefore, this bump will advance with velocity (c: + wm). Similarly the bump B will ad- vance with velocity (@ + uw). Now the fact that Di > pz, implies, as shown above for the acoustic approximation, that wu: > u,; moreover, the equa- tion of state of all ordinary fluids is such that this fact also implies ¢ > c. Thus (q + wm) > (@ + wm) and bump A advances faster than B, so that after a short interval of time the pressure pulse will have somewhat the form shown in Figure 3B. This il- lustrates Riemann’s result, that in an advancing wave the parts of higher amplitude move faster than those of lower amplitude. If continued long enough, this difference in velocity would cause the high pres- sure point A to overtake the low pressure B; how- ever, before this occurs, the curve of p against x will acquire a vertical, or nearly vertical, tangent at some intermediate point C, as shown in Figure 3C. In the neighborhood of this point the pressure gradient and velocity gradient will be very large, and it will no longer be permissible to neglect the effects of viscos- ity and heat conduction, which have been omitted from Riemann’s equations. It turns out that viscosity and heat conduction, by converting mechanical energy into heat, slow up the rate of advance of the high pressure regions, the amount of this slowing up becoming greater the larger the gradient of pressure and velocity. Thus a stage will eventually be reached, as in Figure 3D, where the steepness of the rise from A to Eis just sufficient to keep A from overtaking EF. This state of affairs constitutes a shock wave. Since in practice this limiting value of the time of rise is extremely short, the shock wavebeginswithan almost instantaneous rise of pressure. The importance of this phenomenon of Riemann’s in the understanding of explosive sound is not merely that it explains the origin of shock waves, which could simply be taken for granted, but that it also explains how the characteristics of the disturbance behind a shock front change with the time. This vari- ation will be discussed in Section 8.5. ® Throughout this chapter the symbol ~ will be used to denote “is approximately equal to.” NONLINEAR PRESSURE WAVES AND SHOCK FRONTS biel From what has been said previously it would ap- pear that any mathematical theory of the propaga- tion of a shock front would have to be based on hydrodynamical equations of sufficiently complicated form to include the effects of viscosity and heat con- duction. Fortunately, however, a very simple analysis based on the laws of conservation of mass, momen- tum, and energy suffices to determine the relation between pressure, particle velocity, temperature, and similar factors, just behind the shock front, and the velocity of propagation of the front. A detailed ap- plication of the laws of viscosity and heat conduction turns out to be necessary only if we are interested in phenomena in the shock front itself, that is, phenom- ena taking place in the very thin layer of water within which the abrupt rise of pressure takes place. The simple analysis just mentioned, due to Rankine and Hugoniot, will be described at length in Section 8.4.2. It will be shown there, among other things, that in ordinary fluids a negative shock is impossible, in other words, that a discontinuity in pressure and density can only be propagated toward the region of lower pressure, and that the velocity V with which a shock front advances, relative to the undisturbed fluid in front of it, is greater than the velocity of sound [the value of (dp/dp)*] in the undisturbed fluid ahead of the shock front. The thickness of the region within which the pres- sure rises from pp to pi is of course determined by the dissipative phenomena, namely, viscosity, heat con- duction, and any other sources of dissipation, such as bubbles, which may be present in sea water. A precise mathematical treatment of these factors would be difficult, but order-of-magnitude considerations to be given in Section 8.4.4 indicate that close to the ex- plosion this thickness should be very small; at a distance where the pressure jump is 100 atmospheres, for example, as is the case at a range of about 1 ft from a Number 8 detonating cap, the shock front should be less than 0.001 cm thick, perhaps much less. In this region the thickness of the shock front is a function only of the magnitude of the pressure jump and decreases as the latter increases. It might be supposed that the time of rise would be connected with the time required for the detonation wave to travel across the explosive charge, but this is not the case unless the charge is extremely elongated, since the Riemann ‘overtaking effect’’ will succeed in making the shock front vertical before the shock wave has advanced an appreciable distance from the charge. With increasing distance from the charge the pres- sure amplitude becomes small, and eventually the overtaking effect will become negligible in comparison with the dissipative processes which tend to smooth out the abrupt rise in pressure. In homogeneous sea water, however, the thickness of a shock front should remain quite small until it has traveled a considerable distance. Thus, for example, it can be shown that the thickness of the shock front at 50 yd from the ex- plosion of a detonating cap should probably be only a fraction of a centimeter, corresponding to a few microseconds or less for the time of rise of the pres- sure at a given point. Experimental information on the thickness of shock fronts, or equivalently the time of rise of the pressure at a given point, must be treated with cau- tion. The measured value of time of rise can easily be completely falsified by inadequate frequency response characteristics of the hydrophone and re- cording equipment; in particular, the finite size of the hydrophone seems to have rather more effect on the time of rise than one would at first suppose. A very careful series of experiments has been conducted at NRL? on shock waves from detonating caps con- taining about half a gram of high explosive, at ranges from 1 to 30 ft. In these experiments the time taken for the pressure in the shock wave to rise to its maximum value was measured under the best conditions as about 0.3 microsecond (usec) at all ranges, and since this was about the same as the estimated resolving time of the apparatus used, these experiments support the theoretical expectation of the preceding paragraph that the time of rise should be exceedingly minute. Other experiments supporting this expectation have been made at the Underwater Explosives Research Laboratory at Woods Hole,’ using 14-lb charges and ranges of the order of 10 ft; however, the resolving time of the apparatus for these experiments was only about 4 usec. Measurements made at ranges of the order of hundreds of feet, however, seem to show quite an appreciable time of rise;‘-* this effect, which is probably instrumental but may possibly be related in some way to oceano- graphic conditions, will be discussed at length in Section 9.2.1. 3.4 THEORY OF NONLINEAR PRESSURE WAVES AND SHOCK FRONTS The four following sections will be devoted to a mathematical discussion of some of the topics which 178 have been treated briefly in the preceding sections. Although this material is essential to a complete understanding of explosive phenomena, the con- tinuity of the chapter will not be impaired by omis- sion of these sections provided the reader is willing to accept on faith those results from them which have already been quoted. A more complete account of the theory of waves of finite amplitude and shock waves is given in a report issued by the Applied Mathematics Panel,’ and in textbooks on hydrodynamics.’ 8.4.1 Riemann’s Theory of Waves of Finite Amplitude In Sections 2.1.2 and 2.1.3 the equations of motion of a perfect fluid were derived on the assumption that the amplitude of the disturbance was small, so that the acceleration of a particle of the fluid could be ap- proximated by the partial derivative of the velocity with respect to time. Since we wish in this section to treat disturbances for which this approximation will not be valid, we must start from a more exact form of the equations of motion. As before, let x,y,z be three rectangular coordinates in space, ¢ the time, and uz,uU,,w, the three components of particle velocity in the fluid. As shown in Section 2.1.2, the z com- ponent of the acceleration of a particle of the fluid is OUn OUz i OUx a OUz ne OUz = Ux U, Uz ot at ox ” ay dz’ (1) and this, when multiplied by the density p, must equal f,, the x component of force per unit volume. According to Section 2.1.3, f, is related to the distri- bution of pressure p by Thus the exact equation of motion for the x com- ponent of oe is OUx Uz 1 dp at us = + we =i ule => — 9 Oz p 0x (2) and ae the same reasoning to the y and z com- ponents, we get the remaining equations of motion OUy OUy OUy OUy lop a SS 3 ye oe ie Free) oe Ou; Oz 1o + ars ak ty ae = eels (4) Oz p 02 Let us now consider a cae in which the pres- sure and velocity are functions only of x and ¢, inde- EXPLOSIONS AS SOURCES OF SOUND pendent of y and z. For such a disturbance the equa- tion of continuity [equation (2) of Section 2.1.1] becomes Op _ O(puz) 24 =0 5 are aaa ; (5) and equation (2) becomes OUz OUr lop i— = == =o 6 a 0 ae a ae (6) Riemann discovered that the two equations (5) and (6) could be put into a symmetrical and useful form by introducing the variable wor where pp is a reference value of oe density which is most conveniently chosen equal to the density of the undisturbed fluid. By using this equation and the abbreviation ¢c = (dp/dp)', equation (5) becomes (7) ie dp ow dp ov Ouz WS FER YR oe Oem ay ay dy du, dus = [— 3S Sp — 8 Pe ot we Ox Wits Ox Ox (8) and equation (6) becomes OUr OUr ldpdp oy + Ux ==: | meena EN Graes ot ax pdp dy dx ‘ (9) v = => . Ox Adding equations (8) and (9) gives C) : OW + Uz WE) tu $9 PE 0, C10) and subtracting equation (9) from equation (8) gives similarly a(y — uz) ay — uz) $e — (11) Equation (10) states that the quantity (J + uz) is propagated in the x direction with velocity (uw, + c) while equation (11) states that the quantity (W — uz) is propagated with velocity (u, — c), that is, since ordinarily c > uz, is propagated in the negative x direction with velocity (c — uz). These are Rie- mann’s results. The significance of these equations can be seen by considering a disturbance which is initially confined within the range a < x < 6b. For such a disturbance both (W + uz) and (W — uz) are initially zero below x = aand above x = b. The region in which (y + wz) = 0. NONLINEAR is different from zero will advance toward increasing x while the region in which (W — u,) differs from zero will recede in the opposite direction. Eventually these two regions will separate and leave between them an interval within which both (y + w,) and (Wy — uz) are zero so that the fluid is at rest at its normal density po. The original disturbance has thus been split up into two progressive waves traveling in opposite directions. In the wave which travels in the positive x direction (¥ + wz) is finite while (W — wz) is zero; in this wave, therefore, Y = u, and both the density and the particle velocity are propagated for- ward with the velocity (uz + c) = (W + c). Since for all ordinary fluids both y and ¢ increase with in- creasing pressure, this velocity of propagation will be greater the greater the pressure, and any disturbance will ultimately develop as shown in Figure 3. After the wave front becomes very steep, of course, the basic equation (2) or (6) is no longer valid and must be modified to take account of the effects of viscosity and heat conduction, which are negligible for waves of more gradual profile. Most practical applications, such as pressure waves produced by explosions, involve spherical waves di- verging from a source rather than plane waves of the type we have been discussing. It can be shown, how- ever, that spherical waves have properties very simi- lar to those just established for plane waves in that the high-pressure regions travel faster than the low- pressure regions and tend to overtake them. This overtaking effect becomes slower and slower as the wave advances farther from its source because of the decreasing amplitude of the disturbance due to spherical spreading. For this reason, a pressure pulse radiating from a small source has to be extremely intense if it is to develop a shock front by means of the overtaking effect; rough calculations? have shown that pressure pulses of the amplitudes ordinarily ob- tained from echo-ranging transducers will be only very slightly distorted by the overtaking effect and will not develop shock fronts.* However, in the case of transmissions at supersonic frequencies this slight distortion of the wave profile might be detectable by a receiver tuned to twice the original frequency. 8.4.2 The Rankine-Hugoniot Theory of Shock Fronts We have seen in the preceding sections and Figure 3 how any pressure wave of sufficiently large amplitude ultimately develops an extremely steep shock front PRESSURE WAVES AND SHOCK FRONTS 179 within which the motion of the fluid will be strongly influenced by factors such as viscosity and heat con- duction which do not appear in the equations of motion of perfect fluids. Certain characteristics of such a shock front can be predicted only by a theory which takes account of these additional factors ex- plicitly; one such characteristic, which will be dis- cussed in Section 8.4.4, is the thickness of the region within which the abrupt rise in pressure takes place. Fortunately, however, it was discovered by Rankine and Hugoniot in the last century that many valuable conclusions could be drawn merely by applying the laws of conservation of mass, momentum, and energy to the motion of the fluid, without bothering at all about the details of phenomena in the shock front. To show how this can be done, let us consider the mass of fluid contained in a flat cylinder having unit cross-sectional area and having its end planes parallel to, and respectively ahead of and behind, the shock front. A side view of such a cylinder is shown at a particular time ¢ by the full line ABCD in Figure 4, HIGH PRESSURE REGION UNDISTURBED FLUID Py Oy» Y Po» » U=0 Fieure 4. Cylinder of fluid traversed by a shock front. AB and CD being projections of the end faces. At a time dt later, the fluid which was originally in ABCD will occupy the cylinder A’B’CD, shown with a dotted boundary. Now if the pressure variation in the shock wave is similar to that shown in Figure 3D, the pressure, density, and other factors will change very rapidly in a very thin region near the plane SF, but will change much more gradually everywhere else. If this is the case, we can assume the thickness 180 AD of the cylinder to be very small but still very much greater than the thickness of the shock front, that is, of the region within which the rapid rise of pressure takes place. It will then be legitimate to treat the pressure, density, and velocity as having constant values py, pi, %4, over that part ABF'S of the cylinder which lies behind the shock front. Ahead of the shock front, of course, the fluid is at rest with undisturbed values po, po, of the pressure and density. Remembering that the cylinder has unit cross sec- tion, the mass of fluid in it may be written as a AS + poSD. If we let the boundary of the cylinder move with the water, this cannot change in the course of time. Now, after the interval dt, shown in the figure, the mass is piA’S’ + po S’D SD — S'D = Vat A'S’ — AS = (V — w)dt where V is the speed with which the shock front ad- vances, the two expressions given for the mass can be equal only if pi(V — wu) = po. (12) Similar equations can be derived by applying, in- stead of the law of conservation of mass, the laws of conservation of momentum or energy. Thus, the change in the momentum of the cylinder of Figure 4 in the time dt is and since and pita(V — u)dt and this must be equated to dt times the force (p1 — po) acting on the cylinder which gives pita(V — m1) = pi — Po. (13) For the energy equation, if we denote the internal energy per unit mass of the fluid in front of and be- hind the shock front respectively by « and e« and remember that the moving part of the fluid has kinetic energy Yu? per unit mass, we can write for the change in the total energy of the cylinder during the time dt Ww ale + “Vy — u)dt — poe Vat. This must be equal to the product of the pressure p; by the distance wat through which the rear boundary of the cylinder has been pushed. Thus, we get the final equation 2 ala + Vey =. U1) 7 poeoV = piu. (14) EXPLOSIONS AS SOURCES OF SOUND The three equations (12), (13), and (14), when augmented by known relations between the thermo- dynamic parameters of the fluid, can be shown to determine all the quantities pi,o1,u,V,e. in terms of any one of them, when pp and py are given. The equa- tions may be put in a more explicit form as follows. From equation (12), (p1 po) V. pi U = (15) Inserting this and equation (12) into equation (13), ae = PO vr = ih Sm pi V = eee = 2) (16) Po\ Pi — po whence, from equation (15), Fi |/ (pi = Bo)(o1 = po). «ap Pop1 Finally, if we insert the expression (12) into the first term of (14), and the expression (15) into the right- hand side, 2 u n¥(« =P Ta poV eo = poV whence, using equation (17), Pi(p1 — po) Pop1 Pi — Po ut CT COI aa Pop1 2 Pi — po (A= Bie = a) Dimer ones Se aS GN CRE, Pop1 2 Pop1 4 yl or €& — & = 3(P: + Pol — — —]- Po rat (18) In the discussion leading to these equations, we have spoken of the region behind the advancing shock front as the “high-pressure region,” although all the equations which have been written would still be valid if p; were less than j instead of greater. How- ever, it is easy to show from the energy equation (18) that in ordinary fluids a ‘“‘rarefactional shock wave,”’ that is, one for which pi < po, cannot exist. To prove this, consider the pressure-volume diagrams shown in Figure 5. The state of the undisturbed fluid of the shock front is represented by the point So. If the fluid were gradually and adiabatically compressed to den- sity pi, it would reach the state S;. Now, according to the second law of thermodynamics, any sudden com- pression to this density involving irreversible proc- esses must leave the fluid in a state which is hotter than S;,.in other words, since pressure normally in- NONLINEAR PRESSURE WAVES AND SHOCK FRONTS creases with increasing temperature, the point S, corresponding to the state of the fluid behind the shock front must lie above S}, as shown. That this is entirely consistent with equation (18) for a com- pressional shock wave can be seen from the upper half of Figure 5. The right side of equation (18) represents the area of the trapezoid under the straight line S)So, while the energy difference between S; and So is represented by the area under the adiabatic curve between these two points. If the adiabatic curve is concave upward, as it is for all normal fluids, the area under the trapezoid will exceed the area under the curve, and this is consistent with the known fact that S; has a higher temperature, hence a higher energy, than S;. For a rarefactional shock wave, on the other hand, the energy of S; is lower than that of Sp by an amount represented by the area under the adiabatic curve between these two points in the lower half of Figure 5, and since the area of the trapezoid is again greater than this, equation (18) could not be satisfied unless S; had less energy than S;. Thus, a rarefactional shock is impossible for a normal fluid,» as is indeed to be expected from the fact, proved in Section 8.4.1, that regions of high pressure advance faster than those of low pressure. It is instructive to compare the equations (16) and (17) with the corresponding relations of acoustical theory to which they reduce in the limit. To verify the latter statement we may note that for a disturb- ance of infinitesimal amplitude, equation (18) re- duces to the law of adiabatic compression, while equations (16) and (13) become respectively° dp\? Vw (2) =€ _ Pi — Po my \ pi(V — uu) pe For disturbances of large amplitude, however, it is and Uy b A British report* questions this conclusion on the basis of certain theoretical calculations and of some photographs of rarefactional waves. However, the computed example cited there of a “negative shock front” is merely a normal com- pressional shock when viewed in a system of reference in which the fluid ahead of it is at rest. Moreover, the rarefac- tional waves which have been photographed must be regarded as mere acoustic disturbances; the resolution of the photo- graphs is insufficient to distinguish a discontinuous shock front from a gradual pressure front which has a fairly appre- ciable thickness. ° Throughout this chapter the symbol ~ will be used to denote asymptotic equality; in other words, it implies that the quantity on the left equals the quantity on the right plus other terms whose ratio to the quantity written approaches zero in the limiting process being considered. 181 easily seen from the top half of Figure 5 that for all ordinary fluids the value of V given by equation (16) is greater than the value c@ of (dp/dp)’ in the undis- turbed fluid. For the quantity under the radical in equation (16) is just 1/p¢ times the slope of the straight line SySp while c is 1/p5 times the slope of the tangent to the adiabatic curve at So. By a similar argument it can be shown that for a fluid such as PRESSURE SPECIFIC VOLUME V/p PRESSURE RAREFACTIONAL SHOCK WAVE (IMPOSSIBLE FOR THE FLUID SHOWN) SPECIFIC VOLUME 1/p /p, Vip, Figure 5. Pressure-volume diagram of the changes occurring in a shock front. water, for which S; and S, are very close together (V — um) is less than c, the value of (dp/dp)? im- mediately behind the shock front. These results mean that small disturbances created behind the shock front may overtake it, but that no small disturbance can be propagated from the shock front into the un- disturbed fluid ahead. In fluids such as water and air, the dissipative phenomena taking place in the shock front gradually convert the mechanical energy of a shock wave into heat, causing the amplitude of the wave to decrease as it advances by an amount additional to the familiar decrease due to spherical divergence. A sufficiently intense shock wave traversing a high ex- plosive, however, can maintain itself indefinitely be- cause of the energy supplied by the chemical con- version of the explosive into gaseous products; such a shock wave would constitute a detonation wave. 182 EXPLOSIONS AS SOURCES OF SOUND 8.4.3 The Law of Similarity According to the theory outlined in Sections 8.4.1 and 8.4.2 the disturbance created in the water by an explosion is uniquely determined by: 1. The forces which the explosive gases exert on the water near them. 2. The Hugoniot equations (16), (17), and (18), which hold across the advancing shock front. 3. The equation of continuity [equation (2) of Section 2.1.1]. 4. The equations of motion (2), (3), and (4), which hold true to a very good approximation at all points of the water except points in the shock front. 5. The thermodynamic properties of the water, that is, the relations, such as the equation of state, be- tween pressure, density, and energy. Moreover, from what has been said previously concerning the nature of detonation waves, it is likely that the course of events within the explosive material itself is deter- mined by a similar set of equations, so that factor 1 can be derived from laws of the same type as 2, 3, 4, and 5. Now suppose a disturbance to be given which satisfies all these equations, and is described by = p(x,y,2,t) Pe p(x,Y,z,t) Ur od Uz(x,Y,2,t) Uy = Uy (x,y,z, t) Uz(2,Y,2,t) s | ~ N ll Then it can easily be verified by substitution that all the equations mentioned in 2, 3, and 4 outlined previously and the laws 5 as well, will be satisfied at all points except those in a thin layer at the shock front, by a disturbance described by p’,p’,uz,u,,u; where p'(x,y,2,t) = p(Bx,By,62,Bt) p (x,y,z, t) a (8x, By,Bz,Bt) Uz (x,Y,z,t) al uz(Bx,By,Bz,8t) etc., that is, by a disturbance identical with the first except that the distance and time scales have been changed by a factor B. A scale relationship of this sort may be expected to hold both in the explosive material and in the water, and if the linear dimension D’ of the explosive in the second case is made equal to 6 times the corresponding dimension D in the first case, the disturbances in both the explosive and the water can be scaled together. This law of similarity relating to disturbances pro- duced by different quantities of the same explosive has been fairly accurately verified experimentally at ranges for which the peak pressure in the shock wave is of the order of an atmosphere and above.*® Provided the shape of the explosive charge and the position at which the detonation is initiated are the same on the two scales, an appreciable departure from the simi- larity law could be caused only by failure of the equa- tions of motion (2), (3), and (4) to hold behind the shock front in the water, or by a failure of either the water or the explosive to have thermodynamic prop- erties independent of the scale of times involved. A phenomenon of the latter type might conceivably occur, for example, in an aluminized explosive, if the reaction of the grains of aluminum with the hot gases were so slow that the reaction occupied an appreciable part of the volume behind the detonation front. However, the fact that significant departures from the scaling law have not been observed at the ranges mentioned indicates that such phenomena are not serious. As for the possibility of failure of the basic assumptions to be fulfilled in the water, such a failure could be caused only by bubbles or other extraordi- nary dissipative mechanisms; as far as is known, the effect of these only becomes appreciable at very long ranges (see Section 9.2.1). Ordinary viscosity and heat conduction can be shown to have a negligible effect, on the scale used in experimental work. It should be remembered, of course, that the derivation we have given of the similarity rule does not apply in the very thin region of the shock front in which the abrupt rise of pressure takes place; as will be shown in the next section, the thickness of this region does not ordinarily scale proportionally to the factor 8 used previously. Theoretical Thickness of a Shock Front 8.4.4 We have seen in the preceding sections that it is for many purposes unnecessary to consider the de- tails of phenomena occurring in a shock front, and that many useful conclusions caa be drawn by think- ing of a shock front merely as a surface in the fluid across which the pressure and other quantities change discontinuously. However, these conclusions will be valid only if the thickness of the region in which the pressure rise takes place is very small; and to make our theoretical discussion complete we should verify that this is the case. Moreover, a study of the factors NONLINEAR PRESSURE WAVES AND influencing the thickness of a shock front can be valuable in that it may help us to evaluate and interpret experiments which purport to measure the time of rise of the pressure at the front of a shock wave. As has been explained previously, the Riemann overtaking effect tends to make any pressure pulse develop an infinitely steep front in the course of time, and this tendency can be counteracted only by factors neglected in the equations of motion of a perfect fluid, on which Riemann’s analysis is based. These factors, of which viscosity and heat conduction are the most obvious, must include the dissipative phe- nomena responsible for the fact that the internal energy of the fluid behind the shock front, as given by equation (18), is greater than that which the fluid would have if it were compressed reversibly and adiabatically to the density p;. This fact provides a clue which we can use to get a rough estimate of the thickness of the shock front. For, the amount of energy dissipated into heat per unit time by any given dissipative mechanism will be dependent on the thickness of the shock front, in other words, de- pendent on the rapidity with which the pressure changes from py to pi. This dissipated energy must be equal to the product of the mass of water crossing the shock front per unit time by the amount of energy dissipated per unit mass; this quantity being for all practical purposes simply the difference in energy between the states S, and S; in Figure 5. As will be shown later, the latter quantity can be calcu- lated from equation (18) in terms of the pressure Jump (p: — po) and the known properties of water; for small amplitudes it is proportional to the cube of the pressure jump. Since all reasonable dissipative phe- nomena create heat more rapidly the more suddenly they are made to take place, the greater the pressure jump the thinner the shock front must be in order to dissipate the required amount of energy. Thus if we can show that the shock front should be quite thin for a fairly weak shock wave, it must be even thinner for a strong one. We shall therefore begin by calculating the ap- proximate value of the dissipated energy for a weak shock wave. Referring to the upper diagram in Figure 5, we wish to calculate the energy difference (é: — e,) between the states S, and Sj. Since by equation (18) the difference (« — «) equals the area of the trapezoid under the line S)S, while the difference (€; — e) equals the area under the adiabatic curve from S; to So, we must have SHOCK FRONTS 183 (e«. — e, )= area of region between SSp and adia- batie curve = area of segment between S;Sp and adia- batic curve + area of triangle S,SoSi. Now the area of the triangle S{SoS; is equal to half the product of its altitude (1/p)9 — 1/p:) by its base (p: — p;). Since the latter quantity is in the limit of small amplitudes proportional to (« — «&), we can make the area of the triangle as small as we like com- pared to (e, — ¢,) by taking the amplitude of the shock wave sufficiently small. Thus, for sufficiently weak shock waves (a — «&) ~ area of segment between S;Sp and adia- batic curve stay res ~~!) Po Pl where the constant k is proportional to the curvature of the adiabatic curve and has the numerical value 1.5 X 101° in egs units for pure water. Let us now consider the mechanism by which dis- sipation of energy occurs in the shock front. For any assumed mechanism the rate of this dissipation can be calculated, at least roughly, as a function of the thick- ness of the shock front and the magnitude of the pressure jump. Of the two most obvious mechanisms, viscosity and heat conduction, the former gives much the greater dissipation, and accordingly we shall carry through the calculation only for the case of dissipa- tion by viscosity. According to the theory of viscous fluids, the mechanical energy converted into heat per unit time in a fluid having a coefficient of shear viscos- ity is given, for one-dimensional motion such as that in a plane wave traveling in the x direction, by (19) Dissipation per unit volume per unit time duz\" € = = — ) = 2y(-— 20 2u( Ox ) f p dt eo by the equation of continuity. If 6 is the thickness of the shock front — the thickness of the region within which most of the change in density from the value po to the value p; takes place — we have, roughly, for a weak shock wave, 1dp Me C(p1 — Po) | pdt pod Multiplying equation (20) by the thickness 6 gives a rough value for the dissipated energy per unit time per unit area of the shock front: Dissipation per unit area per unit time ie 2uc?(p1 — po)” (21) 5pp 184 EXPLOSIONS AS SOURCES OF SOUND This must be equated to the product of the expres- sion (19) by the mass of water which unit area of the shock front traverses in unit time, that is, since we are assuming the shock wave to be weak, by pic: ( - 2uc?(p1 — po)? pas = ==) 3 po pr 5po By solving this for 6 rs 2ucpipo ae 2ucpo ‘ k(p, — po) K(p, — po) With the value given above for k and the value uw = 0.01 cgs unit characteristic of pure water at room temperature, equation (22) becomes 2X 107 (e1 — po) (in gm/cc) # 4.5 X 108 (Pp: — po)(in dynes/cm?) (22) 6(in em) ~ (23) More refined calculations of this type have been made!! and indicate that, in fact, nearly all the calculated jump in pressure occurs within an interval of the thickness given by equation (23).8 If we are to regard the relations (22) and (23) as giving a valid estimate of the order of magnitude of the thickness of a shock front in sea water, we must assume three things: 1. That the Hugoniot equation (18) is sufficiently accurate. 2. That the curvature of the adiabatic for sea water is roughly the same as for pure water. 3. That the rate of dissipation of energy is of the same order as that due to shear viscosity. The last of these assumptions is known to be true for sinusoidal disturbances in pure water at fre- quencies from 1 to 100 mc.” Since, according to Sec- tion 5.2.2, the absorption in sea water seems to ap- proach that in pure water at frequencies near 1 me, it is reasonable to expect this assumption to hold in the sea for values of 6 between say 0.1 and 0.001 cm, and perhaps for much smaller values. The second assumption would fail if the water contained many bubbles, but calculations show that this would happen only for an unreasonably large concentration of bubbles. The Hugoniot equation depends for its validity on 6 being very small. Although no reliable calculation of its range of validity has been made, it is not hard to show that the equation (19) derived from the Hugoniot equation should be correct as to order of magnitude for explosive waves from ordinary sources when the pressure amplitude (pi: — po) is greater than about 100 atmospheres. Unfortunately, at 100 atmospheres amplitude the value of 6 given by equation (23) is 4.5 X 10—> cm, a value so small that it is conceivable that assumption (3) might fail. Thus, about all that can be concluded from the pre- ceding analysis is that with a typical explosive source the thickness of the shock front at a distance where the pressure amplitude is 100 atmospheres is probably not greater than about 0.001 cm and may be much smaller. At greater distances from the explosion, a probable upper bound to the thickness of the shock front can be set by neglecting the Riemann overtaking effect, which tends to make the shock front steeper, and imagining the pressure pulse to be propagated out- ward according to the laws of acoustics, subject to the same attenuating mechanisms as sinusoidal sound. By the method of Fourier analysis (see Sec- tion 9.2.4 and Figure 13 in Chapter 9) it can be esti- mated that to avoid inconsistency with the values given in Section 5.2.2 for the attenuation at high frequencies the thickness of the shock front should not exceed a fraction of a centimeter after it has traveled 50 yd through homogeneous sea water. A greater thickness could be produced only by in- homogeneity of the medium or by a highly nonlinear absorption, that is, by some mechanism which would be much more effective in absorbing energy from a disturbance of large amplitude than from a weak disturbance. 8.5 STRUCTURE AND DECAY OF SHOCK WAVES When a shock wave from an explosion passes a given point in the water, the initial behavior of the pressure as a function of time consists ordinarily in a roughly exponential dropping off, as shown schematically in Figure 1. The time required for the pressure to fall to 1/e times its value just behind the shock front is of the order of 15 usec for a Number 6 detonating cap and 600 usec for a 300-lb depth charge. This decay time depends somewhat upon the type of explosive being used, and it may depend slightly upon the range; for any given explosive it varies as the cube root of the charge weight, in accordance with the similarity rule given in Sections 8.2 and 8.4.3. After the pressure has fallen to {5 or 49 its peak value, however, the decay of pressure is much more gradual than would correspond to an exponen- tial law. Theories of the shock wave™ predict a “‘tail”’ STRUCTURE AND DECAY OF SHOCK WAVES 185 in which the pressure dies off with the time ¢ approxi- mately as ¢~‘/5, This law cannot of course hold in- definitely, since the momentum integral /pdt must be finite; the theory of bubble motion to be discussed in Section 8.6 predicts that the excess pressure should eventually go through zero and become weakly nega- tive as the gas bubble expands. For many purposes this tail is unimportant, but its contribution to the momentum integral may exceed that of the earlier part of the shock wave. Detailed experimental infor- mation on the tail is almost entirely lacking since spurious signals due to the impact of the shock wave on the cables leading to the pressure gauge usually mask the latter part of the tail. SSeS IN THOUSANDS OF ATMOSPHERES LoS! owe be) [a eda a Pos.r Pmax vo Vv IN FEET PER SECOND Figure 6. Peak pressure and speed of the shock front near a spherical charge of cast TNT. Under some conditions small secondary peaks or fluctuations may appear in the measured pressure- time curve. These may be due to any of a variety of causes. Sometimes the effect is spurious, being due to instrumental factors — for example, diffraction of the pressure pulse around the hydrophone and its supports or shock excitation of vibrations in the hydrophone.” Irregularities genuinely present in the explosive wave itself have been observed, however. Sometimes these occur under exceptional circum- stances, such as for shots fired on the bottom, for cylindrical charges detonated at one end rather than in the center,"'® for charges surrounded by an air pocket,'*"” and for charges which fail to detonate com- pletely because of inadequate boostering.’® More- over, even for spherical charges detonated at the center, the tail of the shock wave shows a small but reproducible hump or shoulder, whose magnitude de- pends upon the type of explosive. In accordance with the theory of Section 8.4.2, it is to be expected that the speed of advance of a shock wave at great distances from the explosion will be the normal speed of sound, but that at close distances the speed of advance will be considerably greater. Moreover, we should not be surprised to find other departures from the usual laws of acoustics. The upper diagram of Figure 6 gives some experimental values of the peak pressure in the shock wave as a function of the distance r from an explosion and shows fitted to these points a theoretical curve which, though only approximate, should give a reasonably reliable extrapolation of the peak pressure for smaller values of r.!° If the disturbance followed the ordinary laws of acoustics, the curve would be a horizontal line. The lower diagram of Figure 6 shows the velocity of the shock front as a function of distance from the charge, this curve being related to that of the upper diagram by equation (16) of Section 8.4.2. It will be seen that as r increases the pressure becomes more and more nearly inversely proportional to r, as it should be in the acoustic approximation. It has been shown theoretically, however, that even in a prac- tically ideal fluid the peak pressure in a shock wave is not asymptotically proportional to 1/r at large distances, but that instead Constant Pmax ™ 1 fowls LO) ry where 7 is a quantity of the same order as the initial radius 7 of the explosive charge.!® This deviation from acoustical laws is due to the dissipation of energy in the shock front. The relation (24) has been confirmed experimentally? at NRL for No. 6 detonat- ing caps at ranges of 1 and 31.3 ft. The ratio (TPmax)1 ft /(Tpmax)31.3 ft was found in these experi- ments to be 1.31 + 0.04, while the ratio E | 313 ft a) [tog | ft ua) (24) 186 is 1.32. It is not to be expected, however, that equa- tion (24) will hold true indefinitely as the range is in- creased, for its theoretical derivation assumes the rise in pressure at the shock front to be instantaneous, and neglects any dissipation of energy behind the shock front. At large ranges neither of these assump- tions is valid, and we should expect the decrease of pressure with distance to obey a law similar to the attenuation of sinusoidal sound (see Sections 2.5 and 9.2.1). The theory just mentioned also predicts that the duration of the pressure in a shock wave should slowly but continually increase as the range increases. This effect is due to the fact that, according to Rie- mann’s theory, the more intense front part of the wave should travel faster than the less intense tail. Specifically, the theory predicts that if at large values of the distance r from the charge the pressure in the wave is approximated by an exponential PD = Pmaxe ’, then the duration parameter 6 should be given by O~ constant| log | (25) rT where as before 7; is of the order of the radius 79 of the explosive charge. The experimental verification of this relation is less conclusive than for the preceding relation (24). Although a decided decrease in @ has been observed when r is decreased below a value corresponding t0 Pmax = 1,000 atmospheres,” the ex- periments of reference 2, which covered a range from about 3 to 80 atmospheres, showed no measurable increase of duration; yet an increase of the amount given by the relation (25) should have been measur- able. Under most conditions, given complete detona- tion, the peak pressures and pressure-time curves ob- tained at a given range from a charge of a given size are quite accurately reproducible from shot to shot, the deviations of individual peak pressures from the mean being of the order of 2 per cent. However, for asymmetrical charges, such as long cylinders with detonation initiated at one end, both the peak pres- sure and the duration of the shock wave, when measured at a given distance, are different in dif- ferent directions. Experiments on long cylindrical charges have shown that the peak pressure is greatest approximately at right angles to the axis of the cylinder and is least on the axis off the cap end. Dif- ferences as large as 40 per cent have been observed.!® The duration of the shock wave varies in the op- EXPLOSJONS AS SOURCES OF SOUND posite sense from the peak pressure, and in fact the impulse / pdt contained in the early part of the shock wave is slightly greater off the cap end, where the peak pressure is least, than in any other direction. As one would expect, charges fired with an air cavity on one side also show an anisotropy. When a charge is fired on the sea bottom, the peak pressure received at a point above the charge is somewhat greater than it would be in the absence of the bottom. For sand and gravel bottoms this in- crease in peak pressure has been found to be 10 to 15 per cent.'* In directions near the horizontal the amplitude tends to become smaller, as one would expect from the shadowing effect of irregularities on the bottom. The pressure-time curves from shots fired on the bottom are not only likely to be rather irregular, as mentioned previously, but tend to be less consistent from shot to shot than is the case for explosions in free water. SECONDARY PRESSURE WAVES In Section 8.2 it was mentioned that because of the inertia of the water which has been pushed radially outward by an explosion, the gas bubble undergoes radial oscillations. Many features of this oscillatory motion can be explained by a very simple theory which treats the water as an incompressible fluid and the radial flow as spherically symmetrical.” Let u(r,t) be the radial velocity of the water, measured positive outward at distance r from the center and at time #. Tbe volume of water which passes outward in unit time across the surface of a sphere of radius r is 4rr*u. At any given instant the flux across any two concentric spheres must be the same, since otherwise the amount of water in the shell between these two spheres would be increasing or decreasing. We must therefore have 8.6 4zr?u = function of ¢, independent of r. If 7(@) is the radius of the gas bubble at time ¢, we must have dr, 4 u(r,t) = ae = 7 and so u(r,t) = ape : (26) r We are now ready to apply the principle of con- servation of energy. The energy of the water and gas bubble consists of three parts, kinetic energy, po- tential energy due to compression of the gas in the bubble, and potential energy representing work done SECONDARY PRESSURE WAVES 187 C} ieee nO nO eck cao gon ere at gee INCHES RADIUS OF BUBBLE rp ° fs} 8 ° = no uw b a a aS eS FEET fo} = nN Gl + oO FEET TIME IN MILLISEC FiGuRe 7. sure of gas in bubble to be given by Ideal radius-time curve for the gas bubble from an underwater explosion. Full curve computed assuming pres- 4 p = 0.064 (=) atmospheres. 6 Dashed curve computed assuming pressure of gas in bubble to be zero. Scales: (a) No. 8 cap at 50-foot depth; (b) 1 lb TNT at 50-foot depth; (c) 300 lb TNT at 50-foot depth. against the surrounding hydrostatic pressure, which we shall denote by p., since it represents the pressure in the water at a great distance from the bubble at the same level. Consider the kinetic energy first; since the mass of the gas in the bubble is negligible, practically all the kinetic energy resides in the water, and the amount per unit volume is %pu?. The total kinetic energy is thus f Apu?-4ar2dr = Qmaprérz (27) T by equation (26). The potential energy of the gas is a function of its volume, and can be represented by a function G(r); it is a negligible fraction of the total energy except when the radius of the bubble is small. The work which has been done against the external pressure is represented by p.times the volume of the bubble, or (4/3)rpar3. The sum of these three terms must be constant in time in the approximation we are using Qmprer? + srP att + G(r) = W. (28) The behavior of r, as a function of the time is thus determined by solving equation (28) for 7 and integrating. The result can be expressed in a form which is independent of the amount of explosive in- volved, manifesting a similarity rule of the same form as that given in Section 8.2 for shock wave pressures. For if, as before, we let ro be the radius (or equivalent linear dimension) of the original charge of explosive, G will be 73 times a function of the ratio 7/70, and W, which represents that part of the origi- nal energy which is not dissipated or carried away by the shock wave, will be proportional to 73. Using these facts,.it can be seen from equation (28) that 188 the solution obtained for one quantity of explosive will be valid for any other quantity if the scales of 7, and t are changed in proportion to 79. The full curve of Figure 7, taken from a report issued by the David Taylor Model Basin,” shows the form of the radius- time curve obtained from integration of equation (28), using a function G(r,) comparable with that which would obtain for a charge of high explosive; while the dotted curve is the one which would result from equation (28) assuming the same maximum value of 7, but setting G = 0. Several scales are given appropriate to several sizes of charge. The variation of 7 with ¢ near the minimum of the contraction is too rapid to show on the scale of the figure. It would not be worth while to show this portion in greater detail, however, since, as will be explained presently, the motion of the bubble in this stage is strongly in- fluenced by gravity and other asymmetrical factors, and also, although less strongly, by the finite com- pressibility of the water. These effects prevent the present simple theory from being even approximately correct near the minimum. When 7 is greater than three or four times the minimum radius shown in Figure 7, G(r,) becomes small and the motion is practically the same as it would be if there were no gas in the bubble at all, as can be seen from the agreement of the dotted curve with the full one. For this portion of the curve a change in pq is equivalent to a change in W combined with a change in time scale; thus, over most of the period of the oscillation Figure 7 applies not only to charges of different sizes, but to charges at different depths, provided suitable time and radius scales are used; these scales can be deduced from equation (29) below. The period of the motion, in the approximation neglecting gas pressure, is easily deduced from equa- tion (28) and turns out to be fo = 1.135p'pa'W* = 1.829p'p5'7s max ( 3W ) qT max = ATD and is the radius of the bubble at its maximum size. This expression, in spite of its neglect of gas pressure and the other effects to be discussed later, has been found to agree with measured values of the period to within a few per cent, provided the explosion takes place in open water well away from bounding sur- faces, which exert a perturbing effect discussed later. With the same proviso, the variation of the period with depth and size of charge agrees with the ex- ponents in equation (29) to within the accuracy of (29) where EXPLOSIONS AS SOURCES OF SOUND measurement. The measured values of the bubble period are found to be reproducible from shot to shot to within a per cent or less.?*.4 The simple theory just outlined ignores the com- pressibility of the water and any influences which may make the motion asymmetrical. The compressi- bility of the water is important near the minimum of the contraction, when the pressure in the bubble is very high. During this stage of the motion the compression of the water near the bubble initiates a pressure wave which travels outward as an acoustic pulse. This is known as a “secondary pulse’ or “bubble pulse,” to distinguish it from the original shock wave. The acoustic energy carried away by this secondary pulse is usually small but may, under exceptionally symmetrical conditions, be apprecia- ble compared with the total energy W of the oscilla- tion. Loss of energy through this effect and through turbulence has the consequence that the next oscilla- tion, although conforming generally to the theory of the preceding paragraph, is of lower amplitude than the first one, since it has an energy W, which is smaller than W. Energy is radiated in a similar man- ner at each succeeding contraction. Several causes may act to prevent the motion from having true spherical symmetry. Most important of these, because it is always present, is gravity. The bubble, being buoyant, tends to rise; the rate of rise is limited by the inertia of the water around it. The rise is usually rather slight during the first expansion of the bubble, but as the bubble contracts again the rise is enormously accelerated and may result in a large portion of the total energy W being retained as kinetic energy in the water at the time the radius of the bubble is a minimum; since this energy is not available to compress the gas, the minimum radius of the bubble will not be as small as it would be if gravity were absent, and the secondary pressure pulse will be correspondingly weaker. Another effect of the rapid rise is to produce turbulence in the con- tracted stages; this turbulence dissipates energy and is probably the most important factor in the damping out of the oscillations. This rapid rise in the contracted stages has been the object of many theoretical studies.22°—*? The explanation of the phenomenon rests on the fact that a spherical cavity moving through a fluid possesses an “effective inertia’”’ equal to half the mass of the water it displaces. The buoyancy of the gas bubble causes it to acquire an ever-increasing amount of vertical] momentum, and to conserve this momentum SECONDARY PRESSURE during the contraction, when the effective mass is greatly decreased, the velocity of rise must increase. Besides gravity, other effects such as proximity to the free surface of the water or to the bottom can cause departures from spherical symmetry. The ef- fects of such surfaces become appreciable when the distance from the bubble to the boundary surface is a few times the maximum radius of the bubble, and are of two sorts. In the first place, the period of the oscil- lation is shortened by proximity to the free surface and lengthened by proximity to an unyielding sur- face; secondly, a free surface repels the bubble and a rigid surface attracts it. This translational motion, which becomes very rapid in the contracted stage, weakens the secondary pressure pulse for the same reason that the rise due to gravity does. Much theoretical work has been done on the period and migration of the bubble,?”? and the results are in generally good agreement with experiment.” If gravity can be ignored, asymmetrical motion due to proximity to free or rigid surfaces obeys the scaling law of Section 8.2, the distance from the surface being changed in the same ratio as other linear di- mensions. The gravity effect, however, does not scale in the same manner; gravity is relatively more im- portant the larger the charge and the smaller the external hydrostatic pressure p.. Most features of the motion as affected by gravity can be approxi- mately expressed in a form which is independent of the size of the charge, by using a unit of length A = (W/gp)*, a unit of time V A/ g, and a unit of pressure pg A.” From the foregoing it can be seen that the form and strength of the secondary pressure pulses depend greatly on gravity and on proximity of surface, bot- tom, or objects to the explosion. The peak pressure in the first bubble pulse, for example, has been measured at values as large as 0.25, and as low as 0.06, times the peak pressure in the shock wave.?3242829 By contrast, the impulse / pdt contained in any one of the secondary pulses is not very sensitive to these factors. The amount of impulse contained within a few half-widths of the main pressure peak is of the same order as that in a corresponding portion of the shock wave; however, just as was the case with the shock wave, this impulse is considerably less than the amount contained in the “‘tails,’’ which in the case of the secondary pulses extend to both directions in time. The total impulse in a secondary pulse is probably roughly equal to the amount which would be calculated from the simple theory which assumes WAVES 189 the water to be incompressible. It can be shown ?! that this impulse is 1/6,.2 Z Tomax a ae [= aE « V pp. (30) 250 GRAMS OF TETRYL 10 FEET DEEP 32 INCHES FROM GAUGE IN ATMOSPHERES iS) IN ATMOPHERES PRESSURE IN ATMOSPHERES PRESSURE (e) Ds o 300 GRAMS OF ay, 7 FEET DEEP 4 FEET FROM GAUGE PRESSURE oO 300-LB DEPTH CHARGE 80 FEET DEEP ABOUT 100 FEET FROM GAUGE PRESSURE IN ATMOSPHERES TIME IN MILLISEC Figure 8. Typical pressure-time records for the first bubble pulse. At all ordinary depths this is five or ten times as great as the impulse in the exponential part of the shock wave; however, one would expect from theory that the impulse in the tail of the shock wave would 190 EXPLOSIONS AS SOURCES OF SOUND be about half of the quantity (80). In between the end of the tail of the shock wave and the beginning of the tail of the first bubble pulse there is a long period during which the pressure is below normal; this period occupies most of the time consumed by the first oscillation, and the negative impulse delivered during it is just equal to the expression (30). For most ap- plications, however, this negative pressure and the tail parts of the shock and bubble pulses can be neglected. In cases where migration of the bubble is slight, the bubble pulses show a fairly regular rise and fall of pressure. When migration is rapid, irregularities are more apt to occur, and sometimes two or more fairly SURFACE OF SEA x DISTANCE ESH= EH=r, DISTANCE EH = fF, Ficure 9. Superposition of direct and surface- reflected pulses. well-separated peaks are observed.” It has been sug- gested that these multiple impulses may be due to breaking of the bubble into several separate bubbles, which emit distinct pressure peaks in the contracted stage, but which coalesce when they expand again. A few typical oscillograms of bubble pulses, taken from references 23, 24, and 29, are shown in Figure 8. The first bubble pulse is usually by far the strongest; for small charges as many as eight or ten pulses have been counted, but for large charges usually only two or three bubble pulses are measurable. For some reason, a charge fired on or very close to the bottom usually gives a very weak bubble pulse, and the number of measurable pulses is less than for shots in open water. A caution should be added concerning the inter- pretation of oscillograms of bubble pulses; because of the relatively long duration of these pulses the nega- tive pulse reflected from the free surface of the water will often overlap the direct pulse, making the re- corded pressure appreciably different from that due to the direct pulse alone. The statements given previously apply to the latter only. 8.7 SURFACE REFLECTION AND CAVITATION When a hydrophone is placed in the water at some distance from an explosive charge, the shock wave and secondary pulses received are modified by reflec- tion at the free surface of the water. This reflection is most conveniently described by the principle of images, which we have encountered in another ap- plication in Section 2.6.3. This principle is applicable whenever the pressure amplitudes are small enough for the laws of ordinary acoustics to apply. Referring to Figure 9, the pressure produced at any instant at EXCESS PRESSURE PRESSURE -TIME CURVE WHICH WOULD BE OBSERVED IF WATER DID NOT CAVITATE PRESSURE- TIME CURVE AS MODIFIED BY CAVITATION Ficure 10. Modification of a surface-reflected pulse by cavitation. a hydrophone H by an explosion or other source of sound at H is the sum of the pressure due at that instant to the direct wave HH and the pressure due at the same instant to the reflected wave HSH. Ac- cording to the image principle, the latter pressure is exactly equal to the negative of the pressure which would be produced in the absence of a surface by a source EH’ which is the mirror image of E in the sur- face and which has the same time variation. If z, is the depth of the explosion, z, the depth of the hydrophone, and x the horizontal range, the path difference between these two waves is Veta +e—- Ve — a) +2 42.2, a 31 m1 + 1 eh) When the distance HS from the explosion to the portion of the surface at which the reflection takes place is so small that the incident pressure at S is ap- preciably in excess of one atmosphere, the simple i = TL SURFACE REFLECTION AND CAVITATION image law just stated has to be modified; for, sea water is apparently incapable of sustaining a tension of any appreciable magnitude, and cavitation will therefore set in at any point where the pressure be- comes negative. At short ranges, therefore, the pres- sure-time curve for a shock wave and its reflection usually looks like the full line in Figure 10, instead of following the dotted curve as it would if there were no cavitation. 191 Experiments on explosion waves in sea water* strongly suggest that the water begins to cavitate as soon as the pressure becomes negative, and that the cavitation can develop sufficiently rapidly to prevent negative pressures from persisting even as long as 10 usec. This is to be expected if there are even a few tiny bubbles in the water. A theory of the propaga- tion of cavitation fronts has been given by Ken- nard.31,32 Chapter 9 TRANSMISSION OF EXPLOSIVE SOUND IN THE SEA 9.1 INTRODUCTION Fees ESTABLISHED in Chapter 8 the nature of explosions as sources of sound, we are ready to consider the results of experiments showing how pulses of explosive sound are affected by transmission through moderate and long distances in the ocean. The amount of experimental data available in this field is scanty by comparison with that which has been accumulated on sinusoidal sound, and since ac- curate recording of explosive pulses is possible only if very careful precautions are taken, one must be cautious in drawing conclusions from this work. Nevertheless, these experiments demonstrate strik- ingly the utility of explosive sound as an aid to funda- mental research on the nature of the ocean as an acoustic medium. Whereas in experiments using long pulses of sinusoidal sound the signal received at the hydrophone is usually inextricably compounded out of directly transmitted sound, scattered sound, and sound reflected from the surface or bottom, the ex- tremely short duration of explosive pulses makes it possible, in many cases, to distinguish between the contributions of these different mechanisms by virtue of the differences in time of arrival. Another charac- teristic difference between explosive and sinusoidal sound is that dispersion effects, which depend upon the phases of the various component frequencies in the arriving sound, can easily be studied with a transient disturbance, but are practically impossible to measure with single-frequency sound. The disper- sion accompanying the transmission of sound through sea water alone, although doubtless present, is very minute, and has never been detected; in shallow- water transmission, on the other hand, dispersion phenomena are important and can be made to give useful information about the bottom. Other ad- vantages of explosive sound which are significant in certain types of experiments include the high in- tensity attainable and the fact that explosive sources are relatively easy to manipulate and can be fired at great depths. 192 The experiments to be discussed in this chapter shed light on a variety of problems of sound propa- gation in the ocean. For example, the variation from shot to shot in the sound intensity received at a dis- tance is found in Section 9.2.5 to be much smaller than that which is observed for sinusoidal sound, especially when the path of the sound lies entirely in isothermal water. This suggests that most of the variation observed with sinusoidal sound is due to some sort of interference phenomenon. Another ex- ample is provided by the estimates of attenuation of low-frequency sound, or at least of an upper bound to it, given in Section 9.3.2; these estimates are made on the basis of experiments which include ranges up to hundreds of miles. Other interesting results of ex- periments with explosive sound up to the present in- clude the occurrence of a reflection coefficient near unity for the free surface of the ocean at directions of incidence surprisingly close to the horizontal (Sec- tion 9.2.1), the comparison of observed intensities with the predictions of ray theory (Section 9.2.2), the occurrence of diffraction (Section 9.2.3), the deduc- tion of details of the bottom strata from shallow- water experiments (Section 9.4), ete. Before discussing the experimental material in de- tail it will be worth while to say a few words regard- ing the technique of measuring and recording ex- plosive sounds. A systematic discussion of experi- mental techniques would be beyond the scope of this volume; but to enable the reader to form a balanced opinion on past and future experiments with ex- plosive sound, some of the pitfalls and stumbling blocks in this field should be pointed out. In the first place, if actual pressure-time curves are to be re- corded, special attention has to be given to uni- formity of the response of the hydrophone and re- cording circuit, both in amplitude and in phase, over a wide range of frequencies. Because of the very short time scale when small charges are used, trouble may be caused by the finite time of transit of the sound wave across the hydrophone, and by diffrac- tion around the hydrophone and its supports. Changes SHORT-RANGE PROPAGATION in the orientation of the hydrophone sometimes have a surprisingly large effect on the form of the recorded pressure-time curve. Tiny quantities of gas occluded on the face of the hydrophone or included in water- proofing or insulating materials can slow up the response to a steep-fronted pulse, and make the be- havior of the hydrophone nonlinear. Natural reso- nances in the hydrophone can be shock-excited by a steep-fronted pulse, causing spurious wiggles in the pressure-time curve, and in some cases making the pulse appear to last many times longer than it actually does. At short ranges, where relatively in- sensitive hydrophones may be used, emf’s due to the impact of the pressure wave on the connecting cable may give spurious signals. With long cables, im- pedance matching and dielectric losses may have to be considered. These and many other points are dis- cussed at length in other reports. In the following sections we shall first consider propagation of explosive pulses through the water alone, and later, in Section 9.4, shall take up pulses reflected from or transmitted through the bottom. 9.2 SHORT-RANGE PROPAGATION IN DEEP WATER Attenuation and Change in Form of the Pulse 9.2.1 As the earlier chapters of this volume have shown, the most important single factor affecting the shape and strength of a sound pulse of given frequency traveling through the ocean is the variation of the velocity of sound from point to point, due chiefly to temperature gradients but produced also to some extent by pressure and salinity gradients. Since to a first approximation the velocity of sound is a func- tion simply of the depth and to this approximation can be calculated from bathythermograph records, it will be convenient to separate, as far as possible, those features of explosive sound propagation which are due to this variation of velocity with depth from those features which are due to other properties of sea water and which would be encountered even when the bathythermograph record indicates no ap- preciable refraction. In this section we shall consider the latter features, recognizing, however, that unde- tected small-scale fluctuations in the velocity of sound may possibly be an important factor in ac- counting for them. One of the most interesting features to be found in IN DEEP WATER 193 the measurements of explosive pulses at ranges from 30 to 2,000 yd is that with increasing range there is an increase in the time required for the pressure in the initial pulse to rise to its peak value. At shorter ranges, it will be remembered, this initial pulse is a shock wave and its time of rise is less than the re- solving time of any measuring apparatus which has been used (see Section 8.3). Unfortunately, the meas- urements which have been made of the time of rise are not sufficiently detailed to establish the cause of this variation with range. Table 1 summarizes the experimental information to date; this information was taken from two NDRC reports.7® In this table “time of rise” is defined as the interval between the first measurable increase of pressure and the maxi- mum of the pressure-time curve. “Resolving time” is defined as the value of time of rise which the system would record for an instantaneous rise in pressure in the water. For the first set of observations this time was measured directly from records of shots at close range; for the other two sets it was merely estimated from acoustical and electrical characteristics of the hydrophone and circuit. The data given in Table 1 have been chosen to exclude any cases where the hydrophone was in or near the shadow zone predicted from bathythermo- graph data. They therefore presumably represent an effect which occurs in the absence of large-scale re- fraction, although it is not impossible that through inaccuracy of the computed ray diagrams some of the shots at the longer ranges may have been close enough to the shadow zone to increase the time of rise by virtue of the shadow-zone effect discussed in Section 9.2.3 and shown in Figure 9. The deep-water data of reference 8, which are given in the table, are plotted in Figure 9 of that section, for comparison with similar time-of-rise data taken in the shadow zone. It is worth noting that in these experiments no marked dependence of time of rise on the depth of the explosion was found for those cases where the hydrophone was not in or near the shadow zone; a slight, increase in time of rise with increasing depth was observed, but this was not significantly greater than the experimental error. Slightly more than half of the observations of refer- ence 8 fell within +2 ysec of the means given in Table 1. For the deep-water shots off San Diego, the varia- tion of apparent time of rise with range is approxi- mately what one would expect if the resolving time of the apparatus were actually about 10 psec and if the SEA TRANSMISSION OF EXPLOSIVE SOUND IN THE 194 SeAINO j-d Jo SOI}Sl19}081BYO peurejdxeun jo L alt ¢g-Ol O&¢ ” ” Lt CINCO) [EEE 9 él g¢c-Ol 98 ” ” G Petepisuces) Gye p 9 al o-06 GLG ” ” 9 (sq) 8 s0Ud1IJOY g él S€-0O1 a ” ” L CANOES EXT ¢ Joqiey] osaIq, weg 1X6 oI-IL 00-02 098‘T-008'T ” ” Il IZ éI-0l 00F—-06 00F‘T-00T‘T ” ” vs quesaid Aqyensn ST GI-It 007-02 006-002 ” ” 6 COLUM) MEU eI #S-1T 008-01 ¥9-01¢ oo 1¢ quelpuss oinye ral ral OF 0ze * 5 z -Iaduie} dAtyeda Ny 8 v0Ua18Jayy II IT 00T—-0z OFZ 9 g @ 10 [ Ajyensn Gc) 0221 IT or 06 €ET-061 ded 8 ON 8 9784S BIS ¢ UBg Jo ay¥M daaq 09-0¢ LOT » » OOF 0g Or 06 L9E INL 4-6 I 0L-09 Ol 06 eI ” I osnes Jo ssuodsa1 resulfuou Aq peoueny [41903 (eduvs 410Ys -ul A[qissod sjjnse1 9so00| ye SqQ) 2 aouerejayy 0¢ Or 06 && wd 00E if (doqUIA.) UMOUZU/) &1 TOqIVE] POH SPOOM spuodeso101UL 309} Ul yao} ul spied ul asreyy 8}0Ys SUOI}IPUOD 1098 A spuodeso.101Ur S}USUIUIOD Ul OSI auoydoipAy ad1eyo asuey jo ON ul wia3sAs pus jo ourly jo yydeq jo yydeq suIpso0va1 CHEV EN poinsveur jo oully ‘u01jzv00T asvIBAV SUTA[OSOY ‘saduvs snowea ye ‘uotsojdxe ue wou asnd [erztut oY} UI eINssaid 94} JO OSII JO saUIT} pornsvapy ‘| Wavy, -RANGE PROPAGATION IN DEEP WATER CAP AT. 6O FT R - i i dapat) on on oe Figure 1. Shock wave pulses received via direct and surface-reflected}paths. Source: No. 8 blasting cap at depths indicated. Hydrophone at depth 11 feet and range 1,100 yards. Date: Feb. .26, 1942. 196 TRANSMISSION OF EXPLOSIVE SOUND IN THE SEA explosive pulse is subject to the same frequency-de- pendent attenuation law as has been observed for supersonic sound (see Section 5.2.2). A more precise comparison of theory with observation would, how- ever, require knowledge not only of the exact response of the hydrophone and recording system to a dis- continuous change of pressure, but also of the slight dependence of the sound velocity on frequency. The WHOI data, on the other hand, are much less under- standable. The increase in recorded time of rise from 13 psec at short ranges to 50 usec at 100 ft implies an attenuation of the high-frequency components of the explosive pulse, which is orders of magnitude greater than that encountered for supersonic sound. The comparative slowness of the increase in time of rise at greater ranges shows that this attenuation is non- linear; and the most plausible suggestion seems to be that it has its origin in the coating of the hydrophone, rather than in the sea water (see Section 8.3). So far we have considered only the direct pulse. The surface-reflected pulse may be expected to have a longer time of rise, or more correctly, time of fall, because of the diversity of possible paths from ex- plosion to hydrophone involving reflection from vari- ous wave troughs. Whether the smearing out of the wave due to this effect exceeds that responsible for the time of rise of the direct wave will of course de- pend upon the geometry and especially upon the roughness of the sea. Some typical oscillograms of shots made in an average calm sea are shown in Figure 1, taken from a report by UCDWR?® and from reference 8. These shots show that the rough- ness of the surface has surprisingly little effect on the first part of the reflected pulse, although the tail shows irregularities which are probably due in part to nonspecular reflection. The most remarkable thing about these records, however, is that as grazing inci- dence is approached the effective reflection coefficient of the surface remains close to unity far beyond the point at which the crests of the waves are in the geometric shadow created by the troughs. Figure 2 shows the variation of the reflected amplitude with angle of incidence. This quantity was estimated for a number of shots, including those shown in Figure 1, by taking the difference between the reflected peak and the estimated pressure in the direct pulse at the same time. For comparison, it may be noted that for a train of surface waves traveling in the direction from source to receiver, half of the surface of the sea in the region where reflection occurs would be in geometric shadow from source or receiver, or both, if the ratio of crest-to-trough amplitude to wave- length is about one-third the angle which the incident ray makes with the horizontal. Since the sea was not unusually calm on the day the shots were made, Figure 2 strongly suggests that the sea surface acts as a flat reflecting plane for supersonic sound even when only the wave troughs are in the direct sound field. An effect of this sort has been predicted theoretically for the case of sinusoidal sound. REFLECTED AMPLITUDE DIRECT AMPLITUDE ANGLE BETWEEN INCIDENT RAY AND HORIZONTAL IN RADIANS Figure 2. Variation of peak amplitude of surface- reflected pulse with angle of incidence. Horizontal range, 1,100 yds; depth of hydrophone, 11 ft. Measurements of the time interval between the direct’ and surface-reflected pulses have been re- ported in a memorandum by UCDWR" and in reference 9, and agree with the values calculated from the geometrical formula (31) of Section 8.7 to within the accuracy of measurement of the depths of cap and receiver. In the absence of refraction, one would expect the peak pressure of an explosive pulse having a time constant 6 to be attenuated at long ranges at approxi- mately the same rate as sinusoidal sound of a suitably chosen frequency, this ‘‘effective frequency”’ being probably a few times smaller than 1/6. A detailed relationship between the attenuation of a weak ex- plosive pulse and that of sinusoidal sound could be worked out by the methods of Fourier analysis; how- ever, such a relationship would be considerably af- fected by dispersion, that is, by any variation of the velocity of sound with frequency. This is a phenome- non which must occur if there is a frequency-de- pendent attenuation. A brief discussion of attenua- SHORT-RANGE PROPAGATION tion data from the standpoint of Fourier analysis will be given in Section 9.2.4. In comparing the attenua- tion of explosive sound with that of sinusoidal sound, however, it must be kept in mind that the mechanism responsible for attenuation of the peak pressure in the initial pulse from an explosion is somewhat different at short and long ranges. At long ranges one may ex- pect that linear absorption and dispersion will ac- count for the decay and change of form of the pulse; while at short ranges, as explained in Section 8.3 to 8.5, the nonlinear Riemann overtaking effect plays the predominant role, causing the time constant of the pulse to increase with time and causing an at- tenuation of the peak pressure whose magnitude is independent of the specific mechanism responsible for the dissipation of energy. The range at which the transition from the latter type of attenuation to the former takes place is probably roughly the range at which the attenuation given by the short-range law (24) of Section 8.5 equals the attenuation computed by Fourier analysis from the linear laws of absorp- tion and dispersion which hold for weak sinusoidal sound. Observations on the variation of peak pressure with range do not suffice to determine the magnitude of the attenuation of this quantity, or even to establish that it is different from zero. For example, the data of Figures 3 and 4 of Section 9.2, taken from UCDWR experiments cited in reference 8, are in good agree- ment with intensity calculations which ignore at- tenuation. It would hardly be reasonable, however, to assume that there was practically no attenuation in these experiments, since the increase of time of rise with increasing range indicates that dissipative processes were appreciable. Measurements on a larger scale have been made at CUDWR-NLL.” These show an attenuation of about 2 db per kyd, that is, slightly more than that predicted by equation (24) of Section 8.5 for shock waves in an ideal fluid. Because ot the difficulty of correcting accurately for the effect of refraction on the intensity, and because of the pos- sibility of nonlinear behavior of the hydrophone in CUDWR-NLL experiments, none of these results can be given much weight. Pulses have been propagated to very long ranges in the strata of the ocean where the velocity of sound is less than at shallower or deeper depths. These will be discussed in Section 9.3.2. Attenuation measure- ments have been made for these pulses with the use of recording equipment responsive to particular bands of frequencies. Because of the limited frequency re- IN DEEP WATER 197 sponse of the equipment, the results cannot be inter- preted in terms of peak pressure; however, as will be seen in Section 9.3.2, they indicate that the low- frequency part of the pulse is transmitted with very low attenuation. Effects of Refraction 9.2.2 This section and the next will be concerned with effects which can be correlated with the variation of the velocity of sound with depth, as determined from bathythermograph data, at ranges up to a few thou- sand yards. At these ranges few if any of the ray paths will cross one another. In Section 9.3, on the other hand, we shall consider propagation over long ranges in a layer of minimum sound velocity where many different ray paths can be found leading from the source to the receiver. Most of the results dis- cussed in this section and the next will be taken from experiments conducted by UCDWR, and described in references 8, 9, and 11. Similar though less detailed results have been obtained in England at His Ma- jesty’s Anti-Submarine Experimental Establish- ment, Fairlie.° In the UCDWR experiments considerable atten- tion was devoted to the securing of bathythermograph data at as nearly as possible the same time as the firing of the shots. From these data ray paths were computed and graphs of predicted intensity as a func- tion of depth were prepared for various values of the ranges, the intensities being computed from the geometrical divergence of the rays by the methods described in Chapter 3. Figure 3 shows a typical comparison between computed and observed peak pressures for a day when the upper layers of water were nearly isothermal. The pressure levels are all plotted in decibels, that is, the abscissas are 10 logy pmax It will be seen that in the direct zone the observations are in reasonable agreement with the ray theory but that they would not agree at all well with an inverse square law. It is a little surprising that the agreement with ray theory should be so good, since the theoretical intensities were computed with- out any allowance for attenuation. Particularly note- worthy is the decrease in intensity as the cap is raised into the shadow zone from below. The 3,600-yd points are all in the shadow zone. Figure + shows a similar comparison for another day. On this day conditions were rather variable. Of the three bathythermograph runs taken during the morning, one showed a very shallow split-beam pattern while the other two 198 TRANSMISSION OF EXPLOSIVE SOUND IN THE SEA PEAK PRESSURE LEVEL IN DECIBELS, ARBITRARY SCALE DEPTH OF CAP IN FEET 300 400 RANGE 1200 YARDS RANGE 2100 YARDS 8 RANGE 3600 YARDS CURVES PREDICTED INTENSITY FROM RAY THEORY, WITH SOURCE STRENGTH CHOSEN TO GIVE BEST FIT TO 1200 YARD POINTS RANGE IN YARDS (o) 1000 2000 9 SSS ae) HYDROPHONE AT 10 FEET 100 DEPTH IN FEET 8 fe) RAY DIAGRAM 3000 a Te Leg TG FEET PER SECOND 4920 4880 4900 Ficure 3. Comparison of observed peak pressures with values calculated from ray theory for a split-beam pattern. showed a weak negative gradient extending to the surface; in the afternoon there was a strong negative gradient at the surface. The ray diagram and theo- retical intensities shown for the morning shots were constructed from an average of the three tempera- ture-depth curves taken during the morning, and thus are only a rough approximation to the truth at any one time; however, the error should not be serious except near the boundary of the shadow zone. It will be seen that the agreement of the theoretical and observed intensities is again fairly good. The reduction of intensity in the shadow zone for this SHORT-RANGE PROPAGATION IN DEEP WATER 199 PEAK PRESSURE LEVEL IN DECIBELS, ARBITRARY SCALE DEPTH OF CAP IN FEET 4 RANGE 800 YARDS, MORNING O RANGE 1400 YARDS, MORNING © RANGE 1900 YARDS, MORNING CURVES PREDICTED INTENSITY FROM RAY THEORY, WITH SOURCE STRENGTH CHOSEN TO GIVE BEST FIT TO 800 YARD POINTS RANGE IN YARDS 1000 2000 400 RAY DIAGRAM Bane ae og i HYDROPHONE 4 w INT IT i eo nen C4 Ve NTENSITY ZONE z = 200 x ¢ = SHADOW BOUNDARY w (AFTERNOON) aK AZZZ7Z FEET PER SECOND 3000 4000 4880 4900 4920 AFTERNOON Figure 4. Comparison of observed peak pressures with values calculated from ray theory for a negative gradient extending to or almost to the surface. case Is, as one would expect, more pronounced than for Figure 3. A particularly interesting variation of intensity with range is shown in Figure 5." This series of shots was made at a single depth at various ranges, on a day when there was a moderate negative temperature gradient at the surface. The velocity-depth curve and ray diagrams are shown in Figure 6. The hydrophone was placed at a depth of 54 ft, just below the knee of the velocity-depth curve, which comes at 48 ft. This 200 causes a peculiar irregularity in the ray diagram, as shown in Figure 6; in this figure rays are drawn for initial inclinations in 0.1° steps from 1.0° to 2.5°. Rays whose vertices lie below 48 ft (1.0°, 1.1°, 1.2°) are bent downward strongly by the strong negative gradient below the knee. Rays rising above 48 ft (1.3°, 1.4°, ete.), however, are bent downward only weakly by the weak negative gradient above the knee, and diverge more rapidly; their divergence is 100 INTENSITY COMPUTED FROM INVERSE SQ LAW aS OB ABOVE | DYNE PER SQ CM o INTENSITY COMPUTED FROM RAY THEORY WITH STANDARD VALUE FOR SOURCE STRENGTH 60 PEAK PRESSURE LEVEL, 1200 1600 2000 RANGE iN YARDS 2400 2800 Figure 5. Observed and calcutated variation of peak pressure with range for a negative temperature gradi- ent. Hydrophone depth, 54 ft; cap depth, 100 ft; sound conditions as shown in Figure 6. further increased when they curve back down through the knee. The result of this “‘double layer effect” is a “hole” in the sound field immediately beyond the 1.2-degree ray, i.e., as a sharp dip in the intensity- range curve, as shown in Figure 5. The theoretical curve for this figure was not fitted to the points, but was computed from the known absolute strength of the source. At its minimum this theoretical intensity is some 14 db lower than that which would be pre- dicted by the inverse square law, and it is therefore quite significant that the observed intensities follow it so closely. As the shadow boundary is approached the observed intensities drop markedly before the computed shadow boundary is reached. This might be due to a departure from ray theory or to a slight error in the assumed temperature distribution near the surface which would cause the computed shadow boundary to be too far out. While data on the time of rise of the pressure to its peak value (see Table 2 TRANSMISSION OF EXPLOSIVE SOUND IN THE SEA on page 204) favor the latter interpretation, the sys- tematic tendency of the observed shadow boundary to lie closer than predicted, discussed in Section 5.4.1, suggests that some other cause must be found for this apparent discrepancy. 9.2:3 Shadow Zones and Diffraction As we have seen in Figures 3, 4, and 5 of Section 9.2.2, signals of appreciable intensity are received in places where no rays on the sound ray diagram penetrate. This phenomenon is familiar in experi- ments with sinusoidal sound and has been discussed in Section 5.4. This and other departures of observed intensities and pulse forms from those computed by applying ray theory to bathythermograph observa- tions may be due to any of several causes. In the first place, the concept of propagation of sound along ray paths is only approximate; a more exact application of acoustical theory predicts that some sound should penetrate into the shadow zone by diffraction, and that in and near the shadow zone the shape of the pulse should be somewhat different from its shape close to the source. In the second place, it is known that the temperature in surface layers of the ocean is not simply a function of the depth, but varies ap- preciably from one position to another in the same horizontal plane. Thus a set of rays which were really accurately constructed would differ in many features from the rays which one computes from the assump- tion that temperature is a function of depth alone. Thirdly, the water is not homogeneous but contains bubbles, fish, ete., which can scatter sound and cause its velocity to vary with the frequency. Finally, the water is not at rest; portions of it may be set in motion relative to the rest by waves and swells, by tidal or other currents, and by the motion of ships, fish, ete. These irregularities in velocity, although small in comparison with the velocity of sound, may easily cause appreciable alterations in the shape and strength of an explosive pulse at ranges of the order of a thousand yards. Unfortunately, the experimental data so far avail- able are not sufficiently complete to enable many sure conclusions to be drawn about the mechanisms re- sponsible for the various effects observed. However, a few tentative conclusions can be reached regarding the origin of the sound which is found in the shadow zone in experiments such as those of references 8 and 9 which have been discussed in the preceding section. Referring to Figure 5, it will be seen that in the SHORT-RANGE PROPAGATION IN DEEP WATER 201 SOUND VELOCITY IN FEET PER SECOND 4900 4920 4940 0 20 40 60 RANGE IN YARDS 80 DEPTH IN FEET HYDROPHONE AT 54 FEET 00 120 rhe INCLINATION “7 te OF RAYS AT ; HY DROPHONE L314" Figure 6. Sample velocity-depth curve and ray diagram for the day on which the data of Figures 5, 7, 8, 9, 12, 14, 15, and 16B were taken. shadow zone the intensity, defined as the square of the pressure at the first positive peak, decreases at a rate of 35 or 40 db per kyd, down to a level which is about 30 db below the value which would be obtained by extrapolating the pressures obtained at short ranges according to the inverse square law. Less com- plete intensity data obtained on other days give com- parable values for the decrease in intensity. This de- crease is of the same order as that which would be ex- pected for diffracted sound in an ideal medium in which the velocity of sound varies with depth in the manner shown in Figure 6.14 However, as has been noted in Section 5.4, a very similar decrease is ob- served for the case of 24-ke sinusoidal sound;! for this case, however, the rapid decrease of intensity with increasing range ceases after the intensity has fallen to about 40 db below the inverse square extrapolation, and beyond this point the decrease in intensity seems once again to be described by an inverse square law. For this and other reasons the supersonic signal received at a considerable distance ins.de the shadow zone is believed to arrive there by some sort of scattering process, rather than by dif- fraction. It does not seem likely, however, that scattering contributes appreciably to the observed intensities of the explosive pulses plotted in Figure 5 or in Figures 3 and 4. For the disturbance produced at the hydrophone by scattered sound is a superposi- tion of the d.sturbances produced by various scatter- ing centers, and since these have different times of travel, the number of scattering centers which can contribute to the disturbance at the hydrophone at a given instant increases with the length of the pulse. The explosive pulse is so short that one would expect the scattered intensity to be lower by 30 db, at the very least, than that from a 100-msec pulse of sinu- soidal sound having the same initial amplitude and a frequency of the same order as that which pre- dominates in the explosive pulse in the shadow zone. It is thus hard to see how the scattered intensity could be comparable with the shadow zone inten- sities observed in Figures 3, 4, and 5. Thus we are forced to consider diffraction as the mechanism by which an explosive pulse penetrates the shadow zone, at least in cases such as Figure 5 and the afternoon shots of Figure 4, where a true shadow zone is pro- duced by downward refraction. For a split-beam pat- tern like Figure 3, the existence of a shadow zone in the ray diagram is due to the fact that the assumed velocity-depth curve has a discontinuity in slope; since the true variation of velocity with depth is un- doubtedly represented by a smooth curve, it is better in this case to speak of a zone of low intensity, rather than of a shadow zone, and the argument just given for the occurrence of diffraction is less compelling. The diffraction hypothesis receives support from a study of the shapes of pulses received in or near the shadow zone. Figure 7 shows the oscillograms for some of the shots plotted in Figure 5. It will be seen that as the shadow boundary is approached, the direct and surface-reflected pulses merge, and that within the shadow zone the pulse is oscillatory. The time of rise to the first maximum begins to increase suddenly at about the position of the shadow bound- TRANSMISSION OF EXPLOSIVE SOUND IN THE SEA RANGE 169 DS SHoTA2IS en Figure 7. Changes in the shape of an explosive pulse on passing from the direct zone into the shadow zone. Source: No. 8 cap at depth 100 feet and at ranges indicated. Hydrophone at depth 54 feet. Date: Apr. 3, 1942; gain of recording system sometimes changed between shots. SHORT-RANGE PROPAGATION PRESSURE AMPLITUDE ON ARBITRARY SCALE FOR EACH CURVE “Er LAr ae | de Se ttt IN DEEP WATER 00 TIME IN i SEC Curves A through C are computed from diffraction theory, assuming the explosive pulse near the source to have the form P = Po exp (—3 X 10% sec) Shown in the curve labeled initial. with those obtaining in the experiment in the following table: The conditions assumed in the calculations are compared Horizontal distance Velocity Depth of Depth of from shadow Curve gradient source receiver boundary A ae sec! 50 ft 100 ft 500 yd B Constant { 0.1 5x0) 50 “ 500 “ C (0.2 50 “ 50 “ 500 “ Observed See Figure 6 iQ) o 100 “ 460 “ Figure 8. Observed and computed pressure-time curves in the shadow zone. ary and continues to get larger and larger the farther one goes into the shadow zone. These features were observed on all occasions when shots were made in a shadow zone produced by downward refraction, and occasional repeat shots showed that the first cycle or so of the oscillatory pulse observed in the shadow zone was quite reproducible (see Figure 16C). It is interesting to compare these oscillograms with theo- retical pressure-time curves for sound diffracted by an ideal medium which has a plane surface and in which the velocity of sound depends only on the depth. Such theoretical pressure-time curves for ex- plosive sound in the shadow zone have been com- puted by CUDWR;;" because of mathematical com- plications in the theory, however, the theoretical cal- culations have not been made for exactly the same conditions as any of the shots of Figure 7. Figure 8 shows the comparison with shot 21 of Figure 7. The agreement is good as regards time of rise, but the amplitude of the negative part of the pulse is much greater for the observed than the theoretical case, and the observed wavelength is much shorter. The discrepancy would probably be reduced if the calcu- lation could be carried through for a velocity distri- bution approximating the observed one more closely. However, it is quite possible that diffraction theories 204 140 120 ° ° o fe} SHADOW BOUNDARIES COMPUTED FROM Ao) BATHYTHERMOGRAPH of DATA a fe) TIME OF RISE IN [ML SEG 40 SHADOW BOUNDARIES DETERMINED FROM INTENSITY-RANGE PLOTS a 20) 1000 1500 RANGE IN YARDS —eCOMPOSITE OF ALL DATA TAKEN OVER A TWO MONTH PERIOD, AVER- AGED BY RANGE GROUPS, AND WITH EXCLUSION OF ALL SHOTS MADE BEYOND THE SHADOW BOUNDARY AS COMPUTED FROM BATHY - THERMOGRAPH DATA —OSHOTS MADE AT 100 FOOT DEPTH, MORNING OF APRIL 3, 1942, WITH HYDROPHONE AT 54 FEET —- 4SSHOTS MADE AT 50 FOOT DEPTH, AFTERNOON OF APRIL 3, 1942, WITH HYDROPHONE AT 54 FEET Figure 9. Dependence of time of rise on range, show- ing influence of the shadow zone. TABLE 2. TRANSMISSION OF EXPLOSIVE SOUND IN THE SEA observed and computed shadow boundaries are marked on the figure. The abrupt increase in time of rise on crossing the observed shadow boundary is quite conspicuous, and was noticed in UCDWR ex- periments on all days when strong downward refrac- tion was present. For comparison, Figure 9 also shows as a full curve the average values given in Table 1 of Section 9.2.1, which at each range represent the time of rise when refraction conditions are such that the hydrophone is in the direct zone. It will be seen that out to the shadow boundary all values agree to within the fluctuations of the data. The sharpness of the increase in time of rise as the shadow boundary is crossed suggests using a plot like Figure 9 to determine the location of the shadow boundary. Table 2, taken from reference 9, gives a comparison of the range to the shadow boundary de- termined in this way with the range as deduced from the bathythermograph measurements, and also with the range as deduced from a plot of peak intensity against range, such as Figure 5. It will be seen that time of rise and peak pressure always give very nearly the same position for the shadow boundary, but that this position does not always agree well with that given by the bathythermograph. This is not sur- prising, since such things as surface waves and small changes of temperature very close to the surface can Comparison of three methods of determining the range to the shadow boundary.* Date, 1942 Depth of explosion in feet April 2 | April 3 | April 3 April 8 | April 9 Range from bathythermograph (yd) Range from time of rise (yd) Range from peak pressure (yd) 1,800 1,170 1,200 | 100 50 100 50 50 1,490 | 1,940 | 1,080 | 1,010 1,100 | 1,430 | 1,150 | 1,100 1,250 | 1,450 | 1,170 | 1,170 * Depth of hydrophone, 54 ft in all cases. based on the concept of a horizontally stratified medium will prove inadequate to explain the ‘experi- mental results, and that some more complicated proc- ess must be considered. Figure 9 shows how the time of rise to the first pressure maximum is affected by crossing the shadow boundary. The lower dashed curve is for the same shots as Figures 5 and 7, while the upper dot-dash curve is for shots at shallower depth at a different time on the same day. Velocities and ray diagrams for this day have been given in Figure 6. Since the shadow boundaries computed from the temperature data do not agree very well with the boundaries de- termined empirically from the behavior of sound in- tensity as a function of range (see Table 2) both the have quite an appreciable influence on the position of the shadow boundary, and the shadow boundary is determined by the distribution of temperature over a large area of the sea, while the bathythermograph measures temperatures on only one vertical line. Many of the oscillographic pressure-time records obtained of explosive pulses are much less simple and comprehensible than the examples which have been selected for discussion in the preceding paragraphs. Some of the irregularities can apparently be explained in terms of multiple ray paths, while others are more. puzzling. Figure 10 shows some typical oscillograms obtained on a day when the bathythermograph showed that there were alternate layers of large and small temperature gradients, which should have pro- SHORT-RANGE PROPAGATION IN DEEP WATER 205 SHOT Il RANGE 630 YARDS SHOT 16 RANGE 1290 YARDS SHOT 15 RANGE 1110 YARDS SHOT 18 RANGE !370 YARDS SHOT 19. RANGE 1450 YARDS a) Figure 10. Records showing multiple path effects. Source: No. 8 cap at depth 100 feet and at ranges indicated. Hydrophone at depth 54 feet. Date: Apr. 2, 1942. Gain of the recording svstem was sometimes changed between shots. duced considerable crossing of the ray paths. The 630-yd oscillogram has very much the form one would expect if there were two ray paths leading from cap to hydrophone. That for 800 yd suggests a number of ray paths, but the arrivals are less sharp. It is diffi- cult to correlate these features in detail with the calculated ray diagram, however, and there appear to be variations which make it hard to pick out sys- tematic trends in the characters of the oscillograms as the range is gradually increased or decreased. 206 RANGE IN YARDS te) [e) 800 |Z Be EE rs) ° | > wW Ss [2 a rege Bir z 80 ° wy > w = aw © 4840 ¢* Oo PJ z e 60) Nw a = 4820 o 2 w = 2 2 a az a wo 40 4800 = 144780 20 4760 fa) 4740 (o) 5 10 15 20 ANGLE AT WHICH RAY CROSSES AXIS OF SOUND CHANNEL, IN DEGREES TYPE I-UNREFLECTED RAYS TYPE IL- SURFACE REFLECTION AND UPWARD REFRACTION TYPE IL-BOTTOM-REFLECTED RAYS — HORIZONTAL RANGE TRAVERSED BY RAY IN EXECUTING ONE COMPLETE CYCLE OF ITS VERTICAL OSCILLATION =*——MEAN HORIZONTAL VELOCITY EQUAL TO ABOVE RANGE OIVIDED BY TIME REQUIRED FOR ONE CYCLE Figure 18. Mean horizontal velocity and range per cycle for rays oscillating about a sound channel. not be true. In any case, however, whether source and receiver are on the axis of the sound channel or not, the number of different rays connecting them increases with increasing horizontal range. The num- ber of such rays decreases, however, with increasing distance of source or receiver from the axis. If either source or receiver is too far from the axis of the sound channel, no ray can get from source to receiver with- out reflection from the surface or the bottom. To study these phenomena quantitatively, and to compute times of arrival for the various rays, curves like those shown in Figure 18 are very helpful. What- ever its point of origin, any ray which traverses the sound channel can be characterized by the angle at which it crosses the axis of the sound channel; any two rays which cross the axis at the same angle must be congruent, differing only by a horizontal displace- ment. Figure 18 shows how the horizontal range per cycle and the mean horizontal velocity, that is, the quotient of horizontal range per cycle by time per cycle, depend upon the angle of crossing the axis. For angles less than a certain critical value, equal to 12.2 degrees in the example shown, the ray oscillates up and down without reaching either the surface or the bottom. For rays of this type (type I in Figure 17) it will be seen that the mean horizontal velocity is least for the axial ray and increases as the angle with the horizontal, and hence the range per cycle, increases. The consequences of this are espe- cially interesting when both source and receiver are on the axis of the sound channel. For this case the first impulse to arrive will come along a ray for which the number of oscillations in depth has the smallest value consistent with avoidance of surface and bot- tom reflections. When the range is small, this ray will have only one half-cycle between source and re- ceiver, but with increasing range more and more half- cycles are required, since the range per complete cycle can never be greater than a certain value, equal to 85 kyd in Figure 18. Rays with more and more oscillations will arrive later and later, and if for the moment we ignore reflected rays, the last one to arrive will be the straight axial ray. Thus, the early arrivals will be separated by considerable intervals of time, but later arrivals will be closer and closer to- gether, finally merging into an unresolvable cre- scendo, followed, if we neglect reflected rays, by a sudden silence. Figure 19A shows the times of arrival of these sound channel rays, as computed from the data in Figure 18 for a particular value of the range. The total time between the first and last of these arrivals can be computed from the spread in mean horizontal velocities for the sound channel rays; for the case plotted in Figure 18 the total time in seconds comes out to be 0.012 times the range in miles. It will be noticed that the early arrivals in Figure 19A come in groups of three. The explanation of this is shown schematically in Figure 20 for the simplest case of the first arrivals at a very short range. Each oscillating ray travels much farther in a lower half- cycle than in an upper one; consequently the mean horizontal velocity of a ray between source and re- ceiver will be principally a function of the amplitude of its lower half-cycles which in turn is principally LONG-RANGE SOUND CHANNEL PROPAGATION TIME IN SECONDS -0.5 “1.5 c°) MT A 12) 0.5 TYPE I OR SOUND CHANNEL RAYS WHICH STRIKE NEITHER SURFACE NOR BOTTOM TYPE IE RAYS WHICH ARE REFLECTED FROM THE SURFACE BUT NOT THE BOTTOM TYPE IZ OR BOTTOM REFLECTED RAYS G Ficure 19. Times of arrival for the various rays connecting two points on the axis of a sound channel. Range, 400 kyd = 197 miles. Velocity-depth curve assumed same as for Figures 17 and 18. The numeral below each arrival gives the num- ber of lower half-cycles in the corresponding ray path. The zero of time is taken as the time of arrival of the axial ray. dependent on the number of lower half-cycles which occur during the passage from source to receiver. In the example of Figure 20, there are four rays which have two lower half-cycles between source and re- ceiver; however, two of these four, namely, the ones with two upper half-cycles, arrive at the same time so there will be only three resolvable arrivals. When source and receiver are at different depths, the rays analogous to those in Figure 20 will all arrive at different times, and the hydrophone will receive pulses in groups of four. For the later arrivals, the upper half-cycles have more nearly the same travel time as the lower half-cycles, and the pulses no longer arrive in clearly separated groups of three. When either the source or the receiver is at some distance from the axis of the sound channel, the piling-up effect shown in Figure 19A will not occur, since only a limited number of rays will be possible between source and receiver. So far we have not considered sound which arrives at the receiver by paths involving surface or bottom reflection. The rays which undergo surface reflection and upward refraction without reaching the bottom (type IJ in Figure 17) have mean horizontal velocities which, according to Figure 18, are slower than the fastest unreflected, or type I rays, but faster than the axial ray. Thus the arrivals for rays of this sort are mixed in with those of type I, but when source and 216 RANGE <= - a. Ww (=) FIRST ARRIVAL OF GROUP RANGE <= be a WW f=) SECOND AND THIRD ARRIVALS, COINCIDENT IF, AS SHOWN HERE, SOURCE AND RECEIVER ARE AT THE SAME DEPTH RANGE <= = a Ww (=) RANGE <= = Qa wW [=] @ SOURCE LAST ARRIVAL OF GROUP ORECEIVER Figure 20. Grouping of arrival times for sound chan- nel rays. The four ray paths shown each have two lower half-cycles, and the corresponding times of travel are therefore close together, so that the four arrivals form a group. receiver are both on the axis they cease before the “piling up” of the sound channel rays. This is shown in Figure 19B for the particular set of conditions chosen for that figure. The bottom-reflected rays (type III in Figure 17) have times of arrival which are also interspersed among those of type I, but which continue after the TRANSMISSION OF EXPLOSIVE SOUND IN THE SEA latter have ceased. Figure 19C shows these arrivals for the example treated. The grouping for these rays is again in threes or fours, according to whether source and receiver are at the same or different depths. Let us now consider the energies and intensities of the system of impulses arriving at the receiver. First of all, it may be noted that all the energy emitted by the source in directions giving rays of type I, i.e., for the cases of Figure 18 in directions within + 12.2° of the horizontal, is propagated along the sound channel and cannot disappear except by volume absorption or scattering in the water. If the latter processes are neglected, the total energy in the system of impulses transmitted by these unreflected rays, that is, the system exemplified by Figure 19A, must be inversely proportional to the horizontal range. This contrasts with propagation in an infinite homogeneous medium, where the energy given to a receiver varies inversely as the square of the distance. The intensities of the individual arrivals can be calcu- lated in the usual way from ray theory, which should be applicable to the earlier arrivals, before there is appreciable overlapping of consecutive pulses. It will be apparent after a little pondering on Figure 17 that in general these individual intensities must vary ap- proximately as the inverse square of the range, the slower rate of decay of the total energy being due to the increase in the number of arrivals as the range increases. For certain positions of source and re- ceiver, however, some of the arrivals may have an anomalously high intensity due to the fact that two rays of infinitesimally different initial inclinations are tangent at the receiver. This condition will be more closely approached for the latest sound channel arrivals to reach the receiver than for the earlier ones, and accordingly these latest arrivals should be the strongest. The energy traveling along paths involving reflec- tion from the surface or the bottom is channeled in a similar manner. The bottom-reflected rays, however, lose a considerable part of their energy at each re- flection, and therefore die out more rapidly with in- creasing range than the others. (See Figure 21 of Section 9.3.2 and Section 9.4.1.) For the same reason successive arrivals of this type have progressively de- creasing intensities. 9.3.2 Experimental Results Two series of experiments on long-range transmis- sion in deep water have been conducted by WHOI.”?. #! LONG-RANGE SOUND CHANNEL PROPAGATION 217 1 RADIO SIGNAL p 2 SHALLOW HYDROPHONE 3 DEEP HYDROPHONE b macs BS create patacn_emaemmae Tae pi, fn, a eS feat en At am, wd ESD 234 ie : 1010 1010 END OF SOUND CHANNEL ARRIVALS FREQUENCY INC se wen one ' beer ta ee 5 RR Babi 2! eee kalo cod See «iy A ae eSB PS iat gusset re ab ge ayes ~ ea ncn mao a Ae abrelveen la A SHOT 34, RANGE 50 MILES F< ane AOR MEMS °F 9 ee met Vy in apse eA Nt td TRAST RET (ANE I ON A ihe sie ee ot cuearies a Coa iN cine aon 5 een tient ‘ t Tear eu Fare tot ¥ Cree ie in rena Kaan bas ss ome — 1010 1010 END OF SOUND CHANNEL ARRIVALS | FREQUENCY B snot 43, RANGE 300 MILES: : 2 SANG : Figure 21. Typical records of explosive sound received at long ranges. Times marked along top of each oscillogram are in seconds. The first series was conducted just outside the Ba- hama Islands and consisted in the recording of im- pulses from shallow explosions on two hydrophones, one at shallow depth and the other at 1,600 ft, at various ranges all less than 30 miles. The second series was made some time later in regions extending northeastward and eastward from the same locality. In addition to the shallow shots, explosions at 4,000 ft, near the axis of the main sound channel, and a hydro- phone near this depth as well as a shallow one were used. The ranges for this series extended out to 900 miles.» Data pertaining to the conditions of these ex- periments are given in Table 5. The velocity-depth curve for the first series is the one shown previously in Figure 17; that for the second series is very similar, and the curves of Figure 18 can be applied with little error to either series. Figure 21 shows some typical records obtained in these experiments along with thumbnail sketches of the frequency response characteristics of the various recording channels used. The following paragraphs point out a number of features of these records which agree with the predictions of ray theory as outlined in Section 9.3.1. > More recent experiments which have not yet been re- ported in full have yielded detectable signals at a range of 2,300 miles. Curves at left give relative amplitude response of each channel to the various frequencies. TasiLe 5. Experimental arrangements used in long- range transmission studies by WHOL. Second series (from refer- ence 20) First series (from refer- ence 21) Depth shallow hydrophone in feet 80 80 Depth deep hydrophone in feet 1,600 3,500 Charge weights and depths 1 Ib, 50 ft V Ib, 50 ft 4 Ib, 4,000 ft 200 Ib, 300 ft Ranges, nautical miles 2.7-26 20-900 Depths of sound channels in feet 75, 4,500 Depths of water in feet 16,000 4,100 (average) 15,000-18,000 (usually) Identification of the paths by which the various pulses arrive is usually difficult because of the large number of arrivals and because the predicted time for any arrival can be appreciably influenced by un- certainties in the depths of source and receiver and by small variations of the velocity-depth curve along the route. However, the general appearance of the records is very much as one would expect from the considerations given in Section 9.3.1. Thus, in Figure 21B the arrivals at the deep hydrophone come in groups of four while for the shallow hydrophone the four pulses show up as two, each of which is pre- sumably double but unresolved because of the short- 218 ness of the interval between any pulse and its reflec- tion from the surface near the bydrophone. In Figure 21A a similar grouping occurs for the bottom- reflected pulses, which continue after the last sound channel arrival, but the range for this case is so short that the sound channel arrivals have not yet had time to form into well-defined groups. In the records obtained when both source and re- ceiver were shallow, each theoretical group of four is of course entirely unresolved and appears as a single pulse. When source and hydrophone are both deep the total duration of the sound channel arrivals, that is, the time interval between the first arrival and the last arrival via the sound channel is found to agree nicely with the duration predicted by Figure 18. At the longer ranges the last sound channel arrival is easily spotted by the conspicuous ‘‘piling-up”’ effect which occurs just before it, as shown in Figure 21B. At shorter ranges, however, as in Figure 21A, the last sound channel arrival can only be identified by its high intensity and the fact that, like all the ar- rivals which do not involve reflection from the sur- face, it is absent from the record of the shallow hydro- phone. For short ranges the number of sound channel arrivals is small because of the short range and the fact that neither source nor receiver is on the axis of the sound channel, and the bottom reflections, which come in both before and after the last sound channel arrival, may mask the abrupt termination of the sound channel arrivals even when a piling up occurs. At ranges beyond about 300 miles, on the other hand, the bottom reflections become so weak that they no longer appear on the records, and the piling up of the sound channel rays is followed by sudden silence. This disappearance of the reflected pulses is of course due to the fact that according to Figure 18 any ray traveling by bottom reflections cannot go more than about 70 kyd between successive reflections; and since an appreciable amount of energy is lost at each reflection, pulses traveling along such rays are much more rapidly attenuated than those which travel in the water alone. According to Table 5 at the time of the first series of experiments there was a shallow sound channel with its axis at a depth of about 75 ft. That trans- mission to considerable distances near the surface was possible at this time is shown by a comparison of the velocity of propagation of the first arrival with the velocity for the bottom-reflected rays. The ratio of these velocities is found to agree nicely with the TRANSMISSION OF EXPLOSIVE SOUND IN THE SEA ratio of the velocity of sound at the surface to the velocity given by Figure 18 for the particular bottom reflection studied, showing that the first arrival does indeed come by a path lying entirely in the region near the surface. The most significant results obtained in these ex- periments have to do with attenuation and with the reflection coefficients of bottom and surface. To study quantitatively the variation of intensity with dis- tance and with number of bottom reflections, some sort of measurements must be carried out on the os- cillograms. The most obvious thing to measure would be the peak pressures or momentums of individual arrivals. However, in many cases the individual ar- rivals of a group were not completely resolved, and in all cases the pressure-time curves may have been distorted by small-angle scattering or off-specular re- flection. For these reasons it was concluded that the most suitable characteristic of the records from which to estimate attenuations and reflection coefficients is the energy of a poke, or of a group of two pokes, rather than the peak deflection, this energy being as- sumed to be a constant times the integral of the square of the deflection. One may hope that this quantity will represent a suitably weighted average of the spectrum level of the pressure pulse in the water in the region of frequencies covered by the re- cording channel being used. It is of course not strictly true that the ‘“energy”’ measured in this way on an oscillogram represents this weighted average, since, for example, the phase of the transient disturbance produced by the first arrival at the time of the second arrival will determine whether the second arrival increases or decreases the amplitude. However, we may expect that the desired correspondence will be valid for an average over many pokes. Because of the very large ranges covered by the second series of experiments, it was possible to meas- ure the very small attenuations suffered by sound at the comparatively low frequencies to which the re- ceiving channels responded, frequencies at which no other measurements of attenuation have been ob- tained. The results, based on the total energy of all the sound channel arrivals taken together, are sum- marized in Table 6. The data beyond about 200 miles are fairly consistent, as the sample plot given in Figure 22 shows. At shorter ranges the measured energies vary erratically, perhaps because the number of sound channel rays is too small to give a uniform spatial distribution of energy. The interpretation which should be given to these BOTTOM REFLECTION. SHALLOW-WATER TRANSMISSION 219 Taste 6. Attenuation coefficients for explosive sound received in various frequency bands. Frequency limits of recording channel 6 db below peak Spread of sensitivity in Attenuation in Number of ranges used Location of shots cycles per second db per kiloyard observations used in kiloyards Line from Latitude 26° N, Longitude 76° W 22-175 0.005 5 400-1,600 to Latitude 39° N, Longitude 67° W 2,300-10,000 0.013 4 200-1,000 Line running east from Latitude 25° N, 14-75 0.025 5 200-550 Longitude 76° W 56-350 0.043 4 300-550 600—4,000 0.035 4 300-550 56-350 0.050 4 300-550 attenuation figures is rather uncertain, since the re- ceiving channels each cover a fairly broad band and since it is likely that the attenuation varies strongly with frequency. Moreover, it can hardly be decided yet whether the attenuation is due to absorption, to scattering, or to variations in the depth of the sound channel with geographic position. The latter factor would have an influence on the observed intensities similar to that of changing the depth of the explo- sion, the important variable being merely the distance of the explosion from the axis of the sound channel. In spite of all these uncertainties, however, the figures in Table 6 probably do give a significant upper limit to the order of magnitude of the absorption at sonic frequencies. Measurements of a similar sort carried out on the bottom-reflected pulses of the first series of experi- ments give values for the reflection coefficient of the bottom which will be presented later in Table 7 (see Section 9.4.1). In these experiments no difference could be noticed between pulses of the same group whose ray paths differed by one in the number of sur- face reflections undergone. This shows that the reflec- tion coefficient of the surface was unity, to within an accuracy of five or ten per cent, at the angles of inci- dence involved, which ranged from nearly normal incidence down to about 10 degrees from the hori- zontal. It has been suggested that triangulation based on the times of the last sound channel arrivals at several stations might be of practical use in the accurate loca- tion of a boat or plane on the ocean. Extrapolation of the intensities so far measured for the crescendo formed by the last sound channel arrivals suggests that a few pounds of high explosive may be heard above background at ranges of ten or twenty thou- sand miles or more, if shoals or land masses do not intervene to cast a shadow.” RANGE IN KILOYARDS QO 200 400 600 800 1000 1200 1400 1600 1800 2000 RECORDED ENERGY x TRAVEL TIME, IN WATT —SECONDS?/ CENTIMETERS- TRAVEL TIME OF LAST SOUND CHANNEL ARRIVAL IN SECONDS Figure 22. Sample plot of the dependence on range of the total energy recorded for all sound channel arrivals. Source: 4-lb TNT bomb. Location of shots: line from latitude 26° N, longitude 76° W, to latitude 39° N, longitude 67° W. Recording channel within 6 db of peak sensitivity in range 22 to 175. Line shov’ cor- responds to attenuation of 0.0050 db/kyd. BOTTOM REFLECTION AND SHALLOW-WATER TRANSMISSION 9.4 The bottom of the ocean can influence the trans- mission of an explosive pulse in several closely related ways. When a pulse traveling through the water strikes the bottom, it is partly reflected and partly transmitted. If the bottom consists of two or more successive strata with different acoustic properties, the transmitted pulse may itself be partially reflected 220 TRANSMISSION OF EXPLOSIVE SOUND IN THE SEA and partially transmitted at the boundaries between the strata, and a complicated sequence of multiple reflections may take place. Finally, the pulse may be transmitted horizontally through the bottom, the disturbance of the bottom at each point being accom- panied by a corresponding disturbance in the water. In this phenomenon the impact of the explosive wave on the bottom below the charge sets the bottom into vibration, and this vibration is propagated radially outward like a surface wave on the water, or, to use a more accurate analogy, like a surface-bound earth- quake wave, its frequency and velocity being in- fluenced, however, by the water overlying the bottom. The three following subsections deal in turn with the simpler and the more complex aspects of these phenomena. Section 9.4.1 treats ordinary reflections, using the concept of sound rays, and discusses arrival time data for certain parts of the “earthquake wave,” since these data can also be interpreted in terms of rays. Sections 9.4.2 and 9.4.3 discuss the detailed form of the pulses transmitted by the ‘earthquake wave,” which can be understood only by abandoning the ray concept and treating water and bottom as a single dynamical system. In the theoretical portions of all these sections it will be assumed for simplicity that the bottom is smooth and horizontally stratified; and an effort will be made to interpret the experimental material in terms of this idealization. It must be remembered, however, that there may often be small-scale ir- regularities in the bottom which will scatter the ex- plosive pulse, and large-scale departures from hori- zontal stratification, which will complicate the trans- mission phenomena. Reflection Coefficients and Times of Arrival 9.4.1 When a pulse of sound strikes a plane boundary between two media of different acoustic properties, the reflected pulse has a lower amplitude than the incident pulse and in general a different phase. A theoretical derivation of the amplitude and phase relations to be expected at the boundary between two ideal fluid media has been given in Section 2.6.2. Actual ocean bottoms may differ in their properties from the ideal media considered there, however. To describe completely the reflecting properties of a given bottom, one should specify the amplitude re- duction and phase shift for all frequencies and all angles of incidence. An equivalent description, which could be related to this by the methods of Fourier analysis (see Section 9.2.4) would be provided by recording the exact form of the reflected pressure- time curve for an explosive pulse for all angles of inci- dence. So far, however, no pressure-time curves have been recorded for explosive pulses reflected from the bottom. The only quantitative data on bottom re- flections which are available are those obtained at WHOI" in connection with the long-range propaga- tion studies discussed in Section 9.3.2. These data will now be described. As mentioned in Section 9.3.2, the series of experi- ments for which analyses of bottom reflections were carried out was made using a shallow hydrophone at 80 ft and a deep hydrophone at 1,600 ft, with shots of 144-lb TNT fired at depths of the order of 50 ft at ranges up to 30 miles. Two recording channels with different frequency responses were used for the shal- low hydrophone, and five for the deep hydrophone. The reflection coefficients of the bottom were deter- mined for each of these channels by making plots similar to that in Figure 22 for the pulses undergoing respectively one, two, and three bottom reflections and then measuring the vertical displacements be- tween the lines corresponding to different numbers of reflections. The values obtained are given in Table 7. TaBLeE 7. Reflection coefficients for the bottom in the region near Latitude 26°46’ N, Longitude 76°25’ W. Frequency limits Average Recording of recording channel reflection Hydrophone channel 6 db below peak response coefficient 80 ft 2 1,900-6,200 0.04 3 240-2,400 0.36 1,600 ft 4 42-230 0.72 5 42-230 0.60 6 160-2,400 0.33 7 200-2,800 0.33 This method of analysis, while probably the best that can be applied to the data available, is rather crude in that the angle of incidence of the rays on the bottom changes with range and also with the order of the reflection; if, as is often the case, the reflection coefficient varies strongly with angle of incidence, only a vague average over a range of angles will be obtained. The values given in Table 7 suggest a decrease of reflection coefficient with increasing frequency, an effect which would not occur at a plane boundary between two ideal acoustic media. Unfortunately BOTTOM REFLECTION. these results cannot be compared with data for sinusoidal sound, since the character of the bottom in the locality of the experiments is at present un- known, and since measurements with sinusoidal sound at low frequencies are not very complete. Information can also be obtained from explosive sound regarding the geological strata beneath the bottom. Figure 23 shows a typical ray diagram for WATER VELOCITY c N smal Figure 23. Ray paths in a stratified bottom. sound originating in the water over a stratified bot- tom, in which each successive layer has a higher sound velocity than the one above it. Without bothering about the detailed form of the pressure pulse received at a distant hydrophone, a subject which will be discussed fully in the two following sub- sections, we may study the way in which the time of arrival of the first measurable disturbance varies with the range from the explosive to the hydrophone. If this range, HH in Figure 23, is sufficiently short, the first disturbance will arrive by a path which lies en- tirely in the water. But if HH is greater than a certain valuero, which depends upon the depths of source and hydrophone and upon the velocity of sound in the top layer of the bottom, sound traveling along the ray HABA will arrive before the direct sound wave through the water; in such a case the fact that the sound velocity over the path AB is greater than that in the water more than compensates for the fact that EABBG is longer than the direct path HH. To find out when this occurs, let c be the velocity of sound in the water (assumed uniform for simplicity), c: the velocity in the top layer of the bottom, r the hori- zontal range, and z the height of explosive and hydro- phone above the bottom, assuming for simplicity that both are at the same level. We shall first show that the positions of A and B, which minimize the time of travel, are those for which the angles HAB and ABH obey the refraction law of ray theory, and SHALLOW-WATER TRANSMISSION 221 shall then derive an expression for the value of r at which the time of travel via HA BH becomes shorter than via the direct route ZH. The time required for a pulse of sound to travel a path such as HA BH in Figure 23 is ia z(ese 0 + ese 6,) a r — z(cot 6. + cot Dn) (5) c Cy If this time has a minimum as the position of point A is varied, this minimum must occur when dé dé, = 0 that is, when ? Zz Zz —- ese 0, cot 6. + — esc? 0, = 0, Cc Cy which is equivalent to cos, = — Cy This is the well-known expression for the angle at which the transition from refraction to total reflec- tion occurs. Similarly, the requirement that ¢ be a minimum with respect to displacements of point B gives c cos 6, = cos 6, = —- (6) Cy Eliminating the angles from equation (5) by use of this relation, we have for the time of arrival by the shortest path through the bottom Ae 2z avy 22 c/a eVia=e@/é a avi zee r 2wVve—e@ = — ————— . 7a FB 4F CE (7) This equals the arrival time r/c of the direct pulse through the water when nes 2/ate, (8) Cc —C Now if the time interval between the explosion and the first signal at the hydrophone is plotted against the range r, the graph will start out as a straight line passing through the origin and of slope 1/c; and at the range given by equation (8) the slope will change abruptly to 1/c¢. Thus all the quantities c, q, and h could be determined from this plot. If the plot is continued to larger values of r, another abrupt change of slope may occur when the travel time via a path HMNQRH lying partly in a denser stratum (medium 2) becomes shorter than via HABH. If the bottom contains still deeper strata with higher sound velocities, further changes of slope will occur. By methods similar to those outlined above, the depths 222 TRANSMISSION OF EXPLOSIVE SOUND IN THE SEA RANGE te) 800 1600 2400 cy oe rlis c, = 5650 FT PERS IN YARDS 3200 4000 4800 s560c TRAVEL TIME OF FIRST DISTURBANCE PROPAGATED THROUGH THE GROUND IN SECONDS Gl (a mite no ae he we a Ba ia A ce) Q.2 04 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.2 2.6 2.8 3.0 3.2 3.4 TRAVEL TIME OF DISTURBANCE PROPAGATED THROUGH THE WATER IN SECONDS Figure 24. Typical plot of travel time against range showing layer structure of the bottom. Location near Solomons, Md., at mean depth of 52 ft, charge and hydrophone both resting on bottom. Lines cross at r/c = 1.08seconds. Depth of upper layer in bottom = r/c Vc: — ci/ce + c: = 1,240 feet, where r = range at which lines cross, c = velocity of sound in water, c, = velocity of sound in upper layer of bottom, and c. = velocity of sound in lower layer of bottom. as well as the velocities of all these strata can be de- termined from this plot of arrival times. This type of analysis has long been familiar in geophysical prospecting. Figure 24 shows a typical plot of arrival times con- structed from some of the data obtained by WHOI,” with the layer depths and velocities deduced from it. The shots were made with both the charge and the hydrophone on the bottom, so the depth of the upper layer of the bottom can be calculated from equation (8) by replacing c by c: and q by c. When more than two layers are involved, the plot of times of first ar- rival will still consist of straight-line segments, but the calculation of the depths of the second and deeper layers involves more complicated formulas in that case. The representation of the plotted points by two straight lines is fairly easy for this case; however, data are often obtained for which the times of first arrival seem to form an almost smooth curve. This may sometimes be due to the absence of any well- defined layer structure in the bottom, as might be the case for example for a thick mud bottom whose compactness increases continuously with depth. It will be shown in Section 9.4.3, however, that there are many cases where there are recognizable layers in the bottom but where fluctuations of one sort or another prevent them from being accurately identi- fied from mere arrival time data. For such cases the proper choice of straight lines to fit a plot such as Figure 24 may sometimes be facilitated by a study of the predominant frequencies in the first and subsequent arrivals (see Figure 32, Section 9.4.3). 9.4.2 Simplified Theory of Normal Modes We have seen in the preceding Section 9.4.1 that if the range is sufficiently long compared with the dis- tances of source and receiver above the bottom, the first sound to arrive must come by a path lying within the material of the bottom over most of the BOTTOM REFLECTION. distance, as shown in Figure 23. One might at first suppose that refraction of this sort would be similar to refraction in the water alone, and that the re- ceived pulse would be a replica of the pressure wave emitted by the source, with an intensity which could be calculated by ray theory. It is easily shown, how- ever, that this is not the case. We shall first show that when the bottom is acoustically uniform, so that rays in the bottom are straight lines, the intensity pre- dicted by ray theory for a ray such as HABH in Figure 23 is zero. Figure 25 shows a ray having inclination @ in the water, 4; in the bottom, together with a neighboring ray. By Snell’s law of refraction we have cos 6; = & cos 6 (9) Cy where cand ¢, are the velocities of sound in water and bottom respectively. Now the energy which leaves the source # in an interval dé of inclinations and in a fixed narrow interval of azimuth is partly reflected and partly transmitted, and the transmitted part is distributed over the region between the rays AP and A’P’ in Figure 25. If R(@) is the reflection WATER VELOCITY c Figure 25. Spreading of adjacent sound rays on enter- ing the bottom. coefficient of the bottom at the angle @, this trans- mitted energy is proportional to [1 — R(@) |d0. If the range r = AP is large compared to HA, the distance between P and P’ will be approximately c sin 6 ds = rd6, = r—-— c, sin 6; dé (10) by equation (9). By introducing another factor r to allow for azimuthal spreading, the energy received at P per unit area is then proportional to WoROiM 1 4 Re). c sin 6; rds 72 (11) c sin 6 SHALLOW-WATER TRANSMISSION 223 As the ray AP approaches the horizontal, sin 6; ap- proaches zero, and according to equation (11) the intensity at P must do likewise. This conclusion is made even stronger by the fact that, according to Section 2.6.2, R(@) approaches unity as @ approaches the angle for total reflection. Thus, ray theory cannot account for the sound received via a path like HABH in Figure 23, when the bottom is uniform. The argument just given to show the inapplicabil- ity of ray theory to arrivals of the type shown in Figure 23 would of course not be strictly correct if there were a gradual increase of the velocity of sound with depth in the bottom, a situation which is quite common, especially for soft bottoms. It will be in- structive to consider briefly the sound ray paths for this case, since the limitations of ray theory can be most clearly seen by studying this case where it is partially applicable. Ficure 26. Ray paths in and over a bottom giving weak upward refraction. Figure 26A shows a family of rays connecting a source and hydrophone, both of which are lying on a bottom characterized by weak upward refraction. According to ray acoustics the first signal to reach the receiver H will arrive via the path I,. This will be followed almost immediately by arrivals along other paths, such as I, which likewise lie in the bottom but which involve one or more reflections at the inter- face between bottom and water. Some time later an- other group of arrivals will be received, each of which comes along a path involving one reflection from the surface of the water. One path of this type is shown 224 in Figure 26A and labeled II,; many other such paths, not shown, are also possible; some of them in- volving additional reflections from the water-bottom interface as was the case for I;. This second group of arrivals will in turn be followed by another group, exemplified by III, in Figure 26A, involving two re- flections from the free surface, and so on. Mixed in with these arrivals along paths which enter and leave the bottom will be those along paths lying wholly in the water, shown in Figure 26B. These paths have for simplicity been drawn for the case where the velocity of sound in the water is uniform. In practice most of the experiments performed so far have en- countered isothermal water with consequent upward refraction; this case, which will be discussed later, is in most respects little different from the uniform case considered here. The first arrival among these water rays will be along the direct path I,,, the second along the surface-reflected path ILI,,, the third along a path III,, involving one reflection from the bottom, and so on. Thus if the predictions of ray acoustics were valid, we should expect the signal received at the hydro- phone to consist of a number of evenly spaced groups of pulses of diminishing strength, corresponding to the “ground rays’? shown in Figure 26A, plus a number of individual pulses starting at a later time and separated by gradually increasing intervals, which correspond to the ‘‘water rays” shown in Figure 26B. Of these various arrivals, some are posi- tive pressure pulses, others negative, according to the number of phase-changing reflections each has suffered. The extent to which the predictions of ray theory can be trusted in a case of this sort can be estimated by resolving the explosive pulse into a superposition of sine waves by use of Fourier’s theorem, as de- scribed in Section 9.2.4, and then applying the criteria given at the end of Section 3.6.2 for applicability of the eikonal equation to sinusoidal waves. It is clear from these criteria that the condition for ray theory to be applicable to a sine wave along a path of the type I,, II,, etc., in Figure 26A, is that the maximum depth of the path, shown as d; for ray I,, should be large compared to the wavelength of the sound in the bottom. This condition will be fulfilled by the highest frequencies in the Fourier resolution of the explosive pulse, but not by the lowest frequencies; moreover, the frequency above which ray theory is applicable recedes to higher and higher values for the successive arrivals I,, II,, III,, etc. As is to be ex- TRANSMISSION OF EXPLOSIVE SOUND IN THE SEA pected, this critical frequency approaches infinity as the magnitude of the velocity gradient in the bottom decreases to zero, since the depths of penetration dy, etc., of the rays approach zero. A similar consideration of the disturbance propa- gated through the water suggests that ray theory should fail for frequencies of the order of c/h and smaller, where h is the depth of the water. This limit has little meaning, since this frequency can be ex- pected to be lower than the frequency at which the ray picture fails for the ground rays, and we cannot make a clear separation between the ground dis- turbance and the water disturbance after we have abandoned the ray concept. We may thus expect the pressure variation which would be recorded by a very high-fidelity receiver at H to consist of the succession of pulses which ray theory would predict plus a correction which is made up almost entirely of low frequencies. For the dis- turbance due to the shock wave from the explosion, the times of the various ray arrivals can, ideally at least, be identified on the oscillogram of the received pressure by the occurrence of sharp jumps in the pressure; these jumps, due to the sudden rise at the front of the shock wave, cannot easily be obliterated by the low-frequency correction (see Figure 13). Since, as explained above, the intensities of the ar- rivals predicted by ray theory are zero for a uniform or downward-refraction bottom and are small for a bottom with weak upward refraction, we should not be surprised to find the disturbance received by the hydrophone to be dominated by the low-frequency portion, with only a few detectable traces of the ray arrivals. To determine the nature of the low-frequency cor- rection just mentioned, it is necessary to study solu- tions of the wave equation similar to those considered in Section 2.7.2. In a report prepared by CUDWR,” it is shown how the normal modes of vibration of water and bottom can be computed and superposed to correspond to the disturbance produced by ex- plosive source. The mathematical details are too complicated to be given here;* however an attempt will be made below to explain in a simplified manner the physical basis for some of the most important results of reference 23. In particular, it will be shown how many characteristics of the signal received at the ¢ The reader who wishes to study the mathematical theory of normal modes will find it profitable to study also the treat- ments devoted primarily to single-frequency sound in deep water® and electromagnetic waves in the atmosphere.” BOTTOM REFLECTION. hydrophone can be interpreted in terms of a simple dispersion law, i.e., a propagation of different fre- quencies with different velocities. The physical reasons underlying the dispersion phenomena just mentioned can be seen by consider- ing the simple case of a progressive wave of a single frequency f. Let us try to construct such a wave by assuming the pressure disturbance to be p= ee DMC), (12) where « is a horizontal coordinate, and z is the depth below the free surface of the ocean. This function p must satisfy the wave equation 0p a 0p o( 2? + 22) = Ox? nr 02? Celi in the water; that is, when z is less than the depth h and must satisfy the analogous wave equation? 0p fe) &p 2 — — az + 32) — ap in the bottom, that is, when z is greater than h. In addition, p must satisfy boundary conditions at the free surface and at the interface between water and bottom. These conditions are (13) (14) p=O0atz=0 (15) Pwater = Phottom at z = h. (16) 1) 1a (2) ES) en ao p 0z water P1 0z bottom Assuming for simplicity that water and bottom are uniform, so that c and c are independent of z, we have, on inserting expression (12) into equation (13), eM | 1 if Aa a E |wtore h. (20) Now, if [(1/d2) — (f?/c{)] is negative, M will be a periodic function of z in the bottom, and according to equation (12) the pressure disturbance in the bot- tom will consist of progressive waves going diagonally up or down. The disturbance created by an explosion will consist in part of a superposition of progressive waves of this type which travel diagonally downward in the bottom; these waves are, however, a relatively unimportant part of the signal received in the water at a great distance, since their energy spreads out in a downward direction and thus decreases fairly rapidly with distance in the horizontal plane. The part of the signal which is most important in the present applica- tion consists, instead, of a superposition of waves of the form (12), for values of } and f which make [d/\?) — (f2/c?)] in equation (20) positive. The two independent solutions of equation (20) for this case will be exponential functions of z, one increasing to infinity as z increases, the other decreasing to zero. The former of these is physically inadmissible; so we may conclude that if a pressure wave of the desired form exists at all, it must be of the form Le if? M = Bexp (-29)/2 = @ and of course of the form (19) for z < h. However, it is easily shown that no matter what values are given to the constants A and B, it is not possible to satisfy both of the boundary conditions (16) and (17) unless d and f are related in a particular way. For, on in- serting expressions (19) and (21) into these condi- tions, and using the abbreviations ‘age ie if ie = ane 1 m0 0” = 20 c me :) forz > h, (21) we obtain A sin ph = Be" (22) B o cos ph = ae (23) p PL Dividing the first of these equations by the second eliminates A and B, and gives the following relation which must be satisfied by f and 2. = * tan Le Leer, y v If this relation is satisfied, a suitable choice of the ratio B/A will insure that both equations (22) and (23) are satisfied. It is easily verified that if « > c (24) 226 TRANSMISSION OF EXPLOSIVE SOUND IN THE SEA FREQUENCY IN C, h = 100 FEET te} 3.0 2500 2.5 2.0 oe. lo < « ~ SECOND MODE r NN 815 =) Ww > A : 7 ab 7) q = Lo MK ——— SSS FIRST MODE W/, A a “i ae fo) fo) 0.25 0.50 0.75 th jf = Frequency. h = Depth of water. sound .in bottom assumed to be 1.5 X c. and pi > p, equation (24) cannot be satisfied if u is imaginary; this justifies the statement made in the second sentence following equation (19). Graphs of the solutions of equation (24) are given in Figure 27 for a typical set of values of c, pi, and h. Typical curves of the variation of pressure along a vertical line are given in Figure 28, corresponding to particular points on the graphs of Figure 27. As was explained in Section 2.7.1, it is customary, by analogy with the terminology used in the theory of vibrating systems of particles, to use the term “normal mode” “1.25 7 Sto= aa fee : c = Velocity of sound in water. = Horizontal wavelength of disturbance. Figure 27. Variation of wavelength and phase velocity with frequency for normal modes in shallow water. Velocity of Density of bottom assumed 2 times density of water. to describe a state of vibration of the water and bottom in which the pressure distribution is given by equation (12); for modes of the present type this is equivalent to a disturbance of the type shown in Figure 28 and having an amplitude represented by a horizontally moving sine wave. It is convenient to identify families of these normal modes by the num- ber of horizontal planes in the water, including the free surface on which the pressure is always zero. This number is called the order of the normal mode, and is indicated by the labels “FIRST MODE,” BOTTOM REFLECTION. SHALLOW-WATER TRANSMISSION 227 M(z) ARBITRARY SCALE M(z) ARBITRARY SCALE 0 2 0 2 0.5 0.5 210 + 1.0 LS 15 3.0 2.0 b MINIMUM FREQUENCY FIRST MODE M(z) ARBITRARY SCALE First move 42-0,5 M(z) ARBITRARY SCALE 2.0 c SECOND move +? LF) THIRD MODE th. 2.0 f = Frequency. h = Depth of water. c = Velocity of sound in water. z = Distance below surface of water. M(z) = Pressure amplitude at depth z. Figure 28. Variation of pressure with depth along a vertical line for various normal modes. Velocity of sound in bottom assumed to be 1.5 X c. Density of bot- tom assumed 2 times density of water. “SECOND MODE,” and so on, in Figures 27 and 28. A noteworthy fact is that for modes of any given order there is a minimum frequency below which no value of X can be found which will satisfy the boundary conditions. At this frequency the quantity vy, which is inversely proportional to the depth of penetration of the disturbance into the bottom, goes to zero; and the pressure distribution takes a form such as that shown in Figure 28B. As the mathemati- cally inclined reader can verify for himself from equa- tion (24), the minimum frequency for the first mode has a half-period equal to the interval which ray theory would predict between the arrivals of types I,, I1,, I11,, ete., of Figure 26. This half-period is given by iy 2 1 en ee 25 2 e ¢ (Ce) Since these ray arrivals are alternately positive and negative, the period of the disturbance given by ray theory is the same as that for the minimum frequency. It is also noteworthy that the minimum frequency for the vth mode is (2v—1) times its value for the first mode. This has the very important consequence that any simple harmonic disturbance of low frequency which is propagated over a large horizontal range can be represented by a superposition of a finite number of normal modes of low order. Let us now consider the velocity of propagation of a disturbance which consists of a superposition of nor- mal modes of a given order but distributed over a narrow range of frequencies. It is easy to show that such a disturbance, considered as a function of hori- zontal distance x or of time t, will form a wave train. For each component normal mode has a phase factor proportional to ¢?@/A-), At any given time ¢ there will be some value of x for which most of the component modes are approximately in phase; in the neighborhood of this value of x the pressure disturb- ance will therefore be large. If x differs very widely from this value, on the other hand, the phases of all the component normal modes will be rather randomly distributed since the different modes have slightly different wavelengths \; for such values of x the pressure disturbance will be small. If we watch the motion of the wave train in the course of time, we shall find that the region of large amplitude moves with a certain velocity, commonly called the ‘group velocity.” Now, if the center of the wave train is to be near x; at time 4, and near 22 at time t, the phase change (2 — 2:/A) — f(t — ti) must be very nearly the same for all the different normal modes contained in the wave train, in order that they may continue to reinforce one another. This implies, in the limit where only a very narrow range of frequencies is in- volved, d La — X ] = =i =H) || SO 4 nN f(t 1) 5) or, if V is the group velocity, (%2 — 1) an df ermegiens) ae) d Now the phase velocity of any single-frequency component, defined as the speed of advance of a point having a given constant value of the phase 27r(x/d — ft), is equal to Af. If this quantity were a constant independent of frequency, as is the case for (26) 228 TRANSMISSION OF EXPLOSIVE SOUND IN THE SEA EE 88S SSS... — ——————————EEEEEEEeEeEEEEEEeEeEeEUy(Cy(C_ ———————E sound propagated in a single homogeneous medium, the expression (26) would simply equal the phase velocity. For the disturbances we are considering here, however, Af is not independent of frequency, as a glance at Figure 27 will show. The importance of the result (26) in shallow-water transmission is that it enables us to understand the dispersion phenomena in the ground and water waves. The initial disturbance can be represented as a superposition of normal modes having a very wide range of frequencies. However, since the group velocity is different for different frequencies, the dif- ferent frequencies in this superposition will get sepa- FREQUENCY ING he lop FEET 2500 5000 7500 GROUP VELOCITY V IN TERMS OF VELOCITY OF SOUND IN WATER c Figure 29. Dispersion in group velocities of normal modes. Frequency in f; depth of water, h; velocity of sound in water, c. rated out somewhat at long ranges, and each band of normal modes of a given order and a given narrow range of frequencies will be propagated with its own group velocity. This effect is shown quantitatively for a typical set of conditions in Figure 29. The curves of this figure are derived from those of Figure 27 by differentiation. Note that the group velocity varies from the ground velocity c at the low cutoff fre- quency to the water velocity c at very high fre- quencies, but has a minimum at an intermediate frequency. The existence of this minimum produces an interesting effect, which will be described later. The main features of the disturbance received at a distance from an explosive source can be explained most simply by concentrating attention on one of these curves, say that for the first mode. This will not only be illustrative of the main characteristics shared by_all the normal modes, but will in fact provide a rough prediction of what some of the actual records to be discussed in Section 9.4.3 should look like. For it has been pointed out that there exists a minimum frequency for each normal mode and that the fre- quency for the first mode is the lowest. Thus if the disturbance produced by an explosion is received with equipment responsive only to sufficiently low fre- quencies, the resulting signal can be interpreted in terms of the first-order modes alone. Even when high- fidelity recording equipment is used, the first mode should dominate the initial or ground wave portion of the disturbance, since it can be shown theoretically that the amplitude of the first mode is greater than the amplitude of higher modes in this region.?4 Let us therefore suppose that we have a source of sound which generates a transient disturbance con- sisting entirely of a superposition of first-mode vibra- tions of various frequencies. Since according to Figure 29, the highest group velocity occurs for the lowest frequencies above the cutoff, the first sound to arrive at a distant hydrophone will be a wave train whose frequency corresponds very nearly to point A of Figure 29. The disturbance arriving a little later will consist of frequencies having a slightly slower group velocity, that is, of slightly higher frequencies. Thus, the frequency of the received disturbance will gradually increase with time until the value corre- sponding to point B is reached. At this moment, the very highest frequencies present in the original dis- turbance start to come in, traveling in the limit with the velocity c. From this time onward, the received disturbance consists of a low-frequency part and a high-frequency part superposed, the former con- tinuously increasing its frequency along the branch BC of the dispersion curve, and the latter continu- ously decreasing its frequency along the branch FG. Eventually these two coalesce, and the disturbance dies out at an intermediate frequency. All these characteristics are apparent in the theo- retical pressure-time curve of Figure 30, which shows the contribution of the first mode to the disturbance produced under a typical set of conditions by a source which emits a single positive-pressure pulse of short duration. The portions of the curve corresponding to the points A, B, C, F, G of Figure 29 are labeled with these letters. Similar curves showing the con- tributions of normal modes of higher order are given in reference 23. These have lower amplitudes than that for the first mode, especially during the “ground wave” phase, that is, the portion of the disturbance which has traveled with a velocity greater than c and thus lies to the left of B and F. According to the present theory, which idealizes the bottom as a homogeneous fluid, the variation of the pressure at BOTTOM REFLECTION. SHALLOW-WATER TRANSMISSION 229 BEGINNING OF GROUND WAVE ° 0.05 0.10 0.15 I SPARAAA TIME FROM BEGINNING OF GROUND WAVE, = BEGINNING OF WATER WAVE NB F 0.20 0,25 IN SECONDS 0,30 WATER WAVES BLENDING TOGETHER pee AND AT END gull MO ZIG PRESSURE, ARBITRARY SCALE + UTX” ‘ Ds “4 0.35 0.40 0.45 0.50 | al | 0.55 0.60 0.65 TIME FROM BEGINNING OF GROUND WAVE, IN SECONDS Ficure 30. Theoretical contribution of the first mode to the disturbance at a distance from an explosion in shallow water. Source and receiver both assumed to be on the bottom. Range: 9,200 yards. Density of bottom = 2 X densitv of water. sound in bottom = 1.1 X velocity in water. indicating beginning of ground wave.) the hydrophone with time should be given by the sum of these contributions from all the normal modes, plus certain additional terms whose magnitude de- creases rapidly with increasing range, so that they become negligible at very long ranges. This complete pressure-time curve would of course show sharp jumps at the positions corresponding to the arrivals predicted by the ray picture. In the following section we shall compare these theoretical predictions with observations. In this comparison certain factors have to be taken into consideration which for simplicity have been neg- lected in this section, such as the modification of the received disturbance by the frequency response char- acteristics of the recording equipment, and the fact that instead of delivering a single impulse, an ex- plosion gives out a shock wave followed by several bubble pulses (see Section 8.6.). 9.4.3 Analysis of Experimental Records The Woods Hole Oceanographic Institution has obtained a large number of oscillographic records of 60 feet. Velocity of should appear at point Depth of water: (Note, A sound from explosions in shallow water at distances between 0.25 mile and 30 miles.” Several series of experiments were conducted at widely separated places with bottoms of mud, sand, and coral. The depths of the water at the sending and receiving positions were usually similar and in the range 40 to 180 ft; some shots were made at greater depths. The hydrophones used were in all cases placed on the bottom, while the charges were usually on the bottom but sometimes at mid-depth. Charges of 4-lb TNT to 300-Ib TNT were used. At all stations the water was very nearly isothermal, so that sound rays in the water were refracted slightly upward. Figure 31 shows some typical oscillograms of the sound received in these experiments. Each record consists of eight traces simultaneously recorded. The first of these, labeled “‘time break,” is used merely to record the instant at which the charge was set off;the others record the disturbance received, as modified by the frequency responses shown at the left for the various recording channels. Most of the interesting features show up best on the two channels labeled “Mark II low frequency.” which record the same 230 RADIO SIGNAL MARK IL HIGH FREQUENCY GEOPHONE MARK Qi) RADIO BUOY GEOPHONE MARK IE LOW FREQUENCY MARK ID RECTIFIED Meh FREQUENCY ING CHARGE: 5 LBS pie RADIO SIGNAL | 2) MARK IL HIGH FREQUENCY 3, GEOPHONE 4 MARK I RECTIFIED 5 MARK I 6 MARK IT LOW FREQUENCY 7 MARK IL RECTIFIED a7 NEES ae SHOT 28, NEAR SOLOMONS, MARYLAND ‘TNT ON BOTTOM DEPTH OF WATER 90 FEET RANGE 3900 YARDS i SHOT 95, NEAR JACKSONVILLE, FLORIDA iN TRANSMISSION OF EXPLOSIVE SOUND IN THE SEA 2a) 4 1010 10:10 FREQUENCY CHARGE: 25 LBS TNT ON BOTTOM sata NGG, DEPTH OF WATER 60 FEET Loe RANGE 4200 YARDS : Figure 31. Typical records of explosive sound transmissions in shallow water. Times marked along the top of each oscillogram are in seconds. Curves at left give relative amplitude response of each channel to the various frequencies. signal at two different amplitude levels. However, the rectified traces show most clearly the times of the various water wave arrivals, namely, shock wave and bubble pulses. The arrival time of the first of these is especially useful, since the range can be determined for any shot by multiplying by c the interval between the detonation and this arrival. The interpretation of records like those of Figure 31 is often complicated by the fact that each ob- served trace represents a superposition of the dis- turbances produced by the shock wave and all the bubble pulses. According to the theory of normal modes, the amplitude of the disturbance produced by any one such pulse of very short duration should be proportional to the impulse Jf pdt of the pulse; since this quantity is of the same order of magnitude for the shock wave and the first one or two bubble waves, the resulting superposition can become very compli- cated. However, as was mentioned in Section 8.6, the bubble pulses are often much weaker when, as was usually the case in the WHOI experiments, the charge is fired in contact with the bottom. Records (A) and (B) of Figure 31 are fairly typical examples of shots on the bottom; the former shows a strong bubble pulse and the latter a very weak one. Note that the separation of the first two high peaks in the ground wave of Figure 31A is just equal to the bubble period as read from the rectified trace. Since the periods of the oscillations are long compared with the duration of the impulse sent out by the explosions, the only noticeable effects of increasing the size of the charge are to increase the amplitude and to alter the time lag in the arrival of the bubble pulse effects. Chang- ing the position of the charge from bottom to mid- depth also seems to have very little effect. Let us begin the detailed discussion of the ob- BOTTOM REFLECTION. SHALLOW-WATER TRANSMISSION 231 RANGE IN YARCS 6000 8000 @ COMPONENT WITH PERIOD BETWEEN .037 AND .043 SECONDS © COMPONENT WITH PERIOD BETWEEN .O5O AND .0S58 SECONDS 4 COMPONENT WITH PERIOD BETWEEN.O6! AND .070 SECONDS TIME FROM EXPLOSION TO START OF GIVEN GROUND WAVE COMPONENT IN SECONDS 2 = 2.55 c,2 12,700 FT PER SEC G3 = 17,500 FT PER SEC ¢ = VELOCITY OF SOUND IN WATER €,,Ce,C3 INFERRED VELOCITIES OF SOUND IN SUCCESSIVE LAYERS OF BOTTOM TRAVEL TIME OF DISTURBANCE PROPAGATED THROUGH THE WATER IN SECONDS Figure 32. Typical plot of travel time against range for components of different frequencies in the ground wave. Loca- tion, near Jacksonville, Florida, at mean depth of 57 feet, charge and hydrophone both resting on bottom. served records by considering the initial or ground wave phase of the disturbance. All the records agree in a general way with the predictions of the theory of reference 23 as outlined in Section 9.4.2 (see Figure 30) in showing a gradual increase of frequency between the beginning of the disturbance and the ar- rival of the water wave. According to equation (25) of Section 9.4.2, when the bottom is uniform the period of the disturbance at its beginning is a func- tion of the velocity c; of sound in the bottom. Exten- sion of the theory to cases where the bottom con- sists of a deep firm stratum overlaid by a slower one gives the result that at sufficiently long ranges the be- ginning of the ground wave should have a frequency dependent in a complicated way upon the velocities in both layers, but that, if the upper layer is suf- ficiently thick in comparison with the depth of the water, a strong new disturbance of distinctly higher frequency will arrive some time later, the arrival time and frequency of this new disturbance being approximately the same as for the ground wave which would occur if the upper layer were infinitely thick. These theoretical predictions suggest that noting the frequencies of the first arrival and any subsequent arrivals in the ground wave may provide useful in- formation about the different strata in the bottom. This is illustrated in Figure 32, which may be com- pared with Figure 24. Here many of the records show a recognizable new arrival of different frequency from the first which comes some time later. Complete in- terpretation of the data shown in Figure 32 is diffi- cult, but there is definite evidence for a layer in which the velocity of sound is about 1.5 times the velocity in water, as well as of one or two layers of higher velocity which determine the times of the first ar- rivals. It is noteworthy that this dependence of the frequency of a ground wave arrival upon the velocity of sound in the layer chiefly responsible for the ar- 232 MARKIE FREQUENCY HIGH Ficure 33. Dispersion in the water wave produced by an explosion in shallow water. TRANSMISSION OF EXPLOSIVE SOUND IN THE SEA Shot 90, near Jacksonville, Fla.; charge, 5 lb TNT on bottom; mean depth of water, 57 ft; range, 7,000 yd; frequency response of channels as shown in Figure 31. rival in question can be observed even in the first arrivals at fairly short range. Thus in the data from which Figure 24 was constructed, the period of the first arrival was between 0.024 and 0.036 sec for the points to the left of the intersection of the two straight lines, and was between 0.050 and 10 sec for the points to the right, except for two very close to the intersection. On some records the ground wave dies out quite noticeably before the arrival of the water wave. The theory of reference 23 indicates that if the bottom is uniform to all depths, the amplitude of the ground wave should increase steadily until the water wave arrives, and that a decrease in the strength of the ground wave in this region implies the presence of layers of different. materials. In the latter case, there may be a secondary ground wave arrival of the sort mentioned in the preceding paragraph, which dies out considerably before the arrival of the water wave. Let usnow consider the disturbance after the arrival of the water wave. Figure 33 shows, in more detail than Figure 31, the dispersion phenomena occurring in this stage. The third trace from the bottom shows most clearly the ground wave just before the arrival of the water wave, and the gradual development of the water wave from a disturbance of very low ampli- tude and high frequency, superposed on the ground wave, to the final, so-called Airey phase where ground wave and water wave fuse at an intermediate frequency and die out. The similarity of this record to Figure 30 is quite striking. The resemblance is not nearly so close for the second trace from the top, since this trace was recorded with more fidelity at the high frequencies, so that many normal modes higher than the first contribute significantly to it. About 0.2 sec after the main disturbance has died BOTTOM REFLECTION. SHALLOW-WATER TRANSMISSION 233 h =Depth of water. f = Frequency of contribution of first normal mode. c = Velocity of sound in water. ¢: = Velocity of sound in bottom, assumed uniform. Az = Depth below the surface of the bottom above which 99% of the wave energy of the first mode in the bottom is included. Figure 34. Typical curves of the frequency dependence of the depth of penetration of the first normal mode into the bot- tom. Density of bottom assumed 2 times density of water. out, another disturbance is recorded, weaker than the first and due to the bubble pulse. On the low-fre- quency trace this second water wave looks similar to the first; but on the high-frequency trace it is very different. Frequencies above about 200 cycles are absent in the bubble pulse disturbance but strong in the primary disturbance. This is probably due to the fact that, as indicated in Figure 8 of Chapter 8, the pressure delivered by the bubble in its contracted stage has a duration of several milliseconds and is thus lacking in high-frequency components. A detailed analysis of the dispersion phenomena in the water wave can provide useful information on the characteristics of the upper layers of the bottom.”* Unless shots are made at very short ranges, the arrival times and frequencies of ground waves furnish infor- 234 TRANSMISSION OF EXPLOSIVE SOUND IN THE SEA CHARGE CHARGE OEPTH IN | WEIGHT IN FEET h = Depth of water. f = Frequency of contribution of first normal mode. t = Time between explosion and arrival of frequency f. t) = Time between explosion and first water wave ar- rival. V = Group velocity for frequency f. c = Velocity of sound in water. Ci, Co = Velocities of sound in bottom layers. AC He ] a a = — ( my) | Ly Theoretical dispersion in the first normal mode for various uniform bottoms all of den- sity 2. — — — Same for layer of thickness 0.1h and ¢,/c = 1.1 underlain by infinite layer with c:/c = 3, both of density 2. ----- Same for layer of thickness h and ¢,/c = 1.1 underlain by infinite layer with c./c = 3, both of density 2. Ficure 35. Theoretical and observed dispersion in the water wave. Shots were made off Jacksonville, Fla., where the depth of water was 115 to 120 feet. Hydrophone was on the bottom for all shots. Observed frequencies are taken from the Mark IT low-frequency record whose response is shown in Figures 31 and 33. mation only on those layers of the bottom which are reasonably thick, in comparison with the depth of the water; the water wave, on the other hand, can supply information on the uppermost layers even when they are much thinner. This is a consequence of the fact that for normal modes of any order the higher the frequency the more rapidly the disturbance dies out with increasing depth. Since the frequencies in the water wave are much higher than those in the ground wave, the water wave will not penetrate so deeply into the bottom, and will thus be less affected by the characteristics of deep layers and more affected by the characteristics of the top layers. Figure 34 shows how the depth of penetration varies with frequency BOTTOM REFLECTION. for the first mode for various types of bottoms. Figure 35 shows the theoretical dependence of the frequency of the first mode on the time, for these same types of bottoms, and shows also the observed frequencies as recorded in a particular region on the low-frequency Mark II channel, whose frequency response has been shown in Figures 31 and 33. Be- cause of its suppression of high frequencies, the record of this channel should consist principally of the con- tribution of the first mode. It will be noticed that at the highest frequencies the slope of the theoretical frequency-time curve is nearly independent of the assumed structure of the bottom, and that the ob- served points show the same slope. Note that the frequency corresponding to a given group velocity over a uniform bottom varies inversely as the depth of the water, a relation which is easily verified from equation (24). This relation is fairly well confirmed experimentally. The observed points in Figure 35 do not follow any of the curves for a uniform bottom but indicate rather that the velocity of sound increases with depth. A rough estimate of the scale and nature of this increase can be obtained by studying Figures 34 and 35 together. Thus at higher frequencies’ than that corresponding to fh/c equal to 4.5, the points of Figure 35 scatter evenly about the curve for c/c equal to 1.1. According to Figure 34, for fh/c equal to 4.5, 99 per cent of'the wave energy in the bottom is confined within a layer of thickness 0.2 times the depth of the water. We may therefore say that the velocity of sound in the bottom is 1.1c down to a depth of the order of 20 ft. A similar reading of thetwo figures at fh/c equal to 1.5 gives the result. that some sort of weighted average of the velocities over a depth in the bottom of the order of half the depth of the water has a value intermediate between 1.3c and 1.5c. These rough conclusions are confirmed by the fact that the points of Figure 35 fall between the dotted and dashed curves. Information ob- tained from study of the ground waves regarding the deeper layers of the bottom should, of course, be borne in mind when studying the water waves in this way. Measurements have been made on the maximum SHALLOW-WATER TRANSMISSION 235 intensity of the water wave as recorded by the various channels. These indicate a decrease of the recorded maximum intensity with distance, which in different regions varies from an inverse fourth or fifth power to an inverse 2.4 power. Theoretically, if there is no absorption or scattering, the energy in any normal mode should vary inversely as the first power of the distance, because the normal modes spread out hori- zontally but not vertically. Since the duration of the signal increases proportionally to the range, the peak intensity of any norma] mode should decrease about as the inverse square of the range; the more refined calculations of reference 24 show an inverse 5/3 power dependence. Thus, although the experimental data are scattered and hard to interpret, there appears to be a discrepancy between theory and experiment. One would, of course, not expect perfect agreement with a theory which neglects absorption and scat- tering in the bottom, especially since most of the ex- periments were conducted over soft bottoms. At one of the stations where shots were made, near the Orinoco delta, it was found that no fre- quencies below about 300 c appeared in the water wave, although a normal dispersion record was ob- served in deeper water in the same locality. The ground wave on the anomalous records was fairly normal. A few shots were made near the Virgin Islands with land between the shot and the hydrophone. These showed a ground wave similar to that which would have been observed in the absence of the land, but the water waves were entirely absent. A related observation is that blasting explosions on land gave weak ground waves at a hydrophone in the sea off- shore, but no water waves. Shots made near Solomons, Md., produced low- frequency disturbances of periods from 0.1 to 0.3 sec which were propagated with a low velocity, about 1,700 ft per sec. These disturbances have been tenta- tively ascribed to the so-called Rayleigh wave, which is a surface-bound wave in the bottom whose propa- gation involves shearing stresses. Such waves fall outside the province of the theory of the preceding Section 9.4.2, which idealizes the bottom as a fluid medium. Chapter 10 SUMMARY ESEARCH ON SOUND transmission during World War II was concerned almost exclusively with the investigation of sound fields which were operation- ally important. More than half of the experimental work was devoted to the sound field of standard echo- ranging transducers operating at frequencies around 24 ke. The purpose of the work was primarily to pro- vide information which could be used to increase the effectiveness of Navy gear already in use on sub- marines and antisubmarine vessels. The instrumen- tation used for research differed as little as possible from standard operational gear; what modifications were made usually represented the minimum neces- sary for quantitative evaluation of the data obtained. Questions which did not seem important opera- tionally, such as the physical cause of the observed attenuation of supersonic sound in the sea, received scant attention in these studies. In the sections which follow, the essential results of these experiments on underwater sound trans- mission are summarized. Section 10.1 lists the defi- nitions of the most important quantities used in describing underwater sound fields. Sections 10.2 and 10.3 summarize what is known concerning the aver- age transmission of supersonic and sonic sound in the sea. In Section 10.4, data on the fluctuationand varia- tion of seaborne sound are summarized. Finally, Section 10.5 provides a brief discussion of probable trends in future research on sound transmission. 10.1 BASIC DEFINITIONS Sound Pressure and Sound Field Intensity A sound wave in a fluid can be described con- veniently in terms of the pressure disturbance which arises in the vicinity of a sound source, travels through the fluid, and is finally received by a hydro- phone. The instantaneous sound pressure is the dif- ference between the instantaneous value of the pres- sure at a chosen location and the mean or equilibrium 10.1.1 236 pressure at the same point. The rms value of the instantaneous sound pressure is usually called the rms sound pressure. Usually, the average is carried out over a time interval which is long compared with the periods of the principal frequencies making up the sound signal. In the case of single-frequency sound, the average is extended over one period (or an inte- gral number of periods). Unless specified otherwise, “sound pressure’’ as used in the technical literature is short for rms sound pressure. Except in the case of standing waves, the rms sound pressure is an ex- cellent measure of the energy carried by the sound wave. At the present time, sound pressure values are uniformly reported in units of dynes per square centimeter. The sound field intensity is defined as the averaged power carried by a sound wave per unit cross section of a wave front. The units in present use are watts per square centimeter. If the radii of curvature of the wave fronts are large compared with the wave- length, then the rms sound pressure and the sound field intensity are connected in excellent approxima- tion by the formula 2 1=1078, (1) pc in which p is the rms sound pressure, p is the density of the fluid in grams per cubic centimeter, c is the sound velocity in the fluid in centimeters per second, and 7 is the sound field intensity. Sound Level The sound field intensity is usually reported on a logarithmic scale. The most common scale for this purpose is the decibel scale. The quantity L, L = 20 log p (2) in which the rms sound pressure p is expressed in units of dynes per square centimeter, is called the sound pressure level or simply the sound level. As de- fined by equation (2), Z is the sound level in decibels above a standard which corresponds to a sound pres- 10.1.2 BASIC DEFINITIONS 237 sure of 1 dyne per sq cm. In the past, sound levels were frequently reported in decibels above 0.0002 dyne per sq cm. 10.1.3 Source Level The source level is a measure of the power output of a sound source on the decibel scale. Briefly, it is the sound level due to a point source at a distance of 1 yd, in decibels above 1 dyne per sq cm. If a point source is located in a homogeneous, nondissipative medium which is infinitely extended in all directions, the intensity of the sound field is inversely propor- tional to the square of the distance from the source, PF 72 (3) This law is called the inverse square law. In terms of the sound level, equation (3) becomes L = S — 20 logr. (4) In these equations, F and S are constants which de- pend on the power output of the source, and r de- notes the distance (slant range) from the source. That S is the source level as defined above can be verified by setting r equal to 1 in equation (4). For real sound sources in real media, equations (3) and (4) are not everywhere valid. Because of the finite extension of an actual sound source, the in- verse square law fails at ranges of the order of the dimensions of the source. Because of absorption of the sound in the medium and because of scattering and reflection from bounding surfaces, it fails at very long ranges. However, there is frequently an inter- mediate range interval for which equation (4) holds. If there is such an interval, then the constant S is considered the source level, even though S may not be the actual sound level at a distance of 1 yd. For a highly directional sound source, such as a standard echo-ranging transducer, the definition of the source level is further specified by the condition that the sound measurements are to be carried out on the axis, that is, the radial line of greatest sound field intensity. Transmission Loss and Transmission Anomaly 10.1.4 The transmission loss H at the range r is defined by the formula H(r) = S— Li), (6) where S is the source level, and Z is the sound level defined by equation (2). The transmission loss de- fined in this way measures the drop of the sound level with increasing distance from the source and has the virtue of being independent of the particular power output of the source. Other parameters of the source, such as operating frequency and directivity pattern, are known to affect the value of the func- tion H(r). The units of H are decibels. The transmission anomaly A is the deviation of the transmission loss from that functional behavior de- manded by the inverse square law of spreading. The defining equation for A(r) ‘is A(r) = H(r) — 20logr = S — L(r) — 20logr. (6) The transmission anomaly vanishes if the inverse square law of spreading is satisfied, and it is positive if the sound level drops off more rapidly than 20 log r. Large positive transmission anomalies, therefore, correspond to poor sound conditions. In sound transmission work, it has been customary to train the projector in a horizontal plane on the re- ceiving hydrophone, but not to tilt the acoustic axis away from the horizontal. Hence, measured trans- mission anomalies will be large for a close deep hydrophone beneath the sound beam. In supersonic transmission work, it has been found that when successive signals are transmitted a few seconds apart over the same transmission path, the received sound intensity is subject to irregular fluctu- ations. Reported transmission anomalies always rep- resent values which have been obtained by averaging over a number of signals received during a brief period so that much of this fluctuation is smoothed out. 10.1.5 Variance of Amplitudes The standard deviation of the individual pressure amplitudes in a sample of signals, divided by the average pressure amplitude for the sample, is called the variance of amplitudes for the sample. This variance is used as a measure of the fluctuation of re- ceived sound intensity. Observed values of the vari- ance are summarized in Sections 10.4.1 and 10.4.2. 10.1.6 Deep and Shallow Water Water is effectively deep when bottom-reflected sound is much weaker than the direct sound; other- wise, the water is effectively shallow. Over the con- tinental shelf (depth less than 100 fathoms) the 238 water is effectively shallow for most situations. Away from the continental shelf, the ocean is always deep when sharply directional sound is used (as in echo- ranging at supersonic frequencies), but may be shallow when listening at audible frequencies to a target at long range. DEEP-WATER TRANSMISSION The transmission loss in the open ocean depends on the way the velocity of sound changes with posi- tion in the sea, since velocity gradients distort the sound beam. These velocity gradients change with time and location, but in any localized region at any given time depend primarily on depth and relatively little on horizontal position within that region. Changes in sound velocity in deep water closely follow changes in water temperature; the effect of pressure changes is relatively slight and usually need not be considered except for transmission to great depths. The following subsections tell of the transmission anomalies expected for various common temperature- depth distributions in the ocean. 10.2 Isothermal Water When the top 50 ft of the ocean are isothermal, transmission anomalies are determined by two major effects, absorption and surface reflection. 10.2.1 Sonic FREQUENCIES At low sonic frequencies, sound is reflected from the sea surface in somewhat the same way as from a flat, perfectly reflecting mirror. The partial can- cellation of direct and surface-reflected sound reduces the sound intensity at long range near the surface. The transmission anomaly at any range may be com- puted from the equation Arhyh —10 log ( — 274 cos = = ”), (7) where h; is the depth of the sound source, he is the depth of the receiving hydrophone, F# is the range from source to hydrophone, and \ the wavelength. The quantity ya, called the effective reflection co- efficient of the surface, is a semi-empirical param- eter; its average value for different frequencies is given in Table 1. A= TaBLE 1. Effective reflection coefficient of the surface. Frequency in cycles 200 600 1,800 Ya 0.8 0.7 0.5 SUMMARY Absorption has little effect on sound transmission at frequencies below 2,000 c. Hicu Sonic AND SUPERSONIC FREQUENCIES At frequencies above 2,000 c, the value of ya to be used in equation (7) is seldom greater than 0.5 in the open sea and is frequently so small that image inter- ference can scarcely be said to exist. Absorption plays an increasingly important role as the frequency in- creases. The transmission anomaly A may be com- puted from the relation. ar ~ 1,000’ where r is the range in yards and where a is the at- tenuation coefficient in decibels per kiloyard. Average values of a at a number of frequencies are given in Table 2. At frequencies above 1,000 ke, the attenua- (8) TaBLeE 2. Attenuation coefficient in the sea. Frequency in ke a in db per kiloyard 3. 4 20 24 30 40 50 60 80 100 500 1,000 6 10 13 18 26 35 150 300 tion coefficient is about three times the value pre- dicted from the viscosity of the water. At frequencies of 24 ke and below, a is more nearly 100 times this theoretical value. Thermocline below Isothermal - Layer 10.2.2 When sound from an isothermal layer passes at grazing angle into a thermocline or temperature layer, where the temperature decreases sharply with increasing depth, the sound rays are bent downward and become more spread out. The increased distance between sound rays in and below the thermocline reduces the sound intensity; this phenomenon is known as layer effect. The transmission anomaly below the thermocline, at ranges out to 4,000 yd, may be computed from the equation 2Ac — ) ar coh? 1,000’ where Ac is the change in sound velocity in the top 30 ft of the thermocline. (If several thermoclines lie above the hydrophone or if the gradient in the ther- mocline increases with depth, Ac is the velocity change in the 30-ft interval giving the maximum value of Ac/h?.) cy is the sound velocity in the surface A = 5 log (: ap (9) DEEP-WATER TRANSMISSION layer; h; is the height of the sound projector above the top of the thermocline, that is, above the top of the 30-ft interval in which Ac is measured; r is the range from projector to hydrophone. The last term on the right is taken over from equation (8); the values of the attenuation coefficient used are given in Table 2. Equation (9) has been checked in detail at 24 ke only, but presumably gives an approximate indication of the anomalies expected below the thermocline at all frequencies above a few hundred cycles. At increasing depths below the thermocline, the anomaly decreases somewhat, the decrease being most marked for the shallower thermoclines. 10.2.3 Temperature Gradients near Surface When temperature gradients are present in the top 50 ft of the ocean, the transmission loss from a projector at 16 ft to a distant hydrophone is corre- lated with the following variables: the sharpness and depth of the gradients (for practical purposes, the decrease of temperature from the surface down to 30 ft); and Ds, the depth at which the temperature is 0.3 F less than the surface temperature. For a deep hydrophone, the temperature gradients at intermedi- ate depths are also of importance. SHARP SURFACE GRADIENTS When the temperature change in the top 30 ft is more than 1/100 times the surface temperature, the sound beam is bent downward by the decrease of sound velocity with increasing depth. The plot of transmission anomaly against range usually shows three different regions as follows: 1. The direct sound field from the projector out to the shadow boundary. The anomaly within the direct sound field is primarily the result of absorption, and equation (8) is applicable. 2. The near shadow zone. Beyond the shadow boundary, the sound intensity decreases very rapidly for some distance. Representative values for this de- crease are 50 db per kyd at 25 ke and about one- third this at 5 kc. These coefficients of attenuation in the shadow zone are apparently about half the values estimated from the theory of diffraction by a smooth velocity gradient. The range to the shadow boundary increases with depth in accordance with ray theory, but seems to be systematically somewhat less than predicted. 3. The far shadow zone. With standard echo-rang- ing gear and pulses 100 msec long, the transmission 239 anomaly of scattered sound at ranges of several thousand yards is about 50 db. Thus when the trans- mission anomaly of the direct or diffracted sound in the shadow zone exceeds about 50 db, the observed sound is scattered sound, with an anomaly which does not depend strongly on further increases in range. This scattered sound is incoherent. For short pulses the intensity of this scattered sound is propor- tional to the pulse length; it becomes negligibly small for explosive sound. To predict the anomalies expected under given temperature conditions, it is simplest to use curves of average anomalies for such conditions. Since un- explained deviations are frequently found between individual anomalies and the predictions of ray theory, use of average curves gives results about as accurate as the more elaborate methods. An example of this approach is Figure 40 of Chapter 5, where average curves are given for different values of Dz, the depth at which the temperature is 0.3 F less than the surface temperature. WEAK SURFACE GRADIENTS When the temperature change in the top 30 ft is less than 1/100 of the surface temperature, but gradi- ents are present in the top 50 ft, the division of the sound field into the three regions described previously is usually not observed. Since a small change in such temperature conditions may lead to a large change of transmission anomaly, the observed anomalies are highly variable and can neither be compared with theory nor predicted practically with much accuracy. Average anomalies for different values of D2 are given in Figure 49 in Chapter 5 for a shallow hydrophone. For a deep hydrophone, below the thermocline, equa- tion (9) may be used for approximate results. Sound Channels When the velocity of sound above and below the sound source is appreciably greater than the velocity at the source, the sound rays which leave the source with small inclinations will propagate out indefinitely without surface or bottom reflections, bending back and forth but always remaining within some fixed in- terval of depths. 10.2.4 SURFACE SOUND CHANNELS When the sound projector lies below a sharp nega- tive gradient and above a sharp positive gradient, sound channel effects should be marked, with regions of alternately high and low anomaly found out to 240 SUMMARY considerable ranges. When a sharp gradient lies just above the projector and a layer of nearly isothermal water 100 ft or more in thickness lies below, ray theory predicts that the sound bent back up by the positive velocity gradient in the isothermal layer should be focused at shallow depths and long ranges, thus giving anomalously high intensities. Observa- tions made under these conditions show that the transmission anomaly on a shallow hydrophone is sometimes as much as 40 db less than normal over a narrow range interval several thousand yards away. The details of these observed effects are not in good agreement, however, with the exact predictions of ray theory. Derr SounD CHANNELS At a depth of several thousand feet there is usually a deep sound channel. The effect of pressure on sound velocity increases the velocity at greater depths, and a thermocline usually present closer to the sur- face increases the velocity at shallower depths. Sound of frequency less than 200 cycles, for which the ab- sorption is very low, has been observed to propagate out for several thousand miles in such a deep channel. With small explosive charges, the arrivals of the different pulses agree with the different rays pre- dicted theoretically. The largest number of arrivals, with the highest observed intensity, occur just be- fore the observed sound stops entirely; these last arrivals are the rays coming almost straight along the axis of the channel. 10.3 SHALLOW-WATER TRANSMISSION In shallow water, the transmission of underwater sound is determined primarily by the character of the bottom, and by the frequency of the transmitted sound. The state of the sea is a much more important factor than in deep water. Temperature gradients are of secondary importance. There are two situations in which sound conditions do not differ appreciably from those found in deep water: (1) soft MUD bot- tom; (2) strong positive velocity gradients (PETER pattern) below a directional sound source. In both these cases, transmission is very nearly the same as in deep water with the same thermal conditions. 10.3.1 Sonic Frequencies Most of the information on the transmission of sonic sound in shallow water was obtained in harbor surveys. The data obtained may be summarized as follows. No systematic difference was found between dif- ferent types of bottoms, with the exception of soft MUD, which turned out to be a poor reflector. All other bottoms apparently reflect equally well. Transmission over sloping bottoms in the presence of downward refraction tends to be poor, in agree- ment with theoretical predictions. Over flat bottoms, at ranges greater than the water depth and out to several thousand yards, average sound transmission can be best represented by an inverse 1.5th power law of spreading plus an attenua- tion which appears to increase roughly linearly with the frequency up to about 20 ke. The transmission loss is, thus, given roughly by a formula ies Males —= a? Dott GO) 4,000 my where r is the range in yards, f is the frequency in kilocycles, and C is a constant independent of the range r. 10.3.2 Twenty-four Kilocycles In moderately shallow water, and in the presence of any bottom but MUD and soft SAND-AND-— MUD, the transmission anomaly can usually be represented in fairly good approximation by a straight line. For wind forces 0 to 2, transmission anomalies increase with the range at a rate of 5 db per thousand yards over STONY and SAND bot- toms, and at a rate of 6 db per thousand yards over ROCK bottoms. About half of all the runs carried out yield values which differ from these average values by no more than 2 db per kyd. The following special results are also worth noting. (1) For heavy seas, transmission is somewhat worse than for light seas. For wind force 3, about 1 db per kyd should be added to the attenuation coefficients given above. (2) Over sloping bottoms and in the presence of negative gradients, transmission is poor. The transmission anomaly may increase with the range at a rate exceeding 10 db per thousand yards. (8) In shallow isothermal water, transmission is at least as good as in deep isothermal water. (4) In very shallow water (5 fms deep), a series of experiments carried out over SAND gave very poor transmission; the anomaly increased at the rate of about 16 db per kyd. FUTURE RESEARCH 10.3.3 High Supersonic Frequencies As far as is known, transmission in shallow water at high supersonic frequencies is similar to that at 24 ke, except for greatly increased absorption losses; transmission anomalies in the presence of negative gradients are linear and their slopes are somewhat higher than those in deep isothermal water. 10.4 FLUCTUATION AND VARIATION The transmission loss measured at any instant in the ocean will usually differ from the value found several seconds earlier. This rapid change of sound level is called fluctuation. Measured transmission losses and transmission anomalies are averaged to smooth out this fluctuation. Fluctuation is invariably observed in the trans- mission of single-frequency supersonic signals trans- mitted over a path atleast 100 yd long. Fluctuation is negligible over transmission paths of the order of 5 yd. Little is known concerning fluctuation over inter- mediate path lengths. For frequencies of 5 ke and less, fluctuation appears to be less pronounced than at frequencies of 10 ke and higher. The summary which follows is concerned only with the fluctuation of supersonic signals transmitted over paths at least 100 yd in length. 10.4.1 Variance with Shallow Projector For a projector at a depth of 16 ft, the direct sound from an echo-ranging projector cannot be distin- guished from the surface-reflected sound. The fluc- tuation is large and inexplicably variable. Observed values of variance average 40 per cent with an inter- quartile spread of about 20 per cent. The variance at 24 ke is significantly correlated with the variance at 16 ke or 60 ke, the coefficient of correlation being about 0.7. 10.4.2 Variance with Deep Projector For a deep projector and a deep hydrophone, the direct signal can be resolved from the surface-re- flected signal. The observed fluctuation of the direct signal is small; observed values of the variance at 24 kc lie between 5 and 10 per cent and may result from the variability of the measuring equipment. The surface-reflected pulse is highly variable with a variance between 50 and 70 per cent. With explosive pulses, the direct sound can be re- solved from the surface-reflected pulse even at shal- 241 low depths. The observed variance for the direct pulse is about 1 or 2 per cent if the transmission path lies wholly in an isothermal layer, but up to 20 per cent if part of the transmission path lies in the thermocline. 10.4.3 Rapidity of Fluctuation The time during which the sound level is not likely to change appreciably is also variable, but seems to increase with increasing range. At a fixed range of less than a few hundred yards, the transmission loss for a shallow sound projector changes by about 20 per cent on the average during 0.5 sec. At a fixed range of several thousand yards in the direct sound field, the average time for a 20 per cent change might be 2 sec; while in the shadow zone, this average time is likely to be nearer 0.02 sec. 10.4.4 Variation Slow changes in the (averaged) transmission of sound in the sea, which take place in several minutes and which cannot be explained in terms of observable changes in the vertical temperature pattern, are called variations. It has been found that at 24 ke the variation between two transmission runs about 20 minutes apart has an average value of about 4 db.if only pairs of transmission runs are considered in which the bathythermograph pattern is significantly the same. This average value for the variation does not appear to depend significantly on range. 10.5 FUTURE RESEARCH During World War II research on the transmission of underwater sound has been largely devoted to the empirical investigation of certain practical problems. A wealth of detailed information has been accumu- lated on the transmission loss of sound from a stand- ard echo-ranging projector under conditions likely to be observed in practice. Although this information has been useful in subsurface warfare, it has not led to any complete understanding of the physical proc- esses involved in underwater sound transmission. For example, the average attenuation in deep iso- thermal water near San Diego has been extensively measured, but the causes of this attenuation are completely unknown. In the years to come, research in this field will probably change its character. The quest for empiri- cal data on some particular situation has been carried 242 about as far as usefulness requires, and future studies will most profitably be directed to a more funda- mental investigation of the basic factors underlying the observed data of underwater sound transmission. Without such a reorientation of the basic research program, it will be impossible to predict the behavior of underwater sound under new and unexplored con- ditions. Suppose, for instance, that sound gear using a nondirectional supersonic projector were to be pro- posed. The transmission loss for the sound from such a system could not be predicted definitely from pres- ent data, which are all obtained with directional supersonic sources. To make such predictions would require some knowledge of the importance of the scattering of sound through small angles. Similarly, the attenuation of sound transmitted from a deep projector to a deep hydrophone cannot be predicted from the present empirical data taken with shallow projectors, but might be estimated if the basic causes of attenuation were known. In principle, the answer to any practical question about underwater sound transmission could be ob- tained by a program of measurements planned wholly for the purpose of answering that question. When haste is required, this is frequently the quicker method. When time is available, however, such answers can most efficiently be provided by a broad program designed to yield a physical understanding of what is happening. Such a program makes it ulti- mately possible to answer not one but a large number of practical questions. Thus, in the long run, im- proved technology can best be based on a foundation of long-term fundamental research. This final section gives a brief discussion of some of the basic physical factors that may be expected to be important in underwater sound transmission and also treats the type of observations that might be expected to give meaningful information on these different factors. 10.5.1 Basic Factors The wave equation, equation (27) of Chapter 2, presumably governs in good approximation the prop- agation of sound waves in the interior of the ocean. It appears reasonable at first to investigate solutions of the simple wave equation, taking account of the presence of velocity gradients in the sea and of the reflections from sea surface and sea bottom. If the results are in flagrant disagreement with observa- tions, then the effects of the approximations entering SUMMARY into the derivation of the wave equation must be in- vestigated in detail. Apart from the validity of the wave equation as such, it is known that the body of the ocean contains scatterers (their nature uncertain) which deflect a fraction of the sound energy from its original direction of propagation. Furthermore, the observed absorption at supersonic frequencies far exceeds the value predicted on the basis of viscosity dlone, necessitating the assumption of additional dissipative processes. The most important problems of underwater sound transmission may thus be summarized under the following four headings. 1. The effects of velocity gradients in the sea. 2. Absorption and scattering in the volume of the sea. 3. Surface reflection. 4. Bottom reflection. Each of these topics is discussed in the following subsections. Sounp VELOCITY The velocity of sound is known as a function of temperature, pressure, and salinity and thus can be calculated at any point in the ocean where these physical quantities are known. The refraction effects produced by smooth vertical changes of temperature have been extensively investigated theoretically, and the results are in general qualitative agreement with the observations. Since the agreement is not com- plete, however, other effects must also play an im- portant part. While the pressure is known as a func- tion of depth, changes in temperature and salinity over distances of a few feet have not been extensively measured, and the acoustic effects to be expected from such changes have not been thoroughly ex- plored. Microstructure of temperature and perhaps also of salinity may have an important effect on sound transmission, especially when the smoothed vertical gradient of sound velocity is small. Also, microstruc- ture probably accounts for some part at least of the observed fluctuation of transmitted sound. ABSORPTION AND SCATTERING The attenuation observed in deep isothermal water is presumably the result of absorption, that is, some dissipative process which converts sound energy into heat. Since the attenuation observed at 24 ke exceeds by a factor of about 100 the valve predicted on the basis of shear viscosity alone, the principal cause of the observed attenuation must be some other mech- FUTURE RESEARCH anism. Among the dissipative mechanisms consid- ered are compression viscosity (which, however, cer- tainly is not the principal factor at 24 ke), gas bub- bles present in the water, fish bladders, plankton, and thermodynamically irreversible chemical reactions, such as the hydrolysis of dissolved salts. Gas bubbles and other inhomogeneities would not only absorb but also scatter sound. That scatterers are present in the sea is known. Scattering may ac- count for part of the attenuation of highly collimated beams and also is probably responsible for most of the sound observed in predicted shadow zones in the presence of negative velocity gradients. All hypotheses concerned with the cause of the absorption of sound as well as with the role of volume scattering on sound transmission are at present largely speculative. Until further experimental and theoretical work has provided a scientific under- standing of the mechanisms involved, it will not be possible to predict with confidence the attenuation under many different conditions. SurFacr REFLECTION The change in density at the sea surface is known and is so large that for most practical purposes the density of air may be set equal to zero; that is, the surface is almost a perfect reflector of sound. The complexity of surface-reflected sound arises from the complicated form of the ocean surface. In principle, it is simply a mathematical problem to compute the sound reflected from any surface of known properties. In practice, observations are unquestionably re- quired. A thorough understanding of this topic would be important in studies both of fluctuation and of the average transmission anomaly in the surface layer. Bortom REFLECTION The ocean bottom may have a topography equally as complicated as the ocean surface. In addition, the relative change in the elastic parameters and in density across the interface is much less extreme than across the ocean surface, and the detailed values of these changes must be considered. Since the physi- cal properties of the bottom may vary with position, both vertically and horizontally, the problem of bot- tom-reflected sound can be very complicated physi- cally as well as mathematically. In certain regions, where the bottom is flat, and of uniform composition, the acoustic phenomena are perhaps capable of being understood. Bottom-reflected sound is obviously im- 243 portant in many situations, especially when the direct sound is weakened by temperature gradients. Methods To understand the physics of underwater sound transmission, each problem must be given separate consideration. The following methods may be ap- plicable, however, to the investigation of a con- siderable variety of problems. 10.5.2 OcEANOGRAPHIC MEASUREMENTS An important part of any basic research on sound in the sea must be the investigation of the physical properties of the medium in which the sound is trans- mitted. It is in terms of these properties that the acoustic data are presumably to be interpreted. In the first place, detailed measurements of the factors influencing sound velocity seem desirable, especially temperature measurements showing the full detail actually present in the sea. In the second place, detailed measurements of the shape of the ocean surface are required before any attempt can be made to explain surface-reflected sound; in par- ticular, statistical information on the spectrum of the surface water waves present during any interval seems desirable. In the third place, complete physical data on the ocean bottom (on topography, composi- tion, porosity and compactness, etc.) are required to interpret physically the data on bottom-reflected sound. Finally, it may be necessary to make a variety of physical measurements on ocean water as part of the attempt to identify the cause of absorption. CoNTROLLED Acoustic MEASUREMENTS The experimental techniques of underwater acous- tics research will probably be developed in a number of directions. Greater emphasis may be expected on detailed accuracy of the acoustic data; probable errors of several decibels for a transmission anomaly can presumably be considerably reduced. Measure- ments involving smaller samples of the ocean may perhaps be anticipated with relatively complete oceanographic data obtained for the small samples investigated. Some such experiment might be devised for measuring the sound absorption in a relatively small volume. Another possible development is along the lines of multiple measurement, in which many different items are measured almost simultaneously. For example, the inclination of the wave front might be measured at the same time as its intensity with 244 simultaneous recordings at a number of different frequencies. Increasing complexity of the necessary equipment may probably be anticipated. It is possible that explosive sound may be useful as a research tool. As pointed out in Chapter 9, short explosive pulses provide resolution of the direct and reflected pulses even at nearly grazing angles and also reveal clearly any multiple ray paths that may be present. By means of Fourier analysis, it is possible also to obtain with explosive sound many of the re- sults which could be obtained by simultaneous trans- mission of many single frequencies over the entire spectrum. Finally, the high sound intensities possible with explosive pulses can provide data at longer ranges than are possible with standard sound pro- jectors. Thus, explosive sound would appear to be a SUMMARY valuable tool of underwater sound research, deserving wider application than it has had in the past. Regardless of what specific technique is used, the primary requirement for any basic experiment is that it be devised to give answers to certain physical questions rather than to operational problems. To satisfy this requirement, the theory underlying each experiment must be studied in detail before the ex- periment is actually performed to make sure that the results obtained will be significant. Considerable ingenuity may be required to find means for isolating the effects of the different factors involved in order to investigate them separately. It is only by such carefully designed experiments that our general un- derstanding of sound in the sea can be continually increased. PART II REVERBERATION ie, ae a, Mer a Gea Rey Pe ices hie ne Lee. dang I e- : ; ‘AD \ ee ne _ » Sr SOB ewy @ * é Clue Ue pa at kai? on Poe ae bed Se a ; \ i om : \ ea he We “® jana wees li, ya) ia eal ey = ae “ TIME ———> Figure 1. Decay of reverberation intensity. Interference arises because of the wave-like char- acter of sound. Because of interference, the reverbera- tion from n similar scatterers illuminated by the ping does not always have n times the intensity which would be observed if only one scatterer were present. At a particular instant, the sounds returning from the n scatterers may interfere destructively at the hydrophone so that the n sounds annul one another completely. Or, the n individual sounds may all com- bine constructively, so as to give n? times the in- tensity which would have been due to one of the scatterers alone. These are two extreme cases; but in general the n scatterers together may produce com- posite intensities ranging all the way between these two limits. The resultant intensity that occurs in a given situation depends in a critical way on the exact positions of the scatterers relative to one an- other. Since in the actual ocean the separations of the scatterers change from one portion of the ocean to another, it is plain that the reverberation from a ping should not change smoothly with time; rather, it should change irregularly, with bursts where the interference of the individual tiny echoes is primarily constructive, and relative silences where the inter- ference is primarily destructive. There is yet another complication. If the echo- ranging ship and the scatterers were all fixed in posi- tion, the pattern of bursts and silences, although com- plex, would not change from one ping to the next. PREVIEW However, the echo-ranging vessel is usually rolling and pitching; and the scatterers in the ocean, that is, the air bubbles, the suspended solid matter, the tur- bulent regions, and the regions of temperature fluctu- ation are all free to move. Thus the interference pat- tern will vary widely from one ping to the next. Since the exact distribution of the scatterers in the ocean cannot be predicted, the irregularities of reverbera- tion can be described adequately only by statistical methods. That is, if we are to be realistic, we can attempt to assign values only to the average rever- beration intensity, and the average reverberation variability. For example, the inverse square law for the decay of reverberation from the volume of the sea, which was developed previously, could only be valid for the average reverberation from a series of pings; it would be nonsense to expect the reverbera- tion from an individual ping to decay smoothly in exact agreement with this law. In order to make clear the meaning of reverbera- tion, the scatterers in the water, such as air bubbles, suspended solid particles, and the like, have been assumed to be very nearly uniformly distributed throughout the sea volume. We have really been describing what is known as volume reverberation, that is, reverberation due to scatterers in the body of the water. However, there is also reverberation due to scatterers at the ocean surface and ocean bottom. These three types of reverberation, volume, surface, and bottom reverberation, behave quite differently on the average. For example, they set in at different times. Volume reverberation is evident at the mo- ment the ping is put into the water, and surface and bottom reverberation do not set in until the sound has had time to travel to these bounding surfaces and re- turn to the transducer. There are other differences as 249 well, which are discussed in the main body of the text. 11.3 PREVIEW The next six chapters summarize the reverberation studies carried out under the auspices of NDRC through the spring of 1945. Chapter 12 derives theo- retical formulas for the average reverberation in- tensity on the basis of assumptions which are ex- plicitly stated and whose validity is critically ex- amined. In that chapter, the expected intensities of reverberation from the volume, surface, and bottom are examined separately for their theoretical depend- ence on many other variables besides time; some of these other variables which play a major part in de- termining the reverberation intensity are the direc- tivity characteristics of the transducer, the trans- mission loss between the projector and the scatterers, the intensity and duration of the projected signal, and the scattering power of the portion of the ocean under. consideration. Chapter 13 describes the field and laboratory methods which have been developed for the measurement of reverberation intensity and the analysis of the resulting data. Chapters 14 and 15 summarize the observational information on volume, surface, and bottom reverberation which has been obtained by use of these experimental and analytical techniques and compare these results with the theoretical predictions of Chapter 12. In Chapter 16 the variability of reverberation is examined, both theoretically and in the light of the observed data, and the frequency characteristics of reverberation are described. Finally, in Chapter 17 the most im- portant results presented in the body of Part II are summarized. Chapter 12 THEORY OF REVERBERATION INTENSITY es AMOUNT and character of the sound heard or recorded as reverberation depends not only on the properties of this sound in the water, but also on the nature of the gear in which the reverberation is received. The intensity of the reverberation actually heard or recorded, after the sound in the water has been converted to electrical energy by the receiver, amplified, and passed to the ear or recording scheme, will be called the “reverberation intensity,” and will be given the symbol G. As so defined, G equals the watts output across the terminals of the receiving gear. Although in practice the reverberation may be measured in terms of volts, or the height of a line on a motion picture film, or some other convenient quantity, these measurements can always be con- verted to watts output by the use of known param- eters of the receiver system. In general, the reverbera- tion intensity G is a function of time and is related to the sound intensity in the water by such parameters of the receiver system as receiver directivity and re- ceiver gain. Since G depends on the gear parameters, its abso- lute magnitude is usually not of great significance in research. For this reason it is customary to relate G to the reverberation intensity which would be regis- tered under certain standard conditions. This stand- ard reverberation intensity, in decibels, is called the “reverberation level’? and will be defined precisely later. Reverberation levels are more useful than re- verberation intensities for comparing measurements made with different systems. This chapter is devoted to a theoretica] analysis of expected reverberation intensities and reverberation levels. Formulas will be derived giving the depend- ence of these quantities on various gear parameters, oceanographic conditions, and elapsed time since emission of the signal. First, however, we must dis- cuss the scattering of sound, since scattering is usually regarded as the fundamental source of re- verberation. 250 12.1 SCATTERING OF SOUND The analysis in this chapter is based on some very definite assumptions about the nature of reverbera- tion. It is assumed that not all of the sound in the outgoing ping proceeds outward in accordance with the elementary theory in Chapters 2 and 3, but that some of the sound is “‘scattered” in other directions. The reverberation is thought to be that part of this scattered sound which returns to the transducer. Volume, surface, and bottom reverberation are as- sumed to result, respectively, from scattering in the volume of the ocean, at the surface of the ocean, and at the ocean bottom. In an ideal unbounded fluid in which the sound velocity is everywhere the same, it is shown in Chapters 2 and 3 that sound always travels outward from its source along straight lines. In such a medium, then, scattering never occurs and no reverberation should be heard. There is no reason to doubt the validity of this theoretical result. Scattering arises because the ocean is not an ideal unbounded medium with constant sound velocity. It can be shown theo- retically that whenever a sound wave travels through a portion of a fluid where the density or sound velocity varies with position, some energy is radiated in direc- tions differing from the original direction of the wave. Similarly, whenever a sound wave in the ocean im- pinges on a new medium, for example a bubble, in which the density or sound velocity differ from their values in the surrounding water, energy is radiated in directions differing from the original wave direc- tion. Whether or not this deviated energy is called “scattered energy” is to some extent a matter of definition. If the inhomogeneity in density or sound velocity extends over a large region of space, a sound beam traveling through the medium may be changed in direction with little or no loss of energy from the beam; if this happens the sound wave is not regarded SCATTERING OF SOUND as scattered. Such changes in beam direction occur, for example, when the beam is refracted by a tem- perature gradient which is a function of water depth only. Similarly, reflection from an infinitely smooth plane surface sharply changes the direction of the beam; but since all the energy theoretically remains in the beam the process is not termed scattering. However, there are inhomogeneities of small size, such as air bubbles, or small irregularities in the ocean surface, which cause energy to be “detached” from the main sound beam, that is, to travel in a dif- ferent direction from that of the main beam. This detached energy, which differs in direction from the main beam and which results from local inhomogenei- ties in the ocean or bounding surfaces, is called scattered energy. It is apparent from this discussion that the distinction between scattered energy and nonscattered energy is not always too clear; for ex- ample, bottom reverberation received from under- water cliffs might more properly be called reflected energy rather than scattered energy. This possible confusion in nomenclature is of no immediate con- cern. The important point is that the existence of reverberation is predicted by theory from the known inhomogeneity of the ocean. The magnitude of the reverberation reaching the water near the receiver is calculable, in principle, by solving some differential equation which, with ap- propriate boundary conditions, takes into account the inhomogeneity of the ocean. Since temperature gradients and density gradients are small in the body of the sea, the differential equation would be the wave equation 0p 2 0p =) = GS 4 SS 1 0 Ox? a oy? + ; (1) where p is the sound pressure, and c is the velocity of sound at the time ¢ at the point whose coordinates are (x,y,z); this equation was derived and its applica- tion discussed in Chapters 2 and 3. The presence of solid particles and air bubbles, and the nature of the roughnesses in the ocean surface and bottom, are described by the boundary conditions; these condi- tions give the positions at each instant of all the surfaces at which the density and sound velocity change discontinuously and the amounts of these changes. Because of the complexity of the ocean, neither the function c(a,y,z,t) nor the boundary conditions are precisely known. The bathythermograph gives the broad outlines of the temperature distribution. How- 251 ever, the locations and magnitudes of small local temperature gradients cannot be determined with the bathythermograph, although such “thermal micro- structure” is known to exist.! Furthermore, the posi- tions of bubbles, solid particles, waves, etc., change continually and unpredictably. Even if the sound velocity and the boundary conditions were specified exactly, it would be an insuperable mathematical problem to solve equation (1) rigorously for p. Thus, theoretical formulas for reverberation cannot be de- rived by solving equation (1) with boundary condi- tions. Instead, we shall base our mathematical analysis of reverberation on several simplifying assumptions. Since a great deal is known about the general proper- ties of solutions of equation (1), reasonable assump- tions can be made about reverberation, even though a complete solution of equation (1) cannot be ob- tained. The principal assumptions which we shall use are: 1. Reverberation is scattered sound. 2. Scattering from an individual scatterer begins the instant sound energy begins to arrive at the scatterer and ceases at the instant sound energy ceases to arrive at the scatterer. 3. Multiple scattering (rescattering of scattered sound) has a negligible effect on the intensity of the received reverberation. In other words, all but a negligible portion of reverberation is made up of sound which has been scattered only once. 4. The intensity of the sound scattered backward from a small volume element dV is directly propor- tional to each of the three following quantities: the volume occupied by dV, the intensity of the incident sound, and a “backward scattering coefficient”’ desig- nated by m, which depends only on the properties of the ocean in the neighborhood of dV. 5. The average reverberation intensity, which is a function of the time elapsed since the emission of the ping, is the sum of the average intensities received from the individual scatterers in the ocean. To ex- press this assumption in mathematical form, let g(t)dV represent the average intensity, ¢ seconds after the emission of a ping, of the reverberation re- sulting from scattering in the volume element dV only. Then the average intensity of the reverbera- tion received from the entire ocean, at the time in- stant ¢, is given by G(t) = fowav (2) where the integral is taken over the entire ocean. It 252 will be seen later that the function g(t) is zero every- where in the ocean except inside a thin, roughly spherical shell; this shell has the projector as its center, an average radius depending on the value of f, and a thickness depending on the length of the emitted signal. With these assumptions, it is possible to derive theoretical formulas for the reverberation intensity as a function of gear parameters and oceanographic conditions. This chapter will be concerned only with the average reverberation intensity to be expected in a series of pings. No attempt will be made in this volume to predict the level of the reverberation from one specified ping. The observed levels of the rever- beration from pings only a few seconds apart often differ by many decibels. Discussion of the average value of this fluctuation will be deferred until Chapter 16. Although the above assumptions can be defended, they are by no means obvious and require elabora- tion. In particular, it is necessary to specify carefully the meaning of the terms ‘backward scattering coefficient” and ‘‘average reverberation intensity,” which are introduced in assumptions 4 and 5. The average reverberation intensity is defined as the average from ping to ping. That is, if we measure the reverberation intensity on a succession of n pings with each measurement performed at a definite time ¢ after midsignal, then the average reverberation in- tensity at time ¢ is G@ = SG (3) where G,(t) is the reverberation intensity measured on the zth ping; the symbol 2 means summation over all the pings. The number of pings averaged must be large enough to smooth out the effects of fluctuation, yet not so large that such external factors as wave height, water depth, and amount of suspended matter in the ocean can change materially during the series of pings. In practice, the number of pings averaged has usually been between 5 and 12, with not more than about 60 seconds between the first ping and the last. Some discussion of the validity of this averaging procedure is given in Chapter 16. Also, we must specify more exactly the meaning of the backward scattering coefficient m. If we con- sider a volume V made up of many small volume elements dV, then, strictly speaking, dV can scatter sound only if sound energy reaches it and if it con- THEORY OF REVERBERATION INTENSITY tains some scattering substance. Thus, if dV lies en- tirely within some rigid scatterer, such as a bit of metallic dust, practically no sound reaches dV be- cause almost all the sound impinging on the scatterer is scattered at the surface of the scatterer. Another difficulty is that there is no way to predict the loca- tions of the scatterers on any one ping. For these reasons it is impossible to predict how much scat- tering from a specified volume element dV will occur on any one ping. We can, however, speak of the average scattering power of the ocean in the neigh- borhood of dV. The backward scattering coefficient m for a volume JV, in the neighborhood of and in- cluding dV, is defined as follows. Let V be insonified by a plane wave of unit intensity n times in succes- sion. Let b; be the energy scattered per second per unit solid angle in the backward direction, during the 7th trial, by the volume V. Then m for V is defined by m m 1 al Te dV = noe (4) The factor 47 is introduced so that in cases where the scattering is the same in all directions, the average amount of energy scattered per second in all direc- tions will be just mV. With the definition of m given by equation (4) that the average energy scattered by dV per second per unit incident intensity per unit solid angle in the background direction is just (m/42)dV, it also follows that (m/4m)dV is just the intensity of the scattered sound from dV at unit distance from dV when the incident sound has unit intensity. Evidently the volume V in equation (4) cannot be chosen arbitrarily if the definition of m is to have any significance. V must be chosen small enough that m can vary with position in the ocean and can thereby indicate the variation with position of the average number and strength of the scatterers. However, since it is desired that m not vary discontinuously from point to point, V must not be chosen too small. Because so little is known about the scatterers re- sponsible for reverberation, it is difficult to formulate the conditions on V any more precisely than this. Some further discussion of the significance of as- sumption 4, as well as the other previous assump- tions, is given in Section 12.5. That section is not a complete treatment of the problems involved, but may assist the reader to understand the physical ideas underlying the derivation of the theoretical formulas for reverberation. VOLUME REVERBERATION VOLUME REVERBERATION Volume reverberation is defined as sound scattered back to the transducer by scattering centers in the volume of the sea. Let a transducer be located at O in deep water far from both the sea surface and sea bottom. This transducer sends out a pulse of sound, or ping, of duration r. Because the ping is of finite duration, a large part of the sound energy at a time t/2 seconds after midsignal will be contained between two closed surfaces S;, S:, portions of which are shown in Figure 1. If the sound velocity c in the 12.2 Portion of ocean scattering sound at time Figure 1. t/2. ocean is everywhere the same, these surfaces are spheres centered at O, with radii ct/2 — cr/2 and ct/2 + cr/2. With refraction, however, the surfaces may be far from spherical. The reason for saying that a “large part” instead of “‘all’’ the energy in the ocean lies in the volume between S; and S; is that the very existence of reverberation shows that some sound, scattered earlier out of the ping, does not lie between these two surfaces at the time ¢/2. However, accord- ing to assumption 3 in Section 12.1, the amount of the previously scattered sound which is rescattered back to the receiver is negligible. Therefore, because of assumption 2, the only scattering taking place at time ¢/2, which is important in producing reverbera- tion, occurs at those scatterers located within the volume SS, defined by the transmission laws of ray acoustics. 253 Now consider the sound scattered at time t/2 by the volume S;)S,. Obviously, all this sound will not return to the receiver at the same instant since the sound scattered near S; travels a shorter distance than does sound scattered near S». It is shown in Section 12.5.5 that it can be assumed as a conse- quence of Fermat’s principle,” that the average travel time of sound along the path from the transducer O to any point X in S,S, equals the average travel time from X back to O. Using this result, we can readily delimit the region where the sound returning to the transducer at the time ¢ is scattered backward. If 7 is the signal duration, and if all times are meas- ured from the middle of the signal, the signal emis- sion starts at time —7/2 and ends at time 7/2. The sound emitted first, at time —7/2, and returning as reverberation at time ¢, is scattered backward at some definite time which we shall call ¢’. The corre- sponding travel time 7, out to the scatterers must be ’ + 7/2; the travel time back to the receiver must have an equal value because of our assumptions; and the sum of these two travel times and the time of emission —7/2 must equal ¢, the time at which the reverberation is received. Thus, T Te t T 2 t’ er v=-—- (v+5 2 : 2 4 and therefore t T hy S =ab =o 5 = a (5) Similarly, the sound emitted last, at time r/2 and re- turning as reverberation at the time ¢, must be scattered at a time ¢” given by pee aatig Se 2A and the corresponding travel time 7? is t’’ — =) or t T T, =-——- 6 D= 5-4 (6) Thus, all the scatterers effective in producing the reverberation at time ¢ must lie between a pair of surfaces out to which the one-way travel times are, respectively, {/2 — 7/4 and ¢/2 + 7/4. These sur- faces are indicated in Figure 2 by the labels S; and S83. If there is no refraction, these surfaces S; and S are spheres with radii c(t/2 — 7/4) and c(¢/2 + 7/4) respectively; the volume S;S; is thus a spherical shell with average radius ct/2 and thickness cr/2. In general, even with refraction present, the volume S,S; is about half the volume SiS; that is, the 254 volume in which the effective scatterers lie is about half the volume illuminated by the ping at time t/2. We shall now determine the intensity of the rever- beration which reaches O at time ¢ from the volume element dV, located at X in Figure 2. We use the Fiaure 2. Diagrams used in developing volume rever- beration‘formulas. system of coordinate axes indicated in Figure 2; the origin is at O, and the ray from O to X leaves O with spherical coordinates (9,¢) defined by the tangent to the ray at the origin. As drawn in Figure 2, 6 is the angle made by the ray OT with the horizontal plane; thus 6 is the complement of the polar angle made by OT with the vertical direction OH. The amount of energy which the projector radiates per second into the solid angle dQ in the direction (6,¢) is just Fb(6,¢)dQ, where F is the emission per unit solid angle in the direction of maximum emission and b(6,¢) is the pattern function of the projector defined in Part I, Section 12.4.4. The sound intensity at unit distance (1 yd) from O, along this ray, is therefore just Fb(6,¢). If J, is the intensity at the point X, the “intensity diminution” between the point one yard from the projector and the point X may be denoted by h and defined by THEORY OF REVERBERATION INTENSITY it ”= 76,8) The quantity h is simply related to the (positive) decibel transmission loss H between the point 1 yd from the projector and the point X by the formula H = —10logh. (8) The small tube of rays emitted by the projector into the solid angle dQ will have, at the point X, a cross-sectional area, perpendicular to the sound rays, which may be denoted by dS. Let the volume element dV at X be defined as a cylindrical element whose base area is dS and whose height is ds, an infinitesimal extension along the direction of the wave propaga- tion (see Figure 2). The volume included by dV is therefore given by dV = dSads. (9) On the average, the sound which returns to O from X traverses the same ray traced out by the sound which was incident on dV; this assertion, a conse- quence of Fermat’s principle, will be defended in Section 12.5.5. Therefore, the scattered sound giving rise to reverberation has been scattered directly “backward.”” By the definition of the backward scattering coefficient m, the intensity at a point, 1 yd from dV, of the sound which returns to O from dV is just m/42 times the incident sound intensity at X, times the volume of dV, or (7) ”" nFb(0,6)dSds. 4a If we now define h’ as the intensity diminution be- tween a point 1 yd from dV and the receiver at O, the intensity of the sound reaching O from dV is m —hh'Fb(6,¢)dSds. (10) 4a The expression (10) gives the intensity of the sound scattered backward from dV in the water at the re- ceiving hydrophone. Let F’ be the output of the re- ceiver in watts when a plane wave of unit intensity is incident on the receiver in the direction of its maximum response. A plane wave of unit intensity from some other direction (6,¢) will stimulate the re- ceiver to an output of F'’b’(6,¢) where b’(8,¢) is called the pattern function of the receiver. Finally, a plane wave of intensity J incident on the hydrophone from the direction 6,¢ will cause a watts output at the terminals of the receiver of J -F'b'(6,9). (11) VOLUME REVERBERATION With customary receivers of ordinary dimensions, the scattered sound returning from the ranges of interest (say greater than 50 yd) is for all practical purposes a plane wave at O. Thus, using the results (10) and (11), we have Watts output from dV = LHF -F'b(0,6)b'(6,6)4Sds. (12) 470 In equation (12), h’ has been set equal to h. This as- sumption that the transmission loss is the same on the outgoing and returning journeys will be justified on the basis of the laws of acoustics in Section 12.5.5. In addition, b(6,¢) and 6b’(@,¢) are very nearly equal for most transducers. Using assumption 5 of Section 12.1, we next ob- tain an expression for the average reverberation in- tensity G(é) at time ¢. Integrating equation (12) over the volume between the surfaces S; and S3 of Figure 2 gives F.-F’ An In equation (13), the dependence on range is con- tained principally in dS, h, and m; these quantities also depend on the direction 6,¢ at which the ray which reaches a particular volume element leaves the projector. However, equation (13) can be simplified as follows. To a good approximation, the extension of any ray between the surfaces S; and S; can be con- sidered equal to c7/2, where co, the average sound velocity along the ray, is always only slightly dif- ferent from the sound velocity at O. Equation (13) can therefore be rewritten as G(t) = J fnnne0(0,6)0'@,0)a8es, (13) Gy = EE (mieo(a,e)'(0,6)a8, (14) 2 Ar where the integral is to be evaluated on some average surface perpendicular to all the rays. It can usually be assumed that this representative surface is the surface reached at time t/2 by the sound emitted at midsignal; in Figure 2 this surface is labeled 83. This assumption for the surface of integration in equation (14) will not be valid if the average value of mh?b(0,)b’(@,¢)dS along any ray does not occur near S;. For example, this assumption fails when the ping length is not small compared to the range of the reverberation. By using the simplifying assumption that the sound intensity decays inversely as the square of the distance, it is easy to show directly from equation (13) that at close range (¢ not much 255 greater than 7/2), the reverberation intensity may not be regarded as proportional to cor/2; rather it is proportional to the factor COT saline (15) Another situation for which the average value of mh?bb’dS may not occur near 8; occurs when the rays are curving very sharply. For most oceano- graphic conditions, this error introduced by ray bending is not appreciable. However, when the layer effect discussed in Section 5.3 is present, the error might be significant. In that oceanographic situa- tion, the ping travels out of an isothermal layer into an underlying region of sharp temperature gradient; and the sound scattered from parts of S;S, below the isothermal layer has a higher transmission loss to the transducer than sound scattered from above the layer. Although equation (14) cannot be used as it stands for the calculation of volume reverberation levels, it nevertheless is significant. It implies that irrespective of the directivity pattern of the transducer, and of the oceanographic conditions, the average intensity of the received volume reverberation should be propor- tional to the ping length. This important conclusion is based, of course, on the various assumptions made previously. In equation (14), write dS = (dS/dQ)dQ, where dQ is the element of solid angle in the direction (6,9). Then equation (14) can be further simplified if it is assumed that the transmission loss in the ocean de- pends only on the distance traversed by the ray en- tering or leaving the transducer, and not at all on the direction of the ray. Then h and dS/dQ are inde- pendent of (6,¢), and equation (14) can be written as cor FFT? dS 2 4r dQ The term dS/d@ is placed in front of the integral sign in equation (16) because it is a measure of the trans- mission loss due to refraction; d@S/dQ is, in fact, Just the reciprocal of the intensity diminution due to normal inverse square divergence plus refraction, ac- cording to Chapter 3. Finally, if it is assumed that scattering in the ocean is independent of the initial ray direction (0,9), the backward scattering coefficient m can also be re- moved from under the integral sign. This yields as our end result for the average reverberation intensity Gi) = frn(0,9)0'(0,9)20. (16) 256 Gi) = SPERMS [o0,0)0'@)00. (17) 4a These latter assumptions are not always realistic. The assumption that transmission loss in the ocean depends only on the range and oceanographic condi- tions, and not at all on the initial ray direction, is probably a poor one for volume reverberation in a nondirectional transducer, because transmission loss in a vertical direction may differ appreciably from transmission loss in a horizontal direction. Even in a highly directional transducer of the sort used by the Navy for echo ranging at 24 ke, this assumption may be in error, since at long range rays leaving the projector only a few degrees apart may travel along widely separated paths; such a divergence occurs, for example, when the refraction theory predicts a split- ting of the beam. Moreover, when split: beams occur, m will not be independent of (6,9) if the scattering strength of the ocean is not independent of depth. For, if the overall scattering strength of the sea is not the same at all depths, then a pair of rays which become widely separated by the prevailing refraction may reach portions of the ocean with different scat- tering strengths; in such a case m evidently will de- pend on (6,¢). Of course m may always be regarded as an average over the entire volume of the ping, and thereby may be removed from under the integral sign in equation (16). But if the scattering strength and transmission loss in the ocean really vary with angle within the main transducer beam, this type of averaging procedure makes the value of m depend on the directivity pattern of the gear; in this event re- moving m from under the integral sign has little significance. In equation (17) the dependence of reverberation on the directivity pattern of the gear is contained wholly in the integral, which can be evaluated from the known directivity patterns of the transducer as a projector and a receiver. If the transmission loss obeys the inverse square law, that is, if the losses due to refraction, absorption, and scattering are neg- lected, then lL aS La aan sate ee h coe and equation (17) becomes Go = 22 foosweed, 8) where r, the range of the reverberation, is equal to Cot/2. In this ideal case, then, the average intensity THEORY OF REVERBERATION INTENSITY of the received volume reverberation varies inversely as the square of the time following midsignal. In general, however, the ocean is far from ideal and this simple law would not apply. To compare the observed time variation of the received reverberation in the general case with that predicted by the ideal formula (18), the general formula (17) is written as _ 7 FF’ Gt) = ae —(r w(4 = ~ fovran (19) or, in decibels 10 log G(t) = 10 log (= ") + 10 log (F-F’) + 10log m — 20 logr + J, + 20 log (r’h) dQ — 10 log r-— Feo (20) where 1 J, = 10 log ~ fove,a)0'0,6)a0. (21) TT The transmission anomaly A is defined (see Section 3.4.1) by A =H — 20logr. By comparing with equation (8), it is evident that A = —10 log (r’h). By substituting this expression for A into equa- tion (20) 10 log G(t) = 10 loz (% 2) + 10 log (F-F’) + 10logm — 20logr+J,—2A+ A; (22) where — 10 log r2(dQ/dS), the transmission anomaly which would result if the normal inverse square divergence were disturbed only by the effect of re- fraction, has been replaced by A:. It is apparent from the preceding discussion that the quantities A, Ai, and m in equation (22) must be interpreted as averages over that portion of the effective scat- tering volume which lies within the main transducer beam. The quantity A,, which depends on refraction alone, cannot be measured directly. In principle, A1 could be computed from the known temperature structure of the sea, according to the methods out- lined in Section 3.4. However, this computation is difficult, frequently inaccurate, and often totally im- practical because the observed bathythermograph [BT] pattern may not extend to a sufficiently great VOLUME REVERBERATION depth. Alternatively, A; could be inferred from the observed transmission loss, if the losses due to at- tenuation and scattering were known. However, it is clear from the discussion in Chapter 5 that these losses are also uncertain. Equation (22) is the theoretical expression for the average intensity of received volume reverberation and is the one usually used in the comparison of theory with observation. Also, the computation of the scattering coefficient m from observed reverberation intensities is usually done by the use of this equation. It will be.remembered that equation (22) was de- rived on the assumption that the transducer was infinitely far away from the ocean surface or ocean bottom, and therefore that sound traveled from O to X on only one path. In actual measurements, the transducer is always near the ocean surface and may be near the bottom as well. The presence of these surfaces increases the number of paths by which scattered energy from any point in the ocean can reach the transducer. Therefore, equation (22) will give erroneously low values for the reverberation intensity if alternative paths from O to X, of very nearly equal travel time, exist in the ocean. In the following paragraphs, we shall consider the error in equation (22) caused by the existence of such alternative paths. It should be stressed that we are considering here only the increase of volume rever- beration due to these additional paths. Surface or bottom reverberation will result from scattering when the sound impinges on one of the bounding surfaces, but we are interested here only in the reverberation resulting from the scattering of sound by the volume elements in the interior of the ocean. Possible combinations of alternative paths are pictured in Figure 3. If the ocean surface is calm, the case of Figure 3A, energy will reach the point X from O not only along the direct path OBX, but also along the path OAX as a result of specular (mirror-like) reflection from the ocean surface at A. If the ocean surface is rough, however, energy may reach X from O along a large, perhaps infinite, num- ber of paths, as indicated in Figure 3B. Because of the principle of reciprocity, the energy returning from X to O also travels along these additional paths. The existence of these extra paths tends to increase the reverberation intensity received at O at time t. To estimate the amount of increase, we note that for every possible path from O to X and back, there will exist an effective scattermg volume of the type of S,)S2 in Figure 2, bounded by two closed surfaces from 257 which scattered energy traveling along that path re- turns to O at time ¢. In Figure 3A, illustrating specu- lar reflection, there are four such volumes. One is A A REFLECTION FROM MIRROR SURFACE B REFLECTION FROM ROUGH SURFACE VS et QQ QW '"7/e7c1Ii CG REFLECTION FROM ROUGH SURFACE AND ROUGH SEA BOTTOM FIGURE 3. scatterer. Alternative paths from transducer to SS; defined in preceding paragraphs, corresponding to the path OBX BO. The others correspond respec- tively to the paths OA X BO, OBX AO, and OAX AO. The volumes corresponding to the paths OA X BO and OBXAO are identical, because the travel time does not depend on the direction of travel along the ray; but the volume corresponding to OA X BO, the vol- ume corresponding to OA X AO, and the volume S,S, corresponding to OBX BO are in general all different. For each of these volumes there will be an integral similar to that of equation (13), expressing the con- tribution of the volume to G(¢). Each such integral can be simplified to a surface integral multiplied by (oT /2, as in equation (14). It follows that the average intensity of the volume reverberation should be proportional to the ping length, regardless of whether 258 energy travels between the transducer and a scat- terer along one path, or along many paths. Evaluation of the various integrals of the form (14) corresponding to each possible route from O to X and back, is very difficult. These integrals depend on the reflecting power of the surface, on the depth and orientation of the transducer, and on how the back- ward scattering coefficient varies with the direction of the incident sound. Also, the possibility must be considered that for small values of t some of the integrals should not be included, since there may not be time for energy to reach the ocean surface along any path and return to O by the time instant ¢. High seas, with the possibility of a great increase in the number of alternative paths, further complicate the problem. To avoid these difficulties, the customary pro- cedure has been to assume that equation (22) fully describes the volume reverberation intensity, despite the complications introduced by the ocean surface. The quantity 10 log m then becomes an adjustable parameter which measures not only the actual back- ward scattering power of the ocean for incident plane waves, but also the effective increase in the volume of scatterers caused by the existence of a number of alternative paths. If the water is deep and the echo-ranging gear is directional, the ocean surface can complicate the problem only if the main transducer beam strikes the surface. If the transducer beam is directed down- ward at a sufficiently large angle, the predictions of equation (22) should not be put in error by the presence of the surface. Using a depressed beam has proved to be one of the most convenient ways of studying volume reverberation. Most reverberation studies, however, have been made with the transducer near the surface, and the beam horizontal. Under those circumstances it is shown in Section 12.5.6 that the value of 10 log m computed from measured volume reverberation in- tensities and transmission anomalies by means of equation (22) will usually be about 3 db greater than the true value of 10 times the logarithm of the back- ward scattering coefficient. If the water is shallow enough for rays reflected from the bottom to be im- portant, no simple relation exists between the in- ferred value of 10 log m from comparison of equation (22) with experiment, and the actual value of the backward scattering coefficient. However, when the bottom is close enough to affect the validity of equa- tion (22), the volume reverberation will almost al- THEORY OF REVERBERATION INTENSITY ways be masked by bottom reverberation so that the failure of equation (22) is of only academic interest. We shall next define the concepts of ‘‘reverberation level”’ and ‘standard reverberation level,’ which facilitate the comparison of reverberation measure- ments performed with different gear and different ping lengths. From equation (13), the average value of the volume reverberation intensity is proportional to the product F'-F’ where F is the power output of the projector and F’ is the receiver sensitivity. It is convenient to eliminate these variables in comparing the reverberation received on different gear. To this end we define the reverberation level R’(t) as R(t) = 10 log G(t) — 10 log (F-F’). (23) For volume reverberation, we have specifically, from equation (22), R’(t) = 10 log <> + 10 log m — 20logr+ J, i See, C22) In words, R(t) is the level of the received reverbera- tion in decibels relative to the power output which would be produced at the terminals of the receiver by an incident plane wave, parallel to the acoustical axis, of intensity equal to the projected intensity on the axis at 1 yd. It is often convenient to go one step further. Since the intensity of reverberation is in principle propor- tional to the ping length, it is both desirable and practical to convert all reverberation levels to the same ping length. We define the standard reverbera- tion level for the reverberation at the ping length 7 as that which would have been received if the ping length had been some standard value 7. Let the standard reverberation level be denoted by R(é). Then we have R(t) = 10 log G(t) — 10 log (F-F’) + 10 loe(“). (25) The predicted standard level of volume reverberation is therefore given by R(t) =10 log 5 + 10 log m — 20 logr +J,—2A+ A: (26) The standard ping length 7 is usually chosen as 100 milliseconds. It is also frequently useful to convert reverberation levels to reverberation strengths. This is done by adding 40 log r to the computed reverbera- tion levels in equations (24) and (25), thereby ob- SURFACE REVERBERATION 259 taining respectively the reverberation strength or standard reverberation strength. The quantity J,, which specifies the relevant di- rectivity characteristics of the transducer, is known as the volume reverberation index. For standard Navy gear at 24 ke, J, is very nearly —25 db. It is shown? that for transducers which are circular pistons J; can be very closely approximated by J, = 20 log y — 42.6, (27) where y is defined as in Figure 4. Numerically, y is , half the angle in degrees in the plane 6 = 0 between those two rays of the composite directivity pattern TRANSDUCER Ficure 4. Half-width y for circular piston trans- ducers. for which the product bb’ is 0.25. Thus for a trans- ducer in which b = 0’, y is half the angle between the two rays for which the response as a projector or re- ceiver is 3 db less than the response on the transducer axis. The angle y is known as the “half width” of the composite directivity pattern bb’. Reference 3 also gives methods for calculating J, for transducers which are not circular pistons. 12.3 SURFACE REVERBERATION Surface reverberation is defined as the totality of sound scattered back to the transducer by scattering centers in or near the ocean surface. This simple definition is not completely adequate, since it would make surface reverberation a particular part of vol- ume reverberation. We differentiate between these two types of reverberation by assuming that the sur- face reverberation arises from a thin surface layer of scatterers. The scatterers in this surface layer are assumed to owe their existence to the proximity of the surface and therefore differ in character from the volume scatterers which supposedly may be found anywhere in the volume of the ocean. The strength of these surface scatterers would be expected to be a function of the state of the sea sur- face, increasing with increasing agitation of the sea surface. In practice, surface and volume reverbera- tion are frequently distinguished from each other in just this way; surface reverberation is regarded as that part of the received reverberation which seems to depend on the sea state. We now derive an expression for the intensity of surface reverberation as a function of range and gear parameters, with the aid of Figure 5. We may pro- Figure 5. Coordinate system used in derivation of surface reverberation formula. ceed exactly as in the development for volume rever- beration, and arrive finally at an equation similar to equation (13). This equation for the surface rever- beration intensity G(é) is = F.F’ Go ==" {micvoao@gav, (28) where the integral is taken over that section of the volume S;S; which contains the surface scatterers. This section need not be of uniform depth, although it is drawn so in Figure 5. The factor m in equation (28) is the backward scattering coefficient in the surface scattering layer and is very probably a func- tion of the depth below the surface. In equation (28), reflection from the surface is explicitly neglected; that is, the ray paths are assumed to go directly from 260 THEORY OF REVERBERATION INTENSITY O to the volume elements at Y without hitting the surface. To evaluate the expression (28), we must first specify the volume element dV. It is not convenient to define the element dV in the same way as was done for volume reverberation. Instead, we set up a cylindrical coordinate system (p,¢,2) whose axis OP is the vertical line through the transducer, as in Figure 5; and we define dV as the infinitesimal vol- ume lying between p and p+ dp, ¢ and ¢ + dd, z and z+ dz. Then the value of dV is dV = pdddzdp. We integrate over the intersection of the volume S,S3 with the surface scattering layer, which is as- sumed to have depth d. This volume is an annulus (ring-shaped figure) determined by z varying from zero to d, ¢ varying from 0 to 27, and p varying from S; to Sj. Then equation (28) becomes im F-F' ¢¢ Qn Sz! GO {| dz f dd f pmh2bb'dp, (29) 4r JO 0 Si where p is integrated from S; to S3. In the ocean, it can be assumed that on the average sound rays are bent only in a vertical direction. Then, the distance in the p direction from S; to S; is independent of the polar angle ¢, but may depend on the depth z. In order to put equation (29) in a form suitable for calculations, we shall have to make several additional simplifying assumptions. First, we assume that the only factor in the expression (29) which depends on the depth coordinate z is the scattering coefficient m, and that m depends only on z. Then equation (29) becomes ; F.-F' d Pi Se! ( it mdz) ii dp ih phbb'dp. (30) Ar 0 0 Si’ Gt) = This assumption is readily defended. Since the depth of the surface scattering layer is usually small com- pared to the horizontal range from the transducer, there is little difference in initial ray direction be- tween the ray which reaches a point Y’ in the ocean surface and the ray which reaches the point Y” a depth d below Y’ (Figure 5). Therefore, the product bb’ is practically independent of z in our volume of integration. Again, since the depth of the surface scattering layer is usually small, the horizontal dis- tance traversed by a ray in its passage from S; to S; changes but little from the top to the bottom of the layer. For the same reason it can be assumed that there is usually little difference in transmission loss among the various paths from the transducer to points in the volume of integration.* There is little reason for the scattering coefficient to vary with any- thing but depth, as long as the grazing angles of the rays on the surface do not vary appreciably over the volume of integration; this will be the case if the ping length is sufficiently short compared to the range of the reverberation. We may rewrite equation (30) as Qn ‘S2’ G0) = =n’ [a6 | oteob'dp, d m’ = [i mae 0 It should be remarked that the disappearance of d in equation (31) is of little consequence. There is ordi- narily no way to accurately estimate d in any par- ticular case; it is Just the depth down to which scat- terers which depend on sea state appear in significant quantity. Without committing ourselves as to the exact size of d, we may give the factor m’ real physical meaning by redefining it as oo m’ -f mdz 0 where m is the backward scattering coefficient of the scatterers causing surface reverberation, is dependent on sea state, and is negligible below some unspecified depth. It seems likely that the depth at which m be- comes negligible is usually small enough so that the lack of dependence on z of h?bb’ can be assumed. If not, or if for any other reason the assumptions used to derive equation (31) from equation (28) are not satisfied, the first integral in equation (30) cannot be regarded as a separate factor, and the concept of an overall surface scattering coefficient m’ has no mean- ing. One situation in which equation (81) does not apply, while equation (28) does, is for surface rever- beration in the presence of sharp negative tempera- ture gradients near the range where the limiting ray leaves the surface. This situation is pictured in Figure 6. Strictly speaking, in this case the surface S, does not intersect the ocean surface at all; but S; may be drawn to intersect the surface, as in Figure 6, with the understanding that the transmission loss is in- finite to the shaded volume. Under these circum- (31) where (32) ® Tt must be noted that this assumption ignores the possi- bility of the image effect described in Section 5.2.1. The effect of image interference on the behavior of surface reverbera- tion is briefly discussed in Section 14.2.1. SURFACE REVERBERATION stances, the transmission loss varies rapidly with depth in some portion of the volume of integration, and the assumptions used to derive equation (31) from equation (28) are not satisfied. We next make assumptions which enable us to integrate out the variable p in equation (31). The integral in equation (31) is taken over an annulus whose horizontal cross section is the ring-shaped area between the circles of radii PU and PV in Figure 5. LIMITING RAY fe) Ficure 6. Region of surface scattering in presence of sharp downward refraction. If the ping length is assumed to be sufficiently short compared to the range, then h?bb’ varies but little in the distance from U to V, and equation (31) be- comes F-F’ Sz CM) = Tee ‘5 " ebblds i: pdp (83) Bele i Be 0 Qn Se’ b(0,4)b’ (0,6)de I Pde. (34) The step from equation (83) to equation (84) is justified only if the transmission loss h is inde- pendent of the polar angle ¢. With rays bending only in the vertical direction, there is ordinarily no reason why the average transmission loss should depend on this variable. In equation (34), 9 is some average angle of elevation of rays which strike the surface between the two circles of radii PU and PV in Figure 5. If the ping length is sufficiently short compared to the range, this average value of @ may be assumed to be the angle of elevation of the ray which leaves the projector at midsignal and hits the surface after a travel time ¢/2. In other words, this average value of @ is the angle of elevation of that ray which passes through the curve of intersection of the ocean sur- face and the surface S; defined in Figure 2. Now, by simple calculus, 261 Si! 2 2 eG Si’ 2 Se’ 2 Si’ DUS EUO Es (EWE oe 2 2 = uv(eu + ag) = 2(UV) in Figure 5 (35) where, if the ping length is sufficiently short com- pared to the range, p, the mean value of p in the annulus, may be assumed equal to the value of p where S; intersects the ocean surface. UV is the dis- tance on the surface from S; to $3. p olZa\ Figure 7. Expanded drawing of ray between projector and ocean surface. Next, we evaluate p(UV). Referring to Figure 7, if PW = p, if ds is the increment of arc along the ray from O to W in the time interval dé, and if dp is the corresponding increment of horizontal range, then dp = dscosa and 7 t/2 t/2 p= PW= f ds(t) cos a(t) = {l c cos adt, since cdt is always equal to ds. Also, since the bending is in the vertical direction only, we have by Snell’s law Cos a’ Cc @ Cos @ where c’ is the velocity of sound at the surface, and a’ is the angle of elevation of the ray OW at W. It follows that (36) where c is some average sound velocity. To calculate 262 THEORY OF REVERBERATION INTENSITY UV, the second factor in the expression (35), we notice in Figure 7 that if UH lies in S; and if the ping length is sufficiently short, then UWH is very nearly a right triangle with the right angle at H. Thus, HW since the perpendicular distance from S; to 3 is c’r/4. With our assumptions, UV ON oe Thus cr V — U 2 cos a’ Substitution of this expression for UV and of ex- pression (36) for p into equation (35) gives if & q Ctcosa’ c'r (D0 Pe ape Sy 2c’ 2cosa’ Ctr 4 In equation (37), c, the average sound velocity, can be replaced with little error in any actual situation by c, the sound velocity at the transducer. Substi- tuting expression (37), modified by this replacement, in equation (34) gives UA G@) = se hs An The subsequent procedure is similar to that adopted following equation (17); it is convenient to rewrite the theoretical expression (38) for reverbera- tion in terms of decibels as a function of range. As before we define the range r of the reverberation as Cot/2; this differs negligibly from the distance along the ray path OW of Figure 7. Proceeding as in Sec- tion 12.2, we find 10 log G(t) = 10 log (“) + 10 log (F-F’) (37) Gtr (7" cai Ne b(6,¢)b’(0,6)dd. (38) + 10 log (~) ~ 30logr + J.(0) — 2A, (39) where 20 1 J.(9) = 10log=—]} b(0,4)b'(6,¢)d@ —- (40) 2nJ0 and 2A is the two-way transmission anomaly along the ray path. Equation (88) indicates that, the intensity of sur- face reverberation, like that of volume reverberation, should be proportional to the ping length if the as- sumptions used to derive equation (38) are satisfied. If these assumptions are not valid in a particular situation, however, the surface reverberation inten- sity may not be proportional to the ping length. Frequently, these assumptions are not satisfied; thus the proportional dependence on ping length pre- dicted in equation (38) is not as general as the same dependence for volume reverberation predicted by equation (14). For example, if refraction is sharply downward, surface reverberation from ranges near where the limiting ray leaves the surface will not obey the theoretical law (38). Qualitatively, it can be seen from Figure 6 that at a range cof/2 somewhat greater than the limiting range, a halving of the ping length may lead to more than a halving of the sur- face reverberation intensity, since most or all of the shorter ping may be too far from the surface to be effective in scattering. It will be recalled that this situation in which the proportional dependence on ping length predicted by equation (38) is invalid is just the type of situation which had to be ruled out in order to obtain equation (31), which in turn led to the result (38). In deriving equation (38), surface reflections were explicitly neglected. As explained in Section 12.2, the surface reflections make for alternative ray paths from the transducer to the scatterers. It is shown in Section 12.5.6 that these alternative paths usually cause the value of 10 log m’, computed from measured surface reverberation intensities and transmission anomalies by means of equation (39), to be about 6 db greater than the actual value of the backward scattering coefficient of the surface scatterers. The quantity J,(9) in equation (39) is called the surface reverberation index. In general, this index de- pends on the orientation of the projector relative to the vertical and on the range of the reverberation, and is difficult to calculate for arbitrary transducer orientations. When the transducer beam is nearly horizontal, however, the expression (40) can be evalu- ated approximately. It is shown 4 that if the trans- ducer is a large rectangular piston in an infinite baffle then, approximately, a b( — £,0)b'(@ — £,0) cos 6 where Qn , 1(0,9)6"(0,6)46, (4a) Q(0) = SURFACE REVERBERATION 263 and £ is the angle of tilt of the transducer axis relative to the horizontal plane. The relation (41) is probably not valid for angles @ greater than 30 degrees, at which angle the directivity pattern of any actual transducer is likely to differ appreciably from the ideal. Equation (41) was proved in reference 4 only for rectangular pistons. However, even for the circu- lar pistons used in most Navy gear, the use of equa- tion (41) is probably legitimate as long as the correc- tion factor b(@— £,0)b’(@ — £,0)(cos 6)— does not dif- fer too much from unity. For a horizontal trans- ducer (¢ = 0), typical values of the correction b(0,0)b’ (6,0) 10 log ————— oS cos 0 are given in Table 1. This table was computed by the TABLE 1 @in degrees 10log ~ b(4, _ & 0) in eacati 0 0 2 —1.0 4 —2.0 6 —6.0 8 —12.0 use of values of b and b’ measured for the EB1-1, which is similar to standard 24-ke Navy gear.® The use of corrections greater than —12 db is probably not justified. Some further discussion of the use of this correction is given in Chapter 15. Formulas for J,(0) = 10 log Q(0)/27 are given in reference 3. For transducers which are circular or rectangular pistons J.(0) = 10 logy — 23.8. (42) As was done with the expressions for volume rever- beration, we may define the surface reverberation level R’(£) as R(t) = 10 log G() — 10 log (F-F’) = 10loe (F "+ 10105 (™) — 30 logr + J.(6) — 2A. (48) Also, we define standard surface reverberation level R(é) as the level of the average reverberation which would have been received at the time ¢ if the ping length had had some standard value 7 instead of r. Then, RQ = 10 log Gt) — 10 log (F-F’) + 10 log (2), a (44) so that R(t) = 10 og ( =) + 10 log (” ") — 30logr + J.(6) — 2A. (45) Reverberation strengths can also be defined in a manner similar to that in Section 12.2. When using equations (39), (43), and (45) it is necessary to remember that A has been defined as the transmission anomaly along the actual ray path to the surface. This transmission anomaly may differ from A’, the value of the transmission anomaly measured in the usual experimental determination of transmission loss. Consider specifically that the pro- jector axis is horizontal and that a ray leaving the projector with the angle of elevation 6 and an azimuth angle ¢ of zero reaches the surface after covering the slant range r. Then from the definition of A, = 10 log F + 10 log 6(6,0) — 10 log J — 20 log r, (46) where J is the measured intensity in db at the point where the ray strikes the surface. The measured anomaly A’ is usually determined from the equation A’ = 10 log F — 10 log I — 20 log r’ (47) where 7’ is the horizontal range from the projector to the point where the ray strikes the surface. Neglecting the difference between r and r’, we have from equations (46) and (47) A = A’ + 10 log b(6,0). (48) Further, from equations (40) and (41) we have for a horizontal transducer J.(0) = J.(0) + 10 log b(9,0) + 10 log b’(6,0) — 10logcos 6. (49) Substituting equations (48) and (49) in equation (45) gives Ri) = 10 og (%° 24 10 log (“) — 30logr + J,(0) — 2A’ — 10 log 6(6,0) + 10 log 6’(6,0) — 10 logcos 6. (50) Under most circumstances cos 0 is sufficiently near unity and the projecting and receiving patterns are sufficiently symmetrical, with the result that the last three terms in equation (50) can be neglected. Thus, if the measured transmission anomaly A’ defined by equation (47) is used in the analysis of surface reverberation, the correct expression for the standard reverberation level under most circumstances is 264 THEORY OF REVERBERATION INTENSITY R(t) = 10 og ( % =) + 10 10g(™~) — 30 og +J,(0) —2A’. (51) In other words, if A’ is used instead of A in equation (45), then the proper surface reverberation index to use is J,(0) rather than J,(0). The same rule applies, of course, in the evaluation of equation (389) or equa- tion (43) 12.4 BOTTOM REVERBERATION Bottom reverberation is defined as reverberation arising from sound scattered back to the tranducer by scattering centers in the ocean bottom. These scattering centers are thought to usually lie in a very thin layer on the bottom. Thus the formulas for the dependence on range of bottom reverberation will be similar to those formulas for surface reverberation, and can be derived from them by simple changes of notation. We have then, from formulas (39) through (45), 10 log G(t) = 10 log ( ‘) + 10 log (F-F’) + 10 log (=) 30 logr (52) + Jz(6) — 24, with Jz(@) = 10 log ~ "9(0,6)8'(0,8)de (53) R’(t) = 10 log G(@) — 10 log (F- F’) = 10 log (2 ") + 10 log Ss — 30 logr + Jz(@) — 2A. (54) R(t) = 10 log G(t) — 10 log (F-F’) + 10 log = = 10 log ( 2) + 10 log ies — 30 logr + Jx(6) —2A. (55) In these formulas m’” is the backward scattering coefficient of the bottom per unit area of the bottom. The coordinate system in equation (53) is similar to that in Figure 5; however, the transducer is usually directed downward instead of upward as in Figure 5. Equation (41) may be used to evaluate J5(0) in equation (53); and as before Jz(6) should be re- placed by Js(0) if the transmission anomaly as usually measured is used instead of the actual trans- mission anomaly along the ray path. The quantity m’’ is in general a function of the range since it probably depends on the angle of incidence of the ray on the bottom. It should be noted that the similarity of the formulas for bottom and surface reverberation does not imply that bottom and surface reverberation arise from similar mechanisms. The bottom scatter- ing originates in irregularities in bottom contour; these irregularities may vary from the fine separa- tions between grains of sand to such macroscopic irregularities as large rocks and underwater cliffs and valleys. 12.5 EXPLICIT AND TACIT ASSUMPTIONS The preceding derivations of the theoretical for- mulas for volume, surface, and bottom reverberation were based on many assumptions, not all of which were stated explicitly. In this section we shall discuss briefly the significance and probable validity of these assumptions. Because of present uncertainties re- garding scattering sources and the infinite complexity of the ocean, this discussion is partly qualitative and not clear-cut. It was pointed out in Section 12.1 that though equation (1) governs the propagation of reverbera- tion in the ocean, a complete solution of equation (1) for the reverberation received under given conditions is not obtainable. However, certain general properties of solutions of equation (1) are known, and will be of use here. The strength of an acoustic disturbance can be ex- pressed either in terms of pressure amplitude or sound intensity. In practical applications, the sound in- tensity is a more convenient quantity; while in theoretical discussions based on the wave equation (1) the acoustic pressure is more convenient. In equa- tion (1), the sound intensity does not appear ex- plicitly; in fact, it is impossible to derive a simple differential equation, whose dependent variable is the sound intensity, which like equation (1) ex- presses the fact that the disturbance is a wave traveling through the ocean with the velocity c. In discussing the implications of equation (1), we shall, therefore, be directly concerned with the sound pres- sure p. To tie in the discussion with the preceding sections of the chapter, we must relate the sound in- tensity to the sound pressure and also to the voltage generated across the terminals of the receiving circuit. To simplify the discussion. we shall assume, to be- EXPLICIT AND TACIT ASSUMPTIONS gin with, that the instantaneous voltage across the terminals of the receiving circuit is exactly propor- tional to the instantaneous pressure in any plane wave which is incident on the transducer. This as- sumption is equivalent to assuming that the re- ceiver introduces no phase shifts, or in other words, that it behaves like a pure resistance or like an ideal infinitely wide band-pass filter. Of course, actual re- ceivers never behave in this way, but it is convenient to postpone temporarily consideration of the effects caused by departure from ideal response in the re- ceiver circuit. Now suppose a plane wave of pressure p is incident on the transducer from a direction de- fined by the angles (0,¢) of Figure 2. Then the voltage E across the receiver terminal is E = f'6'(8,9)p, where £’(0,¢) is the ‘‘pressure pattern function” of the receiver, and f’ is the voltage across the receiver terminals when a plane wave of unit pressure is incident on the receiver in the direction of its maxi- mum response. Since there is no phase distortion, all the quantities in equation (56) may be assumed real. The rms power output resulting from EH, in watts across the receiver terminals, will be FTPs Tamera (56) (57) where the receiver is assumed terminated in the pure resistance Z, and the bar indicates a time average over many cycles of the wave. Now, in a plane wave, the relation between the sound pressure p and the average sound intensity 7, from Section 2.4.3, is just (58) where pp is the density of water, c is the velocity of sound, and the bar again indicates a time average over many cycles of the wave. Equation (58) re- mains valid even if the plane wave is being refracted by velocity gradients in the ocean. From equations (57) and (58), we see that the rms power output across the terminals of our ideal receiver, caused by a plane wave incident on the transducer, is proportional to the average sound intensity in the water. Further- more, by comparing equations (57) and (58) with the definition of F’ and 6’(6,¢) in Section 12.2, it is evident that 12 Tove (6,6) = 6'2(0,0). i RAG; (59) 265 However, the scattered sound which produces re- verberation reaches the transducer from all direc- tions, and therefore cannot be regarded as a plane wave. For this scattered sound the pressure in the water at any instant is fo DPi; where p; is the pressure in the 7th plane wave which arrives at O. The voltage generated across the re- ceiver terminals, by equation (60), is E = f'218'(6:,.)pi (60) (61) where the angles 0;,; define the direction from which the 7th plane wave reaches the transducer. The rms intensity resulting from equation (61) is given by os FY Ee'@.00p.|] De'e.05 Yh, 5 iG D> 12,2 »> > Ore 5 5 ="7 ORT : DD tA (62) where the double sum includes terms for all values of 7 and 7 except 7 equal to 7. 136.1 Average Reverberation Intensity One of the basic assumptions made in Section 12.1 was that the average reverberation intensity is the sum of the average intensities of the individual scattered waves reaching the transducer. Because equation (62) represents the average reverberation intensity, while equation (57) represents the in- tensity of the individual scattered wave, it is clear that this assumption will be strictly valid only if the double sum on the right-hand side of equation (62) vanishes. The average in equation (62) is the average over a large number of cycles. All the waves reaching the transducer have very nearly the same frequency when ordinary single-frequency (CW) pings are used. Thus, the value of the double sum on the right of equation (62) depends on the relative phases of the various waves arriving at the transducer. On any one ping the value of this double sum may be positive or negative, and its absolute value may be appreciable or near zero, depending on the phases. Thus, if the expression (62) is averaged over a number of pings, and if the phases vary in a random way from ping to ping, the double sum in equation (62) can be neg- 266 THEORY OF REVERBERATION INTENSITY lected, and we can conclude that the rms reverbera- tion intensity, averaged over a number of pings, will equal the sum of the average intensities received from the individual scatterers in the ocean. We may expect the phases to vary in a random way because of tbe properties of equation (1). This equa- tion implies directly that sound propagates through the ocean as a wave with a definite velocity, and that the phase of the returning wave depends on the travel time of the wave from the transducer to the scatterer and back. At 24 ke a phase shift of 27, amounting to a shift of a complete cycle, results from a relative displacement between two scatterers ot about an inch, or a difference in travel time of about 40 ysec. Such phase changes or changes in ray path from ping to ping could result from thermal fluctuations, from the rise and fall of the transducer in the ocean, from wave motion, and from drift of the scatterers and the projecting ship. A relative dis- placement of one inch in five seconds (the approxi- mate time between pings) corresponds to a relative drift of only 60 ft per hr. It is worth stressing that the phase shifts discussed in the preceding paragraph are relative phase shifts between waves from different scattering points in the ocean. At any instant sound is being received from many different points on any one scatterer; but if the scatterer is a rigid sphere, for example, the phases of the waves arising at different points on the spherical surface always bear a definite relation to each other. These waves from the different points on the spherical scatterer will always combine to give the same result in equation (62), irrespective of relative displacement between the scatterer and the transducer. Thus, the likelihood that the double sum in equation (62) will average to zero over a number of pings is connected with what might be called the “correlation” between conditions at various points in the ocean. If the ocean were rigid, so that the relative positions and orienta- tions of the scatterers never changed, knowledge of the phase of a returning wave from one point in the ocean would completely determine the phases of re- turning waves from all other points. In this event the assumption that the double sum in equation (62) averages to zero would be more difficult to maintain. However, since the ocean is not rigid and the posi- tions and orientations of the scatterers change with time, knowledge of the phase of a wave returning from one point determines the phases only of those waves from the immediate neighborhood of the par- ticular point. The averaging to zero of the double sum in equa- tion (62) is made even more probable by our averag- ing procedure, which focuses attention on a definite instant relative to the midtime of the emitted signal. As the transducer drifts or otherwise changes its posi- tion in the ocean, the reverberation received at a definite instant comes from different points of space on different pings. In many cases, the phases of the scattered waves returning from these two portions ot space will be almost completely uncorrelated ; in such cases, random phase relations on successive pings are even more likely. In the absence of definite knowledge about the scatterers responsible for reverberation, it is not possible to make this argument about equation (62) more precise. However, it is apparent from this dis- cussion that in all types of reverberation there are a number of mechanisms that can cause random varia- tions in the phases of the individual returning scat- tered waves. When these phases are truly random the assumption involved in averaging the individual scattered intensities, to get the average reverberation intensity, is justified. Definition of Backward Scattering Coefficient 12.5.2 The backward scattering coefficient was defined for a volume small enough so that its relevant properties do not change too sharply with changes of position inside the volume, and large enough that it contains a reasonable number of scatterers. After an explicit assumption that the scattering by such a volume is proportional to the volume, the scattering coefficient of V was defined by formula (4). It is easy to see that this assumption should be valid if the double sum in equation (62) vanishes. If this term vanishes, the total reverberation intensity will be just the sum of the average intensities of the waves from the individual scatterers; therefore, it should be proportional, on the average, to the size of the scattering volume. 12.5.3. Duration of Scattering by a Scatterer In Section 12.1, it was assumed that scattering from an individual scatterer begins the instant sound energy begins to arrive at the scatterer and ceases the instant sound energy ceases to arrive. In considering this assumption, we must recognize EXPLICIT AND TACIT ASSUMPTIONS that the discussion to this point has glossed over the fact that neither the outgoing ping nor the scattered sound which reaches the transducer is really single- frequency sound. No sound of finite duration can bea pure single-frequency sound since the latter theo- retically lasts an infinite time; it can be shown that the relation between the time duration 6¢ of a ping and the width 6f of the frequency band making up the pulse is approximately fst = 1. (63) In general, the relation between any time-dependent signal F(t) and the frequency spectrum of the signal is given by Fourier’s integral theorem.® That is, the signal can be written in the form 1 . PO =e i Awede, (64) where A(w) is determined by the equation 1 - 5 A Se f F(t)e" “dt. 65 @) =e J Five (65) Equations (64) and (65) are just generalizations of corresponding equations applicable to Fourier series of periodic functions. In these equations F(t) and A(w) are generally complex, and w can be interpreted as equal to 2zf where f is the frequency of the spectral component corresponding to w. It is possible, therefore, to make a frequency analysis of any given ping, using equation (65). This frequency analysis can then be used to obtain a formal solution of equation (1). For, because of the linearity of equation (1), if the scattered sound reaching the receiver as a result of emission of the continuous sound e“ is B(w)e™, then the pressure of the scat- tered sound reaching the transducer as a result of any given ping is 1 cy A BO f A@Be)*eds, where A(w) is given by equation (65) in terms of the pressure variation F(é) of the outgoing ping. It is necessary to qualify equation (66). Because the boundary conditions and the velocity of sound at any point are changing with time, the scattered sound which reaches the transducer is not a pure sound, even though a pure sound e“” is emitted. Thus al- though the pressure of the scattered wave can always be presented as a Fourier integral of the form (64), equation (66) is not rigorously true, if B(@w) and A(w) are defined as above. However, for the purpose of investigating the validity of the assumption under (66) 267 consideration, these effects of time variation can be neglected, and equation (66) accepted as valid. In addition, for simplicity, the velocity of sound ¢ in equation (1) can be assumed constant. By using equation (66), the dependence on time of the pressure p(t) of the scattered sound reaching the transducer can be calculated for pings of any length and for various types of scatterers. If p(t) is plotted as a function of the time following the emission of the ping, it turns out that p(é) is always zero until the sound has had time to travel to and from the nearest point on the scatterer. In other words, scattering from an individual scatterer actually does begin at the instant that sound energy begins to arrive at the scatterer. It is not possible to show in general that scattering ceases at the instant the sound energy ceases to arrive at the scatterer. However, for the special case of an infinitely rigid spherical scatterer, it is possible to show that the duration of the scat- tered sound received from the sphere is the same as the duration 7 of the outgoing ping, as long as the relation D Kr (67) c is satisfied, where D is the diameter of the sphere. The significance of equation (67) is simple; it means that the scattered sound will have the same duration as the initial ping, provided that the travel time of the sound across the sphere is negligibly short. compared to the duration of the initial ping. This result is not unexpected. The reason why no general proof can be given for the validity of the second part of the assumption under discussion, namely, that scattering ceases at the instant sound energy ceases to arrive at the scatterer, is easily understood. Any real not infinitely rigid scatterer, such as a bubble, will have definite resonant frequencies which will be excited by the incident sound, and the scatterer may continue to radiate sound at its resonant frequencies long after the ping has: passed by. Also some sound may enter the scatterer and be reflected back and forth inside the scatterer a number of times before it is scattered back toward the transducer. If the scatterer is large and many such reflections are possible, the duration of the scattered sound will be longer than r. Despite these difficulties it can be argued that the assumption can be regarded as valid. There is good reason to doubt that resonant bubbles or other resonant scat- terers play a large part in reverberation; in any case, 268 THEORY OF REVERBERATION INTENSITY the reradiated sound from such scatterers would die out in a very short time compared to the duration of even a 1-msec ping. For most scatterers equation (67) will be satisfied for pings of ordinary length al- though it may sometimes not be satisfied with scat- terers such as rocks on the bottom, for 1-msec pings. In the light of the discussion in Section 12.5.1, the distance D in equation (67) must be interpreted as the diameter of the volume within which there is ap- preciable correlation between the phases of waves re- flected from different points in the ocean. Scattering volumes separated by, so large a distance that there is little correlation may be considered unrelated scat- terers. In the absence of definite knowledge about the scatterers, it is difficult to make the argument pre- cise, but it seems unlikely that there will be apprecia- ble correlation over a distance as great as a yard, which is about the length of 1-msec ping. Along with assumption 2, Section 12.1, it has been tacitly assumed, in the derivation of the theoretical reverberation formulas that the outgoing ping is square-topped (that is, that the intensity rises abruptly to its steady-state value at the beginning of the ping, remains constant until the end of the ping, and then drops suddenly to zero), and that the wave received from any scatterer reproduces the shape of the outgoing ping. It is possible to make the outgoing ping very nearly square-topped, but it is apparent from equation (66) that the shape of the waves returning from each scatterer is not necessarily the same as the shape of the outgoing ping. However, if the ping does not include too wide a frequency band (that is, is not too short) and if the scattering co- efficients of the various types of scatterers do not vary too rapidly with frequency, then in equation (66), if sound is being received from only one scat- terer, B(w) is nearly independent of frequency, and the returning scattered wave does very nearly repro- duce the wave form of the outgoing ping. It will be seen in Chapters 4 and 5 that even in a 1,000-c band, scattering coefficients in the ocean apparently change very little, so that, from equation (63), square-topped 1-msec pings should result in square-topped scattered waves. We can now see the significance of the assumption, made at the beginning of Section 12.5, that the in- stantaneous voltage induced in the receiving circuit is exactly proportional to the instantaneous pressure of the sound arriving at the transducer. If this is not the case, that is, if there is phase distortion or ampli- tude distortion in the receiver, then the reverbera- tion resulting from any one scatterer will not have the square-topped shape of the outgoing ping, and the formulas which have been derived will be in error. Thus, if measured reverberation intensities are to be comparable to the theoretical formulas of Sections 12.2 to 12.4, it is necessary to use flat wide-band systems which have little transient response to the sudden changes of reverberation intensity. The use of narrow-band systems with high transients will usually cause the reverberation received from any scatterer to last longer than the outgoing ping and have a shape different from that of the ping. Since this distortion will decrease with increasing ping length 7, deviation will result from the predicted proportionality of R’(¢) on ping length 7 in equations (24), (43), and (54). In order to derive appropriate theoretical formulas for such systems it would be necessary to write, using equation (66), 1 i ; Ej) = Se J ABW) eed for the voltage induced across the receiver terminals by each scattered wave where C(w) describes the frequency response of the gear. The expression for the average reverberation intensity would then involve an integration over the frequency band included in the ping and passed by the equipment. It may be re- marked that there is an inherent dependence of re- sponse on frequency in any directional transducer, because the pattern functions 6(0,¢) and 6’(6,¢) are always functions of frequency. Thus, it may be neces- sary to consider further the effect of directivity on the theoretical reverberation formulas for situations in- volving very short pings and highly directional transducers. 12.5.4 Neglect of Multiple Scattering We have assumed that the sound reaching the transducer as reverberation has been scattered only once. It is easy to see that the validity of this assump- tion depends on the range of the received reverbera- tion. For, as the ping proceeds out from the trans- ducer, it loses more and more energy by scattering; the scattered waves are of course no different from any other sound waves and are themselves scattered. Eventually, therefore, a range is reached at which the ratio between the singly scattered sound returning to the transducer and the multiply scattered sound returning is no longer large. The value of this range depends on the amount of scattering which takes EXPLICIT AND TACIT ASSUMPTIONS place in the ocean. Thus, the validity of this assump- tion, like that of all the other assumptions we have made, depends on the properties of the scatterers in the ocean. In considering the validity of this assumption, we may confine our attention to multiple scattering in the body of the ocean, since there is little likelihood of direct multiple scattering from one surface scat- terer or bottom scatterer to another. Experiments on volume reverberation’? have shown that at short ranges (up to a few hundred yards) multiple scat- tering in the body of the ocean probably can be neg- lected. If scattering in the body of the ocean is isotropic, that is, if the backward volume-scattering coefficient m is really the average amount of energy scattered in all directions per unit intensity per unit volume, then it can be concluded from the magnitude of m that multiple scattering is certainly negligible at all ranges of interest in echo ranging. However, it is possible that forward scattering in the ocean is appreciably greater than backward scattering. It has been suggested that the high at- tenuation of sound in the ocean at supersonic fre- quencies results from forward scattering of sound by the temperature microstructure in the ocean. On present evidence, it seems unlikely that forward scattering alone can account for attenuation in the ocean, but if appreciable wide-angle scattering does occur, then at long ranges the neglect of multiple scattering in the theoretical reverberation formulas is not justified. The predicted dependence of volume reverberation on range [equation (24) ] would be changed if volume reverberation contained much multiply scattered sound. The evidence discussed in Chapter 14 suggests that multiple scattering in the ocean probably can be neglected at ranges of opera- tional interest in echo ranging. However, more evi- dence is needed before any definite conclusions can be reached. 12.5.5 Fermat’s Principle and the Principle of Reciprocity It was important to show that multiple scattering can be neglected in the computation of reverberation intensity since that assumption enabled us to deline- ate the volume SS, in Figure 1 within which ap- preciable scattering is taking place. The determina- tion of volume SS; (Figure 2) was then based on an application of Fermat’s principle. Fermat’s principle b This point is discussed in Section 5.4.1. 269 is a theorem about the properties of equation (1); it states that when a sound travels between two given points, it always follows a path such that its travel time is a maximum or a minimum.” This maximum or minimum value is the same no matter which of the two points is the starting point and which the finish- ing point. Thus, provided the refraction conditions and boundary conditions are not changing with time, the ray paths and travel times from the transducer out to a scatterer, and from the scatterer back to the transducer, are exactly the same. However, refraction and boundary conditions in the ocean are not con- stant with time. The existence of thermal fluctua- tions, and the fact that BT patterns vary from one hour to the next, show that refraction conditions change with time; and surface waves are an example of changing boundary conditions. Complete elucidation of the effect of these chang- ing refraction and boundary conditions would be highly complicated, and, as usual, lack of information about the scatterers would make it difficult to be precise. However, it seems justifiable to assume that short-term fluctuations in thermal microstructure, or such variations in boundary conditions as waves or random movements of the scatterers, do not modify the average equality of the travel times and ray paths. Moreover, the long-term variations evi- dent on the BT trace are too slow to affect the aver- age equality in a series of pings lasting about a minute. Thus, it appears that the use of Fermat’s principle was justifiable in Section 12.2, in the delineation of the effective scattering volume S,S3. Another theorem about the properties of equation (1) is the principle of reciprocity. In the ocean, trans- mission loss is thought to be a combination of ordi- nary inverse square spreading, refraction, absorp- tion, and scattering. According to the principle of rec- iprocity,’ if the refraction and boundary conditions are not changing with time, then that part of the transmission loss which is due to inverse square spreading and refraction will be the same for trans- mission from a nondirectional projector at O to the point X.as for transmission from a nondirectional projector at X to O. Of course, the source at O (the echo-ranging projector) is not nondirectional; and the source at X (the scatterer) may not be nondirec- tional, since scattering is not necessarily the same in all directions. For directional sound sources, the reciprocity theorem requires modification, because of the possibility of reflections and multiple paths be- tween O and X. However, along any definite ray path 270 the principle still holds that the transmission losses due to inverse square spreading and refraction from O to X and X to O are the same along that ray, ir- respective of the directivities of the sources. In addi- tion, the principle of reciprocity applies also to ab- sorption losses ° if absorption in the ocean arises from so-called linear processes. Actually, studies of trans- mission loss show that the processes involved in the transmission of sound in the ocean are only imper- fectly understood; but there appears to be no justi- fication at this time for ascribing the absorption losses of sound waves of ordinary amplitudes to non- linear processes. Because the scattering coefficients are so small, transmission losses due to scattering may be neglected at all ranges of interest in echo ranging. It follows therefore from the preceding paragraph, and from the definitions of the quantities h and h’ in Section 12.2 as cransmission losses along the ray, that the principle of reciprocity may be applied to transmission be- tween the points O and X of Figure 2 if refraction and boundary conditions are constant. It is still necessary to consider the effects of the variation of these con- ditions with time; but by an argument similar to that made in the discussion of Fermat’s principle, it seems valid to assume that these time variations will not affect the relation between the transmission losses of the outgoing and incoming sound on a series of pings lasting about a minute. In other words it appears justifiable at this time, in the light of the principle of reciprocity and our present understanding of absorption losses, to assume that on the average the one-way transmission losses h and h’ are equal. 12.5.6 Effect of Surface Reflections The complications induced by such variations in boundary conditions as rough seas are exemplified by the difficulties encountered in extending equation (20), derived for an infinite unbounded ocean, to the more nearly realistic semi-infinite ocean. In rough seas (Figure 3B), with the transducer horizontal, it is very difficult to determine exactly the effective vol- umes from which reverberation is being received at any instant. In calm seas (Figure 3A), however, the volumes corresponding to the various alternative paths should be very nearly identical at ranges greater than a few hundred yards, since at these rather long ranges the path differences between OA X and OBX are usually very small. These volumes will all be about half of the original volume S;S, because THEORY OF REVERBERATION INTENSITY of the presence of the ocean surface; at long ranges and shallow transducer depths the initial angle of ray elevation @ is restricted almost completely to values between —7/2 and 0 in the integral (14), instead of varying from —7/2 to 7/2, as it does when the ocean surface is far away. Since all four of the scattering volumes described in Section 12.2 are very nearly equal, all of the integrals of the form (13) correspond- ing to these volumes should also be nearly equal, be- cause at these ranges the rays OA X and OBX (Figure 3A) leave the transducer in almost the same direction, and because in a calm sea the reflection coefficient of the surface is very nearly unity. Thus, in a calm sea, with the transducer near the surface and the beam horizontal, the total intensity of the received volume reverberation is obtained by adding up four integrals of the form (13), with the region of integration for each just half the volume 5,83. Therefore, the presence of the surface increases the received volume reverberation under these conditions to about double the value predicted by equation (17), or to a value about 3 db greater than predicted by equation (22), with the important proviso that the quantities A and A, used in that equation are the transmission anomalies that would have been meas- ured if there had been no reflected rays. The trans- mission anomaly is usually obtained by measuring the transmission loss from a point about 100 yd from the transducer. If so, it is easily seen that with shallow transducers the inferred transmission anomaly is about the same as the transmission anomaly that would have been measured if the surface was far away. It follows, therefore, that in calm seas, with shallow transducers and horizontal beams, the value of 10 log m computed from measured volume rever- beration intensities and transmission anomalies by means of equation (22) will be about 3 db greater than the true value of 10 times the logarithm of the backward scattering coefficient. For rough seas it is not possible to make so precise an analysis. However, it can be argued that the dif- ference between the computed and actual value of 10 log m will be about 3 db in rough seas also since on the average the existence of many paths (Figure 3B) will be compensated for by the loss of reflecting power of the surface. For surface reverberation the existence of surface-reflected paths causes the value of 10 log m’ computed from measured surface rever- beration intensities by means of equation (39) to be about 6 db greater than the actual value of the back- ward scattering coefficient of the surface scatterers. EXPLICIT AND TACIT ASSUMPTIONS The reason for the 6-db value for the case of surface reverberation is that in equation (28) the volume of integration explicitly includes only the semi-infinite region below the ocean surface. Thus in calm seas the total intensity of the received surface reverberation is obtained by adding up four integrals of the form (28) without any reduction in the volume of integra- tion. However, this conclusion depends on the properties of the surface scattering layer. If the sur- face scattering layer absorbs sound very strongly, sound may never be able to penetrate the layer to reach the actual air-water interface. In this event, the inferred value of m’ from equation (39) will equal the true value of the backward scattering coefficient of the surface scatterers. A situation of this sort in which the surface layer is assumed to consist of a dense layer of resonant bubbles is discussed in Section 14.2.5. If the water is shallow enough for rays reflected at the bottom to be important, the situation becomes more obscure; as shown in Figure 3C, some of the possible paths between O and X involve both re- flection at the sea surface and reflection at the sea bottom. It has already been pointed out that in this situation no simple relation exists between the in- ferred value of 10 log m and the true value of the backward scattering coefficient. In fact, it may be remarked generally that equations (20), (89), and 271 (52), for reverberation from the volume, surface, and bottom, respectively, are invalid when ray paths in- volving several reflections between the projector and the scatterer become important. 12.5.7 Overall Evaluation This section has been concerned with the physical ideas behind the assumptions which have been used to derive the theoretical formulas for reverberation. It has been seen that most of the assumptions used are probably justified, but that no definite proof of their validity is possible at this time. If the assump- tions are not satisfied, reverberation may not depend on range and ping length in the manner predicted by the theoretical formulas (24), (43), and (54). None of the considerations of this section affect the possibility of using these formulas as an empirical means of in- vestigating reverberation and computing in each case a value of the backward scattering coefficient from comparison of the theoretical formulas with experiment. However, if the assumptions which have been made are not justified, the magnitude of the backward scattering coefficient deduced in this way will not have the simple physical significance implied in assumption 4 of Section 12.1. Chapter 13 EXPERIMENTAL PROCEDURES A ies CHAPTER describes the principal methods which have been used in the gathering and analyzing of reverberation intensities. It will be seen that the techniques for the study of reverberation have been greatly improved since the first studies. A great many systems have been conceived for such studies; but only those which have actually been put into operation and used extensively in the gathering and processing of data will be discussed here. Refer- ences to sources which give more detailed informa- tion are included in the body of the chapter. The experimental determination of the frequency characteristics of reverberation will be discussed in Chapter 16. EQUIPMENT AND FIELD PROCEDURES 13.1 Reverberation measurements have been made under a wide variety of oceanographic conditions, over many different types of bottoms, and at water depths between 10 and 2,500 fathoms. The most common projector depth has been 16 ft, but oc- casionally various other projector depths have been used. Most of these reverberation studies have been made by UCDWR;; quite recently, however, WHOI has undertaken a reverberation program of its own. Although differing in details, the field procedures have in all cases been similar in broad outline. In the early measurements off San Diego, made aboard the USS Jasper (PYc13), this procedure was followed. Upon arrival at the chosen location, the main engines of the vessel were shut down, and the Jasper was permitted to drift freely. The rate of drift during the working day varied between 0.5 knot and 2 knots, depending on the wind velocity and ocean currents. The transducer units with their supporting frames were hoisted by means of an electrically driven winch and boom, given the de- sired orientations, swung over the rail, and lowered into the water to the working depth. With the sound 272 gear overside, the projector and hydrophone cables were led through a doorway into the wardroom, and attached to the respective pieces of equipment. In later studies, the depth to which the transducer was to be lowered and the angle the transducer was to make with the vertical were “built in” to the equipment at the time of installation. Hoist train mechanisms were provided, which lowered the trans- ducer to the working depth from sea chests recessed in the keel. The bearing of the transducer in the horizontal plane was adjusted by means of a remote control training system. In these later modifications, the transducers were permanently wired to terminal boards, from which they could be connected to the regular electronic equipment or “‘sound stack,” or to specially constructed research stacks. After all connections are made, a sound pulse of controllable duration is sent into the water by means of a keying arrangement. As a result of this pulse, scattered sound returns to the transducer and gener- ates a voltage in the receiving circuit. This voltage, after amplification, is recorded as reverberation. Somewhat different methods are used by UCDWR and WHOI for recording the reverberation. Systems for measuring reverberation intensities are discussed in detail below. Transducers and Electronic Equipment 13.1.1 Most reverberation measurements have been taken at a frequency of 24 ke, which is the prescribed frequency for most Navy echo-ranging gear. Several types of transducer units have been employed at San Diego. Most of the early data ! were obtained with a pair of similar magnetostrictive units (QCH-3), one used as a projector and the other as a receiver; some of the data reported there were obtained with a crystal transducer, the QB. The QB crystal unit proved to be superior to the QCH-3 units for rever- beration studies because of its higher power output EQUIPMENT AND FIELD PROCEDURES =a=n=-~—— FILTER PROULGTOR bcscseeeeed ——_ = JUNCTION BOX b= | MECHANICAL KEYING CONTROL ATTENUATION KEYING BOX Figure 1. when projecting sound and its better response when receiving. Most of the more recent reverberation studies off San Diego have been performed with crystal transducers. The transducer alone is not capable of sending out pulses and detecting incoming sounds. There must also be equipment which delivers electrical energy to the projector and amplifies and modifies the small electrical impulses at the terminals of the receiver, thereby converting them into a detectable form. In the projector circuit of this auxiliary electric equip- ment there is an oscillator which generates an elec- trical signal of the desired frequency, and a power amplifier. In the receiver circuit, there is usually a preamplifier which takes the output at the terminals of the hydrophone and amplifies it somewhat, and then another amplifier, whose output is connected to the recording mechanism. Somewhat different re- cording techniques have been used by UCDWR and WHOI. At UCDWR, the voltage developed by the returning reverberation is usually fed into a cathode- ray oscillograph so that the instantaneous deflection on the cathode-ray screen is proportional to the instantaneous voltage developed in the receiver. The cathode-ray deflection as a function of time is re- corded in permanent form by the use of a camera with continuously moving film. In the technique used until very recently at WHOI, the current gener- ated in the gear by the reverberation activated a galvanometer, which in turn threw a light beam on 273 p=] SELF— EXCITED SIGNAL DRIVER BEAT FREQUENCY 4 LOUD AMPLIFIER SPEAKER | ee ee KEYING BOX TIMING TRACE 400 CYCLE CONTROL cIRCUIT | FOR Schematic arrangement of apparatus employing QCH-3 units (eauipment A of text) a moving roll of sensitized paper. The newest WHOI equipment uses a cathode-ray oscillograph and a camera, but is different from UCDWR equipment in a number of other features. Usually inserted some- where in the receiving circuit is heterodyning equip- ment, which converts the incoming high-frequency energy into energy within the range of audible fre- quencies and thus permits listening to the returning reverberation by ear. This heterodyned signal may be recorded, if desired. In the following paragraphs, we shall discuss in more detail the principal electronic setups which have been used in making reverberation measure- ments. For convenience, these setups will be identified by the letters A, B, C, D, E. Setups A and B were used at UCDWR prior to January 1943; in later UCDWER studies, setup C was used aboard the Jasper and setup D abroad the Scripps. Setup E has been used at WHOT. EQUIPMENT A This equipment? employed a pair of QCH-3 transducers, one used as a projector and the other as a receiver. Driven at 23.45 kc, the QCH-3 projector generated a sound pressure on the axis of 88.5 db above 1 dyne per sq cm at 1 yd with a total acoustic power output of 1.4 watts. A block diagram for this system is given in Figure 1. The ping was started and completed by closing and opening the ground circuit in the oscillator-driver stage by use of an electronic 274 EXPERIMENTAL PROCEDURES keying relay. The ultimate keying control was a synchronous motor-driven pair of shafts. Disks affixed to these shafts operated microswitches which, in turn, controlled the circuit containing the keying relay. By adjusting these disks, it was possible to choose any ping length between zero and several hundred milliseconds, and to control the interval between pings. Because reverberation invariably decreases rapidly with time, the receiving system must be specially de- signed to handle a wide voltage range. For this pur- pose, a variable resistor was built into the receiving preamplifier, the amount of gain being controlled by relays. By a keying arrangement similar to that for controlling the ping length, the equipment could be adjusted so that a predetermined amount of resist- ance could be removed from the receiving circuit at any desired time after midsignal. Thus, as the rever- beration intensity decreased, the gain of the pre- amplifier was increased in steps. The output voltage from the receiving amplifiers was fed directly to the plates of the horizontal deflec- tion circuit in a Du Mont Model 175A oscilloscope using a short-persistence screen; the vertical deflec- tion circuit was not connected. A continuous record of the oscilloscope deflections was obtained by the use of a fixed optical system and a camera with moving film. Since the spot on the screen moved horizontally, the film moved vertically downward, taking some time to come up to the desired speed of 12.5 in. per sec. Because the film speed at a given instant was thus not known accurately, an accurate timing record of some sort had to be photographed along with the oscilloscope reflection. The chosen type consisted of the successive images of a slit which were illuminated by the short-duration flashes of a strobotron tube driven by an electrically controlled fork. When operating properly, this system makes a faithful record of all intensity changes in the received reverberation, since the cathode-ray oscilloscope suf- fers from no mechanical inertia effects. However, since it is impractical to run the camera at speeds rapid enough to resolve individual cycles, only the time variation of peak reverberation intensity is dis- cernible on the record. Thus this equipment cannot be used to determine the frequency characteristics of the reverberation. In practice, certain difficulties were experienced with this system. For example, when the projector and receiver were close to each other, difficulty was experienced because of blocking of the receiver ampli- fier by the received ping, during the period when the projector is radiating its sound pulse. If this blocking is not eliminated, it leads to what may be described as a period of paralysis which lasts for a time after the end of the ping. During this period of paralysis there is serious distortion of the amplitude of the received reverberation. Another problem, also in- volving blocking effects, was the elimination of transients originating during the keying-in of gain changes. These transients could not be entirely elimi- nated in the final versions of this equipment. In the derivation of the theoretical formulas for reverberation in Chapter 12, it was assumed that the projected signal was ‘‘square-topped,” or, in other words, maintained a constant amplitude for a definite interval. No actual ping has this ideal rectangular shape, since some time is always required for the ping to build up and die away. Figure 2 illustrates the FicurE 2. Shape of 18 MS ping from QCH-3 pro- jector. shape of a 13-msec ping sent out by a QCH-3 pro- jector with electronic setup A and recorded by a system with flat frequency response. It will be noted that a definite time, about 1 or 2 msec, elapses before the signal reaches its maximum value. This maximum value is the same as the steady state level for a signal of indefinite duration, but is not held for long; 7 msec from the start of signal emission, the signal level in Figure 2 has diminished below its maximum value by about 4 db. After 13 msec, the signal dies away; how- ever, the rate of decay is measurable. Other photographs were taken of longer signals, more than 100 msec in length. In all these, the signal attained its maximum in | or 2 msec, fell to 3 or 4 db below maximum at 7 msec, and held a fairly steady level 3 to 4 db below maximum between 7 and 50 EQUIPMENT AND FIELD PROCEDURES TRANSDUCER CHANGE OVER RELAY UNIT 1OOW GLASS A AMPLIFIER SIGNAL DURATION & ELECTRONIC SWITCH INTERNAL CONTROL OSCILLATOR STROBOTRON TIMER FORK CONTROLLED MODULATOR STROBOTRON TUBE 275 MATCHING TRANS~ FORMER ATTENUATOR & PREAMPLIFIER AMPLIFIER AND ISOLATION STAGE ATTENUATORS WITH ELECTRONIC CONTROL SECOND AMPLIFIER BAND PASS FILTER FINAL AMPLIFIER CRO TUBE PLATES NEON LAMP RECORDING CAMERA Ficure 3. Schematic arrangement of apparatus in system used recently at San Diego (equipment C of text). msec. During the interval 50 to 70 msec, the signal level gradually rose to its fully steady state value, which was maintained for times greater than 70 msec. With the QCH-3 equipment A, the smallest re- cordable reverberation level was usually limited by the level of amplifier noise. On occasion, in noisy areas, the ambient water-noise level exceeded the amplifier-noise level. Equipment B This equipment was devised for use with the QB crystal transducer. Since the QB was used both as a projector and as a receiver, the electronic setup had to be somewhat different from the equipment de- scribed under equipment A. A changeover relay had to be provided to switch the transducer from the pro- jector circuit to the receiver circuit. An improved power amplifier was built for this system, with the result that the projected signal was nearly square- topped in form. Since the receiver circuit is not con- nected while the projector circuit isin operation, no blocking during the interval of projection was en- countered in this system. Even though the receiver circuit is not connected during the ping, a record of the outgoing ping is obtained on the film which re- cords the reverberation; this ‘‘ping record” is due to the electrical cross talk generated in the receiver cir- cuit by the high voltages in the projector circuit 276 EXPERIMENTAL PROCEDURES during the interval of the pulse. Thus, an accurate record of the ping length appears on the film. With this system, the minimum recordable rever- beration signal was limited by amplifier noise during calm water conditions, and by water noise when the sea was choppy. EQuieMENT C In the early part of 1943, new equipment was put into operation by the UCDWR? Reverberation Group. This equipment can be used with a wide vari- ety of transducers and was originally provided with four distinct frequency channels—10, 20, 40, and 80 ke. However, any four frequencies between 10 and 80 ke could he used in the projector circuit by proper ad- justment of the oscillator resonant circuit. Receiver circuit changes to accommodate different frequencies, such as provision of properly tuned input trans- formers and band-pass filters, could also be made easily. It appears from later UCDWR memoranda *6 that this system was altered to include a 24-ke chan- nel and that this channel has actually been used in the majority of the reverberation runs made with this system. IOOMS PING card 10 MS PING i Figure 4. Shape of signals sent out by equipment C of text. The power output with this setup varies with the transducer employed. With the JK transducer at 24 ke, the power output averages about 100 db above 1 dyne per sq cm.® A block diagram for this system, assuming a single transducer unit, is given in Figure 3. This system differs from systems A and B in a number of respects. A major innovation was the use of electronic timing circuits to control the ping length and keying interval, instead of the complicated me- chanical motor-driven schemes described previously. The changeover relay circuit, used with a single pro- jector-receiver, is also electronically timed, as are the step attenuators which vary the gain in the receiver circuit. The pulse projected by this system is practi- cally square-topped. Figure 4 gives photographs of the signal shape for signals of 10 and 100 msec. The receiver circuit was specially designed for stability in operation. Positions of the gain changes were automatically marked on the film by means of a flashing lamp. It is clear from Figure 5 that the transients during gain changes are not marked enough to be troublesome. In this illustration the timing trace can be seen at the top of the film; the positions of the gain changes are indicated by spots on the film below the oscillograph trace, which pre- cede the actual gain changes by a fixed distance on the record. Because of its greater convenience, its suitability for a large number of transducers, its elimination of transients, and the square-topped shape of its emitted signal, this system is in many respects a considerable improvement over systems A and B. EQuiepMENT D This system is described in references 3 and 7. The projector used with this equipment, the EBI-1, generated a pressure at 1 yd, on the axis, of 104 db above 1 dyne per sq cm. The receiver was a pre- liminary model, and had a number of faults. Many of the basic features of this equipment are similar to those in systems A, B, and C. However, the detecting mechanism used was not a cathode-ray oscillograph, but a Miller galvanometer mounted in a modified oscillograph camera. Because the galvanometer could not follow 24-ke vibrations, it was operated at 1,000 c by heterodyning the received reverberation to this frequency. One difficulty with this system is that a small percentage change in the i-f oscillator frequency (rated at 251 ke) caused a large deviation in the out- put frequency from 1,000 c, thereby introducing an error since the response of the recording galvanometer is not wholly independent of frequency. In this sys- tem, another galvanometer element was used to record the current fed to the transducer during each ping, while a third galvanometer element was used to make the timing marks. The Miller galvanometer is naturally resonant at 2,500 c because of its mechanical inertia, and there- fore cannot follow the variations in reverberation intensity with the detail possible with the cathode- ray oscilloscope used in systems A, B, and C’. How- ever, the Miller galvanometer is convenient to use and is certainly capable of following the variations in reverberation intensity with sufficient detail for the EQUIPMENT AND FIELD PROCEDURES 277 Ficure 5. Oscillograph record showing negligible transients produced by receiver amplifier gain changes in equipment C’ (input signa] maintained at steady value). accurate determination of average reverberation levels as a function of time. EQuipMENT HE This equipment used by WHOI is described in reference 9. This reference gives only the details of the recording system; presumably in most details the system does not differ essentially from UCDWR systems A to D. One major difference is the introduc- tion of a logarithmic amplifier which makes it possible to record directly in decibels. In the original version of this equipment? a seismographic galvanometer with a linear response up to 70 c was the recording device. The deflections of this galvanometer were recorded on photosensitive paper. More recently, the galvanometer has been replaced by a recording ar- rangement consisting of a cathode-ray oscillograph and a camera. This system has a linear response up to about 1,000 c; at higher frequencies the overall response, through the logarithmic amplifier, falls off rapidly. With this WHOI equipment, the averaging pro- cedure is simplified by superposing on the same film the records from a number of successive pulses. When the reverberation from a number of pulses is re- corded on one film in this manner, the average rever- beration curve can be drawn by eye through the densest portions of the trace. Since the film records the reverberation in decibels, the resulting plot is the desired curve of average reverberation level versus time. A sample record, showing the superposition of reverberation from 12 successive pulses recorded with the seismographic galvanometer, is shown in Figure 6. In this illustration, time increases from right to left; the particular features of the galvanometer used by WHOI made it more convenient to present the data in this way. Table 1 summarizes some of the important infor- mation concerning the equipment used in various reverberation studies at UCDWR. Most of the items in the table are self-explanatory. The letters A to D refer to the electronic setups described in equipment designations; and the figures in parentheses next to the transducer designations in the column labeled “Transducer references’’ tell where detailed descrip- tions of these transducers can be found in the bibliography. The column labeled “Reference” tells where the results obtained with the indicated equip- ment are discussed. TasLE 1. Equipment used in reverberation measure- ments. Transducer Frequency Electronic Reference references in ke setup 1 QCH-3 24 A 1 B® 24 B 4, 10 GB (0.11) 10 Cc 4, 10 GA,GB 20 Cc 4, 10 GA (9,11) 40, 80 C 7 EBI-1 24 D 5 JK @) 24 C 13.1.2 Calibration of Projector and Receiver In order to properly interpret the recorded values of reverberation, and to convert these recorded ampli- tudes to reverberation levels, it is necessary to know the values of the projector output F and the receiver sensitivity F’. These quantities, which occur in the theoretical formulas of Chapter 2, depend not only on the type of transducer, but also on the electronic equipment used. The procedures used in the determi- nation of F and F’ in the field are called “‘calibra- tion procedures.” The projector is calibrated by measuring the pro- jector output with a standard hydrophone whose response is stable. If an auxiliary projector with stable power output is available, F’ can be deter- mined by measuring the output of the receiver when exposed to a pulse from the standard projector. If the output of the auxiliary projector is not accurately 278 EXPERIMENTAL PROCEDURES Ficure 6. Sample record from Woods Hole reverberation camera (reverberation from 12 successive pulses superposed). known, the auxiliary projector itself may be cali- brated with the aid of the standard receiver, and then used to calibrate the receiver. The echo received from a sphere of known target strength at a known dis- tance has been used by WHOI to determine the product FF’. Present practice at both laboratories is to make a calibration at least once every working day, whenever possible.* Although calibration is simple in principle, experi- ence has shown that there is likely to be considerable inaccuracy in all projector and receiver calibrations. At UCDWR, it was found that the values of F and F’ determined by calibration procedures may change unaccountably with time, sometimes changing by nearly 10 db from day to day, and by somewhat lesser amounts during a single day.6 Some method for detecting calibration errors in the field is desirable since these errors are reflected as errors in the rever- beration levels inferred from the measured intensities. 13.1.3 Typical Reverberation Records Most of the reverberation data obtained by UCDWKR are in the form of oscillograms on 35-mm motion picture film. Figure 7 shows a sample record of the reverberation from three successive pings sent out at 8-sec intervals. For convenience in display, each reverberation record was cut into three sec- tions, as shown in the illustration; the three A’s make up the first record; the three B’s the second; etc. The marks on the upper edge of each record give the time scale; these marks are 2.5 msec apart. The point a represents the emission of the signal; b the onset of reverberation with transients caused by the opera- tion of the changeover relay; and c, d, e, places where the gain was automatically increased by the atten- uators. At f, the reverberation has decreased below background noise. The film speed (12.5 in. per sec) is high enough to show considerable fine structure in the reverberation. However, it is not high enough to resolve individual cycles; thus the trace shown in Figure 7 represents the envelope of the received reverberation. The records shown in Figure 7 are quite typical and illustrate some of the statements which have been made in this volume about the behavior of reverbera- tion. The recorded amplitude at a given time past midsignal is not constant from record to record, even though these pings were sent out and the reverbera- tion was recorded under the same adjustments of the experimental apparatus. In general, however, when- ever reverberation measurements are made, there are major features which persist from record to rec- ord. One such characteristic is the point of onset of bottom or surface reverberation. Another is the in- variable tendency of reverberation of a given sort (volume, surface, or bottom) to decrease with in- creasing time, as is predicted by the theoretical formulas of Chapter 12. This decrease makes neces- sary the provision in the system of gain changes at points such as c, d, and e; without these gain changes it would be impossible to record all the reverberation at measurable amplitudes. Occasionally, successive reverberation records show a systematic increase at certain points. These increases can usually be cor- related with the calculated increase due to the onset of surface or bottom reverberation; sometimes they are ascribed to the existence of local regions of high scattering strength. 13.2 ANALYTICAL PROCEDURES After the field work is done, the films containing the received reverberation records are taken to the ANALYTICAL PROCEDURES 279 ab c a Ot ——f-~{ tt) ——— te poem senna! eit) ert raneavynnebingnnnnnnnni ev eboiveveieaerescenersroensnte OO Ficure 7. Oscillograph records of reverberation from three successive pings. laboratory to be analyzed and averaged. These rec- ords are divided into sets, each consisting of records taken within a short space of time under similar conditions. The reverberation measurements making up a set are then averaged, and the resultant averages are supposed to represent the expected reverberation under the known external conditions for the set. Obviously the averages cannot be computed for every time instant after midsignal. Times are chosen which are spaced closely enough so that the major system- atic changes in reverberation level will be evident. At UCDWR, two methods of averaging have been used: ‘‘point method” and “‘band method.” The point method of averaging is to select a set of points, such as 1, 2, and 3 in Figure 7; measure the amplitude at these points; repeat for all records in the set (usually from 5 to 10); make proper allowance for gain changes and projector-receiver calibration; con- ® In selecting and manipulating the data, places on the records where obviously extraneous noise showed up have customarily been rejected. Examples of extraneous noises are pings from destroyers, echoes from porpoises, and bursts of ship noises. vert the amplitudes to decibels above the chosen ref- erence level; and finally, plot the resulting average reverberation levels as a function of time or range. Outstanding features, such as reverberation from the bottom or from a suspected deep scattering layer, can be emphasized by choosing many points in their vicinity on the records; and uneventful portions of the record can be passed over with but one or two points to set the general level. Usually the points chosen were spaced so as to give equal intervals on logarithmic coordinate paper. The alternative method, the band method, was introduced because of the considerable difficulty in- volved in computing an accurate average with the point method. On some records, the amplitudes are changing very rapidly close to the predetermined point where the amplitude is to be read; and to measure these amplitudes accurately it is necessary to look at the records very closely with appropriate viewing devices. This procedure is both time-con- suming and hard on the eyes, especially when the amplitude at the predetermined time is small. In the band method, a set of points is chosen as in 280 the point method. But the amplitude measured is not the amplitude at that point, but the greatest amplitude in a band three ping lengths long and centered at the designated point. Corresponding amplitudes are measured for all similar records; al- lowance is made for gain changes and projector-re- ceiver calibration; finally, after converting to deci- bels, the average reverberation levels are plotted as a function of time. As a procedure for plotting rever- beration data, the band method seems definitely superior to the point method. The amplitude cor- responding to a particular point is much easier to obtain with the band method, since it is simpler to pick out the maximum amplitude in an interval than to measure the amplitude at a predetermined point. Also, amplitudes obtained with the band method show much less fluctuation than amplitudes obtained with the point method; an analysis of reverberation records consistently showed a standard deviation of amplitude for the band method of less than 50 per cent of the standard deviation for the point method.! It is difficult to see the exact significance of the averages obtained with the band method. Certainly the band method does not closely resemble the aver- aging method which was the basis for the theoretical formulas of Chapter 12; the point method, on the other hand, does resemble it. Thus, in order to com- pare the observational results obtained with the band method with theoretical expectations, the sim- plest procedure is to correct the band method results to what would have been obtained had the point method been used. The amount of this correction was determined experimentally by comparing the results for many records processed by both the point and band methods. Except at very short ranges, it was found that the band method gives results which average quite consistently 7 db greater than results obtained with the point method. Subtraction of 7 db from the band method results thus gives average reverberation levels which are comparable with the theoretical expectations of Chapter 12. At very short ranges, on the other hand, the reverberation is chang- EXPERIMENTAL PROCEDURES ing so rapidly that the band method does not give sufficient detail and does not show any consistent relationship to the point method. Some more details of the present UCDWR analyt- ical procedure may be of interest to the reader. The individual records are analyzed by placing the films in a viewer against a graph paper background. Verti- cal and horizontal distances on the film can be meas- ured by counting squares on the graph paper, which is usually ruled in millimeters. In analyzing a record, the analyst first measures the ping length in terms of squares on the graph paper and converts this to milliseconds by comparing millimeters and the dis- tance between points on the timing trace. The num- ber of timing marks in a fixed film length gives the film speed from which a scale of range from midsignal may be constructed. This range scale is set up next to the film in the viewer. At ranges greater than 250 yd, the band method is used to determine the amplitude at the designated range. At ranges of 100 yd and 250 yd, however, the point method is used because at these short ranges, as explained previously, the reverberation is changing too rapidly for the band method to give accurate results. Despite the simplifications introduced by the band method of averaging, the analysis of a set of UCDWR reverberation records is an arduous and time-con- suming process. In the WHOI system £, the final plot of average reverberation level against range can be obtained immediately from. the photographic paper, by drawing a curve through the densest area on the superposed reverberation traces. This system is highly convenient for recording and plotting aver- age reverberation levels; but it does not permit any detailed measurement of reverberation fluctuation. Probably the best system for recording reverberation would combine the advantages of both the UCDWR and the WHOI types. This equipment would make a permanent record of all fluctuation on one recording element, while on the other recording element a smoothed trace would be made from which the final reverberation levels could be readily obtained. Chapter 14 DEEP-WATER REVERBERATION r IS CONVENIENT to begin the study of observed reverberation levels by describing the experi- mental observations in deep water. In deep water it is usually possible to ignore bottom reflections, thereby facilitating comparison of the experimental results with the theoretical formulas of Chapter 12. Also, in deep water, it is frequently possible to elimi- nate surface scattering and reflections, by directing the beam downward at some angle. When reverberation from the surface and bottom is effectively eliminated, the received reverberation can assuredly be called volume reverberation. The first section of this chap- ter describes the experimental facts about volume reverberation, as determined by such unambiguous experiments. In ordinary echo ranging, with the main trans- ducer beam horizontal, part of the received reverbera- tion is surface reverberation and part volume rever- beration. It is not easy to make a clear distinction be- tween these two components on observed reverbera- tion records. The distinction between the two is usually made by comparing the measured levels ob- tained with horizontal beams with the levels observed in unambiguous volume reverberation experiments, and also by observing the dependence of the rever- beration levels on sea state. The observed levels with horizontal transducer beams are described in the second section of this chapter. TRANSDUCER DIRECTED DOWNWARD 14.1 The experimental method for eliminating surface reverberation in deep water has usually been to point a highly directional transducer downward, away from the surface. In this way the main transducer beam does not strike the surface, and the observed rever- beration levels are then assumed to be due to volume reverberation. Of course this assumption requires verification, since in the absence of any information about the relative values of the surface and volume backward-scattering coefficients, it is not possible to know in advance how much directivity is necessary to definitely eliminate surface reverberation. How- ever, it may be accepted as a working hypothesis that pointing the main transducer beam down 30 de- grees, or more, does eliminate surface reverberation, for standard 24-ke echo-ranging gear. It will be seen later that surface reverberation levels are not usually high enough to contribute to the received reverbera- tion under these circumstances. The following sub- sections describe the various experimental studies of volume reverberation which have been carried out in this manner. mia IN ee -100 © AVERAGE OF 20 RECORDS | = THEORETICAL VOLUME Sie REVERBERATION CURVE 111 {||| REVERBERATION LEVEL R’IN DB { D Co} Qo! 0.05 0.1 O5 4 5 0 TIME IN SECONDS Ficure 1. Volume reverberation levels showing in- verse Square range dependence. 14.1.1 Dependence on Range According to Chapter 12, equation (22), if the volume scatterers are uniformly distributed, and if the transmission anomaly terms —2A + A, in equa- tion (22) can be neglected, then the volume rever- beration intensity should be inversely proportional to the square of the range, or in other words, the reverberation level should decrease 20 db with a ten- fold increase in time. As an example of this depend- ence, we may refer to Figure 1, which is a plot of data obtained on June 3, 1942.1 The QCH-38 transducers, projector and hydrophone, were lowered to a depth of 60 ft and tilted downward 60 degrees. The water depth was 600 fathoms and the surface was moder- 281 282 DEEP-WATER REVERBERATION RANGE IN YARDS -100 “120 @ AVERAGE OF IO RECORDS — @— EMPIRICAL CURVE —— INVERSE-SQUARE DEPENDENCE REVERBERATION LEVEL R'IN 0B ' z fo) aes cna aad [ei ately calm with the wind velocity averaging 10 mph, and long low ground swells but no whitecaps. A signal length of 10 msec was used. Twenty records were filmed, measured, and averaged, to give the points shown in Figure 1. It is seen that the experimental data fit fairly well the straight-line dependence of R’ on log r which is predicted, if all quantities except R’ and r are constant, by equation (24) of Chapter 12. In practice, volume reverberation runs usually show even worse agreement with this simple linear range dependence than do the points shown in Figure 1. In the first place, it is known from trans- mission measurements that the transmission anomaly terms can rarely be neglected at ranges greater than 1,000 yd (see Chapter 5). Thus the inverse square dependence can be expected only at relatively short ranges. In addition, there is no real reason to expect the volume scatterers to be uniformly distributed in the ocean. However, the fact that an approximately inverse square dependence has been observed in at least a few cases is evidence that our fundamental assumptions about volume reverberation are not altogether wrong. In general, volume reverberation tends to decrease rapidly with increasing range, in at least qualitative agreement with equation (24) of 400 iz bales i's) SCATTERING LAYER OCEAN FLOOR Figure 2. Volume reverberation levels with deep scattering layer. Chapter 12. However, the detailed dependence on range is frequently observed to be very different from the simple form of that equation; often the depend- ence of R’ on log r is not linear, and when it is linear, a slope of exactly — 20 is quite unusual. Such observa- tions are described in the following subsection. 14.1.2 Dependence on Depth Measurements off San Diego with the transducer pointed downward have frequently shown sudden increases in reverberation level which seemingly could only be explained by assuming that in certain deep layers of the ocean the backward volume-scat- tering coefficient was much larger than at other depths. Figure 2 was drawn from data obtained on July 28, 1942. The QB transducer was pointed down- ward at an angle of 49 degrees relative to the hori- zontal, in 660 fathoms of water. Ten records were averaged to give the points in this figure. At the re- verberation range indicated by A in the illustration there is a sharp rise of more than 10 db in reverbera- tion level. A comparison with equation (24) of Chap- ter 12 makes it seem necessary to ascribe this rise to an increase in the backward scattering coefficient m; TRANSDUCER DIRECTED DOWNWARD 283 Se SSSSSSSSSSSSSSSSSSssesesesesesees certainly it does not seem possible that any change in transmission anomaly could be sufficiently sudden to account for the rise. The geometry of the experiment is shown in the small box of Figure 2, on the assump- tion that the ray paths are approximately straight lines. The peak at A occurs at a time 0.5 sec after the ping. This corresponds to a reverberatior range of 400 yd; thus, the layer of high scattering power must have been centered at a depth of 400 yd X cos 41°, or about 900 ft. The thickness of the layer, as esti- mated from the thickness of the bulge at A in Figure 2, was not less than 500 ft. The large increase in re- verberation level at B corresponds to the point at which the beam strikes the bottom. The rise at C in Figure 2 could have resulted from scattering of bottom-reflected sound by the deep scattering layer. It could also have resulted from sound which was scattered from the bottom toward the surface, re- flected from the surface back to the bottom, then scattered from the bottom back to the transducer. These various possible paths are shown in the small box in Figure 2. Another record of the many which show the pres- ence of a deep scattering layer is one made August 5, RANGE IN YARDS 80 400 800 4000 8000 ao © -80 z oo inh ks MoI Con a > wW—120 rs 2-140 .-4 ec 4-160 oc Ww m—!80 C4 0.01 0.05 O.I 05 4 5 10 TIME IN SECONDS Ficure 3. Volume reverberation levels with deep scattering layer. 1942. The data, plotted in Figure 3, were obtained with the QB transducer pointed vertically downward in 650 fathoms of water. Figure 3 is an average of 10 pings each 12 msec long. This experimental curve has several important features. The first portion of the curve decreases as 20 log t, indicating uniform distri- bution of scatterers to a depth of about 500 ft. A deep layer of high scattering power is evident in the vicinity of A in Figure 3; this layer appears to have a mean depth of 1,000 ft and a thickness of about 750 ft. At the position of highest scattermg power within the layer, the volume-scattering coefficient is very much greater than its value in the body of the ocean above the layer. If we use equation (24) of Chapter 12 to estimate 10 log m, assuming that the transmission anomaly terms are small, then 10 log m at A is 16 db greater than 10 log m at points on the line denoting inverse square decay; in other words, m at A is 40 times as great as m at points in the first 500 ft of the ocean. Once the beam is out of the layer, the reverberation level falls off abruptly. At a depth of 2,250 ft, the calculated value of 10 log m, neglect- ing the transmission anomaly terms in equation (24) of Chapter 12, is 20 db down from the value of 10 log m in the first 500 ft. This difference could not be ac- counted for by ordinary values of the transmission anomaly terms —2A + Ay. It is possible (though not likely) that the sound suffers an abnormally high transmission loss in its two-way passage through the high scattering layer, or there may actually be a layer of low scattering power at the 2,250-ft depth. Echoes from the bottom are noted at B, D, and E in Figure 3. The distance the sound which produces a reverberation peak has traveled can be estimated by noting the time at which the peak appears; it is easily seen that the sound producing the peak at D has gone from the transducer to the bottom, back to the surface, then to the bottom again, and finally back to the transducer. By a similar computation the peak at C is seen to be sound which traversed one of the following two paths: (1) scattered by the layer up to the surface, reflected from the surface to the bot- tom, and then returned to the transducer, (2) re- flected from the bottom up to the surface, reflected back toward the bottom, and then scattered back to the transducer from the deep layer. Not all scattering layers are at great depths. For example, a scattering layer at a depth of about 200 ft is evident in the reverberation curve of Figure 4. These records were taken with the sound beam di- rected vertically downward, and with a ping length of 10 msec. Occasionally both shallow and deep scat- tering layers are present in the ocean simultaneously. An example is given in Figure 5, which is made up of reverberation from 8-msec pings projected at a de- pression of 60 degrees below the horizontal in 620- fathom water. In that figure, three scattering layers are noted, at A, B, and C. The layer A is at a depth of about 100 ft, B at about 600 ft, and C at about 1,000 ft. Some of these deep scattering layers appeared to persist for relatively long periods of time. In the same area of operation as that for Figure 2, deep scattering layers were observed at, about the same depth over a 284 DEEP-WATER REVERBERATION RANGE IN YARDS -1!00 REVERBERATION LEVEL R! IN DB -140 400 @ AVERAGE OF IO RECORDS -@- EMPIRICAL CURVE we q NOISE LEVEL r] Wolke Watt 0.01 0.05 Ol 0.5 1 5 10 TIME IN SECONDS TEMPERATURE IN DEGREES F OEPTH IN FEET Ficure 4. Volume reverberation levels with shallow scattering layer. period from July 9 to August 5, 1942. On the other hand, on June 16 and 17, a layer was observed at 1,200 ft; but a week later no such layer was detected. Thus the observations indicate that deep scattering layers, in a given area, may sometimes appear and disappear, and at other times persist for periods as long as a month or even longer. Just what these deep scattering layers consist of is not known; they may, for example, be concentrations of fish, bubbles or plankton.* The layer of Figure 4 occurs at the same depth as does a temperature inversion on the bathy- thermograph trace shown in the insert of Figure 4. On the other hand, no inversion is noticed at the depth corresponding to A in Figure 5. 14.1.3 Dependence on Frequency An extensive series of measurements of volume reverberation in deep water,” at frequencies of 10, 20, 8 The observations reported in this chapter were made dur- ing daylight hours. More recent studies show evidence of diurnal migration of the deep scatterers and lend support to the theory of biological origin. 40, and 80 kc have been made by UCDWR. These measurements, described later, were made in water depths ranging from 660 to 1,950 fathoms, in the months of January and February 1943. The area of observations extended southwest of San Diego tc Guadalupe Island, which is about 250 miles from San Diego and 200 miles off-shore. The various posi- tions at which observations were taken are marked by roman numerals in Figure 6. Deep scattering layers of the type discussed previ- ously were observed on this cruise. Figures 7 and 8 are plots of typical reverberation records obtained at three positions shown in Figure 6. These data were obtained with the transducers directed vertically downward, sending out 10-msec pings at the four frequencies 10, 20, 40, and 80 ke. Each point on the curves for positions III and VIII is an average of 5 pings, while points on the curves for position IX are an average of 25 pings. It is evident from Figures 7 and 8 that the effective depth of the deep scattering layer does not seem to depend on frequency. This fact is shown somewhat better in Figure 9, which is a TRANSDUCER DIRECTED DOWNWARD 285 RANGE IN YAROS REVERBERATION LEVEL R' IN DB 0.0 0.05 400 @ AVERAGE OF 10 RECORDS —@— EMPIRICAL CURVE TIME IN SECONDS TEMPERATURE IN DEGREES F 46 48 52 56 60 64 100 200 300 DEPTH IN FEET 400 500 Ficure 5. Volume reverberation levels with scattering layer at several depths. plot of the estimated depth of the deep layer ob- served for each position and frequency along the line connecting positions III and VIII. Figure 9 illustrates the persistence of the layer throughout the area of observations. According to equation (24) of Chapter 12, it should be possible to determine log m from the experimen- tally observed reverberation levels, provided the values of the transmission anomaly term —2A + A, are known. Since horizontal velocity gradients in the ocean are usually negligible, refraction can be neg- lected in measurements with a directional transducer pointed vertically downward, and A, can thus be set equal to zero. Furthermore, if the acoustic properties of the ocean do not change much with increasing depth, the transmission anomaly resulting from ab- sorption and scattering should be a linear function of range (see Section 5.2.2 of Part I). In other words, if the ocean is approximately homogeneous, the term —2A + A, in equation (24) of Chapter 12 should equal —2ar/1,000 where 7, the range of the rever- beration in yards, is equal to the depth of the scat- terers giving rise to the reverberation. It follows that if the “uncorrected” scattering coefficient M is determined from the equation R'(t) = 10 log + 10 log M — 20logr + J», then 10 log M = 101 ah (1) + oiany oekSea ame CTO In equation (1) m, the true value of the scattering coefficient, is constant if the properties of the ocean do not change with depth. Thus, for a homogeneous ocean, with m and a constant with depth, a plot of 10 log M against depth on a linear scale should be a 286 DEEP-WATER REVERBERATION LATITUDE GUADALUPE v ® LONGITUDE Figure 6. Locations where reverberation was meas- ured in 1948 cruise. straight line. The slope of this line will determine a, the attenuation coefficient in decibels per kiloyard; and the intercept of the line at zero range will deter- mine the value of the true scattering coefficient m. However, the very existence of the systematic in- crease in reverberation levels at about 1,000 ft ob- served in Figures 7 and 8 means that the ocean is probably not homogeneous with depth; thus a straight-line dependence of 10 log M on depth could hardly be expected in this experiment. Figure 10 is a plot of the mean values of 10 log M for the nine sets of records observed in the period January 17 to 20, 1943, at the positions shown in Figure 6. It is obvious from Figure 10 that even if the points in the deep layer between 1,000 and 1,500 yd are ignored, no good fit to the data could be obtained with a straight line. The failure to obtain a straight-line dependence in Figure 10 means that either m or a, or both, change with depth. It is possible to obtain further informa- tion from Figure 10 by comparing the dependence on depth at different frequencies. From equation (1), for any two frequencies f; and fo, a M(fy) _ X m(fi) © M(f) (fa) — 2La(fi) — a(fe)] 101 101 ep 1,000 2) If the variations in m are caused only by changes in the number of scatterers per unit volume, then m(fi)/m(f2) should be independent of depth. Thus, if the attenuation is independent of depth, 10 log M(fi)/M (fe) in equation (2) should be a linear func- tion of depth in these runs with the transducer di- rected vertically downward. Figure 11 is a plot of this ratio against depth, from the data of Figure 10, for the six pairs of frequencies involved. Only three of the ratios are independent; the other three can be calculated from the first three. All the ratios are shown in Figure 11 for comparison. Although most of the graphs show general tendencies to slope in the direction of increasing attenuation at higher fre- quencies, systematic deviations from the straight line predicted by equation (2) are noted. It appears then that either the kind of scatterer changes with depth or the attenuation coefficient varies with depth. : At distances less than 250 ft, attenuation is small, even at 80 ke. Thus the scattering coefficients in the upper 250 ft of the ocean can be computed from vertical reverberation runs without knowledge of the attenuation coefficient. Mean values of 10 log M = 10 log m, averaged over seven depths between the surface and 250 ft, are plotted in Figure 12 as a func- tion of frequency, for each of the nine positions of the sending-receiving ship during January 1943 (Figure 6). The solid lines are empirical curves and the dashed lines represent a theoretical relationship discussed later. The shapes of the empirical graphs for the different positions bear little resemblance to each other. However, the two curves for position III, which represent data taken 20 hours apart, reproduce each other almost to within sampling error. The curves for positions I and VIII, which were close to- gether in space but separated by 72 hours in time, are also nearly identical. If these resemblances are not accidental, they suggest that position is a more important factor than time in determining the value TRANSDUCER DIRECTED DOWNWARD Pe Bm evalliltae DEPTH IN FEET DEPTH IN FEET 287 Ball PUTT | SH ard aa sa = tes =z -— aaa SEs] Bae] as] =a uli an real a) erates ees =) REVERBERATION LEVEL R', IN DB FicurE 7. Observed volume reverberation levels versus scattering depth; GB units; sound beam vertical. of the volume-scattering coefficient. It also appears from Figure 12 that the scattering coefficient is not affected in the same way at all frequencies by changes in position. These results, if verifiable, also substanti- ate the hypothesis that volume reverberation is not an intrinsic property of water as such, but results from scatterers in the ocean whose number and type are affected by oceanographic and climatic condi- tions. Certainly, if reverberation were a property of water as such, it is difficult to see how small changes in position could result in the different shapes ob- servable in the curves of Figure 12. Figure 13 shows the mean values of 10 log M averaged at each frequency over all the positions of Figure 12 and plotted as a function of frequency. The vertical lines in Figure 13 represent mean devia- tions from this average of the values plotted in Figure 12. Figure 13 shows that, on the average, there 288 DEPTH IN FEET DEPTH IN FEET NOISE LEVEL eee -170 DEEP-WATER REVERBERATION 150 -130 REVERBERATION LEVEL R’, IN DB Ficure 8. Observed volume reverberation levels versus scattering depth; GB units; sound beam vertical. is a slight increase in 10 log M with frequency. This systematic increase is small compared to the irregular variation from position to position, but according to reference 2, the observed trend is considerably larger than the sampling error of the measurements, and also somewhat larger than the errors which could be introduced by calibration. The straight line shown in Figure 13 was fitted by least squares; its slope indi- cates that M ~ m increases as the 0.9 power of the frequency. It seems safe to say that the results of reference 2 do not exclude the possibility that on the average the scattering coefficient is independent of frequency. They admit the possibility also that m may vary as the second power of the frequency but not that it varies as the fourth power of the frequency. The lines m = Kf? (3) are drawn in Figures 12 and 13 for comparison. This fourth-power dependence of the scattering coefficient on frequency is known as Rayleigh’s scattering law and is true for scattering from particles whose di- mensions are small compared to the wavelength of the scattered sound.’ 14.2 TRANSDUCER HORIZONTAL That short-range reverberation with horizontal pings is often due primarily to scattering from the surface of the sea has been amply demonstrated by experiment. Reverberation intensity has been meas- ured first with the sound beam directed horizontally, and then with it directed vertically downward. In the first transducer position, surface scatterers are ir- radiated by much of the central portion of the beam; in the second position, they are strongly discrimi- nated against by the directivity of the transducer. When the experiment is performed in a choppy sea with whitecaps, the horizontal reverberation is many decibels higher than the vertical reverberation at TRANSDUCER HORIZONTAL DEPTH IN FEET ae a ae ce 289 NAUTICAL MILES Figure 9. Depth of deep scattering layer at various positions. short ranges. Furthermore, the difference is usually much greater in rough seas than in calm seas. Of course, it is possible that the volume reverberation obtained with the beam directed downward is less than the volume reverberation with horizontal beams. However, it is pointed out below that observed values of 10 log m with horizontal beams are at most only about 6 db greater than values of 10 log m meas- ured with vertical beams. Thus, the conclusion is in- escapable that in rough seas the short-range rever- beration from horizontal pings is surface reverbera- tion, that is, reverberation caused by scatterers near the surface of the sea whose number and strength are a function of sea state. 14.2.1 Dependence on Range and Oceanographic Conditions Analysis of reverberation records clearly shows that the range dependence of surface reverberation is itself a marked function of such oceanographic parameters as sea state and temperature gradients. For this reason, it is convenient to treat the effects of all these variables together. The following section summarizes the observational results of studies of surface reverberation, which indicate that surface reverberation tends to fall off with increasing range much faster than predicted by the simple theory of Chapter 12, and that the rate of decay increases 290 DEEP-WATER REVERBERATION ) fa 500 lu w Z x & wy 1000 oOo 2 a WW = & 1500 [S) yn 2000 (0) 500 = WW Ww w z Re 1000 WW a oO 2 x F 1500 & [S) n 2000 -100 -80 -60 -100 -80 -60 10 LOG M FiaureE 10. Mean value of 10 log M for all positions in DEPTH IN FEET 1948 cruise. rapidly with increasing sea state. A later section en- titled ‘Possible Explanations” advances some fairly plausible qualifications of the simple theory which may explain away, in part, the seeming discrepancies between theory and observation. EXPERIMENTAL RESULTS Figure 14 illustrates the results of one experiment comparing the reverberation with horizontally and vertically directed beams. At the time of the experi- ment, the sea surface was confused, with whitecaps present and with a wind velocity of 17 mph. A com- parison of the two reverberation level curves shows that for times up to 0.1 sec the horizontal reverbera- tion is more than 20 db above the vertical reverbera- tion; for times between 0.1 and about 0.4 sec, it is more than 10 db above; and for times greater than 0.4 sec it is less than 4 db above. Two conclusions are obvious from the figure. One is that on the day of the experiment there was a surface layer of scatterers which was very different in nature from the scatterers in the ocean body. The other is that the reverberation due to surface scatterers decays more rapidly than the reverberation due to volume scatterers. The pres- ence of a deep scattering layer can also be noted at B in Figure 14, and also a shallower scattering layer at A. According to equation (389) of Chapter 12, the sur- face-reverberation intensity at short range, where 2A in equation (39) can be neglected, should be propor- tional to the inverse cube of the range, provided Ficure 11. Variation with depth of ratio of scattering coefficients. TRANSDUCER HORIZONTAL 10 LOG M a a TD 2 ne a aman 291 40 80 FREQUENCY IN KC Ficure 12. Variation of 10 log M with frequency (mean values of 10 log M for depths less than 250 feet). Positions III and III, refer to measurements made at Position III on two separate days. 10 LOG M © OVERALL MEAN VALUES OF OBSERVED IO LOG M — LEAST SQUARE FIT TO OBSERVATIONS --- RAYLEIGH'S SCATTERING LAW Jf MEAN DEVIATION FREQUENCY IN KC Figure 13. Variation of scattering coefficient with frequency (mean values of 10 log M at all nine positions for depths less than 250 feet). 10 log m’/2 and J,(6) in equation (89) are also inde- pendent of range. This simple inverse cube depend- ence is observed only rarely. Figure 15 shows the reverberation intensities observed on May 8, 1942, with the QCH-3 transducers. On this date ground swells were long and low, and a few whitecaps were forming. The QCH-3 transducers were at a depth of 20 ft with their long dimensions horizontal and with the transducer axes parallel to the sea surface. RANGE IN YARDS 40 80 400 800 SL] | -souno BEAM HORIZONTAL REVERBERATION LEVEL R' IN DB 0.05 Oel 0.5 4 TIME IN SECONDS 0.01 Figure 14. Comparison of reverberation from hori- zontally and vertically directed beams. In this position the QCH-—3 transducers are prac- tically nondirectional in the vertical plane, so that in equation (41), Chapter 12, the correction factor b(@ — £,0)b’(@ — £,0)/cos @ is very nearly unity at all angles of importance. Since this correction factor is the only part of J,(@) which can depend on range, 292 it is clear that in these experiments J,(@) was inde- pendent of range. With negligible A and constant m’ and J,(@), the theoretical equation for surface rever- beration, equation (48), Chapter 12, leads to a straight line with a slope corresponding to inverse third-power decay. This simple reverberation decay is indicated by the solid line in Figure 15. The points in Figure 15 are the averages of 36 pings each 8 msec long and agree fairly well with the theoretical straight line. Barer a YARDS 400 800 PICU JICLONIII Ess CRI SaEn0n ESE RSs Qc IS8enn HEHE fy] e AVERAGE OF 36 RECORDS — THEORETICAL SURFACE REVERBERATION CURVE FOR NONDIRECTIONAL UNITS -70 -90 REVERBERATION LEVEL R! IN DB 0.05 Ool TIME IN SECONDS Ficure 15. Surface reverberation levels showing sim- ple inverse cube dependence. Projector and receiver effectively nondirectional in the vertical plane. On the same day (May 8, 1942) surface-reverbera- tion measurements were also carried out with the QCH-3 transducers oriented differently. In these ex- periments the transducers were placed at a depth of 20 ft with the transducer axes parallel to the ocean surface, as before; but the long dimensions of the transducers were vertical instead of horizontal. With this transducer orientation, the correction factor b(6 — £,0)b’@ — £,0)/cos 6 cannot be neglected. The values of the correction factor as a function of range were calculated from the known directivity pattern of the QCH-3. A theoretical reverberation curve was then obtained, using equation (43) of Chapter 12, assuming 10 log m’/2 independent of range, and neglecting the term 2A in that equation. This curve is plotted as the solid line in Figure 16. It can be compared with the points which show the actual reverberation levels observed in this experiment; each point represents the average reverberation from 30 pings each of length 8 msec. Evidently the agree- ment between theory and experiment is quite good. It may be remarked that at ranges between 80 and DEEP-WATER REVERBERATION 800 yd the observed points in Figures 15 and 16 are in close agreement. This agreement was not to be expected if the scattering coefficient of the surface did not change with time; it is easily verified that the values of J,(9) are quite different for the hori- zontal and vertical orientations of the QCH-3.> Thus the agreement at ranges greater than 80 yd between the observed points in Figures 15 and 16 means that the value of the surface scattering coefficient must have changed during the interval between the two experiments. The value of 10 log m’/2 estimated from Figure 15 is about 5 db greater than the value estimated from Figure 16. RANGE IN YARDS 40 80 400 800 PRANGD 35, ee A ST CE SSotw Sessa © AVERAGE OF 30 RECORDS i -140 | THEORETICAL SURFACE REVERBERATION | CURVE FOR THESE DIRECTIONAL UNITS 0.5 1 0.05 0.! TIME IN SECONDS 0.01 REVERBERATION LEVEL R' IN DB Figure 16. Surface reverberation levels showing agree- ment with theoretical curve. Projector and receiver directional in vertical plane. Evidently if surface reverberation arises from scat- tering in a thin layer near the ocean surface, a drop in reverberation is to be expected when the sound beam is bent away from the surface, provided, of course, that the surface reverberation is not masked by volume reverberation at the range where the beam leaves the surface. Under conditions of sharp down- ward refraction such sudden drops have been ob- served. For example, Figures 17 and 18 show rever- beration levels obtained with the QB transducer on July 24, 1942. On this day ground swells were almost absent, but the ocean surface was scuffed up with a few whitecaps forming. The wind speed was 12 mph. The QB transducer was placed with its axis parallel to the surface at depths of 20 and 60 ft. Twelve con- secutive pings, each 11 msec long, were averaged at each depth to give the points shown in Figures 17 b The values of J,(6) for both horizontal and vertical orien- tations of the QCH-3 can be obtained by using the QCH-3 directivity patterns given in Section II of reference 1, in equation (42) of Chapter 12. TRANSDUCER HORIZONTAL 293 RANGE IN YARDS 400 IL 760i fli: Pal JS xg REVERBERATION LEVEL R' IN DB a SIS Sy SEE ee o EXPERIMENTALLY OBSERVED POINTS e OBSERVED POINTS CORRECTED FOR (| ABS i ony a a ° ° <== THEORETICAL INVERSE- CUBE DEPENDENCE Sig TRANSDUCER DIRECTIVITY -180 0.01 0.05 Oe 0.5 | 5 10 TIME IN SECONDS Figure 17. Surface reverberation levels with strong downward refraction. Transducer depth 20 feet. RANGE IN YARDS REVERBERATION LEVEL R' IN DB TRANSDUCER DIRECTIVITY — THEORETICAL INVERSE-CUBE DEPENDENCE 400 4000 7] © EXPERIMENTALLY OBSERVED POINTS © OBSERVED POINTS CORRECTED FOR TIME IN SECONDS Ficure 18. Surface reverberation levels with strong downward refraction. Transducer depth 60 feet. and 18. Figure 19 shows the bathythermograph record and the calculated limiting rays for the two depths. The points marked A in Figures 17 and 18 show the range where the limiting ray leaves the sur- face. Just as in Figure 16, the observed reverberation levels at short range can be expected to differ from a straight line with —30 log r slope because of the correction factor b(@ — £)b’(6 — £)/cos @. To facili- tate comparison with this line, the observed levels DEPTH IN FEET iN oa 400 600 800 1000 56 60 64 68 RANGE IN YARDS DEGREES F Figure 19. Refraction conditions for data in Figures 17 and 18. 294 DEEP-WATER REVERBERATION STANDARD REVERBERATION LEVEL R IN DB WIND SPEED IN MPH Figure 20. Standard reverberation level at 100 yards as a function of wind speed. are corrected by increasing the experimental points by the value of the correction factor. That is, in Figures 17 and 18 the solid points are values of b(0 — £,0)b'(0 — £,0) cos 6 R’(t) — 10 log (4) By using equations (41) and (43) of Chapter 12, the expression (4) obviously equals QO) Co / 10 log + 10 log (“) — 30 log r + 10 log ~>— TT — 2A. (5) In equation (5), if A can be neglected and if m’ is independent of range, the only dependence on range is contained in the term —30 log r. Thus, with these assumptions the solid points in Figures 17 and 18 should lie on a straight line of —30 log r slope if re- fraction has no effect. It is seen that the first few solid points in Figures 17 and 18 do lie on a straight line of — 30 log r slope, but that at a range close to that when the limiting ray leaves the surface, there is a sharp drop in reverberation level. This drop does not con- tradict the theory in Chapter 12. It will be recalled from Section 12.3 that equation (43) is not valid and therefore cannot be expected to agree with measured levels at ranges past that at which the limiting ray leaves the surface. In equation (4) the correction factor approaches unity and its logarithm approaches zero as the range is increased, and in both Figures 17 and 18 the correction is practically negligible by the time the sudden drop in intensity occurs. Conse- quently it is not possible to ascribe the position of the experimental points at ranges past the sudden drop in intensity to uncertainty in the value of the correc- tion factor. Thus the conclusion seems inescapable that the sudden drop is due to the sound rays leaving the surface. From the evidence in Figures 15 to 19, it appears that our basic assumption, namely that surface rever- beration arises from scattering in a thin layer near the ocean surface, is probably correct. The observed —30 log r slope in Figure 15 and in the long-range portion of Figure 16 also permit the conclusion that TRANSDUCER HORIZONTAL STANDARD REVERBERATION LEVEL R IN 0B oO CHARLIE 4 NAN WIND SPEED IN MPH FIGURE 21. there are times when the surface scattering coefficient m’ is independent of the angle of incidence of the rays on the surface. However, it is not usually possible to fit the observed reverberation intensities with equa- tion (39) of Chapter 12, if m’ is assumed independent of range. Thus, while the basic assumptions leading to that equation are probably correct, it cannot be said that the factors involved in surface reverbera- tion are completely understood. More illustrations of the dependence of surface reverberation on range may be obtained from a memorandum issued by UCDWER,,‘ where extensive measurements of observed deep-water reverberation levels at 24 ke are summarized. This summary is based on data obtained on 6 cruises in the period from November 26, 1943 to September 1, 1944. About 110 reverberation curves were obtained, each an average of five successive pings. The ping lengths used varied from 16 to 80 yd, but in the following curves all data have been corrected to the standard 80-yd length; in other words, the following graphs are all plotted in terms of the standard reverberation level. All the data were obtained with the JK projector at a depth of 16 ft on the USS Jasper with the transducer axis horizontal. Figure 20 is a plot against wind speed of all the reverberation levels measured at a range of 100 yd. All seasons of the year are represented. Thermal pat- terns were of MIKE, CHARLIE, and NAN types (see Chapter 5), represented respectively by dots, circles, and triangles. In Figure 20 a systematic in- Standard reverberation level at 1,500 yards as a function of wind speed. crease in reverberation level of about 35 db is ob- served as the wind increases in velocity from zero to 20 mph. At wind speeds of 8 mph or less, there is little systematic dependence on wind speed. At greater speeds the level rises sharply, up to speeds of 20 mph or more. Increase of wind speed beyond 20 mph has little systematic effect. This dependence on wind speed is correlated with the roughness of the sea. At 8 mph the wind is strong enough to roughen the surface appreciably; occasionally wavelets may slough over, but no well-developed whitecaps are observed. At about 10 mph small whitecaps begin to appear. When the wind has reached 20 mph the sea is liberally covered with whitecaps. The detailed de- pendence of the appearances of the sea on wind force is described in a Navy manual.® Apparently, as the wind speed increases beyond 20 mph, the resulting increase of whitecaps causes little, if any, additional increase in reverberation. The median values of the standard reverberation level at 100 yd, as a function of wind speed up to 20 mph, are roughly described by the equation R = —118+ 10 log (1 + 2.5 X 10-*u) (6) where wu is the wind speed in miles per hour. This equation is represented by the solid line in Figure 20. Beyond 20 mph, the function is assumed to be con- stant at R = —83 db. The reverberation level at long range has a markedly different wind-speed dependence from that at short range. The data in Figure 21 are taken from 296 RANGE 100 YARDS -100 -10 STANDARD REVERBERATION R IN DB “120 -130 DEEP-WATER REVERBERATION RANGE IS00. YARDS -120 -130 -150 «460 -170 e OBSERVED POINTS LINE JOINING MEDIANS -———— LINE JOINING QUARTILES FiGcurE 22. Dependence of standard reverberation level on sea state. the same reverberation runs as the data in Figure 20, with the exception that the levels shown were meas- ured at 1,500 yd rather than at 100 yd. No significant wind-strength dependence is observed. It seems justifiable to conclude from these data that surface reverberation, which is frequently dominant at 100 yd, has little effect at 1,500 yd; in other words, at 1,500 yd the observed reverberation usually arises from volume scattering. Figure 22 shows the de- pendence of reverberation on sea state at ranges of 100 and 1,500 yd. The 100-yd levels depend on sea state while the 1,500-yd levels apparently do not; thus the qualitative dependence in Figure 22 is the same as that in Figures 20 and 21. The relation be- tween wind force and sea state is given in the NDRC survey report on ambient noise.® Figure 23 shows the reverberation levels as a func- tion of range, for high and low wind speeds. In this illustration are given the median reverberation levels and the upper and lower quartiles at each range for wind speeds less than 8 mph and greater than 20 mph. It appears from Figure 23 that the reverbera- tion is entirely independent of wind speed at ranges greater than 1,500 yd. Actually, the quartiles at ranges greater than 1,500 yd are not precisely the same for wind speeds less than 8 mph and wind speeds greater than 20 mph (see Figure 5 of reference 4), but these differences are not thought to be significant. Consequently, in Figure 23, data for all wind speeds are used to determine the range dependence of the reverberation at ranges of 1,500 yd or more. From Figure 23, for wind speeds greater than 20 mph, the median reverberation level drops 46 db be- tween 100 and 1,000 yd. Thus, for high wind speeds, the reverberation intensities usually drop off nearly as the fifth power of the range, rather than as the third power predicted in Chapter 12 on the assump- tion that 10 log m’ is independent of range. (At 100 yd or more, with the JK at a depth of only 16 ft, the variation of J,(6) with range can be neglected.) Even faster rates of decay than the fifth power are fre- quently observed. For example, Figure 24 shows a plot of the reverberation levels obtained on the Point Conception cruise, on March 2, 1944. The rate of decay in this figure is approximately as the sixth power of the range; that is, R decreases approxi- mately as —60 log vr. The wind speed on this run was 37 mph. Figure 25 shows the reverberation level at 1,500 yd plotted against date and area. The data fall into five STANDARD REVERBERATION LEVEL R IN DB TRANSDUCER HORIZONTAL MEDIAN LINE ——-—— LOWER QUARTILE eooeeee UPPER QUARTILE STANDARD REVERBERATION LEVEL R IN DB 200 300 500 700 297 1000 2000 10,000 5000 7000 3000 RANGE IN YARDS Ficure 23. Dependence of standard reverberation level on range and wind speed. 200 300 500 700 1000 RANGE IN YARDS Ficure 24. Sample plot of standard reverberation level at high wind speed. groups, two pairs of which were taken in the same area at different seasons. The data grouped in this way are summarized in Table 1. On the whole, Table 1 shows that the mean reverberation levels at 1,500 yd are independent of season and area, although the Cedros II and Point Conception data may indicate some systematic variation. Figure 26 is an analysis of the dependence of the observed reverberation levels on a parameter which has been found to correlate significantly with trans- mission studies at 24 ke. This parameter is the depth Dy in the ocean at which the temperature is 0.3 F less than at the surface.’ The levels in Figure 26 are re- ferred, for convenience, to an arbitrary zero level which was, however, the same for all curves. Only data obtained after March 22, 1944 were used for this comparison since most of the earlier runs ended. at about 2,000 yd. The median curves are seen tc be practically the same for D, between 5 and 40 ft, but fall off less rapidly for D, between 40 and 160 ft. Since, according to reference 7, increasing D2. means a decreasing transmission anomaly A in equation 298 DEEP-WATER REVERBERATION CEDROS I NOV 1943 PT CONCEPTION MAR 1944 @Q ° -130 2 4 a Ww > WwW a 2 (o} Ee q [tg w —-140 o hal a Ww > Ww ig a a q (=) 2 q e (7) -150 MEDIANS ———- QUARTILES MAR-APR 1944 UCDWR CEDROS II MAY 1944 UCDWR AUG-SEPT 1944 =160 Figure 25. Standard reverberation level at 1,500 yards for various dates and areas. TABLE 1 Median reverberation level Inter-quartile Number Area Month at 1,500 yd difference of cases All data Dec.—Sept. —140 9 113 Cedros I November —141 10 31 Cedros II May —146 3 12 Point Conception | March —137 2 27 San Diego March-April —140 6 22 San Diego August-September —140 12 21 (26) of Chapter 12, the differences between these curves might be explained as a result of improved transmission to long ranges with larger values of Ds. However, on the whole there is little dependence of the curves on the parameter D2; certainly no de- pendence is apparent at 1,500 yards.° ° In this connection, more recent data obtained by UCDWR are of interest. These data, as yet unpublished, show that with NAN patterns (usually falling in the class The results of this subsection can be summarized — as follows. At ranges less than 1,500 yd, in standard 24-ke echo-ranging gear oriented so that the acoustic 5 < Dz < 10) there is frequently a hump in the reverberation curve at a range which corresponds to the deep scattering layer discussed previously. Presumably these humps are most com- mon with NAN patterns because with strong downward re- fraction the main sound beam usually strikes the deep layer at a well-defined range. TRANSDUCER HORIZONTAL REVERBERATION INTENSITY IN DECIBELS ABOVE ARBITRARY ZERO 0 1000 2000 3000 RANGE IN YARDS FicurE 26. Range dependence of reverberation as a function of refraction conditions. axis is parallel to the surface, the reverberation ob- served at wind speeds greater than 8 mph is pre- dominantly surface reverberation. At ranges greater than 1,500 yd, the reverberation does not depend 299 significantly on wind speed, location, season, or ther- mal structure of the ocean. It seems justifiable, there- fore, to regard the reverberation at ranges greater than 1,500 yd as the characteristic volume reverbera- tion of the ocean. PossIBLE EXPLANATIONS The lack of dependence of reverberation on wind speed at ranges greater than 1,500 yd is well estab- lished, but is nevertheless surprising. The masking of the surface reverberation by volume reverberation at this range is in large part due to the rapid decrease of surface reverberation with range. The reason for this decrease is obscure. The following factors have been suggested, in Section VI of reference 1, as possi- ble causes of this rapid decrease of surface reverbera- tion with increasing range: attenuation, variation of the scattering coefficient with angle of incidence on the surface, shadowing effects of waves, and inter- ference effects in a thin surface layer (Lloyd mirror effect). These possible causes will now be considered briefly. In Figure 23, the drop between 100 and 1,000 yd in median reverberation level at wind speeds greater than 20 mph is 16 db more than would have been predicted from the —30 log r dependence of equa- tion (48), in Chapter 12. If this change is due to transmission loss, the term A in equation (43) must have a median value of 8 db per kyd. While this value of the attenuation is not impossible, it is significantly greater than the mean attenuation coefficient ob- served in transmission studies off San Diego (see Section 5.2.2), especially since with high wind forces the surface layer tends to become isothermal. If the steep slope in the curve of Figure 24 is due to atten- uation, the attenuation coefficient would have to be as high as 15 db per kyd. The possibility that some of the increased loss is due to attenuation cannot be ruled out; but on the whole the evidence from trans- mission studies does not justify regarding attenua- tion as the primary cause of the rapid decrease of surface reverberation with range. If the surface scattering layer is very thin, it can be argued that the scattering coefficient m’ should decrease at least as rapidly as sin 0, where @ is the grazing angle of the ray incident on the surface. For the total volume of surface scatterers irradiated by the ping at any instant is proportional to cr, accord- ing to Section 12.3. All the energy reaching this volume must pass through the surface whose cross section in the plane of Figure 27 is AB. Thus, if I is 300 intensity at AB, the total energy reaching the surface scatterers per unit time is proportional to J(AB) = I(r) sin 6. If the simple assumption is made that the energy scattered in all directions is the same as the energy scattered in the backward direction, it follows from the definitions of m and m’ [equations (4) and (32) of Chapter 12] that the total energy OA= UPPER EDGE OF MAIN BEAM OB= LOWER EDGE OF MAIN BEAM FicureE 27. Energy reaching surface scatterers. scattered per unit time is proportional to m’. Thus, the ratio of the total energy scattered per unit time to the total energy reaching the scatterers per unit time is proportional to m’/sin 6. The former ratio can- not exceed unity; if it is to remain finite at small graz- ing angles, m’ must decrease at least as rapidly as sin 0. If the scattering from the surface obeys Lambert’s law (the law of scattering of light by rough surfaces’), then the backward scattering coefficient will be pro- portional to sin? 6. At ranges of 100 yd or more, with transducers at 16 ft, sin 9 = @ and is inversely pro- portional to the range. Thus, comparing with equa- tion (43) of Chapter 12, if the scattering arises in a thin surface layer, and if we can assume that equal amounts of energy are scattered in all directions, the surface reverberation would be expected to fall off as the fourth power of the range, or faster. Supple- mented by the added loss due to attenuation, such a variation of scattering coefficient with grazing angle could explain the observed dependence on range in Figures 23 and 24. However, before we can accept the variation of m’ with grazing angle as an explanation of the depend- ence of surface reverberation on range, we must de- termine how thin a scattering layer is required for the argument of the previous paragraph to be valid. Figure 28 is a more exact drawing of the situation pictured in Figure 27, drawn so that the layer has appreciable thickness. In Figure 28 the projector O is at depth d, and the scattering layer has thickness h. The scattering volume has a cross section CADE in the plane of the paper, with CD at long range very nearly equal to cz. Energy enters the scattering volume through AE (as in Figure 27) or through AC. From Figure 28 it is easy to obtain a simple criterion DEEP-WATER REVERBERATION for the validity of the argument of the previous para- graph, if attenuation in the surface layer can be neglected. For, with this approximation, the energies entering through AC and AE are proportional re- Figure 28. Expanded view of surface layer. spectively to the solid angles formed by rotating ¢ and y in Figure 28 about a vertical axis (in the plane of the paper) through O. At long ranges, 6 small, these solid angles are proportional respectively to angles ¢ and y. Thus, the calculation in the previous paragraph of the energy entering the scattering volume per unit time is incorrect unless the angle ¢ is very small compared to y. At long range we have approximately, with OC = 1, h = (AC) cos 6 = r¢ cos 8 AB = rp = (AE) tan @ = or tan 0. Thus the condition that ¢ be very small compared to w becomes h tan 0 ——_< (Gor) tan 0 (7) r cos 8 T For small 6, cos 6 = 1, sin 6 = 6 = d/r. Thus equa- tion (7) becomes h& (cor) (8) At 1,000 yd, with 100-msec pings and d = 5 yd, equation (8) gives h < 30 in. There are scarcely any data, but it seems likely that the surface layer might frequently be thin enough to satisfy the relation (8). On the other hand, in rough seas it would not be sur- prising to find that the relation (8) is violated. Equation (8) was derived neglecting attenuation in the scattering layer; if attenuation is taken into account then it can be shown that the expression (8) must be replaced by 1 — en"? < (ara, (9) where the attenuation in the layer, in decibels per yard, is 4.84a. The value of a probably depends pri- marily on the population of bubbles in the surface TRANSDUCER HORIZONTAL layer and is difficult to estimate. If the attenuation is as large as 1 db per yd, a value which is observed in wake measurements (see Chapter 35), equation (9) would be satisfied even with very large values of h. In fact, it is obvious that the relation (9) will always be satisfied if (cor)a > 1. (10) If (8) is not satisfied, it is easy to see that (9) will not be satisfied if (10) is violated. We may summarize this discussion of the validity of the argument that m’ should decrease at least as rapidly as sin 6 by stating that the argument may be correct, but requires further quantitative informa- tion on the thickness of the scattering layer and the attenuation to be expected in the layer. Until this information is forthcoming, the hypothesis that m’ decreases with decreasing angle of incidence on the surface is not a wholly acceptable explanation of the variation of surface reverberation with range. At small grazing angles, the peaks of the water waves are sometimes hidden from the sound source by the troughs. This shadowing effect of waves also causes a reduction of the irradiation of the surface. If the scatterers are largely concentrated in a layer whose depth is small compared with the wave height, a reduction in reverberation might be expected at small grazing angles. Since the grazing angle decreases with increasing range, this effect could account for the rapid decrease of surface reverberation. This hypothesis of the shadowing effects of waves has the added virtue that it explains the increasing rate of decay with increasing sea state, since the larger the waves the more important this effect would be. How- ever, this hypothesis is much too qualitative to be accepted without further study. It can be seen that phenomena in high sea states may actually tend to make the reverberation increase with increasing range rather than decrease. For example, in high sea states, at long range, the sound rays may make large angles of incidence with the wave troughs, thereby increasing considerably the sound returned back to the transducer. A quantitative evaluation of the shadowing effect of waves is difficult and requires a detailed examination of surface roughness. Several papers issued by UCDWR °-¥ are initial attacks on the theory of surface scattering. That the nature of the surface irregularities will affect surface rever- beration seems almost intuitively obvious. Another report “ describes measurements in which definite structure was found in surface reverberation. On this 301 day there were strong swells with a wind speed of 16 to 19 mph. Distinct blobs were observed in the reverberation, and these blobs altered their range at a rate equal to the rate at which the surface swells were moving. These blobs could be identified much more readily on the chemical recorder than on the oscilloscope record, where the wealth of detail con- fused the general picture. In Section VII of reference 1, no difference was observed in reverberation meas- ured with the projector beam parallel to and perpen- dicular to the wave fronts. A wave in water reflected at the water-air bound- ary suffers a change in phase of the sound pressure (see Chapter 2). This change of phase results in inter- ference between the direct and surface-reflected rays; the transmission loss between the projector and points near the surface may be increased to a value much greater than the inverse square loss used in deriving equation (43) of Chapter 12. Furthermore, the increase in transmission loss will be a function of the range and of the distance of the scatterer from the surface, and will increase with decreasing depth and increasing range. Thus if the scatterers are lo- cated in a thin layer near the surface, this interfer- ence between direct and surface-reflected waves may explain the observed rapid decrease of surface rever- beration. As with the previous hypothesis of wave shadowing, it is necessary to make a quantitative investigation of this image interference effect before accepting it as an explanation of the range depend- ence of surface reverberation. For a plane surface, it is shown in Section VI of reference 1 that image inter- ference can lead to a decrease of surface reverbera- tion proportional to the seventh power of the range; consequently, this effect could account for the ob- served slopes of Figures 23 and 24. However, the surface is not plane. An approximate treatment of the effect of surface roughness in reference 1 shows that the image interference effect’ becomes less im- portant as the surface roughness is increased. Thus if image interference is causing the range dependence, the slope of surface reverberation ‘should decrease with increasing sea state, which is contrary to what is observed. Another inference from the theory of the image interference effect is that surface reverbera- tion should increase rapidly with frequency at ranges where the interference is important. Unfortunately, there are no experiments on the variation of surface reverberation with frequency. It may be concluded from this discussion of the range dependence of surface reverberation that the 302 DEEP-WATER REVERBERATION — — —SSSSSSSSSSSSSSSsFshseeee reason for the rapid decrease is not understood, but that there are a number of factors which may play a part. Very likely, all the physical factors which have been discussed previously are included to some de- gree. In addition, there may well be other causes which have not been considered. It should be noted that at ranges less than 500 yd the measured rever- beration for wind speeds less than 8 mph decreases only as the inverse first power of the range in Figure 23. This rate of decay is even slower than the pre- dicted inverse square decay of volume reverberation. No definite explanation has been offered for this feature of Figure 23, but it might be caused by a gradual increase in the value of the volume-scattering coefficient as the deep layer is approached. This effect would, of course, be noticeable only in sea states so low that volume reverberation can be measured at short ranges. 14.2.2 Dependence on Ping Length The theoretical formulas (22), (89), and (52) for volume, surface, and bottom reverberation in Chap- ter 12 all have the reverberation intensity propor- tional to the ping length. The theoretical assumptions required to obtain this result have been discussed in Chapter 12. In this subsection we shall discuss whether or not this strict proportionality may be expected in practice. The only data bearing on this question are reported in reference 1, Section IV, and are summarized later. Unfortunately reference 1 does not state whether the reverberation studied was volume, surface, or bottom reverberation, but rever- beration received from ranges as low as 100 yd and as great as 5,000 yd was included in the analysis. However, ranges less than five times the ping length were not included. Figure 29 shows, qualitatively, that the reverbera- tion intensity increases with increase in the signal length. In that illustration, a record A of reverbera- tion following a 70-msec ping is compared with a record B of reverberation following a 10-msec ping. The attenuator settings were the same for both cases; however, because of the higher level of the 70-msec reverberation, each attenuator step was removed a little later for the 70-msec ping. For this reason, the records are directly comparable only in the intervals 1-1, 2-2, and 3-end in which the amplification is the same for both records. In these intervals, the 70-msec reverberation is.clearly much higher than the 10-msec reverberation. To test quantitatively the predicted relation between reverberation intensity and signal length, sets of data were taken on two successive days of the reverberation following pings of lengths very nearly 10, 20, 40, and 70 msec. Ten pings were measured on each day for each signal length. For each ping length, the average reverberation amplitude was measured at a set of logarithmically equispaced positions, by the band method. The squares of these average amplitudes were assumed proportional to the average reverberation inten- sities, in accordance with the usual procedure de- scribed in Chapter 13. The agreement between theory and experiment is shown in Figure 30. In that illustration, ten times the logarithm of the ratio of any two ping lengths is taken as the abscissa, and the decibel difference between the corresponding measured reverberation levels is taken as the ordinate.’ If reverberation intensity were in fact exactly proportional to the ping length, all the observed points should lie on the 45-degree straight line drawn in the figure. In Figure 30 the points for which the longer ping length of a pair was 20, 40, and 70 msec are designated differently so that any systematic departure depending on ping length can be discerned. On the whole, the agreement in Figure 30 between theory and experiment is satis- factory. For some reason, the agreement is better for the ratios involving the shorter signals (10, 20, 40 msec) than for the ratios including the longest signal (70 msec). The deviations from the straight line in Figure 30 are not too great to be ascribable to experimental error. Thus these data give no reason for doubting the prediction of equations (22), (39), and (52) of Chapter 12, that, under the conditions specified in that chapter, reverberation intensity should be pro- portional to ping length. However, in view of the im- portance of knowing the dependence on ping length for comparison of reverberation measurements made with different ping lengths, and for the determination of scattering coefficients, further investigation of this dependence is desirable. The measurements should be repeated for a wider range of ping lengths and for all types of reverberation. 4 The reason that the abscissas of some of the pairs of points in Figure 30 are not the same is that the signal lengths as measured from the film records were not exactly the same on both days. However, the equipment was set on each day for nominal ping lengths of 10, 20, 40, and 70 msec. TRANSDUCER HORIZONTAL BS a 00-2 A:70 MILLISEC PING 303 “DOS Lee mt ccc ra Bz10 MILLISEC PING Figure 29. Comparison of reverberation from a 70 MS ping with that from a 10 MS ping. 14.2.3 Unimportance of Multiple Scattering The theoretical formulas of Chapter 12 are based on the assumption that multiple scattering can be neglected. Experiments designed to measure the amount of multiple scattering are described in a memorandum by UCDWR " and summarized below. The 24-kc, QCH-3 units were mounted 6 ft apart with the long dimension horizontal in such a way that the unit used as a hydrophone could be rotated about a vertical axis. The unit used as a projector was kept in a fixed position. Observations were made with the receiving hydrophone at bearings of 0, 30, 60, and 90 degrees, relative to the bearing of the pro- jector axis. That is, the receiving hydrophone was rotated so that it faced away from the projector, and the sound received in it was measured. Ping lengths of 15 msec were used, and 5 pings were averaged at each bearing. If the two QCH-3 units had been highly directive, interpretation of the observations would have been straightforward. However, they were not highly directive, so that the portion of the signal pro- jected in the direction in which the receiver was pointing could not be neglected. In order to evaluate the data, therefore, the following procedure was adopted. The expected signal in the receiving hydro- phone was calculated from the known directivity patterns of the hydrophone and projector, assuming single scattering was taking place in the ocean. It is 304 easy to see from the derivation of Chapter 12 that this expected signal depends on the integral {~b(0,¢) b’ (8,6 + a)dQ, where b and b’ are defined as in Chap- ter 12, and a@ is the angle between the projector and hydrophone. tt Ei i i P72 HE BA Zee 12 10 @ LONGER SIGNAL ABOUT 20 MILLISEC X LONGER SIGNAL ABOUT 40 MILLISEC © LONGER SIGNAL ABOUT 7OMILLISEG — LINE REPRESENTING LINEAR DEPENDENCE RATIO OF TWO REVERBERATION INTENSITIES IN DB fo) 2 4 6 8 10 12 RATIO OF TWO PING LENGTHS IN DB Figure 30. Observed dependence of reverberation in- tensity on ping length. The values of the above integral were computed for a equal to 0, 30, 60, and 90 degrees, and the pre- dicted reverberation intensities were then compared with the observed intensities. It was found that the calculated levels were within 1 to 2 db of the observed average levels for ranges up to about 200 yd, beyond which no measurements were made. Since the experi- mental error of the measurements was not less than 1 to 2 db, these results show that at short ranges multiple scattering makes a negligible contribution to the received reverberation. However, these results give no information about the effect of multiple scattering at longer ranges. It is easy to show that multiple scattering can certainly be neglected if the volume scattering in all directions is the same as in the backward direction. For, in this event, the total energy scattered per second per unit intensity at dV is just mdV (see Section 12.1). The loss in intensity d/ in a distance dx of a plane wave of intensity J traveling in the z direction is then dl = mldxz I = Ivex™. Thus under these circumstances the attenuation of a which gives DEEP-WATER REVERBERATION sound wave by scattering is 4.34 < 103m db per kyd. It is shown later that m is rarely greater than 10-°. By using this value of m, the attenuation due to scattering is 4.34 X 10-? db per kyd. Now, with any kind of a transducer, but especially with a direc- tional transducer, multiple scattering will not be im- portant in reverberation until the amount of singly scattered energy in the ocean becomes appreciable compared to the amount of energy remaining in the direct sound beam. Obviously, with a scattered energy loss of only 4.34 < 10~ db per kyd, scattered energy in the ocean is negligible compared to the energy in the direct beam for ranges less than 20,000 yd where the total scattered energy loss is not yet 1 db. Despite the arguments of the preceding paragraph, more experimental evidence bearing on multiple scattering would be desirable especially since the oblique scattering may be appreciably different from the backward scattering. One way to check the im- portance of multiple scattering would be to compare with experiment at long ranges the predicted de- pendence of received reverberation intensity on trans- ducer directivity. If multiple scattering is important, the difference between reverberation levels measured with directional and nondirectional transducers will not be given by J, [equation (21) of Chapter 12]. No such measurements have been reported; in fact, the whole question of comparing with experiment the dependence of reverberation on the theoretical rever- beration indices J, and J, seems to have been neg- lected. Knowledge of this dependence is required for comparison of measurements made with different gear, and also for prediction of the effect on reverber- ation in echo-ranging gear of changes in gear direc- tivity. There is no reason for doubting the validity of the formulas of Chapter 12 for ordinary gear, but with highly directive gear, multiple scattering and other effects may produce deviations from the theo- retical formulas. A UCDWR internal report '° de- scribes experiments in which the measured vertical directivity patterns in the ocean were very different from the patterns obtained at a calibrating station. Pitch and roll of the echo-ranging vessel will also cause deviations from the predicted reverberation in- tensities, especially for surface reverberation.” 14.2.4 Average Levels Figure 31 is a summary of the measured 24-kc re- verberation levels reported in reference 4. The levels shown are standard reverberation levels, as defined TRANSDUCER HORIZONTAL 305 -60 POINTS OHIGHEST REPORTED LEVELS AT EACH RANGE @ LOWEST REPORTED LEVELS AT EACH RANGE —80 AMEDIAN LEVELS AT EACH RANGE i HHH Mea aS o > RR ee aa -100 = a S 4 rr ° = -|120 c w o c Ww > w = -140 oa & o : ao Se oma SETHI Se je Ea TE 50. 70 100 300 500 1000 20 10,000 RANGE IN YARDS FIGuRE 31. by equation (25) of Chapter 12. The dots show the lowest reported reverberation levels at each range, and the circles the highest levels. Median values of the data are shown by triangles. At ranges less than 1,500 yd, the lower triangles are the median values for wind speeds less than or equal to 8 mph, and the upper triangles are the median values for wind speeds greater than or equal to 20 mph. These data are neither an adequate nor a random sample, and detailed analysis of them is not justi- fiable. However, some interesting inferences, which should be reasonably reliable, can be drawn from the data. The lower solid line in Figure 31 is a plot of equation (26) of Chapter 12 against range with 10 log m set equal to —80 db, A set equal to 3 db per kyd, and A, set equal to zero. The upper solid line is drawn with 10 log m set equal to —60 db, with A set equal to 1.5 db per kyd, and A; set equal to zero. Both lines were plotted with J, set equal to — 25 db.'8 If all reverberation at ranges greater than or equal to 1,500 yd is volume reverberation, and if the lower levels at shorter ranges are volume reverberation, then the upper and lower solid curves would represent estimated upper and lower limits to 24-ke volume- reverberation levels. The middle solid curve is drawn with 10 log m set equal to —60 db, and with A set equal to 4 db per kyd. This curve fits the median values of reverbera- Surface and volume reverberation levels. tion for wind speeds less than or equal to 8 mph surprisingly well at ranges from 500 to 3,500 yd; it can probably be assumed that these median values for wind speeds less than or equal to 8 mph represent volume reverberation. The 4 db per kyd value of A is gratifyingly close to the average value measured in transmission studies under good conditions (see Sec- tion 5.2.2). These results apparently indicate that a scattering coefficient 10 log m equal to —60 dbisa typical value for the volume reverberation from horizontally projected 24-ke sound beams. Oc- casionally, however, the scattering coefficient be- comes very small, 10 log m becoming as low as —80 db. It does not seem legitimate to attempt any con- clusions based on these differences in the values of A required to fit the curves of Figure 31. Both A and m are highly variable quantities, and are known to bea function of depth in the ocean. However, comparison of the upper and median curves at ranges past 1,500 yd suggests that differences in the long-range reverberation levels may be frequently due merely to variations in A. It has already been pointed out that the deep scattering layers discussed in the first portion of this chapter will tend to increase the reverberation at long range above the levels otherwise expected.* Studies of bottom reverberation (see Chapter 15) 306 have shown that the main beam from standard 24-ke gear will usually reach a depth of 1,000 ft at a range of about 2,000 yd. The median curve for volume re- verberation in Figure 31 does not show any evidence of an increase in the volume-scattering coefficient at long ranges; it will be recalled that the value of 10 log m in these deep layers frequently exceeded the mean value of 10 log m in the ocean by 15 db or more. However, it appears that this failure to observe the deep layer must have been due to sampling. More recent studies by UCDWR, still unpublished, show that the deep layer is frequently discernible as a very definite bulge on the reverberation curve. These new data also show that the maximum value of 10 log m is — 50 db or perhaps even slightly higher rather than the —60 db value indicated by Figure 31. From Figure 31 and from equation (26) of Chap- ter 12, we may conclude that the backward volume- scattering coefficient m for horizontally projected 24-ke beams varies between 10-* and 10-° per yd with 10~ the typical value. It will be noted that the dimensions of m in equation (26) of Chapter 12 are per yard; values of m expressed per foot will be one- third or 5 db less. Also, from Chapter 12, we recall that this value of m is about 3 db greater than the “true” value of the backward-scattering coefficient of the ocean. Since equation (45) of Chapter 12 does not describe the range dependence of surface rever- beration very well, determination of the surface scattering coefficient m’ from comparison of that equation with Figure 31 is not very meaningful. However, if we make the comparison, with A set equal to 4 db per kyd and J, set equal to —16 db, the median values of 10 log m’ at ranges of 100 and 1,000 vd for wind speeds greater than 20 mph are respectively —22 db and —31 db. (Note that m’ is a dimensionless quantity.) It will be recalled that at the beginning of this chapter we assumed that surface reverberation could be eliminated by pointing the transducer downward. Off the main lobe, the response b(6,¢) of standard 24-ke transducers is usually assumed to average about 30 db less than the peak response on the main lobe,!* but this is only an approximate average value. Using this 30-db estimate, we see from Figure 31 that at ranges of 100 yd, in high sea states, surface rever- beration may exceed volume reverberation even with the transducer directed downward, but only if the volume reverberation level is close to the minimum values observed. At ranges greater than about 500 yd, or with wind speeds less than about 15 mph, DEEP-WATER REVERBERATION pointing the transducer downward should usually eliminate surface reverberation (see Figure 20). This estimate of the wind speeds and ranges at which sur- face reverberation can be eliminated by pointing the transducer downward assumes, however, that the volume reverberation with the transducer pointed downward is the same as when the transducer is hori- zontal. Measurements reported in reference 2 sug- gest that the volume scatterers are anisotropic and, specifically, have a smaller backward-scattering coefficient when the sound arrives from a vertical direction than when the sound arrives from a hori- zontal direction. However, the observed difference in 10 log m was only about 6 db and thus hardly affects the conclusion stated previously that surface rever- beration can almost always be eliminated by pointing the transducers downward. 14.2.5 Scattering Coefficient of a Layer of Bubbles It is of interest to compare the median values of 10 log m’ obtained from Figure 31 with the values ex- pected if the surface consisted of a dense layer of resonant bubbles. The theory of air bubbles in water is given in Chapter 28; the geometry of scattering by B > x 0) Figure 32. Scattering from surface layer of bubbles. such a layer is illustrated in Figure 32. By definition, a densely populated layer of bubbles is one in which the attenuation is so high that there is essentially infinite transmission loss through the layer. For this reason, in Figure 32, energy reaches the scatterer at X and returns to the scatterer at O along the direct path OAX only; scattering along a path reflected from the air-water interface, such as OBX, can be neglected since almost no energy reaches B. It fol- lows from bubble theory that multiple scattering can be neglected as well. Neglecting refraction, the expected reverberation intensity can now be calcu- lated directly from equation (29) of Chapter 12, using (11) (12) S ll = St TRANSDUCER HORIZONTAL where N is the number of bubbles per cubic centi- meter, a, and o, are respectively the absorption and scattering cross sections of a resonant bubble (de- fined in Chapter 28), D is the distance AX, a the usual attenuation coefficient equal to about 4 db per kyd, and r the range. Evaluating the integral in equation (29) of Chap- ter 12 with the aid of equations (11) and (12), and, comparing the result with equation (39) of Chapter 12, it is readily found that (13) where @ is as usual the angle of elevation of the ray from the projector to the scatterer at range r (Figure 7,Chapter 12),and d the projector depth. For a trans- ducer at 16 ft, equation (15) gives m’ equal to —28 db at 100 yd and —38 db at 1,000 yd (using Figure 1 in Chapter 34). These values are about 6 db lower 307 than the median measured values of —22 db and —31 db and are still lower than the highest reported levels at 100 and 1,000 yd (see Figure 31). Since equa- tion (13), derived on the basis of a densely populated surface layer, gives the highest possible value of m’ for scattering by bubbles, it seems that measured surface reverberation levels cannot be explained on the hypothesis of scattering by a surface layer of bubbles. It will be noted that, because of the assumed neglect of scattering along OBX (Figure 32), in this situation of scattering by a densely popu- lated surface layer of bubbles the value of m’ indi- cated by equation (13) is the true surface-scattering coefficient. Thus, although the argument presented in Section 12.5.6, that measured values of m’ are 6 db greater than the true value of the surface-scattering coefficient, is probably valid, it is evident that the validity of this 6-db relation depends on the physical process which gives rise to the surface reverberation. Chapter 15 SHALLOW-WATER REVERBERATION SURFACE ry TTOM YY PROJECTOR >] REFLECTION e 7) z Ww = = TT BOTTOM TIME PROJECTOR DOWN 90 DEGREES SURFACE BOTTOM > [= w” 2 Ww = = BOTTOM | . SCATTERING TIME PROJECTOR DOWN 30 DEGREES SURFACE A SCATTERING __,-~ SCATTERING INTENSITY TIME PROJECTOR UP 30 DEGREES SURFACE > PROJECTOR BOTTOM SURFACE SCATTERING BOTTOM SCATTERING INTENSITY TIME PROJECTOR HORIZONTAL Figure 1. Expected behavior of bottom reverberation for idealized projector. 308 (ee OF THEORY and experiment in bottom reverberation is complicated by the directivity pattern of the transducer and by uncertainty regard- ing the dependence of the scattering coefficient on the angle of incidence. An additional complication is refraction, which is of considerable importance in de- termining bottom-reverberation levels. One way in which bottom reverberation differs from the other types of reverberation we have considered is that bottom reverberation is not heard immediately after the initial ping, but appears some time later, usually coming in as a distinct crash. This delay results from the fact that the bottom, unlike the scatterers re- sponsible for surface and volume reverberation, is usually a significant distance from the projector. 15.1 QUALITATIVE DESCRIPTION OF BOTTOM REVERBERATION Figure 1 illustrates the expected behavior of the bottom reverberation for an idealized projector hav- ing constant sound output within 5 degrees of the axis and zero output outside 5 degrees. For illustra- tive purposes, we may assume there is no refraction. Then for this simple type of sound beam, scattered sound is received, at the time instant ¢, only from those scatterers included within a sector of a spherical shell centered at the projector and bounded by the limits of the sound beam. The mean radius of this shell is cf/2 and its thickness cr/2, as pointed out in Section 12.2. Bottom reverberation is received when- ever this shell cuts off some portion of the bottom. Bottom reverberation will set in at the time cor- responding to the shortest range at which the beam strikes the bottom. Since bottom scattering coeffi- cients are usually relatively large, the total received reverberation will increase sharply at the time of on- set of bottom reverberation. Scattering at the bottom will cease, except for sound which is reflected or scat- tered toward the bottom, at the time the last portion of the beam leaves the bottom. The case of vertical incidence on the bottom is illustrated in the first box of Figure 1. In this case, as shown in the box, all portions of the beam strike the QUALITATIVE DESCRIPTION OF BOTTOM REVERBERATION bottom nearly simultaneously. Thus the reverbera- tion begins and ends very abruptly. The time of on- set of the reverberation equals 2d/c, where d is the depth of the bottom below the projector, and c is the SURFACE psn, RK WN MR 0% B OW Figure 2. Vertical incidence of ping on bottom. sound velocity. Evidently, as shown in Figure 2, all portions of the beam do not strike the bottom exactly at the same time so that the duration of the rever- beration does depend somewhat on the beam width. For narrow beams, it is easily shown that the rever- beration duration is given by dae Reverberation duration = 7 + TE (1) c where a is the beam width, shown in Figure 2, and r the ping duration. Thus, with very short pings inci- dent vertically on the bottom, the duration of the reverberation may appreciably exceed the ping dura- tion. When the beam is incident on the bottom at some slant angle, all parts of the beam do not strike the bottom at the same time. Consequently the rever- beration does not begin or end quite as sharply as in the previous case, and the reverberation duration is greater than the ping duration. This case is illustrated in the second box of Figure 1. In this situation it is easily shown from Figure 3 that the time of onset and duration of the reverberation are given by 2d Time of onset = A esc (0 + =). (2) Reverberation duration = 0 Aes « a a T+ —, ose (0+ 2) cot(o - 2). (3) If the beam is pointed up toward the surface, or is horizontal, surface reverberation will be heard before 309 the bottom reverberation begins to come in. With a horizontal beam, different parts of the beam strike the bottom at widely spaced intervals, so that the reverberation lasts a long time. These statements are illustrated in the third and fourth boxes of Figure 1. Of course, the sound beam is not actually confined within a cone; some sound is sent in all directions. However, most sound projectors commonly in use confine all but a small fraction of the emitted and received sound within small angles from the beam axis, so that the simple description of Figure 1 should be a good approximation to the observed phenomena. Figure 4 is an experimental illustration of the qualita- tive predictions of Figure 1. The data making up Figure 4 were obtained in an area where the water depth was 72 ft. It is noticed that the reverberation recorded with the projector directed vertically down- ward consisted of a single pulse of about the duration of the ping. The reverberation levels recorded with the projector directed 30 degrees down fall on a curve approximating the simple case shown in the second box of Figure 1. There is a rapid rise of reverberation, due to bottom scattering, at the time corresponding approximately to the range at which the main beam first strikes the bottom (easily calculated as 35 yd for a half beam width of 6 degrees) and reaching a peak at about the range where the axis of the beam arrives SURFACE Ww A Figure 8. Slanting incidence of ping on bottom. at the bottom (40 yd). The peak is followed by a rapid decay. The case of the horizontal beam is il- lustrated in the bottom curve of Figure 4, and is seen to correspond roughly to the bottom curve in Figure 1. The initial reverberation recorded at 40 to 50 yd is received from the bottom on the projector side lobe; this type of reverberation decays rapidly for about 310 SHALLOW-WATER REVERBERATION BEAM OOWN 90 DEGREES PING LENGTH =4YDS BEAM DOWN 30 DEGREES PING LENGTH = 4 YDS | alain REVERBERATION LEVEL IN DB VS LEVEL AT ONE YARD 160 320 200 360 RANGE IN YARDS © MEASURED LEVELS — PREDICTED LEVELS Ficure 4. Observed and predicted bottom reverberation levels. 0.01 sec; then there is a rather rapid growth in the reverberation intensity when the main beam reaches the bottom. Further illustrations of the reverbera- tion measured with the beam down 30 degrees are shown in Figure 5. The ocean depth was 48 ft and the projector depth 10 ft so that the peak of the reverberation is observed to come in at a time cor- responding to a range of about 25 yd. The ensuing lesser maxima are the result of successive multiple reflections from surface and bottom. These observa- tions were taken over a bottom thickly covered with boulders of the order of one foot in diameter. An interesting feature of these curves is the change in the duration of the main reverberation pulse as the frequency changes. The duration of the pulse de- creases progressively with increasing frequency; this effect is due to decrease in beam width. In the case of the projector directed upward at an angle of 30 de- grees, it is seen from Figure 5 that the surface rever- beration peaks predicted in Figure 1 are lacking. This absence is due to the fact that the reverberation was measured under conditions which combined very shallow water with a smooth sea surface and a rough sea bottom, so that bottom reverberation masked surface reverberation at practically all ranges. Under other circumstances, with a smooth bottom and a rough sea, or deeper water, the peak of surface rever- beration can usually be observed before the crash of bottom reverberation comes in. These remarks indicate that bottom reverberation, under at least some circumstances, behaves quali- tatively as would be expected from a sitnple geomet- rical analysis of the time required for the sound to reach the bottom. The next step in the analysis is to attempt to make a quantitative prediction of the expected reverberation levels, and then to compare these theoretical predictions with experiments. From formula (54) of Chapter 12, it is clear that the received reverberation depends on the transmis- sion loss to and from the bottom, on the transducer directivity, and on the scattering strength of the bottom. It can be assumed that the transducer REVERBERATION LEVEL IN DECIBELS QUALITATIVE DESCRIPTION OF BOTTOM REVERBERATION BEAM DOWN 30 DEGREES BEAM UP 30 DEGREES PROJECTOR DEPTH = 10 FT PROJECTOR DEPTH = 38FT RANGE IN YARDS Figure 5. Observed reverberation level with beam inclined 30 degrees. 311 312 directivity is known in sufficient detail, although in practice it may prove difficult to obtain even this information. The scattering strength of the bottom is in general a function of the angle of incidence of the rays striking the bottom. Temperature gradients in the ocean are frequently sufficiently large that the sound rays in the main transducer beam have suf- fered appreciable bending by the time they reach the bottom. For this reason bottom reverberation is likely to depend much more strongly on transmission conditions than surface reverberation. To accurately predict bottom reverberation levels, therefore, it is necessary to have detailed knowledge of the ray paths and transmission loss to the bottom. Unfortunately, there are practically no reported bottom reverberation measurements for which the transmission to the bottom can be regarded as known in detail (including knowledge of the ray paths and the transmission loss along these paths). Conse- quently, any comparison of predicted reverberation levels with experimental observations must be based on assumptions about the transmission; and detailed agreement in any single experiment between pre- dicted bottom-reverberation levels and observed levels should not be expected. Rather, because of this uncertainty concerning the transmission from the transducer to the bottom, the comparison of theory and experiment becomes even more a purely statis- tical process than was the case for surface and volume reverberation. This statistical approach, which is described later, is in some respects justifiable. Transmission studies made to date reveal little likelihood that detailed knowledge of the transmission can be obtained aboard an ordinary echo-ranging warship in any practicable way. What is required is a statement of the average reverberation levels to be expected for various broad classifications of echo-ranging gear, transmission conditions, and bottom types. 15.2 EFFECTS OF REFRACTION Some of the results of a statistical analysis of bot- tom reverberation are described in an internal report by UCDWER.! In this report, many bottom rever- beration curves were plotted against range. The data comprising these curves were all taken at 24 ke with standard Navy gear directed horizontally, but were obtained in a variety of regions, over many different types of bottoms, at differing depths, and with vary- ing refraction conditions. Examination of these data SHALLOW-WATER REVERBERATION showed a number of similar features on almost all the curves. In general, the curves showed the following characteristics. 1. A peak which comes in shortly after the out- going signal and results from surface reverberation. 2. A rapid decay of reverberation with the level reaching a minimum at a range of two times the depth of water. 3. A broad rise in level as the range increases, de- veloping a second peak at a range of about six times the depth of water. This rise is due to bottom rever- beration. 4. Beyond the second peak a rapid decrease of in- tensity, approximately proportional to the inverse fourth power of the range. However, very large varia- tions from this type of decay were observed. The range of the bottom reverberation peak de- pends of course on refraction; for this reason, the dependence of the range of this peak on depth be- comes a statistical problem. If there were no refrac- tion, in other words, if the sound rays always traveled in straight lines, then the ratio of the range of the peak to the depth (over plane and smooth bottoms) would obviously always be a fixed quantity depend- ing only on the directivity pattern of the transducer. In fact, for standard 24-ke echo-ranging gear, with a beam width of 5 to 6 degrees, the range of the peak would be 10 to 12 times the water depth, if refraction were absent. In order to judge the usefulness of the statistical study in reference 1, it is necessary to know what kinds of temperature gradients were included, and the extent to which these refraction patterns obtained near San Diego are typical of refraction conditions in other localities. There are reasons for believing that the results of reference 1 may be valid for a wide variety of temperature gradients. Bathythermograph patterns are not completely arbitrary in shape; posi- tive gradients are relatively rare, with the result that most patterns other than isothermal ones show a continuous decrease of temperature between surface and bottom. Furthermore, the effect of refraction is greatest for horizontal or nearly horizontal rays. Once the rays have been bent through an appreciable angle, the amount of additional bending, even by quite sharp negative gradients, is relatively small. For these reasons, the ratio of the range of the peak to the depth may be expected to be relatively con- stant for a wide variety of gradients excluding isother- mal or “nearly isothermal’ types, where, for the purpose of this discussion, ‘nearly isothermal” water EFFECTS OF REFRACTION 313 RATIO OF RANGE TO DEPTH 30 40 PER CENT GREATER THAN INDICATED VALUE 50 60 70 80 90 95 Gi} S) Figure 6. Cumulative distribution of observed ratio: Range to reverberation peak Water depth may be defined as water in which the temperature at the bottom differs from the temperature at the surface by less than five degrees. Actually, examination of the data in reference 1 shows that relatively few of the reverberation curves analyzed were obtained in isothermal or nearly iso- thermal water. Thus, the results of that study do not apply to water in which the top-to-bottom tempera- ture change is less than five degrees. This fact helps to account for the disparity between the observed range of the reverberation peak, characteristically about six times the depth, and the predicted value of 10 to 12 times the depth for isothermal water. After these preliminary remarks, we may examine the San Diego results in more detail. Figure 6 is a cumulative plot, taken from reference 1, of the ratio of the range of the bottom reverberation peak to the water depth. The median point on this curve cor- responds to a range-depth ratio of 6.2. Fifty per cent of all peak ranges were found to lie between 5.1 and 7.2 times the depth, and 80 per cent to lie between 4 and 8 times the depth. These results agree well enough with the results of another study by UCDWR.? In reference 2, which, however, was based on a smaller number of reverberation curves, the average range to the peak was about 5 times the depth. The difference between these two estimates of the ratio of the range of the peak to the water depth is prob- ably due to sampling and to the fact that the rever- beration curves plot only a few isolated points of the measured film. The data discussed in reference 2 are described in somewhat more detail in Section 15.3.1; they were obtained by using a transducer whose beam pattern was similar to that of standard Navy echo-ranging gear. Bathythermograph data and ray diagrams were available, and it was found that the range to the reverberation peak corresponded to about the range where the 6-degree ray reached the bottom. In another internal report by UCDWR,? it was found that the range of the peak usually cor- responded to the range at which the 5-degree ray reached the bottom. For standard Navy gear at 24 ke, the half beam width y defined in Figure 4 of Chapter 12 is close to 6 degrees. Thus, the results described in the preceding paragraph suggest that with standard gear at 24 ke the range where the beam’s edge (5- or 6-degree ray) strikes the bottom is the range of the reverberation peak. These results suggest furthermore that in water which is not “nearly isothermal” the range of the reverberation peak is between four and eight times the depth. For simple temperature gradients, it is easy to estimate the range at which various rays will strike the bottom, as a function of the depth to the bottom.‘ Table 1 gives the results of such calcu- lations, for various initial ray angles, water depths, and surface-to-bottom temperature differences; in computing this table it was assumed that the tem- perature decreased linearly from the surface to the bottom. It is clear from the table that with linear gradients the 6-degree ray always does strike the bottom at a range between 4 and 8 times the water depth, when the temperature difference between the projector and bottom is greater than 5 degrees. 314 SHALLOW-WATER REVERBERATION TaBLE 1. Ranges at which rays leaving the projector at different angles strike the bottom. Temperature Depth Range at which ray strikes bottom Ratio of difference between in yards range at which between projector 6-degree ray projector and bottom Angle of ray leaving projector strikes bottom and bottom in feet in degrees to water depth in degrees 4 6 8 12 5 50 171 132 106 74 7.9 100 345 264 211 148 7.9 200 696 534 425 299 8.0 300 1070 809 640 449 8.1 10 50 141 115 96 70 6.9 100 282 230 194 141 6.9 200 571 464 385 282 7.0 300 863 700 580 425 7.0 15 50 123 103 88 67 6.2 100 244 207 176 133 6.2 200 491 415 356 268 6.2 300 743 628 534 402 6.3 20 50 109 94 82 63 5.6 100 219 188 162 127 5.6 200 440 373 329 253 5.7 300 664 568 496 383 5.7 15.3 BOTTOM SCATTERING COEFFICIENTS Having established the probable limits of the range to the reverberation peak, it is desirable to estimate the height of the reverberation peak. This height has been found to depend markedly on the type of bottom. In general, reverberation is highest over ROCK, less over SAND-AND-MUD or MUD, and least over SAND, although in some cases rever- beration over SAND has been reported to be quite high, especially after a storm when rippling of the bottom may be the cause.! The relative values of the reverberation over these bottoms may be ex- pressed in terms of the bottom-scattering coefficient m”, by using equation (54) of Chapter 12 if the trans- ducer directivity and the transmission to the bottom are known in detail. Detailed information concerning directivity and transmission has not usually been available, but it has proved possible to determine average values for the bottom scattering coefficients by making reasonable assumptions about the trans- mission. In this section, we shall summarize present infor- mation concerning the variation of the’ bottom scat- tering coefficient m”. The three factors which are ex- pected to be most important in determining the value of m” are (1) the grazing angle of the sound incident on the bottom, (2) the frequency of the incident sound, and (8) the nature of the bottom. These factors will be considered in the same order. 15.3.1 Dependence on Grazing Angle Knowledge of the nature of the dependence of m” on grazing angle is necessary for the detailed predic- tion of bottom reverberation as a function of range; in addition, the nature of this dependence is of theoretical interest. It has been shown in Chapter 14 that, with simple assumptions about the law of scattering, for a very thin scattering layer the value of the backward scattering coefficient should de- crease with decreasing grazing angle at least as rapidly as sin 6; and for scattering obeying Lambert’s law the backward scattering coefficient should be proportional to sin? 6. Thus, determination of the dependence of m” on angle can furnish information about the law of scattering at the bottom and can also be used to check the validity of our ideas about bottom reverberation. However, determination of this dependence is not easy. It would appear at first thought that the de- pendence would be an easy by-product of the analysis of ordinary reverberation runs with horizontal trans- ducers if temperature-depth data were also available. BOTTOM SCATTERING COEFFICIENTS 315 RANGE IN YARDS 100 “Ho -120 “130 -140 REVERBERATION LEVEL R’ IN DB REVERBERATION LEVEL R’% IN DB -150 0.1 0.2 0.5 ! TIME IN SECONDS COARSE SAND DEPTH I70 FEET AUGUST 18, 1943 RANGE IN YARDS @ -110 a z | z \ _ = -120 F Nl ne Ww Ww 4 -130 a = ° a S & -140 rs a BACKGROUND LEVEL i o & mn > z -150~ fe TIME IN SECONDS MUD DEPTH 255 FEET OCTOBER 9,1943 CALCULATED RANGE IN YARDS TIME IN SECONDS SAND-AND-MUD DEPTH 315 FEET SEPTEMBER 27, 1943 RANGE IN YARDS 1000 TAT Here lb = “110 — == BACKGROUND LEVEL -150 sr po tla 0.1 “140 TIME IN SECONDS ROGK DEPTH 370 FEET SEPTEMBER 27, 1943 RANGE AT WHICH 6° RAY STRIKES BOTTOM Figure 7. Typical bottom reverberation levels with horizontal transducer. The reverberation level at any range could be trans- lated into the bottom scattering coefficient by use of equation (54) of Chapter 12, and the grazing angle of the sound at this range could be computed from the temperature-depth information. However, it will be seen in the following subsection that this procedure is not workable, briefly because the value of m” com- puted in this way is accurate only for the portion of the bottom struck by the central portion of the pro- jected beam, and within this limited area there is not much variation in the grazing angle. Thus, in order to determine the form of the function m’ (6), specifically designed experiments are necessary. An EXPERIMENT DESIGNED TO MEASURE m’ (6) Data casting light on the angular dependence of m’ were obtained in a series of experiments, made in August, September, and October of 1948, and de- scribed in an internal report by UCDWR.? On each day that measurements were taken, the transducer was set either at 0 degrees (main transducer beam horizontal) or at 30 degrees (main transducer beam pointed 30 degrees down from the horizontal), and lowered to a depth of 9 ft. Bottom reverberation for a number of 10-msec 24-ke pings was then recorded as a function of time, by using the equipment de- scribed as D in Section 13.1.1. By comparing measure- 316 ments made with the two different transducer orien- tations in the same or similar areas, it should be pos- sible to obtain some information about the angular dependence of m”. The following paragraphs describe briefly the analysis of these data made in reference 2; it is convenient to begin by considering the 0-degree data. For the purpose of analyzing the 0-degree data, the reverberation records obtained were segregated into nineteen groups, each group comprising at least nine records of bottom reverberation taken at nearly the same time on a single day over one of six bottom areas. The records were then measured and averaged over the group to give the mean reverberation ampli- tude, and the average reverberation level was plotted against range for each group. Typical curves ob- tained are shown in Figure 7. These curves extend only to the range at which the reverberation becomes comparable to the recording background. On each curve in Figure 7 is shown the range at which the 6-degree ray struck the bottom, as computed from the measured BT pattern. The high levels of the first plotted points at 100-yd range in the curves of Figure 7 are due to surface reverberation. These reverberation curves for horizontal trans- ducers were then compared with equation (54) of Chapter 12, by using 4 db per kyd for the absorption; in addition, the anomaly due to refraction was com- puted from the ray diagram drawn from the BT pattern, according to the methods described in Chap- ter 3. The total correction for the anomaly was found to be small for ranges corresponding to the re- verberation peak. The average magnitude of twice the anomaly correction for those ranges was only 2.5 db; but the uncertainty in the transmission anomaly led to uncertain values of m” corresponding to the reverberation at longer ranges. Nevertheless, by us- ing the data and comparing with equation (54) of Chapter 12, it was possible in this way to obtain for each group of records a curve for m’, the bottom scattering coefficient, as a function of the range of the reverberation. At each range the incident grazing angle of the ray reaching the bottom was computed from the refraction diagram. However, from these curves of m” against range, as explained in more de- tail later, the value of m” was accurately determi- nable only for grazing angles on the bottom very nearly equal to the grazing angle of the central ray of the main beam. This angle, for all the horizontal projector curves studied in reference 2, lay between 9 and 13 degrees. It is clear therefore that determina- SHALLOW-WATER REVERBERATION tion of the angular dependence of m” was not possible from the 0-degree data alone. By using the 30-degree data, however, the value of m” when the grazing angle on the bottom is equal to 30 degrees may readily be determined. With this transducer orientation, the rays in the main beam are only slightly bent by the temperature gradients. Thus they strike the bottom at angles that can be cal- culated directly from the geometry. Furthermore, since the rays are only slightly bent, the transmission anomaly due to refraction can be ignored. By using equation (54) of Chapter 12, then, and by assuming A equal to 4 db per kyd, the values of m” at a grazing angle of 30 degrees were determined from comparison with the observed data. This comparison was made on the assumption that the maximum amplitude of bottom reverberation on the 30-degree records cor- responded to scattered sound returning along the central ray of the main beam; this assumption is justified from the qualitative discussion in Section 15.1. The average scattering coefficients determined from these analyses of the data in reference 2 are shown in Table 2. The 10 log m” values in Table 2 TaBLE 2. Average values of m” for various bottom areas. 10 log m” 10 log m” (beam (beam down Bottom type horizontal) | 30 degrees) | k COARSE SAND —33 —24 2.0 FINE SAND —32 —26 1.5 SAND-MUD —25 —21 1.0 MUD —29 —20 2.0 ROCK —18 -9 2.0 FORAMINIFERAL SAND —26 Bo Bee are of course averages of the values obtained from the nineteen groups into which the original data were subdivided. That is, each group gave a value of 10 log m” for a definite bottom type, and the entries in Table 2 are each averages over all the groups per- taining to one particular bottom type. In interpreting these average scattering coefficients, it should be remembered that the 0-degree data were obtained with a horizontal transducer near the sur- face so that an important part of the received bottom reverberation reached the transducer along paths re- flected from the ocean surface. As a result, the values of 10 log m” inferred from comparison of the 0-de- gree data with equation (54) of Chapter 12 are 6 db greater than the true value of the bottom scattering BOTTOM SCATTERING COEFFICIENTS coefficient. The values of 10 log m” shown in the first column of Table 2 are true values, that is, they are 6 db smaller than the values found from com- parison of equation (54) of Chapter 12 with the ob- served reverberation levels. On the other hand, no such 6-db correction for surface reflection was neces- sary for 30-degree data, since at the latter transducer orientation almost none of the projected sound strikes the surface before reaching the bottom.* Having described the procedure for calculating the 0- and 30-degree columns in Table 2, we now proceed to use these entries for an estimate of the angular de- pendence of m”. It will be recalled that for all the 0-degree entries the grazing angle of the sound on the bottom lay between 9 and 13 degrees; for present purposes the grazing angle for all the 0-degree entries may be taken as 10 degrees. The grazing angle for all the 30-degree entries may be taken as 30 degrees, since at such a great angle of depression the effect of refraction is negligible. Thus for each of the bottom types considered, we have in Table 2 an m” for a 10-degree grazing angle and an m” for a 30-degree grazing angle. By assuming a relationship of the form m” ~ sink (6), it is possible to calculate & for each bottom type. The resulting values of k, rounded off to the nearest half- digit, are displayed in the last column of Table 2. These values of k are not too reliable, since in order to calculate the individual scattering coefficients in Table 2 a number of assumptions were required about such questions as the proper method for averaging data obtained on different days, and the proper comparison between the point and band methods of averaging when the reverberation levels are changing rapidly. These assumptions, described in detail in reference 2, mean that the results of 4 Tt will be recalled that a 6-db correction was argued in Section 12.5 for surface scattering coefficients. It may be thought that a similar correction should be applied to bottom scattering coefficients, to take account. of possible reflections in the layer of scattering material at the bottom. This correc- tion arises, in the case of surface scattering, because the scat- terers are thought to extend an appreciable distance into the water side of the air-water boundary; sound can penetrate the seattering layer and strike the air-water boundary at which most of the actual reflection takes place. For bottom scatter- ing, on the other hand, although the bottom scattering layer is not infinitely thin, most of it does lie on the solid side of the twilight region separating the sea volume from the earth’s crust. Thus, there is no need to introduce a correction to bottom scattering coefficients due to reflection at the bottom; in fact such a correction, if introduced, would have no physical significance. 317 Table 2 may be somewhat in error. Nevertheless, Table 2 does indicate that the value of m” increases at least as rapidly as the first power of sin @ for grazing angles between 10 and 30 degrees. The data of reference 2 give no information on the nature of m’(6) for grazing angles @ less than 10 de- grees. It was assumed in reference 1, from which Figure 4 was taken, that m” is proportional to sin? 6 for angles @ greater than 9 degrees, and was constant independent of 6 for angles @ less than 9 degrees. The solid lines in Figure 4 are the reverberation levels predicted on this basis and fit the observed points very well, even at the extreme range of 360 yd on the lower curve, where the grazing angle is only 4 degrees. The very good fit evidenced in Figure 4 seems to indi- cate that m” is constant independent of grazing angle at angles less than 10 degrees. However, there is al- most no other information on the dependence at angles less than 10 degrees; and a constancy of scat- tering coefficient as the grazing angle decreased be- low 10 degrees would make the law of scattering a very complicated function of angle at these small angles. For these reasons it is probably best to regard the dependence of m” on grazing angle for angles less than 10 degrees as still uncertain. More measure- ments of this dependence are needed; to obtain values of m” at small grazing angles, it will be necessary to make measurements in isothermal water. IMPOSSIBILITY OF DETERMINING m’ (9) WITH HorizonTaL BEaMs We shall now discuss why it was not possible, from the 0-degree data alone, to determine the dependence of m’’ on grazing angle on the bottom. Two factors are involved: (1) uncertainty in the beam pattern correction, and (2) the lack of any large variation in the grazing angle, owing to the effect of refraction. For horizontal transducers, the beam pattern cor- rection as determined from equation (41) of Chapter 12 is small for values of @ less than 6 degrees (see Table 1, Chapter 12). At larger angles the correction increases rapidly because of the sharp decline in the measured beam pattern at the edge of the main lobe. At an angle of 10 degrees, for example, the correction is about 20 db. The application of this large correc- tion appeared to seriously overcorrect the data analyzed in reference 2, giving very large values of m" at close ranges. It is not difficult to find reasons for this inability to calculate m” correctly at angles well off the main lobe. In the first place, the very use of equation (41), Chapter 12, for the beam pattern 318 SHALLOW-WATER REVERBERATION correction is questionable at large angles, as has been pointed out previously. In addition, the ship pitches and rolls; at large angles even a small change in orientation of the projector may make a large dif- ference in the received reverberation. There is the further complication that measured beam patterns in the vertical plane have not always agreed with the measured patterns in calibrating stations.® These arguments, taken with the overcorrections noted in reference 2, suggest that a correction of 10 db or more is about the maximum which can be safely applied to measured bottom-reverberation levels, if accurate values of the bottom-scattering coefficients are to be expected. We can conclude that the use of equation (41) of Chapter 12 to obtain m” for rays leaving the projector at large angles (greater than 6 degrees) is quite questionable. Also, little information about the angular depend- ence of m’’ could be obtained from rays leaving the projector at angles within the main beam, that. is, with initial angles less than 6 degrees. For, in these experiments the incident angle at the bottom of the rays within the 6-degree cone was essentially con- stant; at best, this grazing angle varied only slowly with range. This fact alone would make fruitless any attempt to determine, from the data of reference 2, a detailed degree-by-degree dependence of m” on grazing angle. Furthermore, the value of the scat- tering coefficient itself, at ranges beyond the region where the main beam strikes the bottom, becomes more and more doubtful as the range increases, be- cause of uncertainty in the value of the transmission anomaly. These remarks explain why the data of reference 2 were capable of giving m” accurately only for the angle of incidence on the bottom correspond- ing to the peak of the reverberation. It is worth noting that the virtual constancy of the angle of incidence on the bottom, for rays within the 6-degree cone, should be a rather general result with all types of refraction patterns. This conclusion is deduced from Snell’s law of refraction, as follows. Snell’s law, which was proved in Chapter 2, tells us that c cos 86 = — cos A> (4) Co where @ is the bottom grazing angle, is the angle of the ray at the projector, c is the velocity of sound at the bottom, and c@ is the velocity of sound at the projector. It is clear from equation (4) that the bot- tom grazing angle will be smallest for the ray which leaves the projector at 0 degrees. Thus, the derivative d6/d% equals zero at 0% = 0; and @ necessarily varies but little for all the rays leaving the projector within a few degrees of the projector axis. 15.3.2 Dependence on Frequency A report by UCDWR‘ presents measurements de- signed to determine the dependence of the bottom- scattering coefficient on frequency. These measure- ments were made at 10, 20, 40, and 80 kc, with the transducers directed downward at an angle of 30 de- grees with the horizontal. The measurements were made in two shallow water areas near San Diego, both with rocky bottoms. The ping lengths used were be- tween 4 and 8 msec. Further details concerning the bottom character and the experimental procedures are given in reference 6. From comparison of the measured reverberation levels with equation (54) of Chapter 12, values of 10 log m” were determined at each frequency and at each of the two positions where measurements were made. These values of 10 log m” were obtained assuming the transmission anomaly A in equation (54) as zero; because measurements were performed in very shallow water, this assumption should introduce very little error. The results ob- tained in reference 6 are tabulated in Table 3. An irregular variation of 10 log m” with frequency is noted in Table 3, but according to reference 6 this TasLe 3. Backward scattering coefficients (10 log m”) as a function of frequency at 30-degree grazing angle. Frequency in ke 10 20 40 80 Area I —11 —6 —8 —14 Area IT —22 —17 —21 —15 variation is less than the estimated error of calibra- tion. Also, according to reference 6, change in trans- ducer patterns due to changes in frequency and swinging of the ship at anchor could have introduced errors compared with which the observed variation is not significant. Thus there is no evidence that the bottom scattering coefficient for rocky bottoms at a grazing angle of 30 degrees has any systematic fre- quency dependence for the frequency range 10 to 80 ke. The mean value of 10 log m”, averaged for 10, 20, 40, and 80 ke is —10 + 3 db for position I and —19 + 3 db for position II. The mean values of 10 log m” at a grazing angle of 30 degrees, quoted in the preceding paragraph, should be directly comparable with the 30-degree BOTTOM SCATTERING COEFFICIENTS value of m” for ROCK in Table 2. It is seen that there is very good agreement with Table 2 for Area I, but there is a difference of 10 db between the value of m” (80 degrees) obtained at Area II and the value of m” (80 degrees) in Table 2. This difference could be due to sampling error; it is estimated later that the quartile deviation of m” for areas of similar bottom classification is + 5 db. In this regard it is significant that reference 6 states that the bottom of Area II had patches of SAND-AND-MUD. Thus, it is not too surprising that the mean bottom scattering coeffi- cient in Area II should be lower than in Area I, where, according to reference 6, the bottom was covered with boulders. The results of reference 6 give no information on the dependence of m” on frequency for other types of bottom than ROCK. There is no reason to expect that any marked dependence on frequency would be dis- covered. However, it is necessary to definitely know the frequency dependence, if any, in order to predict the effect on bottom reverberation of varying the frequency of echo-ranging gear. Also, knowledge of this frequency dependence would enable us to assess accurately the present information on the dependence of m” on bottom type, much of which is based on the assumption that m” does not depend on frequency. For these reasons it would be desirable to obtain additional measurements over all types of bottom of the dependence of m” on frequency. 15.3.3 Dependence on Bottom An analysis of bottom-scattering data obtained with horizontal beams is given in reference 3. These data include many more records than are analyzed in reference 2, among which are data at 10, 20, and 24 ke. A portion of the data was analyzed in a manner similar to that used in reference 2, except that absorp- tion was not included in the transmission anomaly; the transmission anomaly A in equation (54) was computed from the refraction pattern alone. This analysis gave the results listed in Table 4 for a graz- ing angle at the bottom of 10 degrees. In Table 4, as in Table 2, the values of m” have been corrected to the true values. Table 4 should, of course, be com- parable with the 0-degree column of Table 2, since the grazing angle on the bottom (10 degrees) is the same for both tables. In reference 3, in addition to this analysis, a more complicated analysis was also attempted to deter- mine the variation of the bottom-scattering coeffi- 319 cient with angle of incidence. Some evidence that m” increased with increasing grazing angle was found, but it was not possible to decide which of the three laws, m” constant, m” proportional to sin 6, or m” proportional to sin? #, was most nearly representative of the bottom scattering. In view of the inconclusive nature of the results, and also because this analysis rested on some questionable assumptions, these re- sults for the angular dependence of m” were not in- cluded in Section 15.3.1. TABLE 4. Average values of 10 log m” for various bottom areas for grazing angle at bottom of 10 degrees. Bottom type 10 ke 20 ke 24 ke COBBLES —16 —16 ROCK —23 —23 MUD —32 —38 MUD —31 —34 MUD —34 —36 Hae MUD its a —36 MUD —37 The value of m” for the ray leaving the projector at an angle of 5 degrees was relatively independent of the assumptions made. The values of m” for this ray are given in Table 5. Table 5 includes, for some of the records studied, the grazing angle of the 5-degree ray as calculated from the measured BT pattern. It is seen that in general the 5-degree ray strikes the bot- tom at-a grazing angle of about 10. degrees; thus Table 5 should be comparable with Tables 2 and 4. From Tables 2, 4, and 5 we can now determine, for each bottom type, the average value of m” for a grazing angle on the bottom of 10 degrees. To do this, we recall that Tables 4 and 5 were obtained on the assumption that the transmission anomaly was due to refraction alone, that is, that the absorption loss was negligible. However, the values of m” in Tables 4 and 5 can be corrected for absorption in the following way. It can be assumed, from previous dis- cussions in this chapter, that on the average the ranges at which the data of Tables 4 and 5 were evaluated were six times the depth of the projector above the bottom. These depths are given for the measurements listed in Table 5; for the items in Table 4 they can be obtained from Table 2 of refer- ence 6. Thus, the average absorption loss can be cal- culated at each frequency for each entry in Tables 4 and 5, by assuming median values of the attenuation coefficient at each frequency (see Figure 17 of Chap- ter 5). If we increase m” by the average two-way 320 SHALLOW-WATER REVERBERATION TaBLE5. Average values of m” for various bottom areas for ray leaving projector at an angle of 5 degrees. Depth of bottom | Calculated grazing angle for 5-degree ray in below projector 10 log m” for 5-degree ray Bottom type in yards degrees Frequency in ke 10 20 24 COBBLES 38 5.0 —23 —19 BOULDERS 10 % —24 —21 ROCK 20 * —29 —27 ROCK 110 8.5 —27 —28 MUD 240 = —36 —35 MUD AND SAND 52 10.2 —32 —35 FINE SAND 10 * —33 —36 MUD 275 10.5 —31 —37 MUD AND SAND 100 2 —38 —38 MUD 195 10.0 —35 —39 ae ROCK 15 * O00 ee —26 MUD AND SAND 53 ¢ S00 ae: —31 MUD AND SAND 75 = are ac —25 MUD 240 11.8 —25 MEDIUM SAND 215 —31 MUD 230 11.8 —25 * Angle not calculated. absorption loss in decibels, the entries in Tables 4 and 5 will be more or less corrected for absorption losses, and the resultant values of m” will be the best estimates which can be made from the data of refer- ence 6. Table 6 shows the mean values of m” determined in this way from the data of Tables 4 and 5. The as- sumed attenuation coefficients at 10 ke, 20 ke, and 24 ke in db per kyd were 1.3, 3.2, and 4.0 respectively. In Table 6 the mean values for each bottom type were Taste 6. Mean values of backward scattering coef- ficient at 10-degree grazing angle. 10 log m” 10 log m’ 10 log m” from from from Bottom type Table 4 Table 5 Table 2 ROCK —17 —24 —18 MUD —27 —25 —29 SAND-AND-MUD —31 —25 SAND —34 —30 determined by averaging the corrected values of m” for all three frequencies 10, 20, and 24 ke, giving each entry in Tables 4 and 5 equal weight. The justifica- tion for averaging m” for different frequencies has been discussed in Section 15.3.2. If data for 10 and 20 kc are not averaged with 24-ke values, a large part of the data of reference 3 has to be omitted. Another reason for including the 10- and 20-ke data is the following. Examination of the cor- rected values of Tables 4 and 5 shows that the as- sumptions which were made concerning the range to the reverberation peak and the value of the attenua- tion coefficient seem to overcorrect m” at 24 ke; that is, the corrected values of m” at 24 ke frequently tend to be abnormally large, especially in deep water where the range to the peak is long. For example, in the last two MUD entries in Table 5, the range of the peak was estimated to be about 1,400 yd, so that the two-way transmission anomaly correction was 11 db. This made the corrected values of 10 log m” for those entries equal to — 14 db, which is much too high for a MUD bottom. The reason that this overcorrection occurred is not clear. It might indicate that the at- tenuation coefficient for the sound returned as rever- beration is less than the attenuation coefficient de- termined in transmission runs. However, the data of reference 2, which appear quite reasonable, are based on an assumed attenuation coefficient of 4 db per kyd; in the analysis of volume and surface reverbera- tion in Chapter 14, median reverberation levels were fitted quite well by assuming 4 db per kyd as the value of the 24-ke attenuation coefficient. Whatever the rea- son, the existence of this apparent overcorrection makes it desirable to include values of m” for 10 and 20 ke in the averages based on the data of reference 3, AVERAGE BOTTOM REVERBERATION INTENSITIES since at the lower frequencies the corrections for at- tenuation are not as large. In Table 6, cobbles and boulders have been grouped under rock; and fine sand, foraminiferal sand, and medium sand have all been grouped under sand. The last column in Table 6 gives the results of averaging a similar grouping of the 0-degree values of Table 2, including coarse sand under sand; the designations sand-mud and mud-and-sand of refer- ences 2 and 3 have been replaced by the customary SAND-AND-MUD. If all the entries of Table 6 are averaged with equal weight, we obtain the over- all averages in Table 7. TABLE 7. Overall mean values of backward scatter- ing coefficient at 10-degree grazing angle. Bottom type 10 log m” ROCK —20 +5 MUD =f se SAND-AND-MUD —28 +5 SAND —32 +5 The values of Table 7 do not differ significantly from other estimates of the mean bottom scattering coefficients, also based on the data of references 2 and 3. In reference 7 it is estimated that the quartile deviations of the mean bottom scattering coefficients are about 5 db for each bottom type. In view of the crudeness of a classification system which includes all bottom types in only four categories, a quartile deviation of this magnitude is not surprising. Thus this estimate of the deviation from the mean has been included in Table 7. It must be remembered that the values of 10 log m” in Table 7 are true values; that is, they were determined by subtracting 6 db from the values of m” inferred from comparison of equation (54) of Chapter 12 with the measured reverberation levels. The expected reverberation levels with horizontal beams will therefore be 6 db greater than the levels that would be predicted by the use of equation (54) and the values of 10 log m” in Table 7. 15.4 AVERAGE BOTTOM REVERBERATION INTENSITIES WITH HORIZONTAL TRANSDUCERS Bottom reverberation levels are a function of range and water depth and in addition depend on refraction conditions, transducer orientation, and bottom type. 321 For most practical echo-ranging purposes, however, the transducer is oriented so that the transducer axis is horizontal, parallel to the ocean surface. Under these circumstances, over level bottoms, the data which have been presented in this chapter can be used to make some prediction of average bottom rever- beration levels. The results of reference 2, discussed in Section 15.3.1, show that under most conditions the transmission anomaly due to refraction is negli- gible at ranges up to and including the range of the reverberation peak. The results of references 1, 2, and 3, described in Section 15.2, all show that in water other than isothermal or nearly isothermal the range of the reverberation peak tends to be about 6 times the depth between the projector and the bottom, and that this peak corresponds approximately to the range at which the 5- to 6-degree ray from the pro- jector strikes the bottom. The data of references 2 and 3, which were discussed in Sections 15.3.1 and 15.3.3, show that the angle at which this ray strikes the bottom usually is about 10 degrees in nonisother- mal water. Thus, knowledge of the average value of m" at a grazing angle of 10degrees, coupled with equa- tion (54) of Chapter 12, enables prediction of the aver- age height and range of the reverberation peak over different bottom types in any water depth. It is of course necessary to know the value of A in equation (54). However a value of A equal to 4 db per kyd is probably a good approximation, and for the short ranges at which the reverberation peaks are usually observed, deviations in practice from 4 db per kyd should not be very significant, except possibly when the water is quite deep. Once the height and range of the bottom rever- beration peak are determined, the most significant quantity for echo ranging is the rate at which the reverberation decays as a function of range past the peak. At ranges less than the peak the reverberation usually decreases with decreasing range; such ex- amples as Figure 7 and other similar figures in refer- ences 2 and 3 show that the bottom reverberation can hardly increase with decreasing range as rapidly as the expected echo level. At some ranges less than the principal reverberation peak (where the main beam strikes the bottom) reverberation from side lobes can be very high. However, it seems on the whole that bottom reverberation is likely to be most troublesome at ranges past reverberation peak. At ranges past the reverberation peak in noniso- thermal water, the results of reference 1 indicate that on the average the reverberation falls off at about the 322 inverse fourth power of the range, but that large variations from this type of decay are observed. Since the expected echo level also falls off at about the inverse fourth power of the range,® the results of reference 1 mean that the possibility of obtaining an echo usually depends on the level of the reverbera- tion peak relative to the expected echo level at the range of the reverberation peak. If the reverberation EVEL IN DECIBELS Le fo) STANDARD REVERBERATION L fo} ---—-SAND AND MUD —-— SAND 140 100 200 RANGE TO PEAK IN YARDS 300 500 700 1000 2000 Figure 8. Expected level of peak of bottom reverbera- tion as a function of range to peak and bottom type. peak is high enough to mask the echo at that range, then the echo is not likely to be detected at any range past the reverberation peak. Conversely, if the echo level is well above the reverberation at the range of the reverberation peak, bottom reverberation will probably not limit echo ranging at any range. The above discussion is not by any means a com- plete treatment of the problem of echo ranging in shallow water. Many factors are involved in deter- mining the echo-to-reverberation ratio at any range. Also, no account has been taken of the fact that at- tenuation at long range makes the echo level drop off SHALLOW-WATER REVERBERATION much more rapidly than the inverse fourth power of the range. However, attenuation also decreases the received bottom reverberation levels, so that even at long ranges the echo and reverberation levels should decrease at roughly the same rate. In general, it can be said that the problems involved in determining the echo-to-reverberation ratio are so complicated that no satisfactory quantitative treatment has ever been given, although qualitative discussions have been presented in a number of places. It appears then from the foregoing discussion that with present informa- tion the best way to characterize bottom reverbera- tion levels is intermsof the level at the principal rever- beration peak; this level is determined using equation (54) of Chapter 12 and the known value of the bottom scattering coefficient for 10 degrees grazing incidence. Figure 8 shows the expected average standard re- verberation level at the reverberation peak, as a function of the range to the peak and the bottom type, for ordinary 24-ke echo-ranging gear sending out a horizontal beam. In preparing this diagram it was assumed that the range to the peak is six times the depth of the bottom below the projector. The absorption was taken to be 4 db per kyd, J was set equal to —19 db for the 5- to 6-degree ray, 10 log m” was taken from Table 7, and finally the reverberation level for a 100-msec ping was calculated by the use of equation (55) of Chapter 12. Although Figure 8 is the best average curve which can be drawn with present information, the likelihood of deviations in practice from Figure 8 cannot be overstressed. In particular, Figure 8 is not valid in isothermal or nearly isothermal water, when the range to the reverberation peak will usually be very different from six times the distance between the projector and the bottom. It is also not advisable to extend the results in Figure 8 to ranges less than 100 yd because at such short ranges it is again unlikely that the average assumed relationship between range and depth will be valid. It should be noted that the curves in Figure 8 incorporate the 6-db correction for surface reflections discussed in Section 15.3.3. Thus, when the echo-ranging transducer is deep, 6 db should be subtracted from the values in Figure 8 to obtain the expected reverberation levels; in such situations surface reflections will not be im- portant in determining the bottom reverberation level. In conclusion, we repeat that on the average the reverberation in nonisothermal water seems to fall off at about the inverse fourth power of the range, at ranges past the reverberation peak, but that large AVERAGE BOTTOM REVERBERATION INTENSITIES 323 variations from this type of decay may be observed. Careful examination of the results of reference 1 shows that the shape of the decay curve is probably not the same over all types of bottoms. In other words, the probability of distinguishing an echo at long range is different over different types of bottoms, even with the same echo-to-reverberation ratio at the range of the reverberation peak. A preliminary study of the shapes of these decay curves over different types of bottoms is being made off San Diego, but unfortunately, the results of that study are not avail- able at this time. Chapter 16 VARIABILITY AND FREQUENCY CHARACTERISTICS i PRECEDING CHAPTERS, we have derived theoreti- cal formulas for the average reverberation inten- sity, and have compared these formulas with average reverberation intensities observed in practice. By means of this comparison, we have obtained average values of the backward scattering coefficient for volume, surface, and bottom reverberation. In this chapter we shall attempt to analyze the differences between reverberations from successive pings sent out under apparently the same circumstances. These differences may be deviations in amplitude, or devia- tions in the frequency spectrum of the received re- verberation. There are several reasons for such an analysis. First, it is desirable to know just how well the average curve may be expected to represent individual rever- beration curves. Secondly, the deviations from the average depend on the type of mechanism giving rise to the reverberation; thus, analysis of the deviations can give valuable information on the sources of re- verberation. Finally, such an analysis may easily re- veal significant differences between the behavior of echo fluctuation and reverberation fluctuation since the mechanisms producing these two types of fluctua- tion are undoubtedly somewhat different. These dif- ferences in behavior, if well understood, may be utilized in methods for improving the recognition differential for the echo against a reverberation back- ground. 16.1 FLUCTUATION In analyzing the amplitude deviations of individual reverberation curves from the average, it is con- venient to distinguish between fluctuation and co- herence. Fluctuation refers to the deviation from the average of the intensity received at a definite time following the initial ping. This fluctuation is usually measured by the variance INP == (Gf = INP = IP = (OP (1) where J is the average intensity at a time ¢ seconds 324 after midsignal, and [? is the average of the square of this intensity. Average values will be designated by a bar throughout the remainder of this chapter. As discussed in Chapter 12, the average intensity at the time ¢ is to be determined by the following proc- ess. A large number of reverberation records are taken under circumstances as nearly identical as possible, and the intensity of the reverberation at a time ¢ seconds after midsignal is read off each record. The average of these intensities is the value referred to by the bar. Evidently, if all the pings were sent out under exactly the same circumstances, the received inten- sity at time ¢ should be constant, and AJ? in equation (1) would be zero. However, no two pings occur under precisely the same circumstances. There are variations in the power output of the projector; varia- tions in the orientation of the receiver because of ship roll; variations in such oceanographic factors as wave height, wind force, temperature-depth distribution, water depth, and type of bottom material; and over- all variations in transmission anomaly. Some of these. sources of reverberation fluctuation can be mini- mized. The power output of the projector can be stabilized to a fraction of a decibel; the ship will not roll on a calm day; and the effects of changing wind force and bottom character can be eliminated by studying only volume reverberation. However, large fluctuations remain no matter how much control is exercised. These remaining fluctuations in volume reverberation are regarded as an inherent property of reverberation. In the derivation of equation (13) of Chapter 12, expressing the time variation of volume reverbera- tion from a single ping, it was assumed that the re- verberation is due to the scattering of sound by a large number of scatterers in the ocean. Fluctuation in the received reverberation is caused by the fact that the total reverberation amplitude is the sum of the amplitudes received from all the individual sources. These individual amplitudes have random FLUCTUATION phases with respect to each other, and the total amplitude is large or small depending on the degree of reinforcement or interference between these in- coming individual amplitudes. An expression for the fluctuation of intensity caused by the combination of a large number of amplitudes of equal magnitude but random phase was derived by Rayleigh.! The probability that the resultant intensity will exceed the value J is given by P = e-(/) (2) where J is the average intensity. A derivation of equa- tion (2) is given in Chapter 7. The actual received reverberation is of course a combination of ampli- tudes of many different magnitudes, because the indi- vidual scatterers are not all of equal strength, because the projector is directional, and because the trans- mission loss to different portions of the ocean may not be the same. However, it can be shown that equa- tion (2) remains valid, even if all the amplitudes are not of equal magnitude, provided only that there are a large number of amplitudes of each magnitude, and that the number of amplitudes of each magnitude re- mains essentially constant. Thus, formula (2) is im- plied by the assumptions used in Chapter 12 to de- rive the expression for the time variation of the volume reverberation from one ping, with the proviso that the transmission does not change from ping to ping. The applicability of the Rayleigh distribution func- tion (2) has been tested by the University of Cali- fornia.? They first chose sets of ten or more typical reverberation records in deep water; all the records in a given set were taken under similar conditions. The ratio of the observed amplitude to the average amplitude was computed for various times on each set. All told, 420 values of this ratio were obtained for the QB transducer, and 500 values of the ratio for the QCH-3. The results are plotted in Figure 1. There is apparently no major deviation of either ex- perimental curve from the theoretical expression (2). It appears from Figure 1 that the shape of the fluctuation curve does not depend significantly on such factors as the directivity pattern of the trans- ducer, nor on the shape of the pulse sent out; both of these factors are different for the two transducers. It will be noted that equation (2) predicts this inde- pendence. Since the times chosen on the records were well distributed, and since no effort was made to distinguish between surface and volume reverbera- tion, it appears that the expression (2) applies fairly 325 well to all portions of the deep-water reverberation versus range curve. It is not surprising that formula (2) fits the ob- served fluctuation of surface reverberation levels about as well as it fits the fluctuation of volume rever- beration. If an assumption that the scattering power of the surface remains essentially constant from ping to ping is added to the other assumptions used in the PERCENTAGE OF RATIOS = INDIGATED VALUE Ww =) a s a uw - (9) where f is the audio output frequency, 7 is the pulse length, and K isa constant. Pulses of length 92 msec and frequency 24 ke produced reverberation, which, after being heterodyned to 800 c, had a mean rms frequency spread of 21.5 c. 5. The frequency spread of reverberation does not seem to depend on the frequency of the outgoing signal. 6. The mean observed reverberation frequency agreed with own-doppler values calculated for the axis of the projector. At moderate ship speeds, the finite width of the projector beam did not, through own-doppler effects, cause marked broadening of the beam. Most of the above results are quite reasonable. The first result indicates that the reverberation arises from a process which is essentially random. The last result indicates that the mean reverberation frequency can be used as a reference for the measure- ment of target doppler. That is, the mean reverbera- tion frequency fi agrees with the following formula for the frequency of an echo received on a moving ship from a stationary target. f= s(1 + * cosa), where »v is the velocity of the echo-ranging ship, @ is the angle between the projector axis and the line of motion of the ship, f is the frequency of the emitted ping, and c is the velocity of sound. This agreement is theoretically expected.1® 1 The second result indicates that estimates of target doppler are not reliable unless the target is moving at speeds of 2 knots or more. The third result, that the (10) FREQUENCY ANALYSIS OF REVERBERATION 331 FIGuRE 5. observed frequency spread in reverberation should be increased by frequency fluctuation in the outgoing ping, is also not surprising. The dependence of the frequency spread on the audio output frequency, described by equation (9), seems at first sight rather difficult to understand. In principle, heterodyning simply subtracts a constant frequency from the frequencies of all components of the incoming reverberation signal; thus, if there is no distortion, the average frequency spread should not depend on the audio output frequency.”°?! However, with a little thought, it can be seen that the difficulty arises because the time intervals between successive zeros of a complex signal are not related in any sim- ple way to the frequency spectrum of that signal. Thus, the assumption that the rms deviation of the instantaneous frequency read from the period- meter should be exactly equal to the true rms fre- quency spread of the reverberation spectrum is cer- tainly not warranted. The complete elucidation of the relation between periodmeter readings and the rever- beration spectrum requires a satisfactory theory of the periodmeter, which has not yet been developed because of mathematical difficulties. A partial anal- ysis of the theory of the periodmeter is given in a report by UCDWR.” There it is predicted that the rms spread of the instantaneous frequency should obey the formula Mf = Kf? mm" (11) which differs somewhat from the observational equa- tion (9). However, the mode of derivation of equa- tion (11) has been subjected to some criticism.”° This difficulty in relating the response of the periodmeter to the spectrum of the reverberation Periodmeter record of a ping and its echo. << BOTTOM REVERBERATION——> : | Ficure 6. Periodmeter records of volume and bottom reverberation. illustrates the difficulty which is always experienced in predicting the response to reverberation of any complex circuit, such as the ear or other doppler discriminator. It can be argued that, for the ear, the instantaneous frequencies measured by the period- meter are more significant than the spectrum which gives the intensities in very narrow frequency bands. However, because of the complexity of the ear, this assertion requires proof; for this reason conclusion 2 above, and other similar conclusions, cannot be relied on if based on evidence from the periodmeter alone. 332 VARIABILITY AND FREQUENCY CHARACTERISTICS REVERBERATION FROM WIDE- BAND PINGS 16.4 Up until now, this volume has been concerned almost completely with reverberation from narrow- band CW pings (in a CW ping the nominal trans- mitting frequency is fixed during the interval of transmission). However, other types of pings have been used, as in FM sonar. In this section we shall examine the reverberation resulting from the use of wide-band pings. Strictly speaking, no ping which has a finite dura- tion can possibly be single-frequency; other fre- quencies than the nominal one must be present in order that the signal can build up and die off. How- ever, 100-msec CW pings at 24 kchave band widths of the order of 10 c, while wide-band pings often have band widths of 1,000 c or more (for example, a 1-msec CW ping has a band width of 1,000 c). It may be expected that this very real difference in the order of frequency spread will be reflected in a difference in the nature of the returning reverberation. According to Section 12.5, widening the frequency band in the ping should not seriously affect average reverberation levels, if the frequency response of the gear is sufficiently flat. In other words, the theoretical formulas in Chapter 12 giving the average time decay of reverberation fronra single ping should be just as valid (or invalid) for wide-band pings as for narrow- band pings. It is true that many of the parameters in the formulas of Chapter 12 are frequency-de- pendent, such as the transmission loss, transducer directivity, and the scattering coefficients. However, these quantities vary smoothly and relatively slowly with frequency and can be replaced by their averages over even a 10-ke wide band without introducing much error. Thus the resultant theoretical average reverberation levels for wide-band pings are simply an average, over the frequency band included in the ping, of the levels predicted for the narrow-band pings. A more quantitative discussion of the qualita- tive ideas in this paragraph can be found in a report by CUDWR.” The average fluctuation, as defined by equation (1), cannot be written simply as the average of the fluctuations of the individual frequency components. Thus, there is reason to expect that the fluctuation of wide-band pings may be different from that of CW pings. The expected magnitude of the fluctuation of wide-band pings will depend on the mechanism hypothesized as responsible for the fluctuation. Con- fining our attention for the moment to those wide- band pings resulting from the use of very short ping lengths, then, if the Rayleigh distribution is valid, it is apparent from the form of equation (2) that the magnitude of the fluctuation as defined by equation (1) does not depend on the ping length. In other words, with square-topped CW pings the magnitude of the fluctuation will be the same for wide-band pings as for narrow-band pings. However, the decrease of the ping length required to widen the frequency band of a CW ping will in- crease the rapidity of the fluctuation, since it de- creases the blob width. This decrease in blob width does not necessarily improve the recognizability of a returning echo since the echo length is decreased correspondingly. Studies of echoes resulting from CW pings (see Chapters 21 and 23) indicate that echoes look very much like reverberation blobs, of width about equal to the ping length. This similarity be- tween echoes and reverberation blobs makes it very difficult to devise means for improving the detectabil- ity of echoes from CW pings against a reverberation background, other than the obvious but not always feasible procedure of reducing the average reverbera- tion intensity. It is true that a time average over a relatively short interval will include a large number of reverberation blobs and this time average will fluctuate much less rapidly than the unaveraged re- verberation. However, because of the similarity of the echo to the reverberation blobs, any averaging pro- cedure applied to the sound returning from a short CW ping is likely to eliminate the echo. Nevertheless, by adjusting the time interval over which the average is taken, some beneficial effect may be hoped for, especially when the echo intensity is much larger than the average reverberation in- tensity. Some studies made with very short (0.3 msec) unmodulated pings support this hope. It was found that the use of these very short pings signifi- cantly reduced the number of false contacts reported and that the maximum range at which a target could be identified was not. affected. When frequency-modulated pings are used, the signal sent into the water has a continuously varying frequency. With such pings, the received reverbera- tion at any instant may include a wide band of frequencies, depending on the band included in the original ping and on the receiving pass band of the equipment. Theoretical analysis of the expected fluc- tuation of the reverberation from such pings is diffi- cult, but it has been shown” that the envelope of the REVERBERATION FROM WIDE-BAND PINGS reverberation from frequency-modulated pings should fluctuate more rapidly than does the echo envelope. Observational evidence that the rapidity of rever- beration fluctuation is increased by the use of fre- quency-modulated pings is quoted in reference 24, and similar observations have been reported in other sources. The echo length resulting from a frequency- modulated ping is equal to the effective ping length, just as with unmodulated pings; the effective ping length is itself determined by the pass band of the equipment. The only accurate quantitative data on the fluctua- tion of reverberation from frequency-modulated pings are those given in reference 25. In that report it was found that with frequency-modulated pings the mean standard deviation of the reverberation amplitude is 33 per cent of the mean amplitude, significantly smaller than the 52 per cent value pre- dicted for a Rayleigh distribution. These measure- ments were made with pings of length 2, 4, and 8 sec, frequency-modulated from 48 to 36 ke. Some results on the coherence of FM reverberation are also re- ported in reference 25. The mechanism by which such a decrease in the magnitude of the fluctuation could be accomplished is difficult to visualize; but the possibility of such an effect must be admitted, because our theoretical understanding of the problems involved is not 333 at all complete. Qualitative reports that frequency modulation is effective in reducing the magnitude of the fluctuation cannot be trusted, since they may be based on the results of time averages performed somewhere in the complicated recording equipment. Recapitulating, in echo ranging the objectionable features of reverberation are twofold. Reverberation masks the echo; also, reverberation simulates the echo, so that false contacts are often obtained. It is apparent from this chapter that a great deal of in- formation still is lacking about such characteristics of reverberation as blob shape, cause and rapidity of fluctuation, and frequency spread. Schemes which have been suggested to suppress reverberation may be combined with proposals for decreasing the average level of the reverberation by decreasing the ping length or by increasing the hydro- phone directivity. Of course, the possibilities of de- creasing the ping length or increasing the hydrophone directivity are always limited by other practical con- siderations. Once the ping length has been decreased as much as practical, and the transducer directivity has been increased to its highest practical value, one must resort to devices which do not reduce the average energy received in the hydrophone, but in- stead reduce the instrumental effects of reverberation in the detecting mechanism. Chapter 17 SUMMARY 17.1 DEFINITIONS 17.1.1 Reverberation EVERBERATION IS A COMPONENT of background heard in echo-ranging gear, and is distinguished from the general noise background by the fact that it is directly due to the pulse put into the water by the gear. The traveling ping meets not only the wanted target, but also myriad small scattering centers or other inhomogeneities, each of which re- turns a tiny echo to the transducer. These tiny un- wanted echoes combine to make up reverberation. Thus reverberation, like the echo, is a sound whose pitch is definite and is determined by the frequency of the projected pulse. Often echoes which would be audible over the remainder of the noise background are masked by reverberation. To the operator of echo- ranging gear, reverberation is evident as a quavering ring which sets in as soon as the period of sound emission is finished. The scatterers producing reverberation may be located near the sea surface, in the main ocean volume, or in the sea bottom. The reverberations produced by these three types of scatterers are called, respectively, volume reverberation,- surface rever- beration, and bottom reverberation. This distinction is physically meaningful, since these three types of reverberation apparently have different properties and can be experimentally differentiated from each other. 17.1.2 Reverberation Intensity The strength of the sound heard or recorded as reverberation depends not only on the intensity of the backward scattered sound in the water near the receiver, but also on the nature of the receiving gear. The intensity of the reverberation actually heard or recorded, after the sound in the water has been con- verted to electrical energy by the receiver, amplified, 334 and passed to the ear or recording scheme, is called the “reverberation intensity” and is given the symbol G. As so defined, G equals the watts output across the terminals of the receiving gear. In general, the rever- beration intensity G is a function of time and is re- lated to the sound intensity in the water by such parameters of the receiver system as receiver directiv- ity and receiver gain. Since the quantity G depends on the gear parameters, the absolute magnitude of G is usually not of great significance in studies of the intrinsic character of reverberation or of the mech- anisms producing reverberation. In these studies the reverberation level, defined under the next heading, is ordinarily employed. 17.1.3 Reverberation Level The reverberation level is the decibel equivalent of the reverberation intensity defined in Section 17.1.2, expressed relative to a standard which makes rever- berations measured with different gear exactly com- parable. Specifically, the reverberation level R’. is defined as Rk’ = 10 log G — 10 log (F- F’) (1) where F is the projector output at 1 yd in decibels above 1 dyne per sq cm, and F’ is the receiver sensitivity in watts of output for a received rms sound pressure of 1 dyne per sq cm. Reverberation levels are much more useful than reverberation in- tensities for comparing measurements made with dif- ferent systems, since under identical external condi- tions two different systems sending out pings of the same length should in principle give the same rever- beration level if correction is made for transducer directivity. Reverberation levels usually refer to the average reverberation intensity G found in a suc- cession of pings. Reverberation intensities are proportional to the ping duration. Often it is desirable to convert rever- beration levels to the levels which would be received DEEP-WATER REVERBERATION LEVELS 335 using a standard ping length. For this purpose we define the standard reverberation level R R = R' + 10 log @ (2) ‘Ts where A’ is the observed reverberation level with ping duration 7, and 7 is a standard ping duration usually chosen as 100 msec. 17.1.4 Backward Scattering Coefficients By “backward scattering” is meant scattering back along the incident ray path. If there is only one ray path from the projector to the scatterers, only sound which is scattered directly backward can give rise to reverberation. Thus the more efficient a portion of the ocean is in backward-seattering, the higher will be the level of the received reverberation. The efficiency of a small volume V of the ocean in scattering sound backward is specified in terms of the backward-scat- tering coefficient m, which is defined by the relation b (3) where 6 is the average energy scattered by the volume V per second per unit incident intensity per unit solid angle in the backward direction. The factor 47 is introduced so that in cases where the scattering is the same in all directions, the average energy scat- tered in all directions per second per unit incident intensity will be just mV. 17.1.5 Fluctuation The reverberations from two successive pings never reproduce each other exactly. This short-term variability is called ‘‘fluctuation.’’ A numerical meas- ure of fluctuation is provided by the variance of the reverberation intensity at a time ¢ seconds after mid- signal. More specifically, suppose that a large number n of successive pings are sent out, that records are taken of the n resulting reverberations, and that the n intensities at a time ¢ seconds after midsignal are read off the records. Then if J is the average of these n intensities, and J;, Iz, ---, In are the n individual intensities, then the fluctuation corresponding to the time ¢ seconds after midsignal is measured by the variance ya — 7). (4) Nni=1 17.1.6 Coherence The term “coherence” applied to reverberation re- fers to a tendency of the received reverberation to occur in the form of pulses or ‘“‘blobs.’’ The possession of coherence means that if at any instant the rever- beration level is high, it is likely that the level will remain high for a little while, and that if the rever- beration level is low, it is not likely to become large in a short time. The degree of coherence can be de- scribed mathematically in terms of the correlation coefficient p between the reverberation intensities at two different times on the same record. » = a) = Te) = Te) Via(a) — Ty PY EL) — Te? The bar signifies an average over many successive records. (5) DEEP-WATER REVERBERATION LEVELS 17.2 In deep water the reverberation heard at ranges past 1,500 yd is almost always volume reverberation. At shorter ranges surface reverberation may exceed volume reverberation, if the sea state is sufficiently high and the transducer beam is horizontal. Pointing a directional transducer downward will usually result in the surface reverberation being less than volume reverberation at all ranges past 100 yd. 17.2.1 Volume Reverberation The following subsections summarize the known information concerning reverberation from the vol- ume of the ocean. The statements apply only to that portion of the received reverberation resulting from scattermg in the ocean volume; the salient facts about reverberation from the sea surface are sum- marized in section 17.2.2. THEORETICAL FORMULA FOR VOLUME REVERBERA- TION LEVEL The expected volume reverberation level R’(¢) at a time ¢ see after midsignal is given by the formula R(t) = 10 log = + 10 log m+ J, — 20logr —2A + Ai, (6) where ¢ is the sound velocity in yards per sec; 7 is the ping duration in sec; m is the volume scattering 336 coefficient; J, is the volume reverberation index; r is the range in yards of the reverberation, r = Yat; A is the total one-way transmission anomaly to the range r; A; is the one-way transmission anomaly to the range r due to the effect of refraction. Because of surface reflections, not taken into ac- count in equation (6), observed volume reverbera- tion levels with horizontal beams will average about 3 db higher than the levels predicted by that equa- tion. DEPENDENCE ON RANGE According to equation (6), if the transmission anomaly terms (—2A + A) can be neglected, and if the scattering coefficient m is constant throughout the relevant portion of the ocean, then the intensity of volume reverberation should decay with the square of the range; in other words, its level should drop 20 db for each tenfold increase in range. In practice, this simple inverse-square dependence is only seldom observed, because (1) the transmission anomaly terms can rarely be neglected at ranges greater than 1,000 yd; and (2) the value of m often depends on position in the ocean. The well-established deep scattering layers off San Diego, which apparently scatter much more strongly than surrounding regions of the ocean, are examples of the dependence of the scattering coefficient on position. Though detailed agreement with equation (6) is almost never observed, volume reverberation does tend to decrease rapidly with increasing range, as predicted qualitatively by that equation. DEPENDENCE ON Pine LENGTH According to equation (6), as the ping length is increased the intensity of volume reverberation should increase proportionally. Although more data are needed, measurements to date indicate that equa- tion (6) does describe the dependence of reverbera- tion on ping length. This proportional dependence is also predicted theoretically for surface and bottom reverberation intensities; for these types of reverbera- tion also, more data are needed, but apparently the theoretical relationship is fulfilled. DEPENDENCE ON FREQUENCY The frequency-dependent terms in equation (6) are the volume reverberation index J,, the transmis- sion anomaly terms —2A + Aj, and the scattering coefficient m. The value of the reverberation index can be determined from the pattern function of the transducer by means of equation (27) of Chapter 12. SUMMARY The transmission anomaly can be estimated by the methods described in Chapter 5. Available data in the frequency range 10 to 80 ke indicate that on the average the scattering coefficient m increases about as the first power of the frequency. However, the data also do not deny the possibility that m is inde- pendent of frequency, or that it increases as the square of the frequency. MAGNITUDE OF THE VOLUME-SCATTERING COEFFICIENT AT 24 kc Observed values of 10 log m, inferred from ob- served reverberation levels, vary between —50 and —80 db, with —60 db asa typical value. The varia- tions of 10 log m have not been correlated, in the stud- ies off San Diego, with variations in any factor other than depth in the ocean. The variation of m with depth off San Diego has not yet been fully explained, but its systematic character seems well established. Using a projector pointed straight down, the meas- ured reverberation off San Diego was found to de- crease down to a range of 600 or 700 ft, but then is frequently observed to rise fairly abruptly, and main- tain a high value for a depth interval of about 700 ft. The inferred values of 10 log m for depths within the deep scattering layer are often 15 db or more greater than the values of 10 log m at other depths. Some of these deep layers of high scattering power persist in a given area for periods as long as a month or even longer. Although the scatterers in these deep layers have not been definitely identified, it seems probable that they are of biological origin. 17.2.2 Surface Reverberation The following subsections summarize the known information concerning reverberation from the sur- face of the ocean. This information applies primarily to reverberation measured with horizontal beams at those ranges where surface reverberation exceeds volume reverberation. THEORETICAL FORMULA FOR SURFACE REVERBERATION LEVEL The expected surface reverberation level R’() at a time ¢ seconds after midsignal is given by the formula R'(t) = 10 log : + 10 log a + J4(0) —380logr—2A, (7) where m’ is the backward scattering coefficient of the surface scattering layer; @ is the angle at the trans- DEEP-WATER REVERBERATION LEVELS ducer between the sound returning at time ¢ and the horizontal plane; J,(@) is the surface-reverberation index corresponding to the angle of elevation 0; and the other quantities have the meanings given in the section entitled “Theoretical Formula for Volume Reverberation Level” with the further specification that A is the transmission anomaly along the actual ray path to the surface. Because of reflections from the air-water interface, not taken into account in equation (7), measured surface reverberation levels with horizontal beams will usually be about 6 db higher than the levels predicted by that equation. DEPENDENCE ON RANGE According to equation (7), the surface reverbera- tion intensity at short ranges, where the transmission anomaly 2A can be neglected, should be proportional to the inverse cube of the range, provided m’ and J;(@) also change negligibly with increasing range. This simple inverse cube dependence is observed only rarely. When refraction near the surface is sharply downward, surface reverberation drops abruptly be- low volume reverberation at the range where the limiting ray dips beneath the surface scattering layer. Moreover, the decay of surface reverberation inten- sity is usually faster than inverse cube even when downward refraction is weak or absent. For high wind speeds (and therefore high sea states) the decay is especially rapid ; for wind speeds greater than 20 mph, the surface reverberation levels usually drop off nearly as the fifth power of the range, and rates of decay as high as the seventh power have sometimes been observed. Factors which may contribute to this unexpectedly high decay rate are: (1) a decrease in the surface scattering coefficient m’ as the incident sound ray becomes more nearly horizontal; (2) at- tenuation; (3) the sound-shadowing effect of surface water waves; and (4) image interference, that is, the interference between direct and surface-reflected waves. DEPENDENCE ON WIND FoRCE The wind-speed dependence of surface reverbera- tion is most marked at short ranges. At ranges of 1,500 yd or more, with horizontal beams, the received reverberation does not depend on wind speed, and for this reason is ascribed to scattering from the volume of the sea. At a range of 100 yd, as the wind speed increases from 8 to 20 mph, the median reverberation level rises steeply in a manner roughly described by 337 equation (6) of Chapter 14. With horizontal beams, little increase in level has been observed as the wind speed increases from zero to 8 mph, or as it increases beyond 20 mph. DEPENDENCE ON FREQUENCY The frequency-dependent terms in equation (7) are the surface-reverberation index J,(6), the trans- mission anomaly term 2A, and possibly the surface- scattering coefficient m’. The value of J,(@) can be determined from the pattern function of the trans- ducer, by equations (40), (41), and (42) of Chapter 12. The transmission anomaly can be estimated by the methods described in Chapter 5 of Part I. Un- fortunately, there are no experimental data on the variation of surface scattering coefficients with fre- quency. MAGNITUDE OF THE SURFACE-SCATTERING COEFFICIENT The magnitude of 10 log m’ can be obtained from comparison of equation (7) with the measured rever- beration at any range. Although this process is open to criticism, since equation (7) does not describe the range dependence of surface reverberation very well, it furnishes us with the only information we now have on the magnitude of the surface scattering coefficient. By using equation (7), it appears that the increase in surface reverberation at 100 yd as the wind speed increases, noted in the preceding subsection, is due to increases in the surface scattering coefficient m’ as the sea becomes rougher. Thus, at a range of 100 yd the median values of 10 log m’ obtained by comparing equation (7) with measured levels are —57 db at wind speeds less than or equal to 8 mph, and —22 db at wind speeds greater than 20 mph. At 1,000 yd, for wind speeds greater than 20 mph, 10 log m’ averages —31 db. It does not seem possible according to Section 14.2.5 to interpret the reverberation meas- ured at high wind speeds as a result of scattering from a dense layer of bubbles. 17.2.3 Deep-Water Levels with Horizontal 24-ke Beams For prediction of deep-water, 24-ke reverberation levels with horizontal beams, Figure 31 of Chapter 14 can be used. This figure shows the highest reported reverberation levels, the lowest reported levels, and the median levels, for various ranges and wind speeds. 338 The main import of this figure is that it indicates both the expected reverberation level and the possible spread in values at any range and wind speed. From the upper and lower limits in the figure were inferred the values of the surface and volume scattering coefficients given in Section 17.2.2. 17.3 BOTTOM REVERBERATION LEVELS The following subsections summarize the known information concerning reverberation from the ocean bottom. This information is mainly concerned with reverberation from horizontally projected beams in shallow water. Under these circumstances, after a sufficient time has elapsed for the beam to reach the bottom, the received reverberation is preponderantly bottom reverberation. 17.3.1 Theoretical Formula The expected bottom reverberation level R’(¢) at a time ¢ seconds after midsignal is given by the formula u R'(t) = 10 log = + 10 log = + Jo(6) — 30logr—2A, (8) where m” is the bottom scattering coefficient, @ is the angle at the transducer between the sound returning at the time ¢ and the horizontal plane, J,(@) is the bottom reverberation index, corresponding to the angle of depression 6, and the other quantities have the meanings given in Section 17.2.1. With horizontal beams and transducers near the surface, observed bottom reverberation levels will average about 6 db higher than the levels predicted by equation (8), on account of surface reflections. 17.3.2 Dependence on Range According to equation (8), the bottom reverbera- tion intensity at short ranges, where the transmission anomaly term 2A can be neglected, should be propor- tional to the inverse cube of the range, provided m” and J;(6) also change negligibly with increasing range. This simple inverse-cube relationship is almost never observed. In the first place, because of the distance between the transducer and the bottom, reverbera- tion from the bottom does not set in until a significant time has elapsed after the emission of the ping. Usually the reverberation then quickly builds up to a peak, corresponding approximately to the time when the edge of the main beam strikes the bottom. After SUMMARY the peak, the reverberation intensity falls off rapidly, usually about as the fourth power of the range; how- ever, very large deviations from the inverse-fourth power decay have been observed. The range to the bottom reverberation peak de- pends on refraction conditions and water depth. In isothermal water, the peak is expected at a range about 12 times the water depth. When the tempera- ture decrease from projector to bottom is greater than 5 degrees, the peak occurs at a range between 4 and 8 times the water depth, depending on the severity of the downward refraction, with a median value of 6 times the depth. In general, the quantities m” and J,(6) in equation (8) are dependent on range. However, at ranges past the reverberation peak, both of these quantities usually depend only slightly on range. 17.3.3 Dependence on Frequency The frequency-dependent terms in equation (8) . are the surface reverberation index J;(6), the trans- mission anomaly 2A, and the bottom scattering coefficient m”. The value of J,(6) can be determined from the pattern function of the transducer, by equa- tions (53), (41), and (42) of Chapter 12, and, as be- fore, the transmission anomaly can be estimated by the methods of Chapter 5. Measurements on rock bottoms indicate no dependence of the bottom-scat- tering coefficient m” on frequency, in the frequency range 10 to 80 ke. It is probable that other bottoms as well would show no dependence of m” on the frequency of the incident sound, although more data are needed to confirm this point. 17.3.4 Dependence on Bottom Bottom reverberation levels are not the same over all types of bottoms. Although wide variations are observed, in general the highest reverberation levels are observed over ROCK, lower values over MUD and SAND-AND-MUD, and the smallest values over SAND bottoms. These classifications of bottom type depend on the particle size in the material com- posing the bottom and are more fully described in Chapter 6. 17.3.5 Bottom Scattering Coefficients at 24 ke The backward scattering coefficient depends on the angle at which the sound is incident on the bottom. FUTURE RESEARCH 339 For a grazing angle of about 10 degrees, a typical value for the angle at which sound in the main beam strikes the bottom, average values for 10 log m” are —20 db for ROCK, —27 db for MUD, —28 db for SAND-AND-MUD, and —32 db for SAND. Over individual bottoms of a given type, deviations of +5 db from these average values may be expected. There is not much information concerning the de- pendence of m” on grazing angle. It appears that for angles between 10 and 30 degrees m” is roughly proportional to the square of the grazing angle. 17.3.6 Bottom Reverberation Levels with Horizontal 24-ke Beams Figure 8 of Chapter 15 shows the expected rever- beration level at the bottom reverberation peak, as a function of bottom type and of bottom depth below the projector. The height of the peak is a significant quantity in assessing the importance of bottom rever- beration in any given situation. For detailed predic- tion of the levels at ranges past the peak, accurate knowledge is needed of the transmission of sound along the various ray paths to the bottom. 17.4 FLUCTUATION AND FREQUENCY CHARACTERISTICS 17.4.1 Fluctuation The measured reverberation is probably the result- ant of a combination of a large number of small amplitudes of random pbase. If so, the probability P that the reverberation intensity will exceed the value T is given by the formula P= e@ (9) where J is the average intensity. For the distribution defined by equation (9), the variance defined by equa- tion (4) is 7.2 Measurements indicate that equation (9) is a fairly good description of the distribution of reverberation intensities. However, the observed fluc- tuation of reverberation intensity must, in some part, be due to variability in such factors as transmission loss and transducer orientation. 17.4.2 Coherence Analysis of reverberation records shows that the reverberation tends to occur in the form of pulses or “blobs” of about the length of the ping. For square- topped pings and the intensity distribution defined by equation (9), the correlation coefficient in equa- tion (5) has the value given by aN? 0-9) fora Sr pl T 0 fora =T (10) where 7 is the ping length, and a = |t; — tJ. 17.4.3 Frequency Spread For many purposes it is desirable to know the frequency spectrum of reverberation, which gives, as a function of frequency, the energy in each 1-c band. If the reverberation is simply the combination of a large number of individual echoes, each with the same frequency spectrum as the emitted ping, then the resultant reverberation should also have the same spectrum as the ping. This conclusion is probably not far wrong, although precise measurements of the frequency spectrum of reverberation have not often been attempted. The distribution of the instantaneous frequencies of the reverberation (defined in Section 16.3) is also useful information. This distribution can be measured by an instrument known as the “periodmeter.” Periodmeter measurements indicate, among other things, that the spread of instantaneous frequencies in the heterodyned reverberation depends on the audio output frequency and the pulse length, but does not depend on the frequency of the outgoing ping. 17.4.4 Wide-Band Pings The fluctuation of the reverberation with wide- band pings is probably not very much different in magnitude from the fluctuation with narrow-band pings. However, the rapidity of the fluctuation is in- creased as the frequency band is widened. In general, the average reverberation levels are not affected by widening the frequency band of the outgoing ping. 17.5 FUTURE RESEARCH Reverberation studies are a powerful tool in the investigation of properties of the ocean. Information from such studies is necessary to determine definitely the nature of the scatterers, is useful in evaluating theories of transmission loss, and can cast light on the temperature microstructure of the ocean. Also, these 340 studies are needed to fill in the gaps in our knowledge of the reverberation levels to be expected under vari- ous conditions. In general, measurements of volume reverberation will cast the most light on the funda- mental properties of the ocean. Experiments of the following sort are indicated: 1. Measurements of reverberation over a very wide range of frequencies, from sonic frequencies up to several hundred kilocycles. 2. Measurements of the dependence of volume re- verberation on transducer directivity, which would help evaluate the importance of multiple scattering. 3. Careful correlation of measured volume rever- beration levels with simultaneous measurements of temperature microstructure. 4. Careful correlation of measured reverberation levels with observed transmitted sound levels, espe- cially with such features as sound penetration into predicted shadow zones. 5. Reverberation measurements with deep pro- jectors to demonstrate any fundamental differences between the upper and lower layers of the ocean. 6. A thorough investigation of the deep scattering layers, including the use of underwater photography. 7. Measurements of reverberation in large fresh- water lakes. 8. More complete studies of the dependence of re- SUMMARY verberation on ping length, especially with very short pings. 9. Investigation of various probability and correla- tion coefficients of the sort discussed in Chapter 16. 10. Measurements of the dependence of surface and bottom scattering coefficients on the grazing angle of the incident sound. 11. Correlation of measured surface reverberation levels with simultaneous measurements of optical transparency and of entrapped air or other ma- terial. Theoretical investigations of various questions are also required, so that the results of these experiments may be correctly interpreted. Most of these theo- retical investigations will be of importance in the subject of transmission as well as reverberation. Typical subjects for theoretical research would be the reflection of sound from a rough surface, and scat- tering of sound by thermal microstructure. A final subject of great importance, which requires both theoretical and experimental research, is the development of instrumental means for recording and computing various time averages which are of interest in reverberation studies. Such instrumental pro- cedures would greatly reduce both the time and ex- pense involved in the suggested experiments listed above. PART Ill REFLECTION OF SOUND FROM SUBMARINES AND SURFACE VESSELS Da La Bs ( ‘% ie yee, a ; Thu nat Mahathir: } ; oy A i i} 4 we? ‘ \ y r f } c eT sh shi On 4 We Beta a yt we Ne + i i F f as hy b ; eel ny : } i j i Pa ; Varah. be x ues Chapter 18 INTRODUCTION (oa MAY BE DETECTED by the echoes they re- turn. In water, sound waves are absorbed and scattered very much less than radio or light waves. Consequently, sound waves are particularly useful in detecting distant objects under water by means of echo-ranging, that is, sending out a sound signal and listening for a returning echo. The loudness of an echo depends on how much sound is absorbed and how much sound is reflected. As a signal is sent out, the energy spreads; some of it is immediately absorbed by the water and is dissi- pated as heat energy. The transmission and absorp- tion of underwater sound have been studied exten- sively in subsurface warfare, and are described in Chapters 1 to 10 of this volume. Some of the energy is scattered at random back to the echo-ranging pro- jector, either by particles or other nhomogeneities in the water, or by the ocean surface or bottom. This scattering gives rise to a phenomenon known as re- verberation, which has also been investigated in de- tail and is treated in Chapters 11 to 17 of this volume. The sound distinctly reflected from an obstacle or target in the path of the sound beam — such as a sub- marine or whale — gives rise to an echo. Chapters 18 to 25 discuss the reflection of sound from vari- ous underwater targets. Many types of targets are encountered in practice. In particular, recognizable echoes have been received from schools of fish, whales, patches of kelp and sea- weed, and from sunken wrecks or prominent irregu- larities on the ocean bottom in shallow water. Certain water conditions give rise to echoes; at very short ranges, echoes from ocean swells have been observed. Wakes, “‘pillenwerfer,” and other types of bubble screens are effective targets. Their acoustic properties have been studied both theoretically and experi- mentally, and are described in Chapters 26 to 35. In addition, icebergs have been detected by echo-rang- ing, although no such echoes have been measured. 18.1 TARGET STRENGTH In subsurface warfare, submarines, surface vessels, and underwater mines are the most important tar- gets. The reflection of sound from submarines and surface vessels has been investigated in terms of target strengths, a quantitative measure of their re- flecting characteristics. Submarine target strengths have been studied as a function of the size and shape of the submarine, its orientation with respect to the echo-ranging projector, the distance from the sub- marine to the projector, and the frequency of the echo-ranging sound. Chapters 18 to 25 summarize all available information along these lines. HKeho-ranging measurements on submerged sub- marines under more or less controlled conditions have resulted in a large collection of target strength data. In addition, submarine target strengths have been computed theoretically and measured in experiments with scale models. Unfortunately, very little is known about the reflection of sound from surface vessels, as no exhaustive series of tests has been made with this sole object in mind. The data describing surface vessel target strengths are few and scattered; con- clusions are tentative and uncertain. No measure- ments have been made of the reflecting characteristics of mines. However, spheres of various sizes have been frequent experimental targets, and the results of echo-ranging measurements on spheres are probably applicable to small-object location and the detection of mines. 18.2 USES Tactically, knowledge of how submarines and sur- face vessels reflect sound is very important. A quan- titative evaluation of the contribution which the re- flecting characteristics of the target make to the re- ceived echo intensity is necessary in order to predict maximum echo ranges accurately. In submarine operations, the reflecting properties of the submarine should be known so that effective evasive maneuvers may be taken to reduce, as far as possible, the chance of sonar contact by enemy antisubmarine vessels. For example, it is known that the strongest echo is obtained when the submarine presents its beam to the echo-ranging vessel. Therefore, keeping the at- 343 344 tacking vessel off the beam of the submarine is im- portant in reducing the maximum range at which the enemy vessel can detect an echo from it. It should also be useful for submariners to know under what conditions a submarine is most vulnerable to contact by echo ranging, that is, in what position, at what aspect, or depth, or speed. In addition, any counter- measures designed to reduce the probability of con- tact, such as by making the submarine a less effective acoustic reflector, require a quantitative knowledge of the reflecting characteristics of the submarine. Similarly, such knowledge would be useful to anti- submarine vessels in suggesting searching or attack- ing operations. It is also required for the efficient design and operation of many underwater echo-rang- ing devices, such as certain decoys or mines. Informa- tion on how much sound mines will reflect is impor- tant both in the design of mine detection gear and in the evaluation of echo-ranging equipment tests carried out with a particular type of mine. INTRODUCTION This report emphasizes how submarines and sur- face vessels reflect sound under field conditions. Chapter 19 introduces the concept of target strength, defines it in terms of quantities directly measurable, and derives an expression for the target strength of a perfectly reflecting sphere on the basis of ray acoustics. Chapter 20 presents the theoretical back- ground of reflection and scattering of sound from bodies of various shapes on the basis of wave acoustics, and reviews the theoretical calculations of the target strength of a submarine. The technique of the direct, field measurements of submarine target strengths are described in Chapter 21, the indirect measurements in Chapter 22, and all the results of submarine target strength measurements are sum- marized and discussed in Chapter 23. Finally, Chap- ter 24 comprises all available information on surface vessel target strengths and Chapter 25 summarizes briefly the reflection of sound from both submarines and surface vessels. Chapter 19 PRINCIPLES Wee A TARGET is in the path of a sound beam, the intensity of the reflected sound measured some distance away will, in general, depend on many factors, such as the intensity of the sound striking the target, the distance from the target to the point where the echo is measured, and the size, shape, and orientation of the target. Often it is desirable to separate these different factors so that the effects of the size, shape, and orientation of the target may be discussed independently of all other factors. Such a separation is possible only when the radii of curvature of the sound waves striking the target and returned to the receiver are both much larger than the dimensions of the target, in other words, when the waves incident on the target and the waves reflected back to the receiver are essentially plane. In terms of ray acoustics, the incident sound rays must be substantially parallel over the area of the target which they strike, and the reflected sound rays must be parallel over the area of the face of the re- ceiver. In this chapter target strength is defined quanti- tatively in terms of the echo level, the source level, and the transmission loss. Then the target strength of a sphere is derived as a function of its radius. Finally, the effect of pulse length on target strength is examined for a simple case. Ray acoustics is em- ployed throughout the chapter, and the arguments are necessarily idealized. No account is taken of the wave character of sound; in other words, all effects attributable to the wave nature of sound such as in- terference, diffraction, and phase differences are ex- plicitly ignored. The conditions under which this ap- proximation is valid are discussed in Section 19.4. A more detailed theory of target strength in terms of wave acoustics is presented in Chapter 20. 19.1 DEFINITION OF TARGET STRENGTH Let I) be the intensity of the incident sound strik- ing a stationary target, and J, the intensity of the re- flected sound measured at some particular point. If I, is doubled, J, will also be doubled, other factors remaining unchanged; that is, the intensity of the reflected sound will be directly proportional to the intensity of the incident sound. For a given value of J, J, will depend on the orientation of the target relative to the incident sound and also on where the echo is measured. This dependence of J, may be quite complicated. In practi- cal echo ranging, however, the problem is simplified because the echo is always measured back at the source — in other words, it is always measured in the same direction as the projected sound, and the target strength depends only on the orientation of the target. Therefore, it will be assumed throughout this chapter that the echo is measured at the source. Although this admittedly is not the most general case, it is the only case of practical importance for echo ranging. 19.1.1 Inverse Square Transmission Loss The dependence on distance, although complicated near the target, becomes very simple far away from the target, if the sound rays are assumed to travel in straight paths in an ideal medium, with boundaries so far away that their effects on sound propagation can be neglected. It has been shown in Section 2.4.2 that the intensity of sound from a point source, in this ideal case, falls off inversely as the square of the distance from the source. This same inverse square law applies to sound reflected from any target at distances much larger than the dimensions of the target, since at such distances the target behaves as a point source of sound. Why the inverse square law holds for the intensity of sound reflected from any target, at large but not at small distances, may best be understood by studying Figure 1. Here are shown rays reflected from a target, A to a point near the target, and B to a point far away from the target. Rays reaching a point near the target come from different points on the target and from various directions, if the surface is irregular. 345 346 PRINCIPLES FIGureE 1. Reflected rays at short and long ranges. Therefore, the way in which the sound intensity near the target varies from point to point is complicated. Rays reaching a point far away from the target all come from essentially the same direction, no matter from what part of the target they are reflected. Thus the target “looks like’? a point source, and the in- verse square law of intensity will hold. This conclu- sion, based solely on ray acoustics, is reinforced by considerations of wave acoustics, mentioned in Sec- tion 19.4 and described in more detail in Chapter 20. Sufficiently far away from the target, then, J, will be not only directly proportional to J) but also in- versely proportional to the square of the distance r, or il I, = ke (1) Here k is a constant which in general depends on the size, shape, and orientation of the target. It does not depend on the strength of the sound striking the target, or on the distance from the target, provided I, is measured far enough away from the target to make certain that the intensity of the reflected sound will follow the inverse square law. Incidentally, this relation is not valid for explosive sound, which is treated in Chapters 8 and 9. Now according to equation (89) in Chapter 2, the intensity Io of the incident sound striking the target is equal to the intensity F of the projected sound 1 yd away from the source, divided by the square of the distance r from the source to the target, provided that r is much larger than the dimensions of the source. Then lips = Q) 2 Substitute equation (2) into equation (1), and PF Equation (3) is particularly interesting because it shows that, for an ideal medium, the intensity of an echo is inversely proportional to the fourth power of the range, as long as the echo is measured at the source and the range is much larger than the dimen- sions of the target or source. If logarithms are taken and equation (8) expressed in decibels, 10 log J, = 10 logk + 10 log F — 40 logr. (4) 19.1.2 General Transmission Loss All these equations are derived on the assumption that the medium through which the sound travels is ideal, that all the sound is transmitted freely without refraction, absorption, or scattering, and that the boundaries of the medium are so far away that their effects on the propagation of sound waves may be neglected. In other words, as the sound travels each way, its intensity falls off according to the inverse square law alone. The drop in intensity each way, in decibels, is the transmission loss H, which for this ideal case is simply 20 log r. The total transmission loss 2H to the target and back again is then 40 log r. Generally, however, the intensity of transmitted sound under water does not fall off according to the inverse square law alone. Sound is absorbed and scattered in sea water. It may be bent by tempera- ture gradients and consequently focused or spread out. Often the surface and bottom of the ocean sig- nificantly affect both transmitted and reflected sound. Therefore, H will seldom exactly equal 20 log 7, and DEFINITION OF TARGET STRENGTH 347 a] a Ba ~ 50 ea al Fria eI es EIZooe | a | (a ae HIoooas LiSoeoe (a HISoeoe SCC PNT GEERNECEOEBBEBEEEEE LRN ih TARGET STRENGTH IN DECIBELS HA 20 30 50 70 100 200 300 500 700 1000 RADIUS IN YARDS FigurE 2. Target strength of a sphere. the two-way transmission loss is more conveniently represented by the more general function 2H than by 40 log r. Equation (4) then becomes 10 log 7, = 10 logk + 10 log F — 2H. (5) The total transmission loss 2H cannot be predicted or estimated very reliably because of widely varying oceanographic conditions. Instead, it must actually be measured during the course of the experiment. 19.1.3 Fundamental Definition By defining the target strength T as 10 log k, the echo level E as 10 log TI,, and the source level S as 10 log F, equation (5) becomes = HE—S-+ 2H (6) where 2H is the total transmission loss in decibels from the source out to the target and back to the source again. Equation (6) is the fundamental defini- tion of target strength. This equation is always used in the computation of target strengths measured at sea since it involves only directly measurable quanti- ties — that is, echo level, source level, and transmis- sion loss from the source out to the target and back to the source. Since J) and J, are measured in the same units, it is evident from equation (1) that k has the dimension of an area and the value of T will depend on the units which are used. Since the yard is used in range- prediction work as the unit of length, the source level is defined in terms of the intensity at 1 yd, and the transmission loss, which enters twice into equation (6), is defined in terms of the intensity drop from a range of 1 yd out to the range of the target. Conse- quently k in equation (3) may be expressed in square yards. Equation (6) was derived from physical concepts in order to express as a sum of separate terms the effects on the strength of the received echo of (1) the size, shape, and orientation of the target; (2) the intensity of the source; and (3) the range of the target. This separation can be realized only at long ranges, where the sound reflected from the target be- haves as if it were emitted from a point source and the target strength becomes independent of the 348 dz PRINCIPLES Figure 3. Uniform reflection from a sphere. range. At long ranges, then, only the transmission loss term depends on the range. At short ranges, however, the target strength de- pends on the range as well as on the size, shape, and orientation of the target (see Section 20.4.4). If the source is so close to the target that different parts of the target are struck by sound of different intensities, or if the receiver is so close that the spreading of the sound reflected from the target to it is not the same as the spreading from a point source, the target strength term will depend on range. Therefore, at short ranges equation (6) does not serve primarily to separate the effects of range, transmission conditions, source level, and target characteristics on the echo level, but rather to define target strength under the particular conditions of that measurement. 19.2 TARGET STRENGTH OF A SPHERE Because a sphere is perfectly symmetrical, the echoes which it returns to a sound source are com- pletely independent of its own orientation. For this reason, spheres are convenient targets and have fre- quently served as experimental targets in echo-rang- ing measurements. In this section, the target strength of a sphere will first be derived simply and intuitively by considering the total intercepted and reflected energy without regard to the angular distribution of energy within the reflected sound beam. Then a more rigorous derivation — within the framework of ray acoustics — will be presented, in which the angular distribution of the reflected energy is considered in detail. 19.2.1 Simple Derivation Consider a plane wave of sound of intensity Io striking a sphere of radius A and cross-sectional area 1A?. Then the total sound energy intercepted by the sphere per unit time will be 7A?J» and, if reflection is perfect, the total sound reflected from the sphere per unit time will also be 7A?Jo. TARGET STRENGTH OF A SPHERE 349 Now assume that this sound energy is reflected uni- formly in all directions. At a distance r from the center of the sphere, it will be spread uniformly over the surface of a sphere of radius r or over the surface area 47. Since the intensity J, of the reflected sound equals the total energy 7A?Jp reflected by the target sphere per unit time, divided by the area 4mr? over which it is distributed, then at a distance r from the sphere TA? A? =— 0 — 4rr 4r? ih Io. (7) But, from equation (1) Ih I, =k, (8) 7 where r is the distance from the target to the point where the echo is measured. Therefore by substitu- tion k=— (9) A and T = 10 log k = 20 log ice (10) 2 where T is the target strength and A the radius of the sphere. With the yard chosen as the unit of length, A becomes the radius of the sphere in yards, and from equation (10) it is evident that the target strength is the echo level of the target in decibels above the echo level from a sphere 2 yd in radius. Target strengths for spheres of various radii are shown in Figure 2. 19.2.2 Rigorous Derivation This derivation explicitly assumed that sound is reflected from a sphere uniformly in all directions. To justify this assumption, consider the same sphere of radius A in Figure 3. Two adjacent rays x and y separated by a distance dz are traveling parallel to OO’ and strike the sphere at X and Y respectively, making angles of 6 and @ + dé with the sphere radii drawn to the points X and Y. From Figure 4, dz = XY cos @ = A cos @dé. (11) Now rotate rays x and y about OO’. These rays will describe circular cylinders and dz will generate an area ds between them, where ds = 21X Wdz = 2r (A sin 8) (A cos @dé@) (12) or ds = 27A? sin 6 cos 6d6. (13) The total energy dJ striking the sphere at angles between 6 and 6 + d6 will be the product of the in- PULSE SS ECHO Wx +2z=% eet en PULSE ee ECHO ex => B wWexe2z= 3x=32z 2 4 PULSE (= ECHO —« e— c w=x+2z22z FieurE 4. Effect of pulse length on target strength, echo length, and echo structure. tensity Jo of the incident sound and the cross-sec- tional area ds, or dJ = Inds = 27A?Iy sin 6 cos 6p. (14) Now consider the reflected rays x’ and y’ making angles of 26 and 26 + 2d6 with OO’ in Figure 3. At a distance r from the center of the sphere, x’ and y’ will be separated by a distance dZ. At a distance much larger than the radius of the sphere, dZ becomes much greater than XY; and 2’ and y’ may be replaced by r. dZ = r 2(d0) = 2rdé. (15) Again rotate the rays about OO’, and dZ will generate the area dS between them, where dS = 2nPQdZ = 2r(r sin 20) (2rd6) (16) 350 or dS = 4rr’? sin 26d6 = 8rr? sin 6 cos 6dé. (17) Since the intensity of the reflected sound equals the energy reflected per unit time divided by the area over which it is distributed, then dJ 2A? sin 6 cos 6d0 A? ees Be To = -[o. dS 8zr* sin 6 cos 6dé ; Thus J, is independent of 6 and therefore is inde- pendent of the direction of the reflected sound, and equation (18) is identical with equation (7) derived from a simpler analysis. Rigorously, then Ae A T = 10 logk = 10 log G) = 20 log el (19) where 7 is the target strength and A the radius of the sphere. Equation (19) applies only to target strengths measured far away from the sphere. Close to the sphere, the target strength will also depend on both the direction 6 and the range r. [, = (18) EFFECT OF PULSE LENGTH So far it has been tacitly assumed that continuous sound strikes the target and is reflected back to the projector. Usually, however, sound pulses of finite length are sent out, and most target strengths are measured with such sound pulses. In general, target strength will be a function of pulse length, and the dependence of echo intensity on signal length must be investigated. Consider a curved surface, such as a sphere or an ellipsoid, each part of which reflects sound specularly as a mirror would. This surface is normal to the inci- dent beam at only one point, and only one ray is re- flected back to the projector in the direction of the incident ray. Therefore, the echo intensity — and consequently the target strength — will be inde- pendent: of signal length, and the echo structure will accurately reproduce the signal structure, unless mul- tiple transmission paths which result from surface- reflected or bottom-reflected sound, for example, give rise to multiple echoes. This result, derived on the basis of ray acoustics, is not valid if very short pulses are used, since the wave character of sound must then be considered. However, this result is correct if the pulse is at least several wavelengths long. On the other hand, consider an extended rough surface, each part of which reflects sound in all direc- tions. A pulse 7 seconds long is sent out from a 19.3 PRINCIPLES projector a distance r from the target which has an extension z in the direction of the incident beam, as illustrated in Figure 4. Now the first part of the signal will reach the nearest part of the target at a time r/c after it was emitted where c is the velocity of sound and will be returned to the projector at a time 2r/c. The last part of the signal will leave the projector at a time 7, reaching the nearest part of the target at + + r/c and the farthest part of the target at a time t + r/c + z/c; it will return to the projector at a time rt + 2r/c + 22/c. The duration of the echo will be the difference between the time when the first part of the signal reaches the nearest part of the target and is returned, and the time when the end of the signal is reflected from the farthest part of the target and is received at the projector. Then if the duration of the echo is a, o = T+ 22/c. (20) 19.3.1 Long Pulses First, let the signal be long compared to the ex- tension of the target (Figure 4A). Then (21) and the echo length will approximately equal the signal length. Assume that the reflected energy is always directly proportional to the incident energy, and therefore to the product of the signal intensity and the signal length. Then the echo intensity will depend on the signal intensity but not on the signal length. Now let the signal length equal the depth of the target in the direction of the beam (Figure 4B). Then oT Z G=7-+-— — 37, (22) @ and the echo length will be three times the signal length. The echo will no longer resemble the signal, as the echo intensity grows to a maximum when the target is illuminated by the entire signal. Short Pulses Lastly, let the signal be short compared to the depth of the target (Figure 4C). Then 2z (io c 19.3.2 , (23) and the echo length will approximately equal twice the extension of the target in the direction of the beam. The echo intensity now will depend on the WAVE CHARACTER OF SOUND signal length as well as the signal intensity, since the reflected energy will be less for a short signal than a long signal and therefore — as long as the echo length remains constant — the echo intensity will be re- duced. For short pulses, then, the echo intensity and therefore the target strength will depend on the pulse length. In practice, however, fluctuations in the course of each echo, and from echo to echo, tend to obscure this relationship for any individual echo. For long pulses, the echo very closely reproduces the signal envelope, which is usually square-topped. For short pulses, however, fluctuations in echo intensity result in a very irregular hashed structure where a sharp peak or group of peaks stands out clearly against a background which is sometimes 10 db lower. The peak echo intensity, which is usually used in computing target strengths, is, in general, different from the average echo intensity. Therefore, for short pulses the peak echo intensity may be considerably different from the average echo intensity and may vary in a different way with signal length. Peak and average echo intensities, and how they vary with signal length, are discussed in Sections 21.6.4 and 23151. 19.4 WAVE CHARACTER OF SOUND Ray acoustics has been used exclusively through- out this chapter in defining target strength, in de- riving the target strength of a sphere, and in dis- cussing target strength as a function of experimental variables, just as ray acoustics was employed in 351 Chapter 3 of this volume in treating the transmission of sound through sea water. Experience shows, how- ever, that sound does not always travel in straight lines, and that, for many purposes, ray acoustics is inadequate in explaining and interpreting underwater sound phenomena. An alternative approach in terms of wave acoustics becomes necessary. Throughout this chapter it has been tacitly as- sumed that sound is propagated along straight lines as sound rays, and that reflection is wholly specular, in other words, that the angle of reflection always equals the angle of incidence. Many modifications must be introduced if allowance is made for the various wave phenomena affecting echo ranging. Sound is diffracted when it strikes a target or parts of a target whose dimensions approximate its wave length. Thus, the previous discussion applies only to targets considerably larger than the wave length. For the same reason, these results apply only to pulses whose length in the water is at least several wave lengths. In addition, sound reflected from one part of a target may interfere with sound reflected from other parts. Much of the fluctuation commonly encountered in analyzing echoes from underwater targets is attributable to interference. The results in the preceding section on echoes from extended tar- gets are valid only if the interference effects arising from constructive or destructive interference can be eliminated by averaging over several successive echoes. The effect of the wave length of sound on target strength for both specular and nonspecular reflection is discussed in greater detail in Sections 20.4 and 20.6. Chapter 20 THEORY iB CHAPTER 19 the concept of target strength was introduced and its meaning defined quantita- tively; then the target strength of a perfectly reflect- ing smooth sphere was derived in terms of ray acoustics. The theoretical background will be pre- sented in this chapter in terms of wave phenomena with a mathematical discussion of the reflection of a sound wave from a target of any shape, and a review of the early theoretical calculations of the target strength of a submarine. In principle, the reflection of sound from a target can be exactly determined by solving the wave equa- tion derived in Chapter 2 of Part I, as long as the proper boundary conditions at the surface of the target are satisfied. In practice, an exact computation along these lines is mathematically very difficult; the difficulties are most marked for targets large com- pared to the wavelength of the incident sound. Even for a sphere the rigorous analysis which has been worked out !:? is rather complicated. Numerical ap- plications of these precise formulas have been pub- lished * for a rigid sphere, whose circumference is from 1 to 10 times the wavelength; the results pro- vide an interesting example of the exact behavior of reflected sound in one simple case. However, even for such relatively small targets the mathematical analysis becomes tedious. 20.1 APPROXIMATIONS To obtain more general results, various approxima- tions must be made, physical as well as mathematical. The mathematical assumptions made in this chapter are fairly standard and are believed to give essentially correct results. The physical assumptions about the nature of the reflecting surface are more important, however, and require some justification. In the first place, most of this chapter applies only to targets which are large compared to the wave- length of the sound, and whose surface is smooth; in other words, the radius of curvature of the surface is 352 also large compared to the wavelength. These re- strictions seem legitimate for most targets of practical interest in echo ranging. In the second place, the material of which the tar- get is composed is assumed to be rigid. In terms of sound reflection, a target is said to be rigid if p1c1/o2c2 is negligibly small, where p; and c, are the density and sound velocity in the surrounding medium, and p, and c are the density and sound velocity in the target. When this condition is not fulfilled the problem be- comes much more complicated. In most cases of in- terest to subsurface warfare, the target is bounded by thin metal plates, inside which there may be air or water. The reflection of sound from such plates has been studied,* * and the results obtained show that even for plates only 14 in. thick, such as generally constitute submarine superstructures, the reflection is practically perfect; transmission and absorption are negligible. Thus, the assumption of perfect re- flection from practical targets seems justified. Some of the additional assumptions which may be made in the discussion of the reflection from targets are discussed in an early British report.° This work is particularly interesting because it presents the most complete available application of theory to the target strength of underwater objects. The essential elements of the theory of target strength, restricted by the physical assumptions which have been made here, are presented in the following sections. First, an approximate but general formula is derived for the pressure of the sound re- flected from a target. In Section 20.3, this result is further simplified to give an equation for the target strength of a reflecting surface in terms of the so- called Fresnel zone theory. This latter equation is then used to find practical formulas for the target strengths of simple geometrical shapes, which are applicable to the major reflecting properties of sub- marines; the application to an actual submarine is described in Section 20.5. All this latter analysis ap- plies only to long pulses. The last two sections are REFLECTED PRESSURE devoted to a qualitative discussion of the reflection from targets small compared to the wavelength, and the echoes obtained with very short pulses. REFLECTED PRESSURE 20.2 Consider first a sound beam striking a surface ele- ment dS of a perfectly reflecting, smooth and rigid underwater target. Since this surface is rigid, the primary effect of the target is to prevent the water from moving perpendicularly to dS at the surface of the target. In other words, at the surface of the tar- get, the velocity wu of the water, measured along a line perpendicular to the surface, must be zero, or uz = 0, (1) where the z axis, at the point of incidence, is perpen- dicular to the target surface. By differentiating equa- tion (1) with respect to the time ¢, uz = 0. 2 A (2) But from equation (17) in Chapter 2 of this volume, Ou, Op vate (3) where p is the density of the medium, :p the pressure of the sound wave, and z the coordinate perpendicular to the surface. 20.2.1 Boundary Condition Substitution of equation (3) into equation (2) gives op Oza ver, which means that for a rigid target the component of the pressure gradient perpendicular to the surface must vanish at the surface. This is the boundary con- dition which the solution of the wave equation must satisfy. In the absence of the target, the sound source will send out a wave whose resulting pressure at any par- ticular point may be denoted by p,; then p; must be a solution of the wave equation [equation (27) in Chap- ter 2.] In the presence of the target, this pressure p1 does not satisfy the resulting boundary conditions at the surface of the target. The actual sound pressure p, which must satisfy both the wave equation and the boundary conditions at the target surface, may be written as (4) p= pit po, (5) 353 where p. constitutes the correction which must be added to the undisturbed sound pressure 7, in order to satisfy the boundary conditions at the surface of the target. By differentiating equation (5) with respect to z and by substituting the result into equation (4) a = = 0; (6) which is another way of expressing the boundary condition. Because the wave equation is a linear homogeneous differential equation, the difference between the two solutions p and p, is again a solution, and 7p, by itself must therefore satisfy the wave equation. In other words, the total sound field may be interpreted as the combination of two sound fields. One of these, whose pressure at any specified point is pi, is called the incident sound; the other, whose pressure at the same point is pe, is the reflected sound. Each of these quantities satisfies the wave equation, but only their sum satisfies the boundary conditions at the target. In some places, the measured sound pressure may oc- casionally consist wholly of one or the other of these two sound fields, depending on whether only the sound projected from the source, or only the sound reflected from the target is measured. The problem tackled in this chapter is the evaluation of the re- flected sound alone, since it is this quantity which is most important in echo ranging. Therefore, an ex- pression for pz must be derived. 20.2.2 Mathematical Formulation To obtain, rigorously, a general expression for p is usually a very difficult problem. It is comparatively easier to obtain an approximate solution by distrib- uting, over the surface of the target, point sources of sound. Then, if the distribution and strengths of these point sources over the area are correctly chosen, these point sources will emit sound in such a way as to cancel the pressure gradient of the wave incident on the surface, thus satisfying the condition (6). For a single point source, the solution of the wave equation is B pri(ft-rl) (7) Tr JO = where p is the pressure of the sound field at a distance r from the source, B is a constant which measures the strength of the point source, 7 is~/ —1, ¢ is the time, 354 THEORY fis the frequency and ) the wavelength of the sound. If the reflecting target itself is considered to be made up of many point sources distributed over its surface, p becomes pz, the reflected pressure; and the pressure dp. produced by all the point sources located in a surface element dS is G grits ag, iP dpz = (8) or the pressure p2 produced by the entire target be- comes (Clee es Si as ; S (9) where G is essentially the average value of B in equa- tion (7) for each individual source multiplied by the number of sources per unit area; the integral is eval- uated over the entire area S. Zz Figure 1. Transformation to polar coordinates. This quantity G is a measure of the number and strength of the point sources over the area; in general G will vary over the target surface. The function G must be chosen so that the resulting sound pressure p2 satisfies the boundary condition (6) on the surface of the target. The value of G at a particular point of the target surface will be assumed to be completely determined by the incident sound pressure at that point. This assumption is not rigorously correct, but it leads to a good approximation if the target has a surface whose radius of curvature is everywhere large compared with the wavelength. First, a relationship between the value of G at any point and the resulting gradient of p, at that point will be derived. Then, the gradient of p. may be re- placed by minus the gradient of pi, because of the boundary condition (6). In this manner, a direct relationship will be obtained between the incident sound field p; on the target surface, and the value of G required to compensate the gradient of pi. Because of the assumption made that the gradient of p. at the point on the target surface is determined primarily by the value of G at that point, a particu- larly simple model may be considered and the result generalized. The pressure gradient at the center of a disk illustrated in Figure 1 will be derived, on the as- sumption that G is constant over the surface; in other words, the density of point sources on the surface of the disk is assumed to be uniform. If polar coordi- nates p and @ are introduced, the integral (9) for the pressure on the z axis can be transformed as follows: E d. y p=0 iP ; or, since r? = p? + 2?, TRB. Pz = 2nG it Ce dr, (10) Equation (10) may be integrated directly and gives for the sound pressure on the z axis mt VR2+22 2ri(ft—z, sak anole ( es )-e i(se oy (1) and by differentiating p. with respect to z, the gradi- ent at p2 perpendicular to the surface becomes dpe a) - enhare| Zz 200 —= = SS 6 = 2 ee (12) For the point on the surface where z = 0, the gradient reduces to 0 Aes (22) = ance dz 2z=0 which is independent of the radius & of the circular surface. This result confirms the assumption that the gradient of p: at any point on the surface is deter- mined only by the value of G in the immediate vicinity of that point; thus G is independent of possi- ble variations in G at other points. Actually it is (13) REFLECTED PRESSURE rigorously correct only for a plane surface, but results in a good approximation for other surfaces as long as the curvature is small over the distance of one wave- length. Consequently, it will be assumed that in general the gradient of p. and the value of G are related to each other at each point on the target surface by the equation a enmift Op2 20 Oz If the boundary condition (6) is to he satisfied, —dp./dz in equation (14) may be replaced by 0p;/dz, and in terms of the incident-sound wave G=- (14) 1 . Opi @e = —2mift— b+ 15 i Qn 0z ¢) Since the incident sound pressure is usually a har- monic wave, it may be locally described by pi = perriti—ar) (16) where b is the amplitude of the wave, f its frequency and \ its wavelength, and gq a coordinate parallel to the direction of propagation. The derivative of p: in the direction of propagation is then = _ 271, onitst—a/n), 0g r The derivative of the amplitude b has been neglected in this equation since this derivative is usually negli- gible at distances from the source of many wave- lengths. In any other direction, the derivative will equal expression (17) multiplied by the cosine of the-angle between the direction chosen and the direction of propagation gq. If the angle between the direction of propagation of the incident sound wave and a line perpendicular to the target surface is 0, then the derivative of p; along a line perpendicular to the target surface is Opi 0z (17) 277. = ——bcospenr tt. ON (18) If this expression is substituted in equation (15), G becomes G= =U cos6e 774’. (19) It is particularly interesting to evaluate the wave amplitude b for the case where the incident wave is caused by a point source of sound at a point P, a distance r’ from the point of the target surface con- sidered. If at unit distance from P the amplitude of 355 the incident spherical wave is B, then the local ampli- tude b equals B/r’; the coordinate q may be replaced by r’, and equation (19) assumes the form iB —2mir’/d —-—cosde . rr’ @= (20) If this expression for G is substituted into equation (9), the resulting integral for the reflected sound pres- sure p2. becomes 1 cos 0 of, _7tr" (p= ~in f tan(e BN ) dS where B is the amplitude of the original point source at unit distance, r’ is the range from the source to a point on the surface of the target, r the range from that point on the target surface to the point in space where pz is to be found, f the frequency and ) the wavelength of the sound. The integration is to be carried out over the whole target surface S of which dS is a surface element. (21) 20.2.3 Physical Interpretation So far the discussion has been wholly mathematical, without the benefit of a physical argument to support and justify the approximations made. Physically, the analysis is based on the fundamental principle that in the vicinity of a rigid surface the fluid motion in a direction perpendicular to that surface must vanish. If the incident pressure wave made the fluid move so as to violate this condition, the rigid surface would exert a force on the adjacent fluid elements just can- celing this motion perpendicular to the surface. This effect may be imagined by replacing each element of area on the target surface by a small piston capable of moving in a direction perpendicular to the surface. In the absence of the boundary condition, each of these pistons would be moved back and forth in rhythm with the motion of the adjacent fluid element. In order to act as parts of a rigid surface, however, these little pistons must each be pushed by a force opposite to that of the motion of the fluid, just sufficient to keep each piston permanently balanced in its original position. This alternating force which each piston exerts on the fluid has the same net effect as the force which a transducer exerts on the sur- rounding fluid, in other words, each acts as a sound source with spherical wavelets emanating from each individual piston. The appropriate amplitude and phase of these wavelets has been calculated above. The total reflected sound field then represents the superposed effects of all these individual wavelets. 356 20.3 FRESNEL ZONES In this section, equation (21) will be applied to compute a general formula for the target strength of a smooth and rigid target. Here, smooth means that the radius of curvature of the target surface is large compared to the wavelength of the sound striking it. Moreover, the target is assumed to have a relatively simple shape, always convex, with no marked bumps or protuberances. While this ideal target hardly re- sembles most actual targets, the consideration of this simple problem gives some insight into the means by which sound waves are actually reflected. Even under these special assumptions, however, it will be shown that the integral in equation (21) can be readily evaluated only by an additional approximation, first suggested by the French physicist, Fresnel. Consider only the case where the echo is observed back at the sound source; this case corresponds to the situation of chief practical interest, as pointed out in Section 19.1, and in addition simplifies the com- putations. Then r = r’ and equation (21) reduces to “1 Qnift iBe™ {zs 6 —arir/nag r r (R= = (22) In the integral, however, both 6 and r vary over the surface of the target. Therefore, the integral cannot be evaluated by elementary methods, except for cer- tain special cases illustrated in Section 20.4 where an exact integration can be carried out. For most practical purposes, however, an expression for the reflected sound pressure p, and, therefore, for the target strength 7’ can be derived by means of an approximate method, which was originally developed in optics and which is known as the method of Fresnel zones. This method is based on the mathematical anal- ysis developed in the preceding section. Physically, according to equation (21), every point on the target surface which is struck by the incident sound pres- sure wave becomes in turn a center of outgoing wave- lets so that the points on the target surface may be considered “secondary sources” of sound. In optics, this is called Huyghens’ principle. Simple addition of the sound pressure in each individual wavelet will give the reflected sound pressure pr. Now, every wavelet has a phase depending on the total distance traveled by the sound out to the target and back. In general, these wavelets interfere both constructively and destructively. Destructive inter- ference leads to cancellations due to the phase dif- THEORY ferences. But a sharp maximum of amplitude — due to constructive interference, where wavelets whose amplitudes are all of the same sign are superimposed — exists in the direction corresponding to specular reflection. This is the direction in which the beam is reflected according to ray acoustics. A quantitative calculation of the amplitudes of the different wave- lets will show exactly how much energy is reflected in different directions. In this way, wave acoustics can be shown to give the same results as ray acoustics when the wavelength is very short. Method To compute the amplitudes and phases of the different wavelets, the surface may be divided into successive areas from which all the wavelets emitted are approximately in phase and thus do not interfere destructively. This is the Fresnel method. According to this method, consider a series of wave fronts pro- ceeding outward from a source at the point P, separated by a distance \/4 from each other, where d is the wavelength of the projected sound. When they strike the target, the surface of the target is intersected by these wave fronts in a series of curves which divide the surface into the so-called Fresnel zones. The phase of each reflected wavelet, measured back at P, is 2rft — 4ar/). Since w equals the product of 47/) times \/4, the distance between two adjacent zones, the wavelets from each zone have an average phase difference of 7 from the wavelets of the ad- jacent zones. But a change of phase by the amount + results in multiplication of the amplitude by —1; hence the wavelets from each zone interfere destruc- tively with those from the two adjacent zones. The advantage of the Fresnel-zone approach, as will be shown, is that most of the zones cancel each other, leaving only the effects of the first and last zones to be considered. While the analysis can be carried out for the Fres- nel zones defined by the wave fronts at any one time, it is simplest to take the zones resulting when one of the particular wave fronts considered is just tangent to the closest point on the target. Let R be the value of r at this point, in other words, let R be the distance from the sound source and receiver at P to the nearest point on the target. The first zone is the area on the surface of the target intercepted by the wave front which is a distance \/4 from the wave front tangent to the target. In general, the position of the nth zone is then determined by the inequality of equation (23) 20.3.1 FRESNEL ZONES 357 R+(n- 1 << e — ” (23) where r is the distance from the source to any point in the zone. If S, is the area of the nth zone, equation (22) can be written as a sum of integrals in which each integral extends over only one zone. Then WB oe cos @ _, . Po = Spey f eas; (24) nN nJSn 1 the sum, denoted by the symbol 2, extends over as many values of n as there are zones. To evaluate the integral in equation (24), define a new variable u, for the nth zone by r — R — (n— 1) d/4 gS SSS r/4 If this equation is substituted in inequality (23), un satisfies the relationship 0 ae =ni pees un) Oa S. 4 nR® aces be (84) (33) j= 358 20.3.2 Application To find the target strength corresponding to the pressure of the reflected wave in equation (34), equa- tion (6) in Chapter 19 may be written in the form T = 20 log |p.| — 20 log |B| + 40 log R (35) where the vertical bars mean that absolute values of the complex quantities involved must be taken. The term 20 log |po| is the rms echo level # where p» is the actual echo pressure; 20 log |B| is the rms source level Sat 1 yd; and 40 log R is twice the transmission loss from 1 yd out to the target at a range R, ex- cluding attenuation losses. Strictly speaking, the rms level is the average value of the square of the real part of the complex quantity rather than the abso- lute value; however, a more elaborate computation along these lines leads to exactly equation (85). If equation (34) is substituted into equation (35) the target strength 7’ becomes l : = iL cos 6 "is| ; 2 Js, where the bars again denote that an absolute value must be taken. The quantity uw in the exponent is defined by T = 20 log (36) uy = “(¢ — R). (37) S; in equation (36) is the area of the target in which ux is less than 1; the integral is evaluated only over those surface elements lying within S;. The evaluation of equation (36) provides the solu- tion of the problem presented at the beginning of this section. TARGET STRENGTH OF SIMPLE TARGETS In this section, equation (36) will be used to com- pute the target strength of relatively simple surfaces, such as spheres, cylinders, and other objects, which have a single highlight. The results obtained may also be applied to more complicated surfaces, as long as the radius of curvature is greater than the wave- length. Whenever several highlights are present, the reflected wave is the sum of the waves reflected from each one separately. In general, they will interfere. However, if an average is taken over a considerable spread of target aspects, and if the highlights are spaced much further apart than the wavelength, the interference will tend to be random; in this situation, the intensity of the echo is simply the sum of the intensities computed for each highlight individually. 20.4 THEORY 20.4.1 Sphere The target strength of a sphere, on the basis of wave acoustics, may be easily derived from equation (36). The results of this analysis may be used not only for a perfect sphere but also for any target sur- face whose first Fresnel zone is essentially spherical. Consider a wave from a source P striking a sphere Q Aa-— Figure 2. Reflection from a sphere. of radius A, illustrated in Figure 2, whose nearest point is a distance R away from the source. If ¢ is the angle subtended at the center of the sphere by Q, which bounds an element dS of area, dS is simply dS = 2rA? sin ddd. (38) By the law of cosines, the distance r from the source P to the point Q is given by r= (R+ A)?+ A? — 2A(R + A) cos SPL UiC & BP sin’ (39) When RF is much greater than A, 7 is approximately 2A(A +R). 46 i? Se r=R+ R (40) The quantity wu; from equation (37) is then 8A(A +R). ¢ =) sine =< 4 Un i sin 9 (41) For short wavelengths, sin ¢/2 will be very small in the first zone and may be set equal to ¢/2; similarly cos 6 in equation (36) may he replaced by one. There- fore, if equations (38) and (41) are substituted into equation (36), the target strength of a sphere becomes T = 20 log at I "eth On Atbdd, (42) 2rJ0 where 2A(A + k) Sched ME ll (43) and zoe = 1. (44) The integration may be carried out and yields TARGET STRENGTH OF SIMPLE TARGETS 359 go =e eo do 1 {) CEP ACI AS rr imate (45) 0 — 2710 0 TUL Thus equation (42) becomes A? PA) log? (46) and if equation (43) is substituted for x, the target strength reduces to T = 20 log (47) A 2(1 + A/R) This expression is valid only when the distance from the source to the sphere is at least several times greater than the sphere diameter. When fF is very much greater than A, equation (47) simply becomes T = 20 log j (48) which is identical to equation (10) in Chapter 19 derived on the basis of ray acoustics. At shorter ranges, equation (36) is still applicable, but must be evaluated more accurately. It may be noted that the value of T in equation (47) is based on the assump- tion that the transmission loss to the nearest point of the sphere is used in equation (35). If the trans- mission loss to the center of the sphere is used in- stead, 7’ must be increased by 40 log (1 + A/R) and increases as the range becomes shorter. As already pointed out, equation (47) may be ap- plied whenever the first Fresnel zone of a reflecting surface is spherical in shape, and has a radius of curvature A much larger than the wavelength A. The result is independent of the wavelength. Equation (36) could be evaluated more accurately to find a dependence of 7’ on wavelength. This dependence would be appreciable only when the wavelength was no longer much smaller than the sphere radius A, in which case the total number of Fresnel zones would no longer be large. Since the accuracy of the Fresnel method is doubtful under these conditions, the wave- length dependence found in this way would not be very reliable unless confirmed by a much more elaborate investigation. 20.4.2 General Convex Surface More generally, the curvature of a surface cannot be described by a simple single radius of curvature. In such a ease, the boundary of the first Fresnel zone will not be a circle, as was the case for a spherical surface. In a more general case, this boundary will be elliptical in shape, and the surface intersected will have two principal radii of curvature A; and Ap, which will usually differ from point to point. These radii may be defined as follows. Let O be a particular point on the surface and let OC be a line perpendicular to the surface at the point O. Any plane containing OC will intersect the surface in some Q Q o Figure 3. Reflection from any convex surface. line QOQ’, as in Figure 3. In the neighborhood of the point O this curve is approximately a circle of radius A. However, as the plane intersecting the target is rotated about the line OC, the radius A of the curve Q0Q’ will vary. It will have a maximum value A; and a minimum value Ag, in general, as the plane rotates through 180 degrees. Furthermore, according to dif- ferential geometry, these two radii will be 90 degrees apart. These two quantities A, and A: are called the principal radii of curvature of the surface at the point O. If they do not change rapidly with position on the target surface — more particularly, if they are ap- proximately constant at all points in the first Fresnel zone — the target strength of the surface may be computed. The derivation is more complicated than that in Section 20.3.1 and will not be given here. The result of the analysis is in the following equation 360 1 A,A, 2(1 + Ai/R)(1 + A2/R)’ which reduces immediately to equation (47) when A, is equal to A». While equation (47) was valid only if A,/R was moderately small, equation (49) is ap- plicable even if A:/R is very large as long as A2/R is still small. Equation (49) cannot be used, however, when either A; or As approaches the wavelength of sound. T = 10 log (49) 20.4.3 Cylinder For an infinitely long cylinder, equation (49) may be applied directly by letting one radius of curvature A, be infinite. The target strength found for this case reduces to V/A AGE i = 10 loe= ( SS = 2 Camara where A>» is the radius of curvature of the cylinder. This equation is valid only when the wavelength of the sound is much less than the radius of curvature of the cylinder, and when this radius in turn is much less than the range. For an actual cylinder equation (50) may be used only if the cylinder is perpendicular to the sound beam at some point, and if the cylinder is long enough to include at least the first few Fresnel zones. The expression may therefore be used only at moderate ranges, since with increasing range the length of the first Fresnel zone increases infinitely. To compute the range beyond which equation (50) cannot be used, let the length of the cylinder be L, and let the sound source lie in a plane which is per- pendicular to the axis of the cylinder and bisects the cylinder. Then the path length r to the end of the cylinder is 1/L\ sree y/ oe BG) The length of the cylinder will include many Fresnel zones if r given by equation (51) exceeds 2 by many wavelengths. Therefore equation (50) may be used only as (50) (51) Fe Iii EK =e ; (52) For example, for a wavelength of 4 in., corresponding to a frequency of about 15 ke and a cylinder 10 ft long, the range must be much less than 100 yd if equation (41) is to be used. At long ranges, R is much greater than L?/\, and the computed length of the first Fresnel zone exceeds THEORY the length of the cylinder. In this case, instead of using the approximation (30) an exact integration of equation (22) over all the zones is possible, provided the variation of cos 6 is neglected, and the target is far away from the source. Thus in equation (86), in- stead of one-half the integral over the first zone, we may take the same integral over all the zones. With the same approximation for wu made in the previous section, the target strength becomes L A BS By) N= A) At these longer ranges, the target strength is again independent of the range, in agreement with the comments made in Chapter 19. However, equation (53) presents one case in which the target strength varies appreciably with changing wavelength, even when the wavelength is much smaller than the target. For intermediate values of the length of the cylin- der, both the first and last zones must be considered. A more exact evaluation of equation (22) can be car- ried through in this special case by use of particular functions called Fresnel integrals, which have been tabulated. 20.4.4 Reflection at Close Ranges (53) The formulas developed so far in this chapter are applicable to many simple shapes provided the sound source and receiver are not too close to the target. The target strength at close ranges may also be found directly from equation (36). Detailed results have been worked out for cases of this nature, but will not be reproduced here. In general, when R becomes much less than the principal radii A; and A», the reflection can best be described as reflection from a plane surface. In the limiting case where R/A: is negligible, 1B eorilft-2R/r) LOT as long as the sound field this close to the target obeys the inverse square law; for a large directional trans- ducer, this condition is not likely to be satisfied at very close ranges. If equations (35) and (54) are combined, the target strength becomes (54) R T = 20 log 5 (55) Formulas for the target strength of various types of objects, such as two cones placed base to base, and a circular disk placed at an angle to the sound beam, are given in reference 6. NONSPECULAR REFLECTION 361 20.5 REFLECTIONS FROM SUBMARINES Expressions were developed in the preceding sec- tion for the target strengths of various surfaces in terms of the reflected pressures. These formulas were employed in a theoretical study 7 in order to calculate mathematically the target strength of a German submarine. From an examination of blueprints of the U570,' a 517-ton German U-boat captured by the British early in the war and renamed HMS/M Graph, the radius of curvature of the surface of the hull and conning tower at different points was obtained. In computing the results, the submarine was approxi- mated by an ellipsoid of revolution, whose semi-axes were 110 and 7 ft. The results were then corrected for the reflections from the conning tower, which was assumed to be a cylinder with a “‘tear-drop” cross section. Target strengths were found from equations (49) through (53) in terms of the range and the radii of curvature for different submarine cross sections; ranges of 8, 12, 16, 200, and 1,000 yd were used. At ranges where the conning tower did not include a large number of zones, the Fresnel integrals [ob- tained when equation (22) is integrated exactly along the length of a cylinder] were used. The calculations were actually carried out in terms of reflection coefficients, which differ somewhat from target strengths derived in this chapter. The results of these computations are presented in Chapter 23 together with the results of the direct and indirect measurements. 20.6 NONSPECULAR REFLECTION So far only reflections from highlights on a target surface have been discussed. These highlights cor- respond to specular reflections in optics and give much the same predictions as those found from the ray theory. In particular, the echo is assumed to come only from that region of the target where the surface is nearly perpendicular to the incident sound wave. This section discusses those cases where such reflection cannot occur and where the observations cannot be explained in this way. At the present time, however, different types of non- specular reflections have not been identified with any observed reflections from actual targets, so that at most this section can only suggest the theoretical expectations. 20.6.1 Rough Surfaces The most simple type of nonspecular reflection is that from a rough surface, that is, a surface whose irregularities are much larger than the wavelength. Practical formulas applying to this kind of nonspecu- lar reflection from various underwater targets are de- rived in reference 6. The wavelength of sound is so much greater than that of light, however, that such reflections, which are common in optics, are not to be expected in underwater acoustics. The presence of bubbles on or near the surface of a target can, how- ever, give rise to a diffuse reflection with sound scat- tered in all directions; the reflection of sound from bubbles is described in detail both theoretically and experimentally in Chapters 26 through 35, which deal with the acoustic properties of wakes. 20.6.2 Diffraction Another type of nonspecular reflection is that from a surface which has no highlights. Consider, for ex- ample, a smooth rigid plane surface in the form of a square, set at an angle relative to the incident rays. This surface will reflect sound specularly, but not back to the sound source. In addition, however, some sound will be reflected in other directions; some of it will be reflected directly backward. This phenome- non corresponds essentially to the diffracted sound observed when a wave passes through a square aperture, and the echo intensity will decrease as (y/d)2, where y is the length of the square and ) is the wavelength. The Fresnel zone theory may again be applied, provided that the effects of both the first and last zones are considered. No results have been worked out along this line, however. 20.6.3 Scattering A third type of nonspecular reflection is that from objects much smaller than the wavelength. The Fresnel zone theory is not applicable to such small targets, and even the basic equation (21) derived in Section 20.1 is no longer valid, since the derivation assumes that the radius of curvature of the surface is greater than the wavelength. Corresponding analyses have been carried out for targets much smaller than the wavelength; these yield, for a rigid target, QnV T = 20 log =a (56) where V is the volume occupied by the reflecting ob- 362 THEORY ject. Equation (56) is the so-called Rayleigh scatter- ing law. The echo intensity is directly proportional to the square of the volume of the target and in- versely proportional to the fourth power of the wave- length; thus, the echo intensity drops off rapidly as the wavelength increases. EFFECT OF PULSE LENGTH All the previous discussion in this chapter has been concerned with sound waves emitted in an essentially continuous fashion. While Section 2.3 discussed the effect of pulse length in terms of ray acoustics, this section will describe the effect of pulse length on the observed target strength in terms of wave acoustics, developed from the analysis in the preceding sections of this chapter. It was shown in Section 12.2 that at any instant the scattered sound energy received back at the transducer from the projected pulse comes from a spherical shell of thickness cr/2, where c is the sound velocity and r is the duration of the pulse. This result is still true on the more accurate wave theory pre- sented in Section 20.2, as long as the fluid is homo- 20.7 geneous and the target is rigid and convex. From a concave target, sound reflected several times may arrive later than singly reflected sound. This thickness cr/2 is known as the pulse length. When the pulse length is so long that it includes many Fresnel zones, the echo level will be essentially the same as that observed for continuous sound, pro- vided the echo is measured at a time when the wave- lets are arriving from all these zones. At the beginning of: the echo, when only the first few zones are con- tributing, and toward the end, when only the last zones return wavelets to the source, the echo struc- ture is more complicated. However, an application of the Fresnel zone theory would probably give correct results in this case. When the pulse is only a few Fresnel zones long, the echo structure is presumably more complicated, and the echo duration, for example, may be expected to exceed the duration of the outgoing pulse. The pulse length cannot be less than the thickness of a Fresnel zone, since in that case the outgoing pulse would consist of less than half a cycle, and the wave- length would cease to have much meaning. Chapter 21 DIRECT MEASUREMENT TECHNIQUES UBMARINE TARGET STRENGTHS have been calcu- lated theoretically and measured experimentally. The theoretical calculations described in Section 20.5 are based on assumptions simplifying the geometry of the hull and conning tower, and the way in which the submarine reflects sound. Actual measurements in the field are necessary to verify and amplify these theoretical predictions and to assess their accuracy. Measurements have been both direct and indirect.! Direct measurements consist of echo ranging, with short pulses of supersonic sound, on a submerged sub- marine at various ranges, depths, and speeds. The in- tensities of the received echoes are then measured and converted to target strengths. This chapter describes in detail the various experimental procedures and techniques employed by different laboratories in the direct measurements of submarine target strengths. Indirect measurements, on the other hand, use con- tinuous sound or light reflected from a scale model of a submarine, and interpret these results in terms of supersonic sound reflected from an actual submarine of the same shape; Chapter 22 describes how target strengths are measured indirectly. The results of both the direct and indirect submarine target strength measurements are presented and discussed in Chapter 23 while both the techniques and results of target strength measurements on surface vessels are treated in Chapter 24. PRINCIPLES OF DIRECT MEASUREMENT 21.1 In order to calculate target strengths, echoes from a submarine may be compared with echoes received at the same time and under the same conditions from a sphere. From the relative intensities of the echoes from the submarine and from the sphere, and from the expression for the target strength of a sphere [equation (10) in Chapter 19], the target strength of the submarine could be readily computed. Since only the relative intensities of two echoes would need to be determined, no absolute measurements or cali- brations would be required. But at sea, a sphere large enough to return a strong echo at ranges normally used in echo ranging is too awkward to handle easily and therefore cannot be used in practice to obtain target strengths. Instead, target strengths are always found by using the fundamental definition [equation (6) in Chapter 19], which defines the target strength of any object in terms of the echo level, the source level, and the two-way transmission loss from the projector to the target and back to the projector again, all expressed in decibels. This expression is simple and easy to use and has the advantage that all the quantities appear- ing in it may, in principle, be measured directly. Only the difference between the echo level and the source level, and the transmission loss which the signal un- dergoes as it travels from the projector to the tar- get need to be known in order to find the target strength. Unfortunately, the difficulties of calibration and other practical problems not yet resolved make the fundamental definition less useful than may be sup- posed. In particular, the calibration of the transducer, described in Section 21.4 as the measurement of its output as a projector and its sensitivity as a receiver, and the determination of the transmission loss, de- scribed in Section 21.5, as well as the large fluctua- tions and variations normally encountered in under- water sound experiments, introduce numerical un- certainties which cannot be accurately evaluated. Nevertheless, the fundamental definition of target strength introduced in Section 19.1.3 has been used in all direct. measurements and has led to reasonably consistent results. 21.2 EXPERIMENTAL PROCEDURES Four groups have measured submarine target strengths directly. They are: University of California 363 364 TRANSDUCER OSCILLATOR LOUD SPEAKER MODULATOR CHEMICAL RECORDER AMPLIFIER OSCILLOSCOPE DIRECT MEASUREMENT TECHNIQUES TRAINING GEAR TRAINING CONTROL REPEATER PELORUS };-- REPEATER GYRO COMPASS } -— REPEATER ——— ELECTRICAL CONNECTIONS “SSS S> > ELECTROMECHANICAL CONNECTIONS Ficure 1. Division of War Research at the U. 8. Navy Elec- tronics Laboratory, formerly the U. S. Navy Radio and Sound Laboratory, San Diego, California [UCDWR]; Columbia University Division of War Research at the U. S. Navy Underwater Sound Laboratory, New London, Connecticut [CUDWR- NLL]; Woods Hole Oceanographic Institution, Woods Hole, Massachusetts [WHOI]; and the Underwater Sound Laboratory, Harvard University, Cambridge, Massachusetts [HUSL]. In addition, various groups at Fort Lauderdale, Florida, have also made measurements of this nature. Widely vary- ing procedures and techniques have been employed by these groups. DMT San Diego Most of the direct measurements by UCDWR have been made off the coast of California aboard the USS Jasper (PYcl3), a converted 135-ft yacht built in 1938, which echo ranged on various S-boats or occasionally on new fleet-type submarines. Square- topped signals from 0.5 to 200 msec long were sent out, usually at a frequency of 24 ke and sometimes at 45 or 60 ke. Early trials used a QCH-3 magneto- Experimental arrangement. strictive transceiver driven at a frequency of about 24 ke; ? a few measurements were also made with an experimental medel of frequency-modulated sonar gear.° Later runs employed standard JK or QC trans- ducers ‘ or specially designed equipment.® Most of the echo-ranging equipment was installed in the wardroom of the Jasper; a schematic diagram of the installation is shown in Figure 1. A pelorus, an open sighting device attached to a dial and employed in determining bearings, was mounted topside on the flying bridge. An observer visually trained this pe- lorus on a float towed by the submarine, and the rela- tive bearing of the pelorus was relayed to a repeater dial in the wardroom below. Here, another observer followed the relative bearings of the pelorus and trained the transducer on them; obviously, the bear- ing accuracy obtainable in this manner was not very high. Then the echoes received by the transducer were amplified and fed into a cathode-ray oscilloscope to be photographed on continuously moving film by a high-speed camera. The echoes were also usually heterodyned and monitored over a loud speaker, and supplementary records were made on the sound- range recorder, where the keying interval was con- trolled manually as the range changed. EXPERIMENTAL PROCEDURES 365 Two types of runs were made. In one, illustrated in Figure 2, the target strength was measured as a function of the aspect of the submarine; the sub- marine, usually at periscope depth, proceeded at creeping speed while the Jasper circled it, trying to maintain a nearly constant range. The other, shown in Figure 3, comprised opening and closing runs, and was used to measure the echo level as a function of USS JASPER SUBMARINE Sa a ae Figure 2. Circling run. range to determine the transmission loss. Here, the submarine proceeded on a straight course while the Jasper followed a divergent course, bearing approxi- mately 60 degrees from the submarine and opened the range until contact was lost; then a closing run was made on a collision course down to a range of several hundred yards. During both opening and closing runs, the speed and course of the Jasper and the submarine were held so that the aspect which the submarine presented remained constant. Since these runs were made, a new type of fre- quency modulation sonar has been set up at the Sweetwater calibration station of UCDWR for meas- uring the target strengths of small objects.® It is be- lieved that measurements may be made more quickly with this system than with the standard pinging system, but no results are available at the present time. 21.2.2 New London At New London, tests were made by CUDWR aboard the USS Sardonyx (PYcl2) which echo- ranged in Long Island Sound on the USS S-48 (SS159), a 1,000-ton S-boat 267 ft long, first com- missioned in 1922.7 The submarine followed a straight course at a keel depth of 80 ft while the Sardonyx cireled around it in an are to maintain an approxi- mately constant range. A device was used automatically to range on center bearings. The amplified echo intensity was kept constant by manual control of the amplifier gain as the echoes were observed on a cathode-ray oscilloscope. Relative echo intensity was obtained by recording the amplifier gain settings and by referring to a calibration curve for the system. The bearing, course, and range of both vessels, and the gain settings were recorded about every half-minute. Be- cause complete calibration and transmission data were not available, absolute target strengths could not be computed. Instead, echo intensity was calcu- lated as a function of aspect in decibels relative to the echo level at an arbitrary aspect and plotted for ranges of 600, 1,000, and 1,200 yd. USS JASPER SUBMARINE Figure 3. Opening run. Woods Hole Target-strength measurements were also made by WHOI observers aboard the USS SC665 just off Fort Lauderdale, Florida.® ® Navy QCU sonar gear was employed, with pulses from 60 to 80 msec long sent out alternately at 12 and 24 ke, at slightly different signal lengths to facilitate separation of the 12-kc data from the 24-ke data. Apparatus was used to range on center bearing. A hydrophone nondirectional in the horizontal plane was mounted above the conning tower of the 210-ft Italian submarine Vortice. Accessory recording equipment was installed aboard the submarine in order to measure the level of the received signals and to determine the transmission loss from the SC665 to the submarine. The submarine proceeded on a straight course, while the SC665 circled the subma- rine to investigate aspect dependence, and opened and closed the range to investigate range dependence. The submarine also traveled at different speeds and different depths in order to ascertain possible varia- tion of target strength with the speed and depth of the submarine. 21.2.3 366 21.2.4 Harvard The target strength of the Italian submarine Vortice was also measured by HUSL workers using a special sonar first in the area of the Bahama Islands, then off the coast of Florida near Port Everglades. Sonar gear mounted aboard the USS Cythera (PY31) echo ranged on the submarine at a frequency of 26 ke. The first series of tests was made near stern aspect as the Cythera and Vortice followed parallel courses at speeds from 2 to 6 knots.!° Cut-ons were obtained by listening to the echoes. Very few data were col- lected; only 114 echoes were obtained on the Vortice during the two days of measurements so that the re- sults cannot be considered conclusive. During the second and more complete series of tests, the Cythera maneuvered around the Vortice in order to determine the dependence of target strength on aspect angle, altitude angle, and range.!' The Vortice maintained a speed of 3 knots on a base course at depths of 100, 300, and 400 ft. Echo intensities were obtained for groups of approximately 10 echoes; the source level was measured by training the pro- jector at a monitor transducer, then feeding the voltage across the monitor transducer into a cathode- ray oscilloscope and finding the voltage that had to be applied to the oscilloscope in order to balance it. The speed of the Cythera was held close to that of the Vortice to prevent bearings from changing too rapidly; training the projector was accomplished by cut-ons. A vertically directional beam from a QHF transducer was used in addition to the original non- directional beam. Aspect angles were estimated at intervals of 5 degrees; ranges correct to about 25 yd were read from the sound-range recorder. Altitude angles were not recorded; instead, they were computed from the range, as read from the recorder, and from the depth of the submarine, measured from the ocean surface to the center of the control room about 12 ft above the keel of the submarine. 21.2.5 Fort Lauderdale Three runs were made off the coast of Florida by observers from groups at Fort Lauderdale. In one series of tests, the YP451 remained stationary and echo-ranged on the USS Pintado (SS387) and the USS Pipefish (SS388), two new fleet-type submarines which ran past the YP451 at prearranged depths, speeds, and ranges. The equipment aboard the YP451 DIRECT MEASUREMENT TECHNIQUES included a crystal transducer, driven at 60 ke, which was suspended on a pendulous pipe so that it was 15 ft below the surface. The platform carrying the transducer was stabilized by an automatic pilot gyro control in one dimension, with its horizontal axis of rotation normal to the axis of the sound beam. In addition, the transducer was automatically trained in elevation. The pendulum and gyro provided a plat- form which was stabilized in the most critical direc- tion, while the elevation control centered the sound beam on the target vertically; the transducer was trained manually on the target in the horizontal place. The beam width was roughly 25 degrees hori- zontally and 10 degrees vertically. A 6-string electro- magnetic oscillograph recorded the echoes. Unfortunately, operations with the YP451 were hampered by mechanical difficulties in the alternating current generator and by failure of radio communica- tion with the escort vessel which maintained sound communication with the submarine. Although this lack of communication resulted in unpredictable maneuvers by the submarine, fairly satisfactory data were obtained on echoes from the submarines. In the second and third runs, signals 30 msec long were sent out at a frequency of 60 ke every 0.6 sec. In the second run, a fleet-type submarine at periscope depth followed a straight course at a speed of 6 knots. The echo-ranging transducer, mounted with accessory equipment in a submerged unit, circled about a fixed point, 230 yd from the course of the submarine, in a radius of 125 ft and at a depth of approximately 35 ft. Aspects were estimated trigonometrically from observed ranges, which had been corrected for the position of the echo-ranging unit in its turning circle. During the third run, an R-boat was the target, at a keel depth of 100 ft and a speed of 6 knots. The range was decreased continuously; cut-ons were em- ployed in training. Since the echo intensities varied, depending on where the beam struck the submarine, a series of echo maxima was obtained and was used to calculate the target strength. These maxima are illustrated in Figure 4, where the echo level — in decibels below the source level — is plotted against the range; each point represents an individual echo. The target strength was computed from the received- echo voltage, as measured on a film continuously ex- posed to a cathode-ray oscilloscope, hydrophone sensitivity, total power output into the water, directivity index of the transducer, and the esti- mated transmission loss. ANALYTICAL PROCEDURES 367 90 ” Pe] w a & -100 a z a) Ww > Ww = 8 2 =110 2 fo} ” ow =) Zz = 4 g > -120 a) fo} =x oO wu -130 RANGE IN YARDS Figure 4. Echo maxima at Fort Lauderdale. ANALYTICAL PROCEDURES 21.3 Target strengths reported here were obtained for the most part from measurements of amplitudes of echoes recorded photographically. Sometimes, how- ever, Operating conditions were so poor that upon examination of the photographs it was difficult to identify individual echoes and to distinguish them from noise signals. Therefore echo recognition is im- portant in the study of target strengths and is par- ticularly relevant to a discussion of analytical proce- dures. Echo recognition depends not on the intensity of the echo alone, ‘but primarily on the difference be- tween the intensity of the echo and the intensity of the background.” At close ranges, echoes are usually strong enough to be easily recognized and are clear and well defined. Distant echoes, however, are often so weak that they cannot be distinguished from the background, and irregular spines and patches may effectively obscure the echo; hence a study of the structure of echoes from submarines and other tar- gets at short ranges may be useful in the recognition and identification of distant echoes from these same targets. Weak echoes may sometimes be attributable to poor training of the transducer or roll and pitch of the echo-ranging vessel, both of which may direct the sound beam away from the target. A high back- ground of reverberation and noise may make an echo hard to recognize. Rough seas and a wide transducer- beam pattern contribute to a high reverberation level, while a surface vessel at moderate speeds or a shallow submarine at high speeds may originate enough self-noise in the transducer dome to mask the echo. Reverberation is treated in detail in Chapters 11 to 17. Once the echo is recognized and definitely identified as the desired echo, the problem becomes one of measurement and analysis. Various analytical pro- cedures have been employed by different groups in processing the raw material from the oscillogram to the computed target strength. _ 368 MB San Diego At San Diego, 35-mm film, running at a speed of either 2.5 or 12.5 in. per sec, was exposed to traces on a cathode-ray oscilloscope and then processed and read on an illuminated viewer. Peak echo amplitudes were measured in millimeters, corrected where neces- sary for the width of the spot of light on the oscillo- scope screen, and averaged over a series of echoes. The average was then converted to mean-square pressure level in terms of the calibration constants of the equipments. The transducer and accessory equip- ment were calibrated before and after each run with an auxiliary calibrated transducer, lowered on a boom from the side of the Jasper; Section 21.4 comprises a discussion of calibration errors. In computing target strengths at San Diego, cor- rection was also made for the deviation of the target from the axis of the sound beam on the basis of beam patterns measured in the laboratory. In addition, the range was found by measuring the distance between the midpoints of the echo and the signal on the film, and referring to index marks recorded every 50 msec at the bottom of the film, corresponding to range intervals of about 40 yd. From calibration data, the source level was calculated, which together with the echo level, and the transmission loss as measured dur- ing the opening and closing runs or, less accurately, estimated from prevailing oceanographic conditions but neglecting possible surface reflections, gave the target strength. Simultaneous sound-range recorder records provided a convenient check on the oscillo- grams. 21.3.2 Fort Lauderdale A similar procedure was followed by the groups at Fort Lauderdale in analyzing the echoes obtained there. Here, however, the film moved more slowly, at a speed of approximately 1 in. per sec; only 50 ft of film could be accommodated inside the camera. Con- sequently, the echoes were compressed horizontally and were less detailed, but were still readily measur- able. The fine detail of the oscillograms made at San Diego enabled close determination of echo length as well as a study of echo structure for short pulses; this information was supplemented by an examination of echoes registered on the sound-range recorder. Target strength determinations from the films recorded at Fort Lauderdale, however, may be more accurate DIRECT MEASUREMENT TECHNIQUES than that recorded at San Diego since the motion of the transducer was better controlled, the fluctuations smaller, and the values more consistent. 21.4 CALIBRATION ERRORS Errors in target strengths measured directly must be due to errors in the echo level, the source level, or the transmission loss, since these target strengths are computed from equation (6) in Chapter 19. Incorrect echo level or source level determinations are usually attributed either to errors in calibration, or to errors in reading the echo level from the trace of the echo recorded oscillographically which UCDWR observers estimate as 2 or 3 db at the most. This section de- scribes errors attributable to calibration of the equip- ment; uncertainties in the evaluations of the trans- mission loss are discussed in Section 21.5. 21.4.1 Purpose of Calibration In target-strength studies, the principal purpose of calibration is not so much the absolute determination of the source level and the absolute determination of the echo level, but rather the measurement of the difference between the two levels. In other words, it is necessary to know only the sum of the transducer output as a projector and response as a receiver if the echo level is measured in terms of the voltage across the terminals of the transducer. Then the difference between the echo level and the source level is simply the difference between (1) the echo level, in decibels above one volt, and (2) the sum of the projector out- put and receiver response of the transducer. The latter sum can be obtained by means of auxiliary transducers, without bothering about actual sound pressures. One scheme may employ an auxiliary hydrophone and an auxiliary projector. As a first step, the hydrophone could be lowered from a boom on the echo-ranging vessel, a few yards away from the transducer to be calibrated, and the transducer out- put measured in terms of the response of the hydro- phone. Then the auxiliary hydrophone and the trans- ducer, close together, are both exposed to sound from the auxiliary projector some distance away. Thus, the response of the transducer could be compared with the response of the auxiliary hydrophone; com- bining the measurements would give the desired cali- bration of the transducer. TRANSMISSION LOSS Calibration Techniques at San Diego 21.4.2 The methods of calibration most commonly used in target strength measurements, however, employ calibrated transducers; at San Diego, an auxiliary transducer is lowered over the side of the Jasper and used with the standard echo-ranging transducer. First one is used as the projector, then the other, and final calibration is accomplished by referring to the con- stants of the auxiliary transducer as calibrated at a separate measuring station. Unfortunately, this system is susceptible to errors at every step, so that too much reliance cannot be placed on the accuracy of the calibration. At San Diego, the greatest error in calibration is believed to be in the measurement of the output of the auxiliary transducer, which is used to calibrate the echo-ranging transducer before and after each run, as mentioned in Section 21.3.1. This auxiliary transducer is calibrated at intervals of roughly four months. Slow drifts of as much as 3 or 4 db have been detected for crystal transducers between calibration checks every three or four months; this drift may be responsible for part of the “variation’’ observed dur- ing target-strength runs, as described in Section 21.6.1. However, since it was not practicable to con- trol or even measure all the factors entering into gear calibration, there is no direct evidence on which to base estimates of the overall calibration error of echo- ranging equipment. 21.4.3 Observed Calibration Errors Recent indirect evidence suggests, however, that calibration errors as great as 12 db may occur. An example of such large calibration errors is evident in the results of San Diego echo-ranging tests on a sphere.— The sphere, 1 yd in diameter, was sus- pended 16 ft below the surface of the ocean at ranges from 24 to 166 yd; echoes from pulses from 0.5 to 7 msec long were received on a JK transducer. Target strengths computed from equation (6) in Chapter 19 varied from —24 to +3 db, approximately 12 db above and below the theoretical value predicted from equation (10) in Chapter 19. Although the very low values are possibly the result of training errors, the very high values seem rather large to be attributed to errors in the estimated transmission loss, especially since the values as high as 3 db were found when the transmission loss was measured directly with a hydro- phone placed (1) close to the projector and then 369 (2) close to the target. However, the possibility that the transmission loss at short ranges fluctuates by 12 db cannot be ruled out at the present time. This large error must result either from large fluctuations in short-range transmission, or from errors inherent in the calibration of the gear, provided that the theoretical formula in Chapter 19 for the target strength of a sphere is applicable to direct measure- ments. To provide a check on the validity of this formula, an auxiliary hydrophone was placed a few yards from the sphere during this series of observations and was used to measure both the outgoing pulse and the re- turning echo. The mean target strength of 350 echoes was found by this method to be — 13.3 db, in unusu- ally close agreement with the theoretical value of —12 db. A similar result was obtained at Woods Hole. Thus, the 12-db discrepancy observed when the JK transducer alone was used is undoubtedly the result of errors in the estimated transmission loss, in calibration, or in both. That large systematic errors in these quantities may sometimes be present, even when careful checks are provided, is suggested by the anomalously high values found at San Diego for the target strength of a submarine at 60 kc, and the similar results obtained by Woods Hole at 12 and 24 ke, both reported in Section 23.6.2. Large errors in calibration may result from (1) large-scale variability of the calibrated auxiliary units employed in methods involving absolute cali- bration at sea; or (2) gross deviations of the sound field from the theoretical inverse square law in cali- bration measurements at close ranges, because of interference with reflections from the hull or from other surfaces nearby. Neither of these explanations seems very likely. So far no really satisfactory ex- planation of the large internal inconsistencies in direct target strength measurements has been ad- vanced. Calibration of ship-mounted gear at sea re- mains one of the most troublesome of all underwater sound measurements. TRANSMISSION LOSS It has already been pointed out that much of the error in the direct measurements of target strength may be due to errors in the estimated transmission loss; probably a large part of the variability in ob- served target strengths arises from variability in the transmission loss. This quantity varies widely from hour to hour and from place to place and is seldom known accurately. 21.5 370 i = 4.5 DECIBELS PER KILOYARD TRANSMISSION ANOMALY IN DECIBELS 2000 3000 DIRECT MEASUREMENT TECHNIQUES 4000 5000 6000 7000 RANGE IN YARDS Figure 5. Typical transmission anomaly at 24 ke for an isothermal layer 70 feet deep. The transmission loss H is defined as the loss in intensity, in decibels, as the sound travels between a point 1 yd on the axis of the sound beam from a small projector, and the target. If the medium through which the sound travels is ideal —if no sound is absorbed, scattered, or refracted, or re- flected from the ocean surface or bottom — then the intensity of the sound varies inversely as the square of the distance from the source, as pointed out in Section 19.1.1, and the transmission loss, in decibels, is simply 20 log r, where r is the range in yards. In this case the total transmission loss 2H as the sound travels to the target and back to the projector again is simply 40 log r. This inverse square loss, however, is only a part of the total transmission loss of sound in water. Sound energy is absorbed by the water and dissipated as heat energy. Small particles in the water scatter the sound in all directions. Furthermore, as the beam is refracted by a temperature gradient, it is bent and the cross section of the beam changes in area, chang- ing the intensity of the sound correspondingly. To account for transmission loss due to absorption, scattering, and divergence arising from refraction, the transmission anomaly is defined as the difference between the total measured transmission loss, and the transmission loss due to divergence according to the inverse square law alone. In decibels, then, A = H — 20logr, en) where A is the transmission anomaly, H the total transmission loss, and r the range. A typical plot of the transmission anomaly against the range is il- lustrated in Figure 5. The transmission anomaly has been found to de- pend rather strongly on the prevailing oceanographic conditions and most particularly on the variation of the temperature of the water with depth. The water in the ocean is usually characterized by a mixed layer of nearly constant temperature down to a certain depth; below that depth, a decrease in temperature with depth, or thermocline, will appear. The trans- mission anomaly depends markedly on the depth to this thermocline as well as on the depth of the hydro- phone receiving the echoes. When the temperature difference in the top 30 ft of water is 0.1 F or less, the transmission anomaly may be considered a linear function of range (see Chapter 5). Hence, it is convenient to define an attenuation coefficient as the change in transmission anomaly with range. As a derivative, (2) where a is the attenuation coefficient, A the transmis- sion anomaly, and r the range. Since in target- strength runs the attenuation coefficient is measured not as a derivative, but as an average over range in- tervals of 500 or 1,000 yd, a is usually taken as A r (3) a TRANSMISSION LOSS E-S +40 LOG r IN DECIBELS 371 O OPENING RUN @ CLOSING RUN Ee as eee eae 2000 Eats pate tote soe 2500 RANGE IN YARDS Figure 6. Target strength plot. In Figure 5, a amounts to about 4.5 db per kyd, at 24 ke. It may be that actually A=ar+b (4) where b isa constant. Present data indicate, however, that 6 is probably negligible." By definition, the transmission anomaly includes the effects of reflection from the ocean surface. So little is known about surface reflection with any de- gree of certainty, however, that no attempt is made to include in equation (4) an additional term to take it into account. If surface reflection is appreciable, it may cause the constant b in equation (4) to be nega- tive, in effect decreasing the transmission anomaly and therefore the transmission loss itself, as described later in Section 21.5.4. 21.5.1 Methods of Measurement Transmission loss may be measured in three ways. First, during an opening or closing run, the echo level in decibels above the source level may be corrected for geometrical divergence by adding 40 log r; then, the result may be plotted as a function of the range, as long as the submarine maintains a constant aspect. A typical plot of this nature is illustrated in Figure 6. Then the slope of the points represents twice the attenuation coefficient, and the intercept at zero range corresponds to the target strength. Such a de- termination presupposes that the target strength does not change over the ranges used. Second, before and after each run on a submarine, a transmission run may be made with an auxiliary surface vessel, in the usual manner, as described in Section 4.3.2. Third, the signals transmitted by the echo-ranging vessel may be received by a hydrophone mounted on the submarine, amplified, and measured. The trans- mission anomaly and attenuation coefficient may then be determined readily by correcting the level of the echo above the source for simple geometric divergence, and measuring the slope of the plot of the echo level against the range. But measuring the transmission loss in any one of these three ways is difficult. Aspects and speeds must be carefully maintained and measured, a pro- cedure particularly difficult for a submerged sub- marine. For reasons of safety, the echo-ranging vessel is advised not to approach the submarine closer than about 300 or 400 yd, and poor sound conditions often limit echo ranges to 1,000 yd or even less, especially off the coast near San Diego. When a transmission run is made with an auxiliary surface vessel, hori- zontal temperature gradients may result in a trans- mission loss between the projector and the hydro- phone suspended from the surface vessel which is different from that between the projector and the submarine. Another disadvantage of measuring the 372 DIRECT MEASUREMENT TECHNIQUES IDEAL SOUND CONDITIONS ISOTHERMAL LAYER 100 FEET OR MORE DEEP ATTENUATION COEFFICIENT IN DECIBELS PER KILOYARD FREQUENCY IN KILOCYCLES Figure 7. Attenuation coefficient as a function of frequency. transmission loss by the latter method is the presence of four vessels in the operating area, that is, sub- marine, escort vessel, echo-ranging vessel, and trans- mission measuring vessel. The use of a hydrophone mounted on a submarine is a definite improvement but introduces new horizontal and vertical directivity problems as well as installation complications. 21.5.2 _Inadequacy of Transmission- Loss Measurements All three methods have been used to measure transmission loss during direct target strength tests. Where ample and consistent data have been taken by any one of these methods, the transmission loss calculated from these data has been used to evaluate the target strength. Often, however, data have not been consistent. During one run at San Diego, for example, the plot of the echo level, corrected for inverse square law spreading against range, indicated an attenuation coefficient of 19 db per kyd at a frequency of 60 ke while measurements aboard the submarine when analyzed showed a value of only 10 db per kyd. An- other identical run the following day gave values for the attenuation coefficient of 11.5 and 16 db per kyd, respectively, as measured by the two methods. Ap- parently the errors were not systematic. This lack of consistency between two methods was not infre- quent. Recent San Diego target strength measure- ments, however, based on transmission loss measured with a nondirectional hydrophone mounted on the submarine, have been more consistent; this method promises to eliminate much of the uncertainty in the evaluation of the transmission loss. However, the measurements reported “— indicate that even this method does not eliminate systematic error in the determination of target strength, possibly because of peculiarities of transmission at short ranges, possibly because of calibration uncertainties. Certain 60-ke measurements on the USS S-37 (SS142) at San Diego gave a beam target strength of 28.7 db with a TRANSMISSION LOSS 373 standard deviation of 8.5 db when an attenuation coefficient of 20 db per kyd was assumed; when the transmission loss measured aboard the submarine was used in the computations, the beam-target strength rose to 40 db with a much smaller standard deviation of 3.5 db. In most trials reported here, it was necessary to evaluate the transmission loss from an attenuation coefficient, estimated for each run from the echo-ranging frequency employed and sometimes from the prevailing oceanographic conditions. 21.5.3 Estimating the Attenuation Coefficient The attenuation coefficient in sea water varies widely and depends primarily on the frequency of the echo-ranging sound and on the prevailing oceano- graphic conditions. For example, in mixed water, or water of constant temperature, at least 50 ft deep, this coefficient is about 5 db per kyd at 24 ke, and in the neighborhood of 15 db per kyd at 60 ke. A plot of the attenuation coefficient against frequency for ideal sound conditions is reproduced in Figure 7 and represents a rough average of observations primarily at 20, 24, 40, and 60 kc; the increase in attenuation coefficient with frequency is quite marked and shows why it is impractical to use very high frequencies for echo ranging. The attenuation coefficient increases markedly with poor sound conditions. At 24 kc, it may be as high as 15 db per kyd under poor conditions, or even 40 db per kyd under extremely bad conditions. Very few data are available at 60 ke on the varia- tion of the attenuation coefficient with oceanographic conditions. Empirical formulas have been derived for the attenuation coefficient at 24 ke, however, as a function of the depth of the thermocline. For a hydro- phone above the thermocline, 17 A vets TO (5) and for a hydrophone below the thermocline (6) where a is the attenuation coefficient in decibels per kiloyard and D is the depth in feet to the thermocline. The probable error is about 2 db per kyd.!* As implied in Chapter 5, these empirical formulas are, in general, less suitable for predicting the attenuation coefficient than other methods based on a more quantitative classification of the variation of temperature with depth, because transmission anomaly-range graphs significantly depart from straight lines under certain conditions. However, equations (5) and (6) are suf- ficiently accurate for the present purposes. Harly target strength measurements showed that different values of the transmission loss were obtained by different methods, as described in Section 21.5.2. Therefore, in most calculations representative values of 5 and 20 db per kyd at 24 and 60 ke respectively were taken for the attenuation coefficient. Much of the time no account was taken of the oceanographic conditions which prevailed at the time of the tests, however, with the result that the reported target strengths varied considerably. Examples of this varia- bility are given in Section 21.6 of this chapter. Surface Reflections Reflection of sound from the surface of the ocean is neglected in all calculations of target strengths. Such an effect would offset, in part, the loss in in- tensity caused by spreading and absorption. Perfect specular reflection from the surface would effectively double the intensity of the sound incident on the target and the intensity of the echo returned to the projector, under ideal conditions. In other words, it would reduce the transmission loss by 3 db each way, or by a total of 6 db from the projector to the target and back again. Thus, in equation (4) the constant 6 would equal —3 db at ranges of a few hundred yards or more. Some evidences of surface reflection have been found experimentally. At San Diego a number of oscillograms of echoes from submarines have shown peaks or ‘‘spines’”’ at the beginning and end of each echo, separated by a relatively smooth echo of lower intensity; an example is shown in Figure 8 for an S-boat at beam aspect.” The first peak is attributed to direct reflection from the hull of the submarine alone when the first part of the pulse strikes the tar- get and is reflected back to the projector along the shortest possible path; the final peak comes from the ray reflected from the submarine to the surface and back to the projector, after the direct echo from the submarine has been received. In other words, the two spines are attributable to reflection along only one path, since there will be a short time at both the beginning and end of the echo when the sound travels only one path back to the transducer. The intensity of the intervening echo is consequently lower because 374 Figure 8. Surface-reflected sound. of a combination of both constructive and destructive interference throughout the duration of the echo, be- tween direct and surface-reflected sound. A different effect produced by surface-reflected sound is also indicated by more recent information from San Diego.!8 During echo-ranging tests on a sub- marine from 90 to 200 ft deep, double echoes were observed, under certain conditions, on the chemical recorder and on the oscillograph — a strong primary DIRECT MEASUREMENT TECHNIQUES echo followed by a faint secondary echo, illustrated in Figure 9. This appearance of double echoes suggests that some of the sound is reflected directly back to the projector to form a primary echo, while some of it is reflected vertically upward to the ocean surface, reflected by the surface back to the submarine and finally back to the projector to form a secondary echo. Quantitative data show that the lapse of time between the primary and secondary echoes is equal to the time necessary for the sound to travel up to the surface and back again, thus confirming this hypothesis. Although almost all the sound striking the surface is unquestionably reflected back into the water at some angle, the perfect specular reflection expected from a flat surface seems unlikely at sea. The nor- mally rough surface of the ocean and the presence of air bubbles tend to scatter the sound rather than allow perfect specular reflection at the surface. Further evidence minimizing the effects of surface reflection on target strength values is seen in the excellent agreement between the results of the direct measurements computed neglecting surface reflec- tions, and both the indirect measurements and theoretical calculations, where surface-reflected sound either does not appear or may be readily eliminated. Partly for this reason, surface-reflected sound is neglected in all target strength computations in Chapters 18 to 25. However, the results shown in Figure 2 of Chapter 9 and described in Section 9.2.1 suggest that reflection from the ocean surface is frequently very nearly specular. More data are needed to clarify the exact importance of surface- reflected sound in practical echo ranging. At present, the resulting uncertainty of 6 db is about the same as the other uncertainties of observation in target strength measurements. 21.6 VARIABILITY OF ECHOES Perhaps the largest source of uncertainty in target strength measurements arises from variability of echo intensity. Observed echoes vary widely in two ways (see Section 21.1). Gradual changes in echo in- tensity over a relatively long period of time from a few minutes to hours are called variations. Superim- posed on these variations are marked changes which occur from echo to echo and are called fluctuations. A large part of the variability of echo intensity is due to variability in the sound-transmitting character- VARIABILITY OF ECHOES 50 MSEC SECOND ECHO PRINCIPAL ECHO S- TYPE SUBMARINE AT 100 FT DEPTH ee MSEC | ieee UL A SE ES S- TYPE SUBMARINE AT 90 FT DEPTH Figure 9. Double echoes. 375 | 376 DIRECT MEASUREMENT TECHNIQUES RATIO OF OBSERVED TO PREDICTED ECHO AMPLITUDE fo) 20 40 60 80 100 120 140 ECHO NUMBER FicurE 10. Variations and fluctuations in sphere echoes. istics of the ocean (see Chapter 7). The remainder may be ascribed to such external causes as changes in the performance of equipment and changes in target aspect. 21.6.1 Variation Variations occurring over a sufficiently long time are very difficult to detect. Sometimes they result from gradual changes in the characteristics of the echo-ranging gear employed and may be detected each time the system is calibrated. More often, however, variation may be most promi- nent during a long run in the course of a single day or on successive days. At long ranges, changes in the transmission conditions in the water may be responsi- ble for some of the variation observed; horizontal temperature gradients may occur and cause changes in the value of the transmission anomaly. This effect may be most conspicuous at long ranges for two in- terrelated reasons. First, if the ranges are long the operating area is much larger, and horizontal differ- ences in temperature may be more likely. Second, since the transmission anomaly increases with range, variations attributable to slow changes in the transmission anomaly will be greatest at long ranges. At short ranges, much less is known about variation. Marked variation in the echo level was observed during the course of a number of runs during early echo-ranging tests on a sphere in San Diego.” The re- sults of one reel of film exposed to the sphere echoes, as shown on a cathode-ray oscilloscope, are repro- duced in Figure 10; pulses were sent out at intervals of 1.2 sec and the range of the sphere was about 109 yd. Here, the ratio of the observed echo ampli- tudes to the echo amplitudes predicted from theory (in which transmission loss is taken into account) is plotted for each individual echo received. The short- term changes are most noticeable, but the slow up- ward slope of the average of the points is evidence of variations as defined here. The cause of thisvariation, however, is not known. Changes in the calibration of the equipment over a period of time, known as “drift,” are also responsi- ble for some of the variation observed. As pointed out in Section 21.4.2, slow drifts of 3 to 4 db have been observed between calibration checks at San Diego, at approximately four-month intervals, in a crystal projector. Just how much of the variation normally encountered can be attributed to drift, however, can- not be estimated very accurately. VARIABILITY OF ECHOES 377 21.6.2 Fluctuation Many factors contribute to the observed fluctua- tions of echoes. Much of this rapid change in echo in- tensity may be ascribed to the roll and pitch of the echo-ranging vessel, which by changing the direction of the sound beam causes the received echoes to vary in intensity. Although gyroscopic stabilization of the transducer was employed at Fort Lauderdale to re- duce fluctuations arising from the roll and pitch of the ship, as described in Section 21.2.5, this system has not been used elsewhere for this purpose. Errors in training the echo-ranging transducer toward the sub- marine have also been responsible for some of the fluctuations encountered; training on the bearing of maximum intensity, by means of cut-ons, is approxi- mate and introduces variability in the received echo intensities by changing the direction of the beam relative to the submarine. In addition, surface reflection and interference phenomena may be expected to account for part of the fluctuations observed, as the sound beam fre- quently follows multiple paths to reach the submarine and return back again to the transducer. Chapter 7 of Part I of this volume discusses the evidence show- ing that transmission fluctuations are very much re- duced when surface-reflected sound is minimized. Correlation has been observed between the depth of the transducer below the ocean surface, and the mag- nitude of the fluctuations observed in echoes from a sphere two ft in diameter; ? at a range of the order of 65 to 75 yd, elevating the transducer from a depth of 50 to 10 ft below the surface increased the standard deviation of the echo intensity from 18 to 39 per cent. In addition, the overall fluctuation appears to de- crease as the signal length increases. At other than beam aspects, interference between echoes from dif- ferent parts of the submarine is undoubtedly respon- sible for part of the fluctuation observed, giving rise to an irregular “hashed” echo structure described in Sections 23.8.2 and 23.8.3. Effects on Echo Level and Echo Structure 21.6.3 Variation affects echo intensity ; fluctuation affects both echo intensity and echo structure. Echo en- velopes never repeat exactly, and successive echoes at the same range, aspect, frequency, and signal length often appear totally different. This diversity of echo structure not only complicates measurement of the intensity of the echo, but also makes it difficult to resolve the length of the echo and the center of the echo, in effect preventing precise measurement of the range of an individual echo. Likewise successive echo intensities seldom repeat. As a result, some sort of average must be taken over successive echoes. If target strength is regarded as a measure of the fraction of the incident sound energy reflected by the target, the total reflected energy should be compared to the total transmitted energy. Such an analysis would require squaring the echo amplitudes to give the echo intensities, then inte- erating the intensities over the duration of the echo to give the total echo energy; this same procedure would be followed with the signal to yield the total signal energy. Such an analysis has not been found practical because of the complex instruments re- quired. In addition, it may be that aural and non- aural detection devices respond more to peak echo intensity rather than to total echo energy, and that therefore peak intensities are more significant. Peak versus Mean Echo Intensity 21.6.4 Since it has not been feasible to compare the re- flected and transmitted energy directly, peak echo amplitudes have been used to compute target strengths. Observations show that these average peak amplitudes do not differ significantly from the rms peak amplitudes, which would correspond to peak intensities. Thus the San Diego results may be re- garded as giving average peak intensities. Not only is this method simple and easy to apply, but also it pro- vides values which may be compared directly with recognition differential measurements where peak- echo intensities alone are considered. Peaks, however, fluctuate enormously, especially for off-beam echoes; a sample survey of 100 oscillograms of submarine echoes at San Diego showed a maximum fluctuation of 25 db between peaks, with fluctuations of 10 db not uncommon. An approximate comparison of reflected and trans- mitted energy might be made by measuring the mean echo intensity, averaged along the entire length of the echo, and correcting this intensity for the pulse length since the echo length generally is longer than the pulse length. Then the ratio of the signal and echo intensities, based on the same pulse length, would be equal to the ratio of the signal and echo energies. This procedure was attempted for six echoes 378 DIRECT MEASUREMENT TECHNIQUES recorded oscillographically at San Diego.! The echo amplitude was measured at small intervals along the length of the echo, squared to give the echo intensity, and averaged over the echo length as closely as the echo length could be estimated; then the enclosed area was calculated. Division of this area by the signal length gave a new intensity, the intensity which presumably would have resulted if the echo length had equaled the signal length. This sample analysis, although based on data not sufficient to warrant definitive conclusions, showed an insignifi- cant difference between peak echo intensities and mean echo intensities corrected for signal length. In general, however, the peak echo intensity differs from the uncorrected mean echo intensity, and this difference is a function of the signal length. It was pointed out in Sections 19.3 and 20.7 that for long pulses, the echo will reproduce the signal envelope while for short pulses fluctuations in intensity will re- sult in an irregular structure, where sharp peaks stand out against a weak background. In the latter case, the peak echo amplitude may be considerably different from the mean echo amplitude, and may vary with signal length quite differently (see Sec- tion 23.5.1). The variability of echoes is responsible for a large part of the uncertainty in the echo level and trans- mission loss values which are used to compute target strengths. Since echoes are often so irregular that visual estimates of peak intensities are, at best, in- telligent guesses, UCDWR observers estimate that systematic errors of as much as 2 or 3 db may result from the difference in personal judgments of different observers. Because in practice fluctuations and variations be- have as very large accidental errors, only a statistical analysis of many echoes may be considered reliable. Hundreds of individual echoes must be carefully aver- aged, corrected, and analyzed to give target strength results of any significance. At San Diego, a camera has been installed aboard the Jasper to record, at the same time as each signal is transmitted, the roll and pitch of the vessel, the true bearing of the ship, the relative bearing of the transducer, and the time and pulse number. Such a record should be useful in analyzing and evaluating each echo, but so far has not been applied to a large number of measurements. So far target strength runs have been analyzed from a reasonably large number of individual observations; first, successive groups of five echoes each have been averaged, then an overall average computed con- sidering changes in transmission loss with range and changes in target aspect. Cumulative distributions and computations of probable errors and quartile deviations have been useful in interpreting the re- sults and assessing their reliability. Chapter 22 INDIRECT MEASUREMENT TECHNIQUES [yal FROM SUBMARINE models have been studied in order to discover the principal reflect- ing surfaces on a submarine and to measure sub- marine target strengths under controlled conditions. Both visible light and supersonic sound have been used in these model tests. Tn the investigation of reflection from submarines, models have many advantages over actual subma- rines. Generally, experimental conditions can be con- trolled much more easily under laboratory conditions than in the field. Laboratory use of carefully con- structed scale models makes possible a reasonably. reliable evaluation of target strength as a funetion of aspect and altitude angles, as well as submarine class, and provides both a theoretical guide and a con- venient check on the direct measurements. PRINCIPLES OF INDIRECT MEASUREMENT 22.1 Three groups have participated in the indirect measurements of reflections from submarine models; University of California Division of War Research at the U. 8. Navy Radio and Sound Laboratory, San Diego, California [UCDWR]; Underwater Sound Laboratory, Massachusetts Institute of Technology, Cambridge, Massachusetts [MIT-USL]; and the Underwater Sound Reference Laboratories, Columbia University Division of War Research, Mountain Lakes, New Jersey [USRL]. Only qualitative results were obtained at San Diego while actual target strength values were measured at MIT and at USRL. DoMal San Diego Early experiments were carried out at San Diego ! on a 1:60 scale model of the U570 or HMS/M Graph, a 517-ton German Type VIIC U-boat which was captured in 1941 off Iceland and served in the British fleet. The model, made of wood and finished with glossy white enamel, was illuminated by a standard projection bulb and photographed in vari- ous positions. The bulb was enclosed in a metal housing with a hole 114 in. in diameter on one side, and was placed as close as possible to the camera lens so that the angle between the incident and the reflected light at the submarine was only about 3 degrees. Photographs were made at different aspect angles, first with the submarine finished with enamel, then with the sub- marine covered in part by horizontal and vertical corrugations, and finally with coarse emery cloth covering certain areas on the model. The corruga- tions and emery cloth were affixed to the submarine in an effort both to reduce prominent reflections and to suggest locations for possible absorption treat- ment. These experiments were wholly qualitative, since no measurements were made. The main purpose was to discover the highlights on a submarine which might be largely responsible for strong reflections. Photographs for different aspect angles of the sub- marine model, without the corrugation or emery cloth, are reproduced in Figure 1. 22.1.2 Massachusetts Institute of Technology Quantitative experiments using visible light re- flected from scale models were conducted at MIT-— USL to calculate target strengths of four different submarines.2* A series of measurements were made on models of HMS/M Graph, an old S-boat, the USS Perch (SS313) and the USS Sand Lance (SS381) ; these models were from 60 to 120 times smaller than the original submarines and were finished with a glossy black enamel. In order to compute the target strength of one of the submarines, light reflected from the submarine model was compared with light reflected from a sphere also enameled in glossy black. The target strength of the submarine was calculated from the 379 380 INDIRECT MEASUREMENT TECHNIQUES 180° 60° Tasha 90° 105° 120° Vicure 1. Refleetion of light from HMS/M Graph. PRINCIPLES OF INDIRECT MEASUREMENT 381 relative intensities of the light reflected from the sub- marine model and from the sphere, the scale factor of the submarine model, and the expression for the target strength of a sphere [equation (10) in Chap- ter 19]. The technique of these optical measurements was not simple. Light from a motion picture projector bulb passed through a polarizing element rotated by a synchronous motor and was focused on the submarine model. As a result, the plane of polarization of the incident light rotated at a high speed. Upon reflection from the model, the light passed through a second polarizing element and fell on a photoelectric cell; this second polarizing element was stationary, but adjustable. In effect, the two polarizing elements modulated the intensity of the light incident on the cell and made possible the use of a-c instead of d-c amplifying and measuring equipment. Moreover, the use of modulated polarized light greatly reduced the error caused by light scattered from the walls and other objects in the room in addition to the desired reflected light. At the same time, the photoelectric cell was also exposed to light from a neon lamp which was supplied with current from both a battery and a step-down transformer. As a result, the neon light contained a small a-c component whose intensity was directly proportional to the alternating current through the lamp, which was measured on a vacuum tube volt- meter. Since the light reflected from the model was adjustable in phase, by use of the second polarizing element, and since the light from the neon lamp was adjustable in magnitude, one was balanced against the other, thus canceling out the a-c component of the light reaching the photocell. When the a-c output from the photocell vanished, this condition of balance was obtained, and the voltmeter reading of the a-c lamp current was then proportional to the intensity of the model-reflected light. The use of this null © method made it unnecessary to rely on a calibration of the photoelectric cell. To compute the target strengths, spheres from 1 to 121% in. in radius were substituted for the submarine models, and a similar procedure was followed. Photo- graphs were also made at different aspect angles and are illustrated in Figures 2 through 5. 22.1.3 Mountain Lakes At Mountain Lakes, |New Jersey, a model of HMS/M Graph, similar to the model used at UCDWR and at MIT, was suspended in water in the path of sound from a supersonic transinitter.* The model, built to a 1:60 scale, was constructed of copper 0.5 mm thick, plated with nickel 0.025 mm thick as a protection against corrosion. The model was sus- pended approximately 21% ft below the surface of the lake by wires at distances between 1 and 17 ft from the transducers, corresponding to full-scale target ranges between 20 and 340 yd. Pulses were not used in the indirect measurements at any of the laboratories. At Mountain Lakes, con- tinuous sound was transmitted by a quartz crystal projector, and the echo was received by a separate similar unit which served as a hydrophone. The model scale was 1:60. Since the importance of nonspecular reflection depends on the ratio of the wavelength to the dimensions of the target, it was necessary to scale the wavelength similarly. Consequently, an actual echo-ranging frequency of 24 ke, which is standard for most Navy gear, would require a frequency of 1,440 ke in tests with a 1:60 scale model. However, since the response of the trans- ducers was somewhat higher at higher frequencies, a frequency of 1,565 ke was used most of the time; the corresponding actual echo-ranging frequency was 26 ke. A beat-frequency oscillator, with a fixed frequency of 15 me, provided signals between 50 and 3,600 kc, which were amplified and sent through coaxial trans- mission lines to the projector. The received echo was amplified by a preamplifier in the hydrophone hous- ing, demodulated by the detector circuit and recorded on a continuous strip of paper as the submarine was slowly rotated about a vertical axis. The known cali- brations of the transducer and receiver were used, together with an assumed inverse square transmission loss to determine the target strength by using equa- tion (6) of Chapter 19. Under the controlled condi- tions possible at a reference station on a lake, the calibration is less difficult than it is for gear mounted on a ship at sea; thus the calibration error in these tests was probably small. Also, at such close ranges, temperature gradients and surface reflections are negligible. At a frequency of 1,565 ke, the attenua- tion coefficient predicted from Figure 7 in Chapter 21 is about 0.6 db per yd. At ranges of only a few feet, this attenuation is negligible and the transmission loss may safely be assumed to obey the inverse square law. At ranges as great as 17 ft, however, this assumption may lead to target strengths which are about 6 db too low. 382 INDIRECT MEASUREMENT TECHNIQUES 70° ° © ro) Ficure 2. Reflection of light from HMS/M Graph. PRINCIPLES OF INDIRECT MEASUREMENT Figure 3. Reflection of light from S-type submarine. INDIRECT MEASUREMENT TECHNIQUES rN) NS) Ne ~“ (2) @ m-) 88° 98° oO “NJ no oO (o) ° Ficure 4. Reflection of light from USS Perch (SS313). PRINCIPLES OF INDIRECT MEASUREMEN liaurE 5. Reflection of light from USS Sand Lance (88381). 386 OP)2) SUBMARINE REFLECTIVITY Since indirect target strength measurements are measurements not on actual submarines but on their scale models, certain possible corrections must be considered before the results can legitimately be compared with the results of the direct measure- ments. One possible source of error is in the reflec- tivity of the models used as compared with the re- flectivity of actual submarines. Since the experiments at UCDWR were qualitative in nature and designed only to determine the principal reflecting surfaces on a submarine, the question of absolute reflectivity is unimportant for those tests. The optical experiments at MIT, however, reported specific target strengths. These results were com- puted from the expression for the target strength of a perfectly reflecting sphere, in other words, a sphere which reflects all the sound striking it without trans- mission or absorption. Since both the submarine models and the spheres were finished in exactly the same way, these target strength results will apply only to perfectly reflecting submarines. At USRL, the reflectivity of the hull itself was found to be perfect, within experimental error, over the range of frequencies used. The hollow model was first tested filled with air, then filled with water. No difference was observed in the intensity of the re- flected sound for all frequencies between 50 and 2,000 ke. Since reflection from an air-filled hull would be almost perfect, regardless of the transparency of the hull to sound, and since reflection from a water-filled hull submerged in water would come solely from the hull, with no air-water interface to reflect the sound, the experimental results did not justify assuming any- thing less than perfect reflectivity. Thus both the optical and acoustical indirect measurements are based on perfect reflection of the sound striking the submarine; transmission through the hull and ab- sorption in the steel are neglected. The steel hull of an actual submarine is also almost perfectly reflecting. Therefore, it appears that the re- sults of the indirect measurements may be inter- preted in terms of sound reflected from actual sub- marines. However, the presence of barnacles, moss, and other marine growth on the hull may appreciably affect the reflectivity. Such an effect would be im- portant for surface vessels or surfaced submarines, where the fouled hulls are exposed to the direct sound beam, but might not be significant for a submerged submarine, since the sound beam might not often INDIRECT MEASUREMENT TECHNIQUES strike the lower part of the hull where such growths attach themselves. No measurements have been made to ascertain the effect of barnacles and moss on the reflection of sound, but it is not believed to be significant. Therefore it appears that reflectivity considerations should not greatly affect any compari- son between direct and indirect measurements. 22.3 WAVELENGTH EFFECTS If the indirect measurements of target strengths with submarine models are to be trusted, the experi- ments must be properly scaled, that is, the dimen- sions of the models, the ranges and depths at which the tests are made and all the wavelengths must be reduced by the same factor. This factor was 60 for the acoustical measurements at USRL, and all the quantities relevant to the measurements were changed by this factor. At MIT-USIL, however, visible light was used. The models used in the optical experiments were from 60 to 120 times smaller than the submarines they repre- sented. Assume an echo-ranging frequency of 24 kc, and the corresponding scaled wavelengths would be reduced to 0.1 cm for a 1:60 scale or 0.05 cm for a 1:120 scale. Since the actual wavelengths employed were much shorter, errors might be expected in the results. Two errors in particular might be introduced. At certain aspects where the surface of the submarine subtends only a few Fresnel zones at 24 ke, as de- scribed in Sections 20.3 and 20.5, the model subtends many such zones, since the wavelength is much shorter compared with the dimensions of the sub- marine. As a result, the Fresnel integrals approach their asymptotic values, especially for surfaces of large radius of curvature, such as planes or cylinders, which subtend many Fresnel zones. Since the conning tower on a submarine is relatively flat, the optical measurements with very short wavelengths may overemphasize the effect of the conning tower. Secondly, nonspecular reflection is less than if the wavelength was properly scaled, by a factor equal to the square root of the ratio of the properly scaled wavelength to the improperly scaled wavelength actually used. This may account for the extremely low target strengths obtained optically at aspects giving very little specular reflection, such as the bow and stern. Diffuse reflection or scattering may be excessively large optically since the wavelength may be con- DIFFERENCES IN METHODS siderably smaller than the surface irregularities. How- ever, this source of error has been minimized by the use of glossy black surfaces. 22.4 DIFFERENCES IN METHODS Certain errors may arise from the differences in- herent between the direct and indirect techniques. As a submarine travels through the water, each sur- face may be assumed to be screened by a wake of some sort, or at least a turbulent condition in the water, and possibly also by air bubbles surrounding the hull and conning tower. Although this phenomenon may be present in the direct measurements of target strengths, it is absent in the indirect tests. In the optical methods, no re- flecting layer surrounded the submarine model; every effort was made to reduce reflection from dust par- ticles and from other objects in the room. In the acoustical tests, the submarine model was stationary throughout the measurements except for a very slow rotation in the horizontal plane, which could not give rise to wakes or air bubbles. The importance of this effect, of course, depends on the extent to which sound is reflected by turbulence in the water, which is negligible (see Section 34.3.2), or by air bubbles in the vicinity of the submarine (see Section 28.3.5). Extraneous reflections also may occur during the indirect measurements. Removal of the models, how- ever, has shown that the background level during both the optical and acoustical experiments is negligi- ble compared with the levels of the echoes from the models. Since continuous signals were employed during both the optical and acoustical measurements, sepa- rate transmitters and receivers had to be employed — a moving picture projector bulb and a photoelectric cell in the optical tests, and two similar transducers 387 in the acoustical tests. The distance between them, however, was minimized, so that the angle of inci- dence and the angle of reflection at the model were as small as possible. At MIT, the bulb and photo- electric cell were approximately 14 in. apart, whereas the model was from 6 to 20 ft distant. At USRL, the two transducers were separated by less than 5 in., while the closest distance of the submarine model was 11 in.; most of the measurements, however, were made with the source and receiver about 17 ft away from the model. Another difference between the direct and indirect measurements of target strength lies in the method of measurement. In the direct measurements, peak- echo amplitudes were used in all cases, since the echoes were short; in the indirect measurements, however, the echoes were continuous and the results were obtained by using rms intensities. The difference between mean intensities and peak intensities, and the dependence of this difference on pulse length are discussed in Chapter 21. Other errors may result from discrepancies in the construction of the models. Considerable difference was observed between the two models of HMS/M Graph, one used at MIT and the other at USRL, so that comparison of the two series of measurements is not completely justifiable. At some aspects, a differ- ence of 6 db in the target strengths of the two models was observed when optical measurements were later made on both models of HMS/M Graph; these differences are described in Section 23.2.2. In addi- tion, rudders and propellers were missing from some of the models used at MIT and the model tested at USRL; at certain aspects, they may give rise to strong echoes. The models of the S-boat, the USS Perch and the USS Sand Lance, however, were sup- plied by the Bureau of Ships and are believed to be accurate. Chapter 23 SUBMARINE TARGET STRENGTHS Pes STRENGTHS of submarines have been com- puted mathematically from the size and shape of a particular submarine, by the Fresnel zone method outlined in Chapter 20. They have also been meas- ured, both directly and indirectly, by use of the pro- cedures and techniques described in Chapters 21 and Y ECHO-RANGING VESSEL Figure 1. Definition of angles. 22, and have been studied in general as a function of orientation, submarine class, speed, range, pulse length, and frequency of the echo-ranging sound. This chapter presents the results of the different methods of determining target strengths of subma- rines and discusses their applicability to practical echo ranging. 23.1 DEPENDENCE ON ORIENTATION Since a submarine is irregular in shape, the echoes which it returns depend markedly on its orientation with respect to the echo-ranging beam. The orienta- 388 tion of such an irregular target is conveniently described in terms of aspect and altitude angles, defined in Figure 1. Consider a system of rectangular coordinates with the origin O at the center of the submarine. The aspect angle is defined as the angle between the x axis and the projection of the echo-ranging beam on the hori- zontal (xz) plane. It is measured in degrees from the bow of the submarine, in a clockwise direction as viewed from above; bow aspect is 0 degree, stern aspect 180 degrees, while beam aspect will be 90 and 270 degrees for the starboard and port beams respectively. The angle between the echo-ranging beam and its projection on the horizontal (xz) plane is the altitude angle. It is measured in degrees, positive when the sound source is above the submarine, negative when it is below the submarine. If the projector is at the same depth as a level submarine, the altitude angle is 0 degree; similarly, if it is directly above a level sub- marine, the altitude is 90 degrees. The vertex of both aspect and altitude angles is placed at the origin O of the coordinate system, which is taken at the geometric center of the submarine. 23uIia Aspect Angle The strongest echo from a submarine is usually found within 20 degrees of beam aspect — between 70 and 110 degrees, and between 250 and 290 de- grees, from the bow of the submarine.! These beam and near-beam echoes average about as strong as the echo from a sphere 35 yd in radius and correspond to a target strength of 25 db. Actually, target strengths as low as 7 db and as high as 40 db have been ob- served at beam aspect, directly and indirectly; most values, however, lie between 20 and 30 db. At other aspects, the target strength is much smaller and averages between 5 and 15 db, depending on the submarine and the altitude angle. At stern aspect, for example, target strengths measured di- rectly with standard gear vary from 4 to 19 db, de- DEPENDENCE ON ORIENTATION pending on the submarine ?~* (see Section 23.2.1). Negative target strengths have been observed in the optical studies at certain aspects and altitudes; for example, at bow and stern aspects the target strength of the German U570 (HMS/M Graph) varies from —4 to —6 db when the echo-ranging beam is below the submarine, at altitude angles between —5 and — 15 degrees.® Since such negative altitude angles are not encountered in practice when echo ranging from a surface vessel on a submerged submarine — because 389 furthermore, the uncertainty in the aspect angle in some of the measurements was rather large. Conse- quently, the beam target strengths do not apply to an aspect angle of exactly 90 or 270 degrees. The altitude angle in all cases was small. In the direct measure- ments reported in this table, the submarine was sel- dom submerged to a keel depth greater than 100 ft, which at a range of 500 yd corresponds to an altitude angle of 4 degrees, while for the indirect measure- ments quoted the altitude angle was 0 degree. TasLE 1. Submarine target strengths. Beam Bow Stern target strengths target strengths target strengths and and and Frequency | standard deviations | standard deviations | standard deviations Submarine in ke in decibels in decibels in decibels Theory U570 (HMS/M Graph) § 25 25.5 11.5 11.5 Tambor class 7 18.5 8.57 1.5 USS 8-28 (SS133) 8 24 18 Bes br USS S-40 (SS145) 2 24 25 + 4 12.5 + 4 12!5 = 4 Fleet type * 24 24 + 5§ 13 + 6 19 +5 Direct * Fleet type 3 24 29 + 3.5|| measurements | USS S-34 (SS139) ° 45 25 USS Tilefish (SS307) ° 45 26 Fleet type 60 25 ents R class 60 4 R class 60 sae Bike 8.1 British 4 18 29 ius z Vortice 26 42 40 40 Vortice 26 Bee 10.5 ° S class ® 30 6 5 mee arett g | USS Perch ($8313) 5 30 6 5 USS Sand Lance (SS381) > 26 5 9 U570 (HMS/M Graph) ® aoe 27 4 1 U570 (HMS/M Graph) # 26 25 6 6 * San Diego measurements at 60 ke are not included here. + Beam focused on conning tower. t Beam focused on screws. the projector is always above the target — these low target strengths are not very significant. Moreover, they may result from errors in the construction of the model (see Section 23.6.2) or from possible systematic errors inherent in the optical method (see Section 29.1). Table 1 summarizes submarine target strengths at beam, bow, and stern aspects for the theoretical cal- culations ° and for the direct 7! and indirect meas- urements.” ” Certain controversial values discussed later in this chapter are omitted, as, for example, cer- tain San Diego measurements at 60 ke. Most values Were averaged in sectors of roughly 10 or 20 degrees; § Average of all values in a 30-degree sector centered at beam aspect || Average of maximum values in the sector for each run. Ranges varied from 200 to 1,000 yd, and the fre- quency from 12 to 60 ke for all tests except the optical studies at MIT, where the full-scale frequency was much higher. Although the results of the mathemati- cal studies are only approximate and the results of the direct measurements highly variable, all the data are generally consistent. The early San Diego values for a fleet-type sub- marine of the Tambor class? and an S-boat ® are not reliable. An early experimental frequency-modulated gear was used to echo range on the Tambor-class sub- marine; since these results are difficult to interpret in terms of standard echo-ranging gear, they cannot be SUBMARINE TARGET STRENGTHS TARGET STRENGTH IN DECIBELS bow Y STERN Ficure 2. Aspect dependence (theoretical). weighted as heavily as other measurements. An ap- preciable variation of target strength with aspect angle was observed, in agreement with other meas- urements. Also, an observed difference in these ex- periments of between 10 and 16 db in target strengths at beam and stern aspects, depending on where the beam was focused, seems confirmed by more reliable results. The actual values, however, must be con- sidered doubtful. The target strengths of the S-class submarine taken from reference 8 were measured by comparing sub- marine echoes with the echoes from a submerged sphere at a much shorter range than the submarine; fluctuations were large (see Figure 10 in Chapter 21), and the transmission loss was not known accurately. Furthermore, no significant variation in target strength with aspect angle was observed for the S-boat; this result alone makes the reliability of these measurements very dubious. The remaining values in Table 1 are in moderately good agreement with each other, especially at beam aspect; values for which no reference is given were found at Fort Lauderdale. The only results out of line are those from the Woods Hole measurements on the Italian submarine Vortzce.” These measurements on the submarine Vortice were made at a range of 1,000 yd and frequencies of 12 and 24 ke; the submarine was proceeding at 6 knots at a DEPENDENCE ON ORIENTATION 391 TARGET STRENGTH !N DECIBELS BOW STERN Ficure 3. Aspect dependence (San Diego). depth of 150 ft. The values reported are of the order of 40 db, so much larger than any reported previously by any method that they appear to be the result of faulty calibration; the systematic error of about 20 db at all aspects is difficult to explain in any other way. It might be pointed out, however, that the trans- mission loss as measured aboard the submarine was about 15 db greater than that expected from the pre- vailing oceanographic conditions. Therefore an at- tenuation coefficient of 4 db per kyd was assumed, in addition to inverse square divergence. In spite of such a transmission loss, however, the target strengths are more than 10 db greater than the highest values previously observed on a similar submarine. Figures 2 to 6 show typical variations of target strengths with aspect angle. Figure 2 is a plot of the theoretical calculations of the target strength of the U570 (Graph) for an echo-ranging frequency of 25 ke and a range of 1,000 yd.° They are based on approxi- mating the submarine by an ellipsoid of appropriate dimensions, with a conning tower of “tear drop” cross section; details of this method are described in Section 20.5. Figure 3 shows the result of a typical target strength run at 24 ke on a fleet-type submarine at 392 SUBMARINE TARGET STRENGTHS RELATIVE ECHO LEVEL IN DECIBELS BOW STERN Figure 4. Aspect dependence (New London). San Diego. The submarine followed a straight course at about 214 knots at periscope depth, while the Jasper circled it at a range of about 500 yd (see Figure 2 in Chapter 21). Each point on the curve is the average of all echoes obtained within the 15-de- gree sector centered at the indicated aspect angle, and represents, on the average, about 40 individual echoes. Figure 4 is a plot of the echo level against aspect angle for a series of tests made on the USS S-48 (SS159) at New London and described in Section 21.2.2.15 Each contour represents measurements made at a different range. Since no correction was made either for the transmission loss or for the calibration Curves (range in yards): (A) 600; (B) 800; (C) 1,000; (D) 1,200. of the equipment, no absolute target strengths are plotted. These echo level values cannot be compared directly with other target strength values, since they are relative to an arbitrary level. However, the differences in echo levels at different aspects correspond to the differences in target strengths at different aspects, as long as the range remains constant. Indirect measurements of target strength are il- lustrated in Figures 5 and 6. Figure 5 shows target strength as a function of aspect angle for a model of the USS Perch (SS313), as measured optically at MIT-~? The altitude angle was 0 degrees and the full- scale range 600 yd. The results of the acoustical tests DEPENDENCE ON ORIENTATION 393 TARGET STRENGTH IN DECIBELS STERN Figure 5. Aspect dependence (MIT). made at Mountain Lakes on a model of the U570 are reproduced in Figure 6, for an altitude angle of 0 de- gree, a full-scale range of 340 yd, and a full-scale frequency of 26 ke.” Bow and stern target strengths in Figures 5 and 6 are considerably lower than in Figure 2, probably because the ellipsoid used in the theoretical calculations was rounded at either end while the optical and acoustical models were pointed. Other discrepancies at bow and stern are discussed in Sections 23.8.2 and 23.8.3. 23.1.2 Altitude Angle Target strength varies with altitude angle, but in most cases this variation does not appear to be im- portant practically. Figure 7 illustrates the theoretical predictions of the target strength of the U570 at beam aspect as a function of altitude angle for a range of about 16 yd; ° since target strengths were found only for certain intervals of the altitude angle, they are represented as sectors in the polar plot of Figure 7. The sharp sectors at particular altitudes are attrib- uted to the sum of two separate reflections — from the blister tank and from the hull itself — neglecting interference phenomena. Figure 8 shows the same plot for the optical measurements made on a model of the USS Perch (SS313) for echo-ranging distances; the peak at 90 degrees, when the projector is directly above the submarine, arises from a strong reflection 394 SUBMARINE TARGET STRENGTHS TARGET STRENGTH IN DECIBELS BOW STERN FicurE 6. Aspect dependence (Mountain Lakes). from the deck of the submarine. The absolute values in Figures 7 and 8, however, are not comparable be- cause the theoretical calculations were carried out for @ projector very close to the submarine, while the optical measurements applied to the ranges of several hundred yards usually encountered in practical echo ranging. Figure 9 is a smoothed curve showing the relative target strength of the Italian submarine Vortice plotted against aspect angle for altitude angles of 0 to 10, 10 to 20, 20 to 45, and 45 to 90 degrees, as measured by Harvard observers; “ for each curve, the relative target strength at beam aspect was arbitrarily set at 25 db. These data were obtained at a frequency of 26 ke, for submarine depths of 100 to 400 ft and ranges up to 1,000 yd. The aspect de- pendence apparently becomes less marked and the curve smoother as the altitude angle increases. It might be pointed out, however, that for altitude angles greater than 20 degrees, and a submarine depth of about 400 ft, the sound beam does not com- pletely cover the submarine at near-beam aspects. Therefore the target strength would be expected to show less aspect dependence. Figures 10 and 11 show target strength aspect curves for different altitude angles, as measured in- directly. Optical measurements on a submarine of the S class are given in Figure 10 for altitude angles of 0, DEPENDENCE ON ORIENTATION 395 30 so ALTITUDE ALTITUDE ANGLE IN ANGLE IN DEGREES 20 10 BEAM TARGET STRENGTH IN DECIBELS BEAM TARGET STRENGTH IN DECIBELS ok 0 (0) fo) (0) 10 20 30 (o} 10 20 30 Ficure 7. Altitude dependence (theoretical), 16-yd FicurE 8. Altitude dependence (optical), 193-yd range. U570 (HMS/M Graph). range. USS Perch (SS313). 30 20 Pee eae coos ag RELATIVE TARGET STRENGTH IN DECIBELS (o) 30 60 90 120 150 180 BOW BEAM STERN ASPECT ANGLE IN DEGREES Figure 9. Target strength-aspect curves at different altitudes, for the Italian submarine Vortice (direct measurements). 396 SUBMARINE TARGET STRENGTHS 30 20 TARGET STRENGTH IN DECIBELS ar iu 2 30 Baaiab ce cane BEAM ASPECT ANGLE IN DEGREES FicurE 10. Target strength-aspect curves at different altitudes (optical). 15, and 45 degrees. Figure 11 is a similar plot for the acoustical measurements on the model of the U570 at much smaller altitude angles of 0, 1.0, and 1.8 de- grees. Occasionally a pronounced maximum of the target strength has been observed at a very small and critical altitude angle, where the conning tower gives a prominent specular reflection or highlight. An ex- ample may be seen in Figure 11, at an aspect angle of about 105 degrees. Another example, at only one aspect angle, is shown in Figure 12, where the target strength of the U570 at bow aspect is plotted for five different altitude angles used at Mountain Lakes; here a strong reflection from the conning tower at an altitude of 1.0 degree results in a target strength of 19 db, while the target strengths at altitude angles only a fraction of a degree different are considerably lower. Such maxima, however, are not common, and are generally confined to such a small sector of alti- tude angles that most of the time they are not likely to be observed in actual echo ranging. Their possible effect on the direct measurements has not been veri- fied because of the wide fluctuations in echo level tending to obscure such fine detail. Negative altitude angles are not encountered for surface vessels echo ranging on submarine targets. The low target strengths at these negative angles, measured in the optical studies, have already been mentioned in the preceding section. At very large positive altitude angles, the differ- ences between beam, bow, and stern target strengths are less marked and the resulting target strength- aspect curve is considerably smoother, as illustrated in Figure 9 for altitude angles between 0 and 90 de- grees, and in Figure 10 for altitude angles of 15 and 45 degrees. Direct measurements on a deep subma- rine were also made at San Diego on the USS Tvlefish (SS307) at a depth of 400 ft,9 and gave results not significantly different from measurements at shal- lower depths except at quarter aspects. Values be- tween 25 and 27 db were obtained at beam aspects in DEPENDENCE ON CLASS TARGET STRENGTH IN DECIBELS ASPECT RAGE Figure 11. good agreement with values at periscope depth. At an aspect angle of 150 degrees, however, a target strength of 27 db was obtained, much higher than any other values reported at that aspect, directly or indirectly. An S-boat, for example, measured in the same series of runs, gave a target strength of 14 db at the same aspect. This high value of 27 db may result from overcor- recting for the transmission loss by assuming an excessively large attenuation coefficient, since the range of the Tilefish was 760 yd compared to 150 yd for the corresponding measurements on the S-boat. The attenuation coefficient assumed was 10 db per kyd of sound travel at a frequency of 45 kc. However, high target strengths may actually be characteristic of deep submarines at certain aspects. It may be men- 397 an DEGREES Target strength-aspect curves at different altitudes (acoustical). tioned that at Fort Lauderdale, one of the strongest series of echoes was obtained when echo ranging at stern aspect on a fleet-type submarine submerged to a depth of 250 ft, at ranges between 220 and 700 yd. 23.2 DEPENDENCE ON CLASS Submarines of different sizes and shapes may be expected to reflect sound differently; the echoes from an R-boat 186 ft long and those from a new fleet-type submarine more than 300 ft long are not likely to be the same. In particular, specular reflection of sound from a submarine would be expected to depend rather critically on the shape of the hull and espe- cially on the different radii of curvature. 398 SUBMARINE TARGET STRENGTHS BOW TARGET STRENGTH IN DECIBELS ALTITUDE ANGLE IN DEGREES 12 db; and for a large fleet-type submarine about 19 db. At beam aspect, the target strengths are more nearly the same. Unfortunately, no comparative measurements have been made on two different vessels by a single group during a single operation. In view of the sys- tematic errors of 5 to 10 db that may be present in target strength determinations, suggested in Sections 21.4, 21.5, and 21.6, the differences shown in Table 2 are not too conclusive. More accurate data are re- quired to allow any conclusion tobe drawn about the variation -of target strength between different sub- marines. 23.2.2 Indirect Measurements Table 3 lists target strengths for different sub- marines measured indirectly at an altitude angle of 0 degrees. These values were obtained under con- trolled conditions and are more self-consistent than the values measured directly; as a result, fluctuations are smaller and the differences between the values are probably more significant than in the direct measure- ments. Beam target strengths may vary between 25 Ficure 12. Altitude dependence (Mountain Lakes). and 30 db; at stern aspect; the limits are 1 and 9 db. Tasie 2. Dependence of target strength on submarine class (direct measurements). Beam Bow Stern target strength target strength target strength and and and Frequency | Length | standard deviation | standard deviation | standard deviation Submarine Reference in ke in feet in decibels in decibels in decibels R class 60 186 Ae 4 R class 60 186 et 8.1 S class 2 24 219 Soi 12.5+4 12.544 S class Average from 24 219 19.7 + 2.5 Aire are Table 4 Fleet type 3 24 304 2445 13 +6 19+5 British 4 18 300 29 ate Oe : The possibility that a fleet-type submarine may re- 23.2.1 Direct Measurements P y yP y Representative target strengths of different sub- marines measured directly are listed in Table 2. S-boats, R-boats, and fleet-type submarines have all been targets in direct measurements, but the experi- mental error is so great that it generally obscures any possible correlation of target strength with submarine class. However, Table 2 shows an apparent depend- ence of target strength on submarine class at stern aspect. The target strength of R-boats at stern aspect varies from 4 to 8 db; for an S-boat it averages about turn a strong echo at stern aspect, which is suggested in Table 2, is confirmed by the optical measurements on the USS Sand Lance (SS381) quoted in Table 3. These measurements give a stern target strength of 9 db, which is less than most similar values measured directly but still considerably larger than for any other submarine tested optically at that aspect. Since the indirect measurements at off-beam aspects are much lower than the direct measurements at those aspects, however, these optical valués are probably not too significant. DEPENDENCE ON CLASS 399 Tas_e 3. Dependence of target strength of submarine class (indirect measurements). Beam Bow Stern Length | target strength | target strength | target strength Submarine in feet in decibels in decibels in decibels S class > 227 30 6 5 USS Perch (SS313) ® 308 30 6 5 USS Sand Lance (SS381) > 310 26 5 9 U570 (HMS/M Graph) ® 220 27 4 1 U570 (HMS/M Graph) ® 220 25 6 6 w rr] o rr a = x - o z Ww c i - w oO c < ie) 30 60 90 120 150 180 BOW BEAM STERN ASPECT ANGLE IN DEGREES Ficure 13. Comparison of theoretical, optical, and acoustical target strength of the U570 (HMS/M Graph). Hach submarine class will often evidence its own peculiar reflecting characteristics at certain aspects and altitudes. A striking example of this is the value of 20.2 db for the target strength of the U570 at an aspect of 120 degrees, reported in the Mountain Lakes measurements. Contrast this value with a maximum target strength of 26.3 db for beam as- pects, 9.4 db astern, and 17.4 db bow-on. This result is attributed to a strong specular echo from the back- plate of the conning tower, but is not confirmed by optical tests, perhaps because of differences in the two models. To test the reliability of these model results for each class of vessel, a comparison may be made be- tween results obtained by different indirect methods on the same type of submarine. Such a comparison is made in Figure 13, which shows target strength plotted against angle for the U570, as calculated theoretically and measured optically and acous- tically. 400 IN DECIBELS TARGET STRENGTH BEAM ASPECT ANGLE IN DEGREES SUBMARINE TARGET STRENGTHS Figure 14. Optical comparison of two models of U570 (HMS/M Graph). The theoretical results at off-beam aspects are not very realistic, since they assume a perfectly smooth and curved reflecting surface. In Figure 13, port and starboard sides were averaged in the optical and acoustical measurements to simplify the illus- tration. A considerable difference is evident between these three curves, which may be attributed in part to the errors in the models as well as to possible dif- ferences arising from the different methods used. Since different models of the U570 were used at MIT and at Mountain Lakes, the target strength of each model was measured optically, after the model used at Mountain Lakes had been finished in the same way as the MIT model. The results are reproduced in Figure 14 for the model originally used at Mountain Lakes and for the model first tested at MIT. Unfortunately, the Mountain Lakes model was not measured at aspects within 30 degrees of the beam, so that the comparison is not complete. Since the beam target strengths measured both optically and acoustically agree quite well, however, the tar- get strengths of both models are probably the same at beam aspect. The differences between the two curves in Figure 14 cannot be ascribed to differences in the methods, since both models were tested in the same way — optically. Instead, these differences, which near the bow are as great as 5 db, must be due to differences in the models themselves. Generally higher values were obtained for the Mountain Lakes model. In particular, diving planes at the bow, and horizontal and vertical rudders at the stern increased bow and stern reflections from the Mountain Lakes model. The MIT model had a knife-edge finish at bow and stern. Furthermore, the difference in the profile of the conning towers and the supporting structures for thetwomodels would be expected to reduce somewhat the reflections of the MIT model at stern aspect. In view of these discrepancies between different models of the same submarine, and between the different indirect methods of determining target strength, too much reliance cannot be placed on the general differences in submarine classes shown in Tables 2 and 3. 23.2.3 Asymmetry To a first approximation, the shape of a submarine may be represented by an ellipsoid which is sym- DEPENDENCE ON CLASS q) SKY We 401 Y STERN Figure 15. Target strength asymmetry. Target strength in decibels. metrical with respect to all three planes in Figure 1. This approximation was the basis of the theoretical predictions described in Section 20.5. If now the conning tower is added to the submarine immedi- ately above its center, the submarine loses its sym- metry about the horizontal (xz) plane. This asym- metry about the zz plane is largely responsible for the asymmetry in target strength-altitude angle plots for positive and negative altitude angles. In addi- tion, the shape of fuel and ballast tanks are fre- quently not symmetrical about the horizontal plane. Since the shape of the bow of a submarine generally differs from the shape of its stern, perfect symmetry about the yz plane is also aksent. However, the port and starboard side of a submarine are, usually, nearly identical, with only very minor differences, to a first approximation, and therefore symmetry with respect to the zy plane may be expected. In other words, plots of target strengths as a function of aspect should be symmetrical about the longitudinal axis of the submarine. In the theoretical calculations of the target strength of the U570 as a function of aspect angle, symmetry about the ry plane was assumed, since the submarine was approximated by an ellipsoid of revolution. A lack of symmetry about the zy plane has been observed in both direct and indirect target strength 402 SUBMARINE TARGET STRENGTHS measurements, and is apparent in Figures 3 to 6. Much of it is due to experimental error, and may not be real. Repeatable asymmetry in target strength- aspect curves has occasionally been found at San Diego and is illustrated in Figure 15 for three runs on an S-boat, where a sharp dip is evident just off the port bow at an aspect angle of about 340 degrees. This dip may be attributable to particular features of the construction of the S-boat, such as the lack of any surfaces normal to the sound beam so that specu- lar reflection cannot occur at that aspect. It is also possible that this decrease in target strength is a characteristic of bow echoes and that the aspect angles were in error by as much as 20 degrees; low bow target strengths are conspicuous in the re- sults of the optical measurements. On-the other hand, the variability of echo intensities at other aspects is so large that this dip may not be real, even though it appeared during all three runs. Horizontal asymmetry in the indirect measure- ments is apparent in Figures 5 and 6. This asym- metry may be attributed largely to the asymmetrical models used, in other words, the port and starboard sides were not the same. Not only were the models asymmetrical, but, as shown in Section 23.2.2, models of the same submarine appeared to differ rather markedly. Because of these model differences, the observed data cannot be used to confirm the exist- ence of asymmetry in the target strength in the hori- zontal plane. 23.3 DEPENDENCE ON SPEED Almost no data are available on the variation in target strength with the speed of the submarine. If echoes come only from the hull and conning tower of thessubmarine, it can be argued theoretically that the target strength of the submarine itself should not change as the speed is changed. But if a layer of air bubbles immediately surrounding the submarine con- tributes appreciably to the echoes received, then the target strength would be expected to depend on the speed and depth of the submarine; the scattering of sound from bubbles is discussed in Chapters 26 to 35. In addition, turbulence in the water adjacent to the submarine, which would depend on the speed of the submarine, may be responsible for part of the reflection of sound. So far, little evidence has been uncovered to iden- tify a layer of air or turbulent water surrounding the submarine as an effective reflector of sound (see Sec- tion 33.3), although bubbles have been observed on a submerged submarine traveling through the water' (see Section 27.1.1). Most direct measurements have been made on submarines at a creeping speed be- tween 1 and 3 knots; none have been made on a stationary, balanced submarine. Some data describe tests at 6 knots, but no significant difference has been observed between these results and results at lower speeds. At 6 knots, however, reasonably strong echoes were received from a wake behind a fleet- type submarine at periscope depth. As the sound beam crossed the submarine from bow to stern, echoes grew stronger, faded and died out completely for a short time. Then strong echoes were received for several hundred yards behind the submarine, which were attributed to reflection from the wake. 23.4 DEPENDENCE ON RANGE At long ranges, target strength is practically inde- pendent of range. Close to the submarine, however, the target strength will decrease with range. This phenomenon has two causes. First, at very short dis- tances from the submarine, the submarine reflects more like a plane or a cylinder than a sphere, and the inverse fourth power law does not apply. In other words, the target is not equivalent to a point source, and the target strength decreases. This effect de- pends on the aspect of the submarine and diminishes as the range exceeds the maximum radius of curva- ture of the submarine. Secondly, at short ranges the effective portion of the sound beam may cover only part of the area ex- posed by the submarine. Such an effect would be expected primarily at beam aspect, since geometric foreshortening reduces the area exposed by the sub- marine at other aspects. For example, a sound beam 12 degrees wide will not cover the entire length of a 300-ft submarine, at beam aspect, at ranges less than 475 yd. However, only those areas on the submarine giving rise to nonspecular reflection would be af- fected, so that if most of the reflection were specular, arising in a small area amidships, the target strength as measured very close to the submarine from per- fectly aimed pulses would not depend on the beam width. From Figures 1 to 5 in Chapter 22, it appears that most of the reflection is concentrated amid- ships. Thus the effect of beam width, though present, is probably not important except possibly at very short ranges, as long as nonspecular reflection is neglected. DEPENDENCE ON RANGE ite SRB Bee nicked | | Be thse | 30 TARGET STRENGTH IN OECIBELS 403 le — eo es 12 YARDS | 8 YARDS ASPECT ANGLE IN DEGREES Ficure 16. Theoretical dependence of target strength on range for the U570 (HMS/M Graph). 23.4.1 Theory In the theoretical target strength studies described in Section 20.5,° plans of the Graph were employed in calculating target strengths.!® This submarine has a maximum radius of curvature of about 560 yd. Since target strengths vary with range primarily for ranges less than the maximum radius of curvature of the submarine, the target strength of the Graph would be expected to approach a limiting value as the range is increased to 500 or 600 yd. Actually, the target strength of the Graph is very near this limiting value at ranges beyond only a few hundred yards, especially at beam aspect. Target: strengths have been computed for ranges of 8, 12, 16, 200, and 1,000 yd on the assumption of a non- directional source of sound and are plotted against aspect angle in Figure 16. The shorter ranges are the distances from the projector to the nearest part of the submarine. It is apparent from Figure 16 that the variation of target strength with aspect changes markedly as the range is reduced from 200 to 16 yd; no intermediate ranges were used. These calculations were based on a nondirectional source; therefore the results neglect the effect of limited coverage of the submarine by a directional beam at close ranges. Such an effect is not important, however, if the reflection is primarily specular and comes from a small area amidships, as discussed in the preceding section. 23.4.2 Observations Verification of a dependence of target strength on range has not been possible in the direct measure- ments because the transmission loss has not been known accurately. Instead, it has been assumed, for the ranges used during the target strength measure- ments — from 200 to 1,000 yd—that the target strength at near-beam aspects remains constant. Such an assumption is necessary in order to calculate the transmission loss; in some of the measurements at San Diego, a constant target strength was assumed at constant aspect, and the transmission loss was computed from a plot of the echo level E plus 40 log r against the range r as the range is opened or closed (see Section 21.5.1). The most recent measurements at San Diego have employed a nondirectional hydro- phone mounted on the submarine to measure the transmission loss; this method, if practical, might 404 SUBMARINE TARGET STRENGTHS reduce the usual fluctuations enough to enable an evaluation of target strength as a function of range. The indirect measurements at MIT, at selected as- pect angles, and at full-scale ranges of about 250 and 630 yd, gave about the same values as those at a full- scale range of 190 yd.” This result is consistent with the theoretical computations shown in Figure 16, where the maximum change in target strength is only about 3 db at beam aspect, as the range changes from 200 to 1,000 yd. Since for these ranges the target strength was found to be independent of range, it was apparent that the intensity of the light reflected from the models and measured at the receiver varied with range at the same rate as that from a sphere, rather than that from a cylinder or plane, in other words, inversely as the fourth power of the distance. The shortest range was approximately twice the length of the submarine. It is likely that this relation would not hold at much closer ranges where target strengths would be expected to depend strongly on the range. Target strength was found to depend on the range in the acoustical model experiments at Mountain Lakes, for full-scale ranges of 85, 170, and 250 yd. The submarine model behaved as a cylinder, not as a sphere or plane. For reflection from a cylinder, the echo level should decrease 9 db when the range is doubled as long as the range is not much greater than the length of the cylinder (see Section 20.4.3); for.a sphere the same increase in range causes a drop of 12 db. It was found that the echo level at beam as- pect actually dropped 8.8 db as the full-scale range was doubled, from 85 to 170 yd. Thus the hull at beam aspect and at short ranges behaves as a cyl- inder, as might be expected from its large radius of curvature in the horizontal plane. In general, the indirect measurements agree with the theoretical predictions of the dependence of tar- get strength on range. This predicted dependence should be most marked at ranges less than 200 yd; experimentally, it was observed and verified only at ranges less than 200 yd. However, too much impor- tance cannot be attached to these results, since the measurements were made only at three particular ranges in each of the two indirect measurements, and since experimental errors were so large. 23.55 DEPENDENCE ON PULSE LENGTH When short pulses are used instead of continuous sound, target strength may depend on the lengths of these pulses. Sections 19.3 and 20.7 discussed in an elementary way the effect of pulse length on meas- ured target strengths. For short pulses — signals whose length in the water is less than the length of the target in the direction of the sound beam — the echo level and therefore the target strength will de- pend on the signal length. The exact variation of target strength with signal length, however, depends on whether peak echo intensities or average echo intensities are used in computing target strengths. 23.5.1 Theory In most direct measurements of target strength, peak amplitudes are measured from the oscillograms rather than average amplitudes, because echo pro- files are so irregular that the average amplitudes over the length of the echo would be difficult to measure. A simple analysis given in Section 19.3.1 shows that for long pulses the average echo intensity is inde- pendent of signal length, while the echo length varies with the signal length. For short pulses on the other hand (see Section 19.3.2), the average echo inten- sity is approximately directly proportional to the signal length, while the echo length remains con- stant. This analysis applies to square-topped pulses striking an extended single target. It is arbitrary whether peak amplitudes or aver- age amplitudes are used to compute target strengths. Peak amplitudes are easier to measure. In addition, most other underwater sound measurements, such. as those undertaken in the investigation of recogni- tion differentials, are based on peak amplitudes. It may be, however, that the ear, or the sound range recorder, or other detection devices may respond to the average echo intensity instead of the peak echo intensity, or to the total energy in the echo. There- fore a comparison of average and peak echo ampli- tudes and their variation with signal length might be a profitable study. The change of average and peak echo intensities with pulse length has been investigated theoreti- cally,!8 as described in Section 21.6.4. First it is as- sumed that the length of each individual peak in an echo is approximately equal to the signal length. Then it is assumed that these peaks are statistically independent, or distributed at random throughout the length of the echo. By assuming that the echo is essentially a group of rectangular peaks, each of which follows the Rayleigh distribution for succes- sive echoes and each of which is independent of the DEPENDENCE ON PULSE LENGTH 405 Tioo MILLISEC -'30MILLISEC Tio MILLISEC -" 10 MILLISEC T100 MILLISEG~ 10 MILLISEG ’ fo) a ° a (c) cos 0 Figure 17. Dependence of target strength on signal length for an S-boat at 24 ke. others, the change in the peak echo intensity, aver- aged over many echoes, turns out to be considerably less than the change in mean echo intensity, aver- aged over the length of the echo and then over many echoes, for the same change in pulse length. For example, it is shown that, for a uniformly re- flecting target whose length is 300 ft in the direction of the sound beam, corresponding roughly to a fleet- type submarine at bow or stern aspects, the average peak echo intensity will decrease only 3 db and the average mean echo intensity will decrease 5 db, when the signal length is reduced from 30 to 10 msec. The assumptions on which this analysis is based are so idealized that the exact numerical results are prob- ably not very significant; nevertheless, the general conclusion that the peak amplitude changes less rapidly with the signal length than the average amplitude seems fairly well established theoretically. Ti00 MILLISEG ~T 30 MILLISEG {o) 0.2 0.4 0.6 0.8 1.0 BEAM BOW STERN cos @ Figure 18. Dependence of target strength on signal length for a fleet-type submarine at 24 ke. RELATIVE TARGET STRENGTH IN DECIBELS fo) 90 180 BOW BEAM STERN ASPECT ANGLE IN DEGREES Figure 19. Dependence of relative target strength on signal length at different aspects for an S-class sub- marine at 60 ke. 406 SUBMARINE TARGET STRENGTHS TARGET STRENGTH IN DECIBELS BOW ae xy STS SSRI ————_ 30 eas — —— — 10 MILLISEG — I MICEISEG a STERN Figure 20. Dependence of target strength on signal length for an S-boat at 24 ke. 23.5.2 Observations Since evidence showing a dependence of target strength on pulse length could be found only in the direct measurements, these measurements have been examined and analyzed from this point of view. Different signal lengths have been employed in direct target strength measurements at San Diego, where pulses from 0.5 to 200 msec long have been used. Peak echoes from 5-msec signals were found to average about 4 db lower than echoes from 33-msec signals, according to early runs at San Diego on an S-class submarine, using JK gear at a frequency of 24 ke.’ Further studies showed a minimum depend- ence of target strength on pulse length at beam aspects, and a maximum, nearly linear, dependence at bow and quarter aspects.”? Later measurements, however, reported no very significant dependence of target strength on signal length for signal lengths of 10, 30, and 100 msec.’ These measurements are illus- trated in Figures 17 and 18 for an S-boat and a fleet- type submarine respectively; the difference in target strengths between 100- and 30-msec signals and 100- and 10-msec signals were greatest at aspects near the bow and stern and are plotted against the DEPENDENCE ON PULSE LENGTH 407 TARGET STRENGTH IN DECIBELS BOW 30 MILLISEG ——— 10 MILLISEG 1 MILLISEG eile EN HN Me SW [| FA IST STERN FIGURE 21. cosine of the aspect angle. The mean curve connect- ing the mean points is also indicated. However, this dependence on pulse length is relatively small and not very reliable in view of the large scatter. Measurements were made at a frequency of 60 ke on another S-boat at creeping speed and a keel depth of 100 ft, with signal lengths of 1, 10, and 30 msec.” A ‘significant variation in relative target strength with signal length was observed and plotted in Fig- ure 19 for bow, beam, and stern aspects where each point at beam aspect represents about 40 echoes, at stern aspect about 20 echoes, and at bow aspect only a few. Surprisingly, a large variation with pulse Dependence of target strength on signal length for an S-boat at 60 ke. length is found at beam aspect; here theory would predict a minimum dependence, since the echoes for the most part accurately reproduce the pulses. An attenuation coefficient of 20 db per kyd was assumed in calculating the transmission loss; ranges averaged about 300 yd. Figures 20 and 21 show target strength as a func- tion of aspect angle for three different signal lengths, 1, 10, and 30 msec, for a submarine of the S class at 24and 60 ke. A definite dependence on signal length is evident from an examination of the three curves in each figure, although the actual target strength val- ues are rather low. The dependence on aspect angle Sp SATIS <— O SQ HABE BEAM Siece= . soue BEAM STERN Figure 22. Dependence of target strength on frequency (San Diego). is somewhat obscured by the fluctuations encoun- tered during these particular runs. In general, target strength depends on the signal length for short signals although it varies less rapidly than the signal length, or rather, less rapidly than 10 log 7, where 7 is the signal length; a decrease in tar- get strength is most marked at signal lengths less than 10 msec and at aspects away from the beam. 23.6 DEPENDENCE ON FREQUENCY Both sonic and supersonic frequencies have been employed in echo ranging. Lower frequencies are de- sirable from the point of view of transmission, since the transmission loss increases with frequency. On the other hand, since high-frequency sound is more directive, the bearings of targets may be located more accurately at high frequencies than at low frequen- cies. As a result, choosing a frequency for echo rang- ing is always a compromise between these two char- acteristics. Target strengths have been predicted and meas- ured directly at frequencies between 12 and 60 ke. Theoretical calculations have been based on a fre- quency of 25 ke. Most of the direct measurements have been made at 24 ke, since this frequency is DEPENDENCE ON FREQUENCY 409 I standard for most Navy sonar gear, although some measurements have been made at frequencies of 12, 18, 45, and 60 ke. The indirect measurements used scale models; at Mountain Lakes, with a 1:60 scale model of the Graph, a frequency of 1,565 ke was em- ployed to simulate an echo-ranging frequency of 26 ke. At MIT visible light was used, and the cor- responding full-scale frequency was very much higher than for any of the other measurements. 23.6.1 Theory The target strength of a submarine depends on frequency according to equation (36) in Chapter 20, especially if nonspecular reflection contributes ap- preciably to the target strength. Specular reflection depends only slightly on frequency, as described in Section 20.4. Beam echoes result largely from specu- lar reflection, as pointed out in Section 23.8.1. Thus the variation of target strength with frequency at beam aspects may be expected to be slight. At off-beam aspects, however, specular reflection is much less important and nonspecular reflection may become appreciable; this effect is discussed in Section 23.8.2. At these aspects, the whole sub- marine appears to scatter sound. If reflections from the superstructure or exterior protuberances on the submarine, such as rails, guns, and periscopes, con- tribute appreciably to the target strength, the target strength may depend on frequency as long as the dimensions of these scatterers are of the same order of magnitude as the wavelength. Section 20.5 de- seribes the origins of nonspecular reflection. 23.6.2 Direct Measurements No reliable direct measurements substantiate this expected dependence of target strength on frequency. Figure 22 shows target strength as a function of as- pect angle plotted for frequencies of 24 and 60 ke for a fleet-type submarine at San Diego at signal lengths of 10, 30, and 100 msec. Each point represents the average of all observations in a 30-degree sector centered at the point indicated. A clear-cut dependence of target strength on fre- quency is apparent in this illustration, as well as in Figures 20 and 21. Figure 21 gives quite reasonable values for the target strength at 60 ke, but Figure 20 shows values about 10 db lower, for a frequency of 24 ke; thus the dependence on frequency is still evi- dent. It is unlikely, however, that the increase in tar- get strength with frequency would be not only so great but also so nearly uniform at all aspect angles, as Figure 22 shows. Furthermore, the measurements cannot be relied on for two reasons: the transmission loss was not known but estimated, and the calibra- tion of the 60-ke gear was less reliable than the 24-ke gear calibration. The estimated attenuation co- efficient of 20 db per kyd used for these computations is larger than that measured elsewhere and is per- haps excessive, since attenuation coefficients of only 10 db per kyd at 60 ke were measured at Fort Lauderdale. However, correcting the high San Diego target strengths at 60 ke by reducing the attenuation coefficient from 20 to 10 db still results in values greater than those obtained elsewhere. The remain- ing discrepancy may perhaps be attributed to faulty calibration of the gear, described in Section 21.4, since this discrepancy is systematic and apparently independent of aspect angle. In addition, the high target strengths measured at 60 ke at San Diego do not seem substantiated by target strength measurements at that frequency at Fort Lauderdale. These measurements gave a target strength of 25 db for a fleet-type submarine at beam aspect at 60 ke, assuming an attenuation coefficient of 12 db per kyd. An assumption of an attenuation coefficient of 20 db per kyd would raise this to only 33 db, compared with the maximum value of 44 db recorded at San Diego for the target strength of a fleet-type submarine. Furthermore, wake echoes measured with the same equipment at San Diego and described in Section 33.4.2 were found to be much higher at 60 ke than at 24 ke, contrary to theoretical expectations. These results support the suggestion that calibration errors are responsible for the high values obtained at San Diego. However, in view of the many uncertainties in this subject, the possibility that submarine target strengths are sys- tematically some 10 db higher at 60 ke than at 24 ke cannot be entirely ruled out, even though there is little if any theoretical expectation of such a varia- tion. No difference was apparent between the measure- ments at 12 ke and 24 ke made by Woods Hole ob- servers; !° both target strength-aspect curves were very similar. However, the target strengths reported are so much larger than all other measurements else- where that calibration errors were probably present. Therefore, since the calibration at 12 ke was quite different from that at 24 kc, and since all the values seem very uncertain, the lack of any frequency de- pendence cannot be considered significant. 410 23.6.3 Indirect Measurements Visible light from a motion picture projection bulb was used in the optical measurements at MIT. Since the frequency, or the band of frequencies, was not properly scaled to correspond with usual echo- ranging frequencies, it was not practical to investi- gate the dependence of target strength on frequency. However, since improperly scaled light was used to give results for comparison with direct and other in- direct measurements, four frequency effects should be remembered in interpreting the results of the op- tical measurements.” First, at certain aspects when the insonified sur- face of the actual submarine subtends only a few Fresnel zones at 24 ke, the surface of the model il- luminated by visible light subtends many zones, since the wavelength of the light was much shorter compared with the dimensions of the model than was the wavelength of the sound used in echo ranging compared to the dimensions of an actual submarine. As a result, for the optical measurements, the ex- pressions for the target strength due to the effects of a group of Fresnel zones approached their asymp- totic values, especially for surfaces with at least one large radius of curvature, such as cylinders and planes. On submarines this effect might apply to the conning tower, keel, and top deck, so that in the optical measurements the effect of the conning tower, and the effect of the deck at an altitude angle of 90 degrees, might be overemphasized. Secondly, nonspecular reflection is too small by a factor of the square root of the ratio of the actual wavelength used to the properly scaled wavelength. In the optical measurements, this factor is about 45, or about 17 db. In other words, nonspecular reflec- tion ‘measured optically is about 17 db too low. This factor may account for the very low target strengths obtained optically at off-beam aspects where nonspecular reflection may be more important. Thirdly, diffuse reflection or scattering may be too great optically because the wavelength may be much smaller than the surface irregularities. An attempt was made to minimize this error by using glossy black surfaces. Finally, where two or more specular reflections occur, the light beams do not interfere, as sound beams do, because they are incoherent. Since the incoherent sum is an average of the interference pat- tern, this sum may actually be more interesting than the detailed interference pattern itself, as the aver- SUBMARINE TARGET STRENGTHS age is more significant, in most applications to prac- tical echo ranging at sea, than the exact pattern. Thus the results of the optical measurements, though suggestive of what might be encountered in practical echo ranging, cannot be compared directly with the other measurements unless these effects of the wavelength are considered and accounted for. In the acoustical measurements at Mountain Lakes, no long-term systematic variation with fre- quency was observed for full-scale frequencies from about 1 to 35 ke. Figure 23 shows relative echo level plotted against frequency for the Mountain Lakes measurements on the Graph at beam aspect, at a full-scale range of about 15 yd. The peaks and dips evident in this illustration are largely the result of interference phenomena arising from two specular reflections from the hull and conning tower, as the frequency is changed, and of the response character- istics of the system. Interference phenomena result- ing from multiple reflections from several surfaces on the submarine are clearly shown in Figure 24, where the relative beam target strength is plotted against altitude angle for a full-scale frequency of 26 ke. The nearly smooth curves at altitudes of 0 and 90 degrees result from direct specular reflection from the deck and hull respectively. The intricate interfer- ence pattern at altitude angles between 10 and 50 degrees and 270 and 350 degrees results from path differences in the sound doubly reflected from the hull and from the blister tank; similar patterns are evident for sound reflected from the bottom of the submarine model. 23.7 OCEANOGRAPHIC CONDITIONS Target strength measures the reflecting character- istics of a target and is computed from the echo level, source level, and transmission loss from equation (6) in Chapter 19. Since it depends only on the target itself, it is theoretically independent of the medium and its characteristics, independent of the transmis- sion characteristics of sound in water, and therefore independent of oceanographic conditions insofar as they affect transmission. In practice, however, reported target strengths have been found to depend markedly on the prevail- ing oceanographic conditions in cases where the transmission loss has not been accurately known. Since target strength is computed from the echo level, source level, and transmission loss, improper appraisal of the transmission loss will appear as an error in the target strength values reported. Por Hcl ow ie) nN ie] S) RELATIVE ECHO LEVEL IN DECIBELS 50 70 100 s ° o2o000 ° ° ° Ci Wey ep Teh Gis O) a, O) 16) To) (oo io a +t Ooo90 fe) fe) fo) fo) fo) a £2888 oP pate a eee Sao (oe Cane ACTUAL FREQUENCY IN KG fo} QQ iro: ° - m (CY [S) i Key bs 2° m 10) IN mio) ir (] NN ow o (te) o So =o ote 13 o fo} om Oo © OO a = = - = - N Nn NN NN MH =m FULL SCALE FREQUENCY IN KC FIGURE 23. Dependence of target strength on frequency (Mountain Lakes). TOP DECK + x = I | KEEL ae J —— _——, ————— — a —- —l —, ——— —— = Ly bok | data Nd til MY 1 ‘ae ATO | 150 180 210 240 270 300 330 360 ALTITUDE ANGLE IN DEGREES Figure 24. Double reflection and interference (Mountain Lakes). uh tM 4 ; t seiko ve] o iy ACTUAL PREC hay a ae Se a 8 FULL. SGALS. “Oey TN KC. apo: BS ian ie aw Trequenoy | Mee bonny Be AT uOR POE: ie gence (oh ancon? a enc | Inieiterense (Menriste Tako — a t b . ie Sy ing ante te Swuiticaut, is most applications to prac- eng at dea, than the exnet pattern, * peaulte of the optical measurements, 1) itive of what night be encountered in 4 caning, Cannot be compared direethy ; iy mbesurements woloss these effete of dit ere considered and accounted for. | epustical measurements ‘at Mountain ifig-term systematic voriation with fre- o] eheerved for full-ecale frequencies from ike, Figure 23 shows relative echo level ‘ TH 41 Bcd frequency for the Mountain Lakes ba | | fi on the Graph af, beam aspect, at, a HH Te of abut 16 yd, The peaks and dips 4 ; iiuetention are largely the result of jE A tenomeris’ arising from two specular | Vy ic “in the hull and conning tower, as the i Veaite | changed, and of the response character- "eg BF > aystem. Interlerense phenomena result- Soe & ai. reflections from several surfaces wirine are clearly shown in Figure 24, “ @ & © itive beom target strength is plotted 2.8 8.8 Gide wngle for a full-eoale frequency of wry sxnooth curves at altitudes of 0 and ult frotn direct specular reflection from Pall: Hite The intricate interfor- OCEANOGRAPHIC CONDITIONS Section 21.5 introduced the concept of an attenua- tion coefficient to describe more accurately the trans- mission loss. This attenuation coefficient varies both with the oceanographic conditions and the frequency of the echo-ranging sound. From a quantity of trans- mission data at 24 ke two empirical formulas were suggested for estimating the attenuation coeffi- cient,” equations (5) and (6) in Chapter 21. They are 170 = 3. — 1) a=3.5+ D ( for the hydrophone above the thermocline, and 260 = 4. — 2) a=45+ 5D ( for the hydrophone below the thermocline, where a is the attenuation coefficient in decibels per kiloyard and D the depth of the thermocline in feet. The prob- able error of this estimate is about 2 db per kyd. 411 measurements at 24 ke on different S-boats, calcu- lated by assuming an attenuation coefficient of 5 db per kyd, showed such deviations that a marked de- pendence on the particular submarine used was sug- gested.’ The particular submarines used are desig- nated in the first column of Table 4, while the target strengths, varying from 7 to 25 db, are reproduced in the second column. Investigation of the oceano- graphic conditions prevailing during the different runs showed an unmistakable correlation between the target strengths and the thermocline depths; when the thermocline depth was only 18 ft, the com- puted target strength was only 7.3 db, while it rose to 25 db for a thermocline 160 ft deep. Accordingly, equations (1) and (2) were used to calculate new and presumably more reliable attenua- tion coefficients, from the thermocline depths, listed in the third column, and the depths of the submarines during the runs, in the fourth column of Table 4. Tass 4. Dependence of reported target strengths on attenuation coefficient. Reported beam |} Depth to Depth of target strength | thermocline | submarine S-boat in decibels in feet in feet USS 8-28 (SS133) 18* 85 45 USS S40 (SS145) 25* 160 90 USS 8-23 (SS128) 13.7* 50 90 USS 8-23 (SS128) 15.4* 75 100 USS 8-33 (SS138) 7.3* 18 90 USS 8-31 (SS136) 22.3f 100 95 Mean beam target strength and 16.9 + 4.8 standard deviation in decibels Computed attenuation coefficient Computed beam in decibels Range Correction | target strength per kiloyard | in yards in decibels in decibels 5.5 350-520 +0.5 18.5 4.6 300—400 —0.3 24.7 9.7 350-500 +4.7 18.4 5.8 330 +0.3 15.7 19.0 415 +11.6 18.9 5.2 440 0.0 22.3 19.7 + 2.5 * Assumed attenuation coefficient, 5 db per kyd. t Measured attenuation coefficient, 5.3 db per kyd. 23.7.1 Effect on Measurements Direct measurements of the transmission loss dur- ing target strength runs on submarines have been difficult and generally unsuccessful, as described in Section 21.5.2. As a result, it has been customary at San Diego to compute the transmission loss from an estimated attenuation coefficient at the particular frequency employed, usually 5 db per kyd at 24 ke and 20 db per kyd at 60 ke. Target strengths computed in this way show enor- mous differences. For example, various series of These new values appear in the fifth column, the ap- propriate ranges in the sixth column, in the seventh column the corrections resulting from the assumed value of 5 db per kyd of sound travel, not range — assuming the maximum range from the sixth column —and the new target strengths in the last column. Two results are noteworthy: the standard deviation is reduced by a factor of almost two from 4.8 dh to 2.5 db, and the mean beam target strength for an S-boat is raised from 16.9 db to 19.7 db. The new value agrees more closely with other measurements. At 60 ke fewer data are available. An assumption 412 SUBMARINE TARGET STRENGTHS ASPECT ANGLE f IN DEGREES — coo SO'MILLISEC INTERVALS = —_—_—s&BEAM NOTE: FIRST ECHO IS A SURFACE WAKE QUARTER | FicurE 25. Oscillograms of submarine echoes (S-class submarine). STRUCTURE AND ORIGIN OF ECHOES 413 of 20 db per kyd for the attenuation coefficient, al- though assumed in the San Diego target strength : ASPECT ANGLE ¢ 7 : z i IN DEGREES computations, is not substantiated by tests made at, \ . ‘| Fort Lauderdale which gave results of 9 to 10 db per ii = : kyd. These latter measurements were made by plot- : | ting the echo level, corrected for inverse square loss, = against the range, as the range continuously de- PORT creased from about 600 to 150 yd; the submarine was —=————- BEAM at stern aspect throughout the run. This discrepancy was mentioned in Section 23.6.2. The high value as- sumed for the attenuation coefficient at San Diego was suggested by early transmission measurements; in these tests, the presence of shallow thermoclines off the coast of California was partly responsible for the high values measured. For deep, mixed water, lower values are more common * (see Chapter 5). Ce es PCr tr Ni ° | } t } wip . “t yf ite Mh in uni, Ha. apa Figure 27. Oscillograms of submarine echoes at beam aspect for 10-millisecond pulses. 23.8 STRUCTURE AND ORIGIN OF ECHOES Most target strengths have been measured from echoes recorded oscillographically on 35-mm motion picture film. At San Diego, this film was run past the oscilloscope at a speed sufficiently high to record the detail of each echo; depending on the signal length and the exact film speed, these echoes may be from Figure 26. Sound range recorder records of submarine 0.2 to 3 or 4 cm long. Accordingly, these echoes have echoes from 30-millisecond pulses. been carefully studied in an attempt to formulate 414 SUBMARINE TARGET STRENGTHS ASPECT ANGLE IN DEGREES (e) BOW : 180 STERN 220 QUARTER 315 345 QUARTER Figure 28. Oscillograms of submarine echoes at off-beam aspects for 10-millisecond pulses. 415 STRUCTURE AND ORIGIN OF ECHOES more precise conclusions regarding the process of reflection of sound from submarines. Examination of hundreds of oscillograms of echoes at San Diego has made possible a separation of echoes into two classes, beam and off-beam, as illus- trated in Figure 25. Each class shows its own char- acteristics and peculiarities, on the basis of which tentative explanations of reflection phenomena have been made. These two types of echoes produce such different traces on the sound range recorder that the appearance of these traces is used tactically to esti- mate the aspect of the target. Typical sound range recorder traces are illustrated in Figure 26. 23.8.1 Beam Echoes Beam echoes are always stronger, on the average, than echoes at any other aspect, both according to the theoretical calculations and the direct and indi- rect measurements. There are four lines of evidence which indicate that reflection is specular and arises primarily from the hull of the submarine. First, theory predicts strong specular reflection at beam aspect.® The theoretical values derived assum- ing only specular reflection are in excellent agree- ment with other values measured directly and in- directly. The effect of the conning tower appears to be negligible for the U570 at beam aspect, since it contributes orily 0.2 db to the target strength at long ranges. Secondly, the oscillograms of beam echoes are clear cut and closely resemble the square-topped pulses sent out; further examples are illustrated in Figure 27, which shows oscillograms of three succes- sive echoes from a submarine at beam aspect. These beam echoes contrast sharply with off-beam echoes, which are illustrated in Figure 28. Occasionally, beam echoes show very sharp and narrow peaks at either end, or a short “tail” of lower intensity, which are attributed to two different types of surface reflec- tions described in Section 21.5.4. A typical sound range recorder record of the double echoes at beam aspect is illustrated in Figure 29. Since beam echoes usually equal the signal in length, for signals 10 or more milliseconds long, the effective reflecting sur- face does not appear to be much extended in the direction of the sound beam; typical oscillograms of beam echoes are illustrated in Figure 30. In other words, the relatively flat area on the hull of the sub- marine is responsible for almost all the energy in the echoes received at beam aspect. This same specular France in yaros q © 200 400 Figure 29. Double echoes recorded on the sound range recorder. reflection may be inferred from Figure 24 where a fine interference pattern is conspicuously absent at beam aspect for altitude angles in the neighborhood of 0 degree. Thirdly, optical measurements on a model of the Sand Lance as well as both optical and acoustical measurements on models of the U570 gave identical target strength at beam aspect with and without the conning tower, over a sector of about 20 degrees, as Figures 31, 32, and 33 show. The conning tower, although important at other aspects, contributes little to reflection at beam aspects. Fourthly, the importance of hull reflections is evident in Figures 1 through 5 of Chapter 22. Al- though these photographs refer only to optical illu- mination of the model and may not apply perfectly to the reflection of supersonic sound from submerged submarines, they may be representative of what hap- pens acoustically. However, measurements made with short pulses at beam aspect show a detailed echo structure which suggests that not all the reflected sound comes from the submarine hull or ballast tanks. Measurements on 416 SUBMARINE TARGET STRENGTHS /2-MILLISECOND PULSE ECHO ‘ SUCCESSIVE ECHOES - IO-MILLISECOND PULSE ECHO | 100-MILLISEGOND PULSE > Figure 30. Detailed oscillograms of submarine echoes at beam aspect, STRUCTURE AND ORIGIN OF ECHOES TARGET STRENGTH IN DECIBELS Effect of conning tower on optical meas- urements on USS Sand Lance (SS381). Figure 31. an S-boat at beam aspect for signals from 0.5 msec long resulted in echo oscillograms showing two dis- crete “blobs” of about equal mean amplitudes with a range separation of about 4 yd.%?6 Although the finer details of the echo envelopes and the relative values of the peak amplitudes did not repeat from echo to echo, the main features consistently suggested the presence of two distinct reflecting surfaces on the submarine at beam aspect. For signals longer than 4 yd (5 msec), the echo envelopes were almost al- ways resolved into three distinct segments, with the central portion corresponding to the overlap or addi- tion of the two primary echoes found for shorter sig- nals. The amplitude of this central portion presum- ably varied according to the initial phase difference and amplitude of the two component signals; the difference between the phases changed little during the period of reflection. Typical oscillograms are il- 417 BOW 30 20 4 my =! ra) w ra) = Z i | FA VF TARGET STRENGTH 20 30 STERN Ficure 32. Effect of conning tower on optical meas- urements on U570 (HMS/M Graph). lustrated in Figure 30; the weak echo following the main one is sound reflected from the submarine up to the surface, back to the submarine and then back to the projector, as discussed in Section 21.5.4. Because the individual echo components in Figure 30 appear to be coherent, at beam aspects the echo is probably specular. Nonspecular or diffuse reflection is appar- ently unimportant at these aspects, although other surfaces besides the hull and ballast tanks may con- tribute to the echo. Off-Beam Echoes Echoes from aspects other than beam are generally very different from beam echoes. Not only is the echo weaker, but it is also less well defined; examples are shown in Figures 25 and 28. If a system of high re- solving power is used, such as the oscilloscope and 23.8.2 418 4 SUBMARINE TARGET STRENGTHS WITHOUT GONNING TOWER TARGET STRENGTH IN DECIBELS Figure 33. Effect of conning tower on acoustical measurements on U570 (HMS/M Graph). high-speed camera at San Diego, the echoes appear to be-a group of fine spikes or peaks, although some evidence points to peaks which are found in the same places in successive echoes. A more complete discus- sion of the detailed structure of off-beam echoes and their origin is postponed to the next section. Here the more general features of off-beam echoes are dis- cussed. The beginning and end of an echo at off-beam aspect are not clearly defined; usually the amplitude builds up and dies away gradually, blending into the background at either end, so that precise measure- ment of the echo length is impossible. However, an examination of off-beam oscillograms shows that these echoes are longer than the signals and there- fore suggests an extended target.” If the entire length of the submarine is effective in reflecting sound, as seems indicated for off-beam echoes, the lengths of these off-beam echoes should vary with the aspect angle, depending on the length of the submarine in the direction of the sound beam. In other words, if a submarine is an extended target and scatters sound throughout its length, the length of the echoes which it returns should depend on the aspect which it presents to the sound beam. Accordingly, the lengths of these off-beam echoes, diminished by the length of the signal used, have been measured or estimated as accurately as possible, then plotted against the cosine of the aspect angle which accounts for the foreshortening of the sub- marine.” Figure 34 illustrates the results of this analysis, where the broken curve connects the meas- ured points, and the solid curve is a polar plot of the cosine of the aspect angle, modified at bow aspect to account for the “shadow”’ which the forward section casts on the stern section. A similar dip is included at stern aspect, where the after part of the submarine shadows the forward part. The maximum elongation — echo length minus signal length — was found to occur at quarter as- pects, roughly 15 degrees from the bow and stern on either side, and amounted to about 85 yd.* The actual length in the direction of the sound beam of a fleet-type submarine 300 ft long, at aspects 15 de- grees from bow and stern, is about 96 yd, which con- firms the suggestion that the entire surface of the submarine scatters sound. At an aspect angle of 135 degrees, when the target had an extension of 49 yd in the direction of the sound beam, the elongation amounted to about 38 yd. At bow aspect, this elongation was reduced to 50 yd. This reduction is attributed to the shadow cast by the forward section on the after section. A similar drop, however, was not observed at stern aspect. These elongation phenomena are apparently inde- pendent of signal length and echo-ranging frequency Apparently they are the result of scattering from the entire length of the submarine, instead of reflection from only one or two major surfaces such as the conning tower or screws. They suggest that in addi- tion to specular reflection from the hull, nonspecular reflection or diffuse scattering also occurs, especially at aspects away from the beam. The exact mechanism by which sound is reflected from the entire length of the submarine is unknown. Similar elongation phenomena were analyzed by British observers in an effort to determine the origin STRUCTURE AND ORIGIN OF ECHOES 419 30 MILLISEG 7 10 MILLISE) 4 100 MILLISEC 180 STERN Figure 34. Dependence of echo elongation on aspect angle (San Diego). 420 SUBMARINE TARGET STRENGTHS 80 60 ~wae ECHO ELONGATION IN YARDS” BOW —.—— PROJECTED LENGTH OF SUBMARINE IN DIRECTION OF SOUND BEAM (DIFFERENCE IN RANGE BETWEEN FURTHEST AND NEAREST POINT OF SUBMARINE) 120 160 BEAM STERN ASPECT ANGLE IN DEGREES FIGuRE 35. of the nearest echo, but the hypothesis that the en- tire submarine reflected sound was not confirmed.” The elongation, plotted in Figure 35, amounted to only about half the calculated exposed lengths of the submarine, after the pulse lengths had been sub- tracted from the echo lengths. Similar results have also been obtained in this country using the same technique. It may be pointed out, however, that a sound range recorder, not an oscilloscope, was used in these experiments, and that records from a sound range recorder might be expected not to show the weaker tail part of the echo. 23.8.3 Source of Echoes Beam echoes originate in large part at the hull of the submarine, as described in Section 23.8.1, with some additional contribution possible from the hull and bilge keel. The echoes are nearly square-topped and result from simple specular reflection from only one or two surfaces on the submarine. Off-beam echoes, however, apparently come from all parts of the submarine. Oscillograms of these echoes are detailed and show a fine microstructure of peaks and valleys, somewhat similar to reverbera- tion, especially for short pulses. Since study of the elongation phenomena suggests that echoes are re- turned from most of the submarine, various peaks in the detailed structure of an echo might be correlated with discrete reflecting surfaces on the outside of the submarine. Only short signals could be used, how- Echo elongation and projected length of submarine as a function of aspect angle. ever; otherwise the signals from individual reflectors on the submarine might overlap. Accordingly a series of echo oscillograms from sig- nals approximately 0.5 msec (0.4 yd) long were studied at San Diego.”” The target was a submarine of the S class at quarter aspect, 135 degrees from the bow. The echoes were recorded oscillographically, as usual, but the film was run at a speed of about 13 in. per sec, five times faster than normally. This high speed lengthened each echo and permitted better resolution of the echo structure. Each echo analyzed consisted of a number of sharp spines, usually between twenty and fifty, which rose clearly above a fuzzy background. The envelope of these spines was roughly cigar-shaped while the en- velopes of the less intense parts of the echo peaks were similarly shaped but only about half the ampli- tude of the spine structure. The distribution of these spines appeared to be random, and no peaks or groups of peaks could be definitely correlated with individual reflecting surfaces, such as the conning tower or ballast tanks. Thus the peaks may be more the result of constructive interference of sound scat- tered at random from the entire submarine, than of strong reflections from discrete surfaces on the sub- marine. Studies of other short-pulse echoes obtained at other aspects from various submarines usually yield somewhat similar results. The echo almost always consists of a succession of peaks rising above the background. These peaks, however, do not always STRUCTURE AND ORIGIN OF ECHOES 421 FIGURE 36. Repeatable peaks in submarine echoes. occur at the same place in successive echoes; rather, they usually appear to be distributed unsystemati- cally though generally nearer the center of the echo than either end. In some cases, however, repeatable peaks seem to be present in submarine echoes. A series of nine con- secutive echoes from 5-msec signals at 60 ke are repro- duced in Figure 36 and show two separate peaks or groups of peaks at the same places in each echo. In these measurements the submarine aspect was held nearly constant at about 330 degrees. Thus no defi- nite conclusions can be drawn at the present time as to how often an echo peak will reproduce itself. It is therefore uncertain whether these peaks represent highlights on the submarine or random interference between several reflected sound waves. In general, the process of reflection of sound from a submerged submarine at off-beam aspects is still im- perfectly understood. The entire submarine appears to contribute to the reflected sound, yet specific, repeatable highlights have not been observed in most examinations of echo oscillograms. It is difficult to understand how nonspecular reflection from the sub- marine hull or from protuberances and fixtures on the outside of the submarine can account for these echoes. Until the origin of these off-beam echoes from actual submarines is satisfactorily explained, the ap- plicability to actual echo ranging of the results ob- tained with the indirect optical and acoustical tests is open to question. Chapter 24 SURFACE VESSEL TARGET STRENGTHS Me LESS 18S KNOWN about surface vessel tar- get strengths than about submarine target strengths. Few measurements have been made of the sound-reflecting characteristics of ships, and much of the available information has been extracted from experiments where the investigation of the target strength was only incidental to other studies. No mathematical analyses or measurements on scale models have been attempted, so that all target strengths reported here are the results of direct measurements. Experimental conditions have been far from con- trolled during these measurements. Ship speed, course, range, and especially aspect angle have been difficult either to estimate accurately or to maintain closely. Many of the tests were made completely at random on vessels happening to pass in the vicinity. Various types of ships served as targets — destroy- ers, freighters, tankers, coal colliers, transports, and Liberty ships — with the result that although many measurements were made, the data on each ship are too scanty to afford a comparison between different ships. Many variables might have significantly af- fected the measured echo levels — ship speed, length, width, draft, hull curvature, course, range, aspect angle, sea state, wind force, temperature gradients — so many that a clear-cut separation of variables is out of the question. Furthermore, the results are so few in number, compared with other underwater sound measurements, and the scatter of values is so wide, that only the most tentative and general con- clusions may be suggested at the present time. One of the most important generalizations that may be suggested is the difference between the re- flecting properties of moving vessels and still vessels. Ships under way are known to entrain air along their sides as they move through the water (see Section 27.3) With still vessels, on the other hand, en- trained air seems less likely. Since small air bubbles are extremely efficient scatterers of sound, it is rea- sonable to expect that sound striking a moving vessel 422 might be scattered diffusely by the air bubbles along the sides, just as sound is scattered by the wake laid by the ship. A still vessel, however, might be ex- pected to reflect sound specularly. Such an hypothe- sis seems to bring some coherence into the observed data. Therefore, it is largely from this point of view that surface vessel target strengths are examined, in this chapter, as a function of aspect angle, range, ship speed, ship type, pulse length, and frequency. 24.1 TECHNIQUES OF SAN DIEGO MEASUREMENTS Surface vessel target strengths have been measured by only two groups, the University of California Division of War Research at the U. S. Navy Radio and Sound Laboratory, San Diego, California [UCDWR]], and Bell Telephone Laboratories, New York, New York [BTL ].! The measurements off San Diego were made from November 12 to 17, 1943, during a program investigating the acoustical prop- erties of wakes laid by ships at various speeds? During these tests the USS Jasper (PYci8) echo ranged on two flush-decked World War I destroyers, the USS Crane (DD109) and the USS Lamberton (DMS82, ex-DD119), which followed straight courses at speeds of 10, 15, and 20 knots in deep water. A standard Navy JK transducer was used, sending out pulses 10 msec long at a frequency of 24 ke. The Jasper ranged on the destroyer as it ap- proached; then, just as the beam of the destroyer passed, the Jasper began to range on its wake. Con- sequently no target strengths were measured at as- pect angles beyond about 110 degrees from the bow. Errors in the estimated aspect angles were quite large because of unknown deviations of the destroyer from its normal course but could not be evaluated. Ranges varied from 112 to 660 yd. Echoes from the destroyer and its wake were re- ceived and recorded oscillographically on moving picture film, with the equipment employed in the TECHNIQUES OF NEW YORK MEASUREMENTS reverberation studies. The average maximum ampli- tude of five successive echoes, together with the calibration constants of the equipment, was used to compute the echo level at each aspect angle; an auxiliary transducer measured the source level be- fore and after each run. However, the transmission loss was not measured directly. Transmission condi- tions were fair, since the water was isothermal to a depth of about 50 ft. Accordingly, inverse square divergence and an attenuation coefficient of 5 db per kyd were assumed in estimating the transmission loss. Each target strength was computed from the average echo level of five echoes; each range was the average range over the five echoes, as measured on the oscillograms. Aspect angles were estimated trigonometrically. 24.2 TECHNIQUES OF NEW YORK MEASUREMENTS Two series of tests were made by BTL on ships in Long Island Sound early in 1944, as part of a specific development project to study the effects of short pulse lengths and receiver bandwidth on echo rang- ing,’ and to measure echoes from surface vessels.* Because little time was allocated to this part of the program, the work was discontinued as soon as enough data were obtained to establish the range of echo intensities to be expected. In the earlier measurements, made in Long Island Sound near City Island, Hart’s Island, and Execu- tion Light, pulse lengths from 0.05 to 150 msec were used at frequencies between 20 and 30 kc.* Echo- ranging gear including a transmitter and a receiving system of adjustable characteristics was mounted aboard a laboratory boat, the Elcobel, which was al- ready equipped with a standard Navy projector dome. Targets of these tests were various freighters in the vicinity. No absolute echo levels or target strengths were measured, since the experiment was conducted largely to investigate the effects of pulse ‘and receiver characteristics on reverberation, noise, and echo character. Relative echo amplitudes were found for different pulse lengths, however, and are reported in Section 24.7. Later studies reported in more detail the reflect- ing characteristics of a total of twenty surface ves- sels.4 In these measurements, a crystal transducer was mounted on the Elcobel, a 65-ft boat, in such a way that it could be carried just below the keel while under way, or lowered to a depth of 10 ft for echo 423 ranging. In the lower position, the transducer could be trained by means of a hand wheel on top of the shaft; however, the speed of the Elcobel could not exceed a few knots without interfering with the satis- factory operation of the transducer. An oscillator aboard the Elcobel delivered pulses approximately 3 msec long to the transducer, at a frequency of 27 ke. The echoes received by the trans- ducer were amplified, observed and photographed on the screen of a cathode-ray oscilloscope, whose hori- zontal sweep was proportional to the time — and therefore to the range of the echo — and whose ver- tical sweep was proportional to the amplitude of the echo. In order to obtain target strengths, the average range and the average peak amplitude of between 10 and 60 echoes were measured on the oscillograms. The transmission loss was estimated on the assump- tion of inverse square divergence and an attenuation coefficient of 7 db per kyd, although such an assump- tion probably was unrealistic since the water was shallow during these measurements and the ocean bottom was an effective reflector of sound. From the average range, the average peak amplitude, and the transmission loss, the diameter of the equivalent sphere was computed — the sphere which would theoretically return the same echo under the same conditions. Then the target strength was readily determined from the diameter of the equivalent sphere by use of equation (10) in Chapter 19. 24.2.1 Tests on Anchored Vessels In the first part of the second series of tests, the targets were ships at anchor in the tideway of Long Island Sound, near City Island, New York, where the water was less than 100 ft deep. Echoes from five freighters, a tanker, a Liberty ship, and a small British carrier were measured. During these tests the Elcobel was kept under way at a very slow speed, so that both the range and the aspect angle of the tar- get varied in almost all the tests. 24.2.2 Tests on Moving Vessels The Elcobel also ranged on moving ships farther out in Long Island Sound, in the vicinity of Lloyd’s Neck, Long Island. These tests were made, without any advance arrangements, on passing ships whose courses brought them close enough to the Elcobel to make them satisfactory targets. Unfortunately, the 424 speed of the Elcobel had to be held down to a few knots when the transducer was in operation, while the target ships were traveling at least several times faster. Consequently a special procedure was de- veloped to fit these conditions. As the ship approached, the Elcobel maneuvered so that it neared the ship at an aspect angle just off the bow of the target ship. When the range was closed to about 600 yd, the test began, and the Elcobel followed a course that kept the transducer constantly aimed at the stern of the ship; the sound beam was wide enough to cover the entire ship even at close ranges. It is possible that echoes were also obtained from the wakes of the ships, although such echoes were probably distinguishable from ship echoes for pulse lengths of 3 msec. Observations were made as frequently as possible, and the range and aspect angle were estimated at the time of the ob- servations. Actually both ships deviated from their nominal courses because of the effects of the wind and sea state as well as inaccuracies in steering, so that the ranges and aspect angles changed rather irregularly. SURFACE VESSEL TARGET STRENGTHS expected to depend on aspect angle in much the same way as the target strength of a submarine depends on its aspect. In fact, since most surface vessels are more nearly flat at beam aspects than submarines, a sharper dependence might be predicted as long as re- flections come exclusively from the hull, in other words, as long as the ship is anchored, or moving through the water very slowly, and gives rise only to specular reflection. If a moving ship reflects sound diffusely, some change of target strength with aspect angle might be expected, but not so marked a change close to beam aspect as results from specula re- flection. Table 1 lists beam and off-beam target strengths for still and moving ships, together with the ranges at which they were measured and the number of in- dividual observations — each comprising at least five echoes — which were averaged to obtain the tabulated results. Here the values given for beam target strength include all measurements at esti- mated aspect angles between 70 and 110, and be- tween 250 and 290 degrees from the bow, whereas off-beam target strengths include measurements at TaBLE1 Aspect dependence. Beam Off-beam target strengths Number target strengths Number Range and of Number | Range and of Number in standard deviations | obser- of in standard deviations | obser- of Test yards in decibels vations ships yards in decibels vations ships Anchored ships (New York) } | 250-587 37.3 + 16.3 8 5 168-508 13.3 + 7.6 23 9 Moving ships (San Diego) ! | 112-640 20.8 + 5.7 42 2 140-660 17.2 + 4.9 35 2 Moving ships (New York) ! | 250-490 16.2 + 6.1 62 12 300-562 13.7 + 5.1 66 12 24.3 ASPECT DEPENDENCE Surface vessel target strengths have been measured at San Diego and New York for different aspect angles, ranges, speeds, types of ships, pulse lengths, and frequencies. A dependence on aspect angle and on range is suggested by the reported data; in addi- tion, the target strength of ships under way appears to be considerably different from the target strengths of still vessels. Sufficient information, however, is not available to permit evaluation of the effects of the class of ship, pulse length, or frequency on the target strengths measured. The target strength of a surface vessel might be all other aspect angles. Ranges, speeds, ship types, pulse lengths, and frequencies are not separated. Table 1 is illustrated graphically in Figure 1. Still Vessels Only for the anchored ships is the difference be- tween beam and off-beam target strengths roughly the same as the scatter of the observations, as repre- sented by the standard deviation. The dependence of target strength on aspect angle for the individual measurements on these still vessels is shown in Fig- ure 2. Two target strengths at an aspect angle of ap- proximately 100 degrees are conspicuously higher 24.3.1 ASPECT DEPENDENCE 425 TARGET STRENGTH IW DECIBELS STERN Test San Diego measurements on moving vessels. — — — New York measurements on anchored vessels. —-—-— New York measurements on moving vessels. Curve Figure 1. Aspect dependence for all tests. than values at any other aspect, almost 20 db higher than the next highest value, and more than 40 db higher than the average target strength at off-beam aspects. Such a peak would be expected theoreti- cally from specular reflection from the broadside of the ship at beam aspect; at a few degrees away from beam aspect, however, the target strength should be markedly reduced. This peak in Figure 2 may be exaggerated for two reasons, so that the actual aspect dependence may not be so sharp as it appears. First, the two observa- tions constituting the peak were made at ranges be- tween 500 and 660 yd, but most of the other observa- tions were made at much closer ranges. Since the sur- TARGET STRENGTH IN DECIBELS le} 20 40 60 80 100 120 140 160 180 BOW BEAM STERN ASPECT ANGLE IN DEGREES Figure 2. Aspect dependence for still vessels (New York). face vessel target strengths were also found to de- pend on the range, increasing as the range increased, especially at beam aspect, these high values may be more the result of the effect of the range on the tar- get strength, than the effect of the aspect angle of the target. The data were too few to permit separa- tion of these two factors and independent evaluation of the effect of each on the measured target strengths. Secondly, so few observations are plotted in Fig- ure 2 that two very marked peaks may not be re- liable. Only 31 values were obtained in the New York tests on anchored vessels, and only 8 of these were at aspects within 20 degrees of the beam. In view of the large scatter of values, the observations cannot be considered conclusive, but are at least gen- erally consistent with the theoretical expectation that still vessels reflect sound specularly. 24.3.2 Moving Vessels The dependence of target strength on aspect angle for moving vessels is much less than the dependence found for still vessels. The small variation in Table 1 is too small to be very significant. For comparison with Figure 2, the individual target strengths for moving ships measured at New York are plotted in Figure 3 and for the destroyers measured at San Diego in Figure 4. The scatter is so large that any possible systematic variation of target strength with aspect angle is largely obscured. Analysis of the San Diego data on moving vessels 426 SURFACE VESSEL TARGET STRENGTHS 40 TARGET STRENGTH IN DECIBELS = i) ol (o) {o) (o) (@) 20 40 60 80 100 120 140 160 180 BOW BEAM STERN ASPECT ANGLE IN DEGREES I'iguRE 3. Aspect dependence for moving vessels (New York). TARGET STRENGTH IN DECIBELS BEAM ASPECT ANGLE IN DEGREES Figure 4. Aspect dependence for moving vessels (San Diego). leads to much the same results, which are illustrated in Figure 4. Since the target strengths measured at San Diego were found to depend markedly on range (see Section 24.4), they were separated according to range in order to examine the dependence on aspect angle. The data were broken down into three groups, for ranges of 100 to 300 yd, 300 to 500 yd, and 500 to 700 yd, and each group was analyzed for a possible dependence on aspect angle. Average target strengths for aspects within 20 degrees of the beam, and for all other aspects are shown in Table 2 for each range group. TaBLE 2. Aspect dependence at different ranges for moving destroyers (San Diego). Beam Off-beam target strength target strength and and Range standard deviation standard deviation in yards in decibels in decibels 100 to 300 12.3 + 1.8 13.0 + 1.7 300 to 500 22.5 + 3.6 16.9 + 4.8 500 to 700 23.9 + 3.6 20.9 + 3.7 The scatter of the individual values from the aver- ages in Table 2 is less than the scatter from the overall averages in Table 1, and a slight change of target strength with aspect seems significantly shown. Some dependence of this nature might be expected from a diffusely reflecting surface, if all the target were in the path of the sound beam. However, any change with aspect angle as great as that found for submerged submarines seems ruled out by Table 2. These results are generally consistent with the hypothesis that bubbles along the side of the ship are responsible for the echoes observed from moving ships. More accurate data would be required, how- ever, for verification of this theory. RANGE DEPENDENCE In Sections 20.4.4 and 23.4.1 it was pointed out. that the target strength of submarine depends on the range at ranges less than the maximum radius 24.4 427 RANGE DEPENDENCE 250 300 350 400 450 500 550 600 RANGE IN YARDS 200 150 t Mm N Figure 5. Range dependence at beam aspects for anchored vessels (New York). $71381930 nN NI HLONSYLS L3A9YVL 400 450 500 550 350 RANGE IN YARDS 200 250 300 ff-beam aspects for anchored vessels (New York). 150 100 Figure 6. Range dependence at o 428 TARGET STRENGTH IN DECIBELS SURFACE VESSEL TARGET STRENGTHS. RANGE IN YARDS Figure 7. Range dependence at beam aspects for moving vessels (San Diego). LS rr) iN OECIBE TARGET STRENGTH RANGE IN YARDS FicureE 8. Range dependence at off-beam aspects for moving vessels (San Diego). TARGET STRENGTH IN DECIBELS 300 RANGE IN YARDS 350 FicurE 9. Range dependence at beam aspects for moving vessels (New York). of curvature of the submarine. The target strength of a still ship would be expected to behave in the same way under similar conditions. Because ship hulls may be flatter and may have a larger radius of curvature than submarines, this dependence on range might extend to much longer ranges than for submarines. On the other hand, the target strength of a moving ship might be expected to increase as the range increases, as more and more of the scattering surface lies in the path of the direct sound beam. Accordingly, target strength was examined as a function of range, for beam and off-beam echoes, for all three sets of data. The results of this analysis are illustrated in Figures 5 to 10, where in each graph the solid line represents the least squares solution based on an assumed linear relation between the tar- get strength and the range. The slopes of these lines are listed in Table 3. It is apparent that in all cases the dependence of target strength on range is most pronounced (1) for still vessels and (2) at beam aspect. Three explanations may be suggested to account for the increase in target strength with range: (1) failure of the sound beam to cover the target at short RANGE DEPENDENCE 429 TARGET STRENGTH IN DECIBELS 200 250 300 350 RANGE IN YARDS 400 450 500 Figure 10. Range dependence at off-beam aspects for moving vessels (New York). ranges; (2) reduced reflection as the range approaches the dimensions of the target; and (3) incorrect evalu- ation of the transmission loss. The first effect applies only to the measurements on moving vessels at San Diego, since the sound beam used during the New York tests was wide enough to cover the target at all ranges. The second applies primarily to measure- ments on anchored vessels where specular reflection seems most likely to occur, and the third applies to measurements on both moving and still ships. Tas Le 3. Range dependence. Slope of target strength-range curve in decibels per kiloyard Test Beam aspect Off-beam aspects Anchored ships (New York) 105.0 68.0 Moving ships (San Diego) 30.5 24.4 Moving ships (New York) 26.3 4.5 At short ranges, how much of the target the sound beam covers depends on the dimensions and aspect angle of the target and on the directivity pattern of the transducer. At San Diego, a standard JK trans- ducer was employed, which had a total beam width of 20 degrees between points on either side of the axis where the response was 10 db lower. If it is assumed that the sound beam was 20 degrees wide and that the destroyer was 300 ft long, then the sound beam did not cover the ship, at beam aspect, at ranges less than about 300 yd. Since many of the beam target strengths were measured at shorter ranges, this failure of the sound beam to cover the ship may account for the decrease in target strength with decreasing range. 24.4.1 Transducer Directivity To evaluate the effect of the transducer directivity on the target strength-range dependence, the differ- ence between the echo level from a destroyer at beam aspect and from a small target always within the sound beam was calculated, as a function of range, from the directivity pattern of the transducer. This difference is expressed as Bg) dx ca 10 log f @- 2 10 log |; (1) where 2; is the length of the target in a direction per- pendicular to the sound beam; b?(¢) is the composite directivity pattern of the transducer; and r the range to the center of the target. This difference in decibels between the echo level from the destroyer and the echo level from a small target of the same target strength, as the range is decreased from 650 to 100 yd, is superimposed on Figure 7 as a broken line. The zero level, where the sound beam effectively covers the entire target and the two echo levels are the same, is placed at a target strength of 23.5 db, which is the average beam target strength measured at San Diego at ranges of 450 yd and greater. The difference calculated from equation (1) amounts to about 7 db at a range of 100 yd, and drops to less than 1 db at ranges greater than 500 yd. This analysis does not take into account the ex- tension of the target by the wake. However, even if the target were assumed to extend infinitely in one direction, the target strength for long pulses would increase only as 10 log r and would not be signifi- cantly different from the broken curve in Figure 7. The increase of target strength with range in such a case would be analogous to the similar increase for the target strength of wakes discussed in Section 33.1.1. This failure of the sound beam to cover the en- tire target, especially at short ranges, is responsible for much of the dependence of target strength on range observed at San Diego at beam aspect. Apparently, however, it is not responsible for all the dependence observed. Significantly, these echoes from destroyers at beam aspect are approximately as strong as echoes ob- served from the wakes directly behind the destroy- ers. In Section 24.1 it was noted that echo-ranging experiments were made on the wakes after the de- stroyers passed; in these measurements, the sound beam was perpendicular to the axis of the wake. The wake echoes showed the same variation with 430 range as the destroyer echoes, and, like them, showed no significant dependence on speed. Differences as great as 10 db were observed between the wake echoes and the destroyer echoes immediately pre- ceding them, but these differences appeared to be quite random and unsystematic. This equivalency between wake echoes and destroyer echoes is con- sistent with the theory that both arise from scatter- ing by small bubbles. The dependence of target strength on range at off- beam aspects from the San Diego results shown in Figure 8 cannot be analyzed very simply. In the first place, the spread of aspect angles covered is very wide; the projected length of the destroyer, measured in a direction perpendicular to the sound beam, varied from 30 ft at bow and stern aspects to 290 ft at an aspect angle 20 degrees from the beam. In the second place, with a pulse length of 10 msec, the en- tire destroyer was not in the sound beam at the same time, especially at bow and stern aspects; for aspects close to the beam, this effect of pulse length may be neglected, but it becomes important at other aspects. When more of the target comes into the sound beam, the observed echo from a very short pulse will not be stronger but instead will last longer, as pointed out in Section 19.3. Thus the change of target strength with range at off-beam aspects cannot be explained even in part by this simple mechanism. Transducer directivity is also relatively unim- portant in the New York measurements on still and moving vessels since a very wide beam was employed. The total horizontal beam width of the combined projector-hydrophone directivity pattern, between points where the response was 10 db lower than on its axis, was about 40 degrees, which even at a range of 168 yd, the shortest range at which measurements during either test were made, still covers the long- est ship at beam aspect. Therefore the decrease in target strength with decreasing range in the New York measurements cannot be explained as a result of the failure of the sound beam to cover the target. 24.4.2 Predicted Dependence The second explanation which might be suggested for the observed dependence of target strength on range is the predicted decrease of specular reflection with decreasing range for ranges less than the maxi- mum radius of curvature of the target, providing the reflection is specular (see Section 20.4.4). This effect would apply only to echoes from vessels sta- SURFACE VESSEL TARGET STRENGTHS tionary in the water, which presumably arise prima- rily from the hull and not from a uniformly scatter- ing layer. However, in the most extreme case, reflec- tion from an infinite plane surface, the target strength will not vary more rapidly than as the square of the range. Such a variation is quite insufficient to ac- count for the large effect observed during echo- ranging trials on still vessels illustrated in Figures 5 and 6. Qualitatively, however, it partly explains the difference in range dependence at beam and off-beam aspects, since the radius of curvature of the ship is greater when it presents its broadside to the incident sound than when it is at bow or stern aspect. 24.4.3 Transmission Loss A third possible explanation of the observed range dependence is a possible incorrect evaluation of the transmission loss. In none of the measurements was the transmission loss measured directly. Instead, an attempt was made to estimate it from the prevailing conditions on the basis of inverse-square divergence and an additional attenuation proportional to the range. At San Diego, it was assumed that the intensity of the echo was inversely proportional to the fourth power of the range, weakened by an additional loss of 5 db per kyd of sound travel. The water was isothermal to a depth of 50 ft, so that an assumption of 5 db per kyd for the attenuation coefficient seems somewhat low. Use of equation (1) in Chapter 23 gives an attenuation coefficient of about 7 db per kyd. An attenuation coefficient of about 10 db per kyd would be required to explain the departure of the plotted points from the theoretical curve in Fig- ure 7. Such a high coefficient does not seem very likely when the surface layer is isothermal down to a depth of 50 ft, but it is not impossible. At New York, the transmission loss was assumed to follow the same inverse square loss with an atten- uation coefficient at 27 ke of 7 db per kyd. The tem- perature conditions of the water were not known; the wird velocity varied from 1 to 23 mph. Whether or not the assumed attenuation coefficient is reliable it is difficult to say. In addition, bottom-reflected sound may have had a marked effect on the trans- mission loss. Conditions were very favorable to bottom reflec- tion during these New York tests. The bottom was composed of sand and mud, a mixture which reflects sound very effectively. In addition, the water was DEPENDENCE ON SHIP TYPE very shallow, from 60 to 110 ft deep. The sound beam was not highly directional; the total vertical beam width between points where the response was 10 db down was 20 degrees. Thus if the transducer were level, the sound beam would strike the bottom at a range of only about 126 yd, for water 60 ft deep. Consequently the bottom undoubtedly reflected part of the incident sound in much the same way as the surface, and contributed to the intensity of the echoes received at the transducer. Assume, for example, that both the surface and bottom reflected sound perfectly, so that at the particular ranges used the sound beam could spread in only one direction — horizontally. In this extreme case, the intensity of the echo would be inversely pro- portional, not to the fourth power of the range, but to the square of the range. This assumption, of course, is not realistic, but the result suggests that for the New York measurements the actual drop is some- where between inverse fourth and inverse square; perhaps the echo intensity actually varies more nearly inversely as the cube of the range over a shal- low reflecting bottom. This relatively slow increase of transmission loss with increasing range may ac- count for much of the range dependence for moving vessels in the New York data. Even an inverse square dependence of echo level on range fails to account, however, for the observed variation on still vessels shown in Figures 5 and 6, where the echo level actually imcreases rapidly with increasing range. Another possibility might account for the depend- ence of target strength on range for stationary ves- sels. Sound incident on the hull of the ship will be reflected downward where the hull is curved slightly downward, then reflected upward from the bottom. It is possible that the curvature of the hulls of the surface vessels measured is such that the rays re- flected to the bottom will strike the bottom and be reflected back to the transducer only at longer ranges, so that target strengths measured at long ranges will be greater than target strengths measured at short ranges. This explanation may account for the stronger range dependence for stationary ves- sels than for moving vessels, although it must be re- garded as highly tentative in the absence of further substantiating evidence. In all, it has been well established that the target strength of surface vessels on which measurements have been made apparently increases with range. This increase is much greater at beam aspects than 431 at off-beam aspects. The exact rate of increase is un- certain because many causes are responsible; meas- ured rates vary from 4.5 to 105 db per kyd at ranges between 200 and 500 yd. The dependence of target strength on range arises from (1) smaller coverage of the target by directive transducers at close ranges; (2) incorrect evaluation of the transmission loss neglecting surface and bottom reflections; and (3) the dimensions and curvature of the target, in so far as they reduce specular reflection at close ranges. Probably none of these effects, however, can explain the enormous observed range dependence for an- chored vessels. Further measurements would be re- quired to show the extent to which this observed effect is generally found. 24.5 DEPENDENCE ON SPEED Very little information is available on the varia- tion of target strength with the speed of the ship, for moving vessels. At San Diego, speeds of 10, 15, and 20 knots were employed; Table 1 lists target strengths without separating the speeds at which they were BSS fe) ©) TARGET STRENGTH IN DECIBELS Ny fo) 0 100 200 300 400 500 600 RANGE IN YARDS Ficure 11. Range dependence at different speeds for beam aspect (San Diego). measured. Figure 11 shows beam target strengths plotted as a function of range for three different speeds, 10, 15, and 20 knots, from the San Diego measurements. The dependence on range is evident, even in only twenty observations, but no significant dependence on ship speed is apparent. Ship speeds were not estimated or measured in the New York tests. As already mentioned, this same lack of de- pendence on ship speed is characteristic of wake echoes. 24.66 DEPENDENCE ON SHIP TYPE No clear dependence of target strength on ship type is indicated by the evidence now available. While a large number of different vessels have been a) w Q o a 2 x= = 2 i = o tS w ae) az oO 20 40 60 80 100 120 140 160 _ 180 © Bow BEAM STERN ASPECT ANGLE IN DEGREES Length Draft Water Range Curve Ship Tonnage in in depth in feet feet in feet yards Navy Transport 12,000 325 25 100 300-510 ——z Liberty Ship 10,000 300 20 60 300-450 Ficure 12. Variation in target strength between simi- lar ships (New York). 40 RELATIVE ECHO LEVEL IN DECIBELS SURFACE VESSEL TARGET STRENGTHS their estimated tonnages, lengths, and drafts; fur- thermore, both were measured at roughly the same ranges. It is possible that the fluctuation and varia- tion normally encountered in underwater sound transmission may be responsible for the 10 db differ- ence between the two curves, or that bottom reflec- tion may be the cause, since the transport was under way in water 100 ft deep while the Liberty ship was under way in water almost half as deep. Even per- fect bottom reflection, however, cannot account for the observed difference between the two curves, which suggests faulty calibration, widely variable transmission, or large unsuspected systematic differ- ences between the two ships. Zane Nl PAL HH mine LENGTH IN MILLISECONDS Range in yards Approximate aspect angle in degrees = 180 2200 180 Figure 13. Effect of pulse length on measured echo levels (New York). made,’ the scatter is so great that any correlation be- tween target strength and ship draft and tonnage is obscured. As an example of the variation in target strength between one ship and another, as measured at New York, Figure 12 illustrates target strength plotted against aspect angle for two large ships of nearly equal dimensions. The difference in their target strengths cannot be attributed to the difference in 24.7 DEPENDENCE ON PULSE LENGTH AND FREQUENCY Although surface vessel target strengths have not been systematically investigated as a function of pulse length, early studies at New York reported a dependence of echo amplitude on pulse length for pulse lengths of 0.05 to 110 msec.* The results of these measurements are reproduced in Figure 13, DEPENDENCE ON PULSE LENGTH AND FREQUENCY where the relative echo level in decibels is plotted against the pulse length for four freighters. Little dependence on pulse length is evident for pulses more than 10 msec long, in qualitative agreement with the results described in Section 23.5.2 applying to submerged submarines. However, for pulse lengths of less than 10 msec, the echo level drops rather sharply. More data are required, however, to show how great this dependence will be for any actual vessel. No information is available on how surface vessel target strengths vary with the frequency of the echo- ranging beam employed. The only tests were made 433 at San Diego at 24 ke and at New York at 27 ke; any difference in the target strengths at these two fre- quencies would probably be very small, from theo- retical predictions, and the actual measured differ- ence is too small to verify any such dependence. For still vessels, if the echo comes from the hull, very little variation of target strength with frequency would be expected (see Sections 20.2 and 20.3). For moving vessels, however, with sound scattered from a layer of bubbles, the target strength would be ex- pected to vary with frequency in accordance with the acoustic properties of small bubbles. Chapter 25 SUMMARY 25.1 DEFINITION OF TARGET STRENGTH Ke THE PURPOSES of discussing the reflecting characteristics of different vessels, the target strength T of a target is defined by T=E—S-4+ 2H, (1) where E is the echo level, S the source level, and H the one-way transmission loss from the source to the target, all in decibels (see Section 19.1.3). For most targets, T is independent of range at ranges much greater than the dimensions of the target (see Sec- tions 20.4 and 23.4), but may change with the chang- ing orientation of the target relative to the sound beam (see Section 23.1). 25.1.1 Echo Level The echo level E is defined by E = 20 log p., (2) where p- is the rms pressure of the echo, in dynes per square centimeter averaged over a few cycles (see Sec- tion 19.1.3). If the rms pressure is not constant dur- ing the echo, E is defined as the peak rms pressure. 25.1.2 Source Level For directional supersonic projectors, the rms pres- sure p of the sound on the axis of a projector is in- versely proportional to the square of the range 7, as long as the range is much greater than the dimen- sions of the target and as long as the range is small enough so that attenuation and surface reflection may be neglected. Under these conditions, the source level S is defined by S = 20 log p + 20 log r, (3) where p is the pressure of the sound on the axis of the projector, in dynes per square centimeter, at a distance r, in yards, from the projector (see Section 19.1.3). 434 25.1.3 Transmission Loss ‘The difference between the pressure level of the transmitted sound at some point, and the source level is called the transmission loss H from the projector to that point (see Section 19.1.2). 25.1.4 Average Values Since both # and H often fluctuate by as muck as 10 db from pulse to pulse, it is customary to use the average echo amplitude in determining FE, and the average pressure amplitude at the range r in deter- mining H, in equation (1), where the average ampli- tude is the average of a number of peak rms ampli- tudes, if the rms amplitude is not constant over each echo (see Section 21.6.4). 25.1.5 Target Strength of Sphere A sphere reflects a plane wave equally in all direc- tions (see Section 19.2.2). The target strength of a sphere is T = 20 log : (4) where A is the radius of the sphere in yards (see Sec- tions 19.2.1, 19.2.2 and 20.4.1). This formula is accu- rate to 0.5 db if the range to the sphere is greater than ten times its radius, and if the wavelength of the sound is less than the radius of the sphere. 25.1.6 Target Strength of a General | Convex Surface The target strength for specular reflection from any convex surface is AA, Cae is where A; and A, are the principal radii of curvature of the target surface at the point where the surface is T = 10 log BEAM ECHOES FROM SUBMERGED SUBMARINES 435 perpendicular to the sound beam, and r is the range (see Section 20.4.2). This formula is valid only if both A; and A» are greater than the wavelength of the sound, and if either A; or Az is much less than r. 25.1.7 Target Strength of a Cylinder For a cylinder, A: is infinite in equation (5) and the target strength becomes A T = 10 log uf Con. | This equation is valid only when the cylinder radius A, is less than r and the wavelength is less than Ai. 25.2 BEAM ECHOES FROM SUBMERGED SUBMARINES At aspect angles within about 20 degrees of the beam, echoes from submarines are produced prima- rily by specular reflection from the pressure hull, the fuel and ballast tanks, and the conning tower (see Section 23.8.1) and are much stronger than echoes at other aspects. Oscillograms show that the echo generally reproduces the outgoing pulse (see Figures 25 and 27 in Chapter 23). 25.2.1 Beam Target Strengths Observed submarine target strengths at beam as- pects and at long ranges lie mostly between 20 and 30 db. About 25 db is the average value (see Section 23.1.1); typical values of A: or A» in equation (4) which would correspond to this target strength would be 500 and 2.5 yd respectively. The observed spread of values may result entirely from experimental errors. Off-beam target strengths, found at aspects 20 degrees or more away from the beam, are reported in Section 25.3. VARIATION WITH SUBMARINE CLASS Observed differences in the target strengths of different submarines measured both directly and in- directly are less than the estimated experimental error in the direct measurements (see Section 25.2.1). Consequently no reliable overall evaluation of the dependence of the target strength on the class of sub- marine can be made. VARIATION WITH SUBMARINE SPEED No significant variation of target strength with submarine speed is expected, since the wake of a sub- merged submarine is a poor reflector of sound (see Section 33.3). No pronounced variation has been ob- served in practice for submerged speeds from 1 to 6 knots at keel depths of about 100 ft (see Section 23.3). VARIATION. WITH RANGE Theoretically, beam target strengths depend on the range at ranges less than the principal radii of curvature of the submarine at beam aspect (see Sec- tions 20.4.4 and 23.4). For a 517-ton German U- boat, approximated by an ellipsoid with principal radii of curvature of 576 and 2.3 yd, the variation of target with range predicted from equation (5) is shown in Table 1. Although no observations are TaBLE 1. Theoretical range variation. Submarine target strength Submarine Range beam aspect target strength in (without conning beam aspect yards tower) (with conning tower) 8 5.5 5.8 12 7.5 74 16 8.9 9.1 200 19.2 22.9 1,000 23.2 25.5 (o-) 25.2 Bane available to confirm this variation with range, the result is believed to be reliable. VARIATION WITH PULSE LENGTH No marked dependence of target strength on pulse length is expected at beam aspect, since the echo approximately reproduces the pulse (see Section 23.5.1). The available evidence is neither very con- sistent nor conclusive, but does not demonstrate any sharp variation in the target strength with the pulse length (see Section 23.5.2). VARIATION WITH FREQUENCY No variation of target strength with frequency is expected theoretically at beam aspects for specular reflection (see Sections 20.4 and 23.6.1). Observa- tions confirm this prediction (see Section 23.6.2), ex- cept for a few measurements at 60 kc; these 60-ke target strengths, however, are so large that calibra- tion errors are believed responsible. 25.2.2 Echo Structure Generally, beam echoes are square-topped and resemble the outgoing pulses (see Section 23.8.1) For very short pulses, beam echoes from submarines 436 SUMMARY reveal a definite structure. For observations on one S-boat, the main echo consists of two components separated by a distance of about 4 yd; the first com- ponent may come from the broadside of the sub- marine, while the second component may be an echo from the bilge keel or conning tower. After this main echo comes a much weaker secondary echo, pre- sumably resulting from sound reflected from the submarine straight up to the surface, back down to the submarine, and then back to the projector (see Figure 9 in Chapter 21, and Figure 29 in Chapter 23). The presence of this echo structure will be expected to modify slightly the conclusions in the preceding section, since for long pulses the different components will combine. Such a combination will increase the average target strength 3 db at most above its value for very short pulses. 25.2.3 Fluctuation The fluctuation of beam echoes may be primarily attributed to the fluctuation in the transmission of the outgoing and incoming sound (see Section 21.6). Much of this fluctuation is apparently due to the presence of surface-reflected sound (see Section 21.5.4). Estimates of the fluctuation of transmitted sound are given in Chapters 7 and 10. In addition, for pulses more than a few milliseconds long, inter- ference between the different components of the echo will somewhat increase the fluctuation. 25.3 OFF-BEAM ECHOES FROM SUB- MERGED SUBMARINES At aspect angles more than about 20 degrees off the beam, echoes from submarines originate along the entire length of the vessel and probably result from both specular and nonspecular reflection (see Section 23.8.2); they are 10 to 15 db weaker than echoes at beam aspect. The echo does not reproduce the outgoing pulse (see Figures 25 and 28 in Chapter 23). 25.3.1 Off-Beam Target Strengths Observed submarine target strengths at off-beam aspects and at long ranges lie mostly between 5 and 20 db for pulses 100 or more msec long, and usually between 10 and 15 db (see Section 23.1.1). The spread of values is apparently real to some extent, since at different aspect angles echo characteristics are markedly different. At certain off-beam aspects and altitudes, strong specular reflections from nearly flat surfaces, such as the conning tower, may give target strengths greater than 20 db (see Section 23.2.2); these reflections depend critically on the particular submarine measured. VARIATION WITH SUBMARINE CLASS No variation in the off-beam target strengths of different submarmes has been observed in either the direct or indirect measurements to be greater than the estimated experimental error in the direct meas- urements (see Section 25.2.1). VARIATION WITH SUBMARINE SPEED No important variation of target strength with submarine speed has been observed at off-beam as- pects (see Section 25.2.1). VARIATION WITH RANGE At off-beam aspects, submarine target strengths decrease with decreasing range (see Section 25.2.1). At ranges less than the length of the submarine, off-beam target strengths are roughly equal to beam target strengths. Under such conditions, a submarine may be approximated by a cylinder at off-beam as- pects except bow and stern, and equation (6) may be used. VARIATION WITH PULSE LENGTH Since at off-beam aspects the echo does not usually reproduce the pulse and the echo length considerably exceeds the pulse length, for pulses 100 or more msec long, some variation of target strength with pulse length may be expected (see Section 23.5.1). Ob- served target strengths decrease with pulse length for signals shorter than 100 msec (see Section 23.5.2). The decrease is most marked for pulses shorter than 10 msec, but even for such short pulses the target strength does not decrease as rapidly as the pulse length, or rather, as rapidly as 10 log 7, where r is the pulse length in milliseconds. VARIATION WITH FREQUENCY No variation of target strength with frequency is expected at off-beam aspects (see Section 23.6.1). This conclusion is contradicted by some target strength measurements at a frequency of 60 ke, which give much higher results than similar measurements at 24 ke (see Section 25.2.1). However, the differences between beam and off-beam target strengths are about the same at 60 ke as at 24 ke, so that if the ob- served frequency effect is real, it is the same at all aspects, ECHOES FROM SURFACE VESSELS 25.3.2 Echo Structure At off-beam aspects, echoes from submarines do not reproduce the outgoing pulses because the entire length of the submarine reflects sounds (see Section 23.8.2). The duration of the echo, measured on an oscillogram, may be given by L T == cos0 +r, (7) where T is the duration of the echo, Z the length of the submarine, c the velocity of sound, 0 the aspect angle measured from the bow of the submarine, and 7 the pulse length. On a sound range recorder, how- ever, the echo length is about half that given in equation (7), perhaps because only the stronger part of the echo would be expected to show on a recorder using chemically treated paper (see Figure 25 in Chapter 23). 25.3.3 Fluctuation The fluctuation of echoes at off-beam aspects is due not only to fluctuations in the transmission of the sound each way (see Section 25.2.3), but also to fluctuations resulting from interference phenomena. The echo obtained from a long pulse will be the re- sult of constructive and destructive interference be- tween echoes from individual reflecting surfaces dis- tributed over the length of the submaring. Changes in this interference pattern as the aspect or altitude of the submarine changes slightly will increase the observed fluctuation of echoes. 25.4 ECHOES FROM SURFACE VESSELS Information on reflection from surface vessels is even more fragmentary than on reflection from sub- marines. The following conclusions are suggested by the data but cannot all be regarded as confirmed. Still Vessels Vessels at anchor seem to behave as targets in the same way as submerged submarines. At aspects close to the beam, target strengths may be very high, as 25.4.1 437 much as 40 db, but at other aspects, for pulses 3 msec long at a frequency of 27 kc, it is usually between 5 and 20 db (see Section 24.3.1). The strong echoes at beam aspect are presumably the result of specular re- flection from the hull of the ship. 25.4.2 Moving Vessels When a vessel is under way, beam echoes are about the same as off-beam echoes (see Section 24.3.2). Ob- served target strengths of moving destroyers and merchant vessels lie between 10 and 25 db, for pulses 3 and 10 msec long at frequencies of 24 and 27 kc; a systematic difference in the target strengths of differ- ent ships is not evident (see Section 24.6). An in- crease in speed from 10 to 20 knots apparently does not affect the target strength appreciably (see Sec- tion 24.5). A decrease in pulse length decreases the resultant target strength, especially for pulse lengths less than 10 msec, but the target strength does not drop as rapidly as 10 log r, where 7 is the pulse length (see Section 24.7). Echoes from moving vessels may arise from scat- tering by bubbles of entrained air along the side of the ship. This implies that the echo from a moving ship may be treated as an echo from a short stretth of wake (see Section 33.1.1). 25.4.3 Dependence on Range Most of the data on target strengths of moving vessels show a marked increase in target strength as the range increases from 200 to 600 yd, in one case amounting to more than 30 db (see Section 24.4). Although some increase is expected from the geom- etry of the ship (see Sections 24.4.2 and 25.2.1) and from the failure of the sound beam to cover the en- tire ship at short ranges (see Section 24.4.1), so marked a change seems greater than can be explained on any simple basis; it is quite possibly a statistical accident. Beyond about 600 yd, it is reasonable to assume that the target strength does not depend on the range, and that its value lies within the spread specified for surface vessel target strengths at off- beam aspects in the preceding section. PART IV ACOUSTIC PROPERTIES OF WAKES Sy sa aS Air MutLY | 7 Chapter 26 INTRODUCTION 26.1 WHAT ARE WAKES? i APPEARANCE of a streak of foamy, churned water behind a ship under way, known as the ship’s wake, is familiar to every mariner. Because the wake extends along the path of the ship over a length many times the ship’s length, it is hard to get a good view of the wake as a whole, even if a some- what elevated vantage point, such as the bridge or masthead, is chosen. Figure 1 shows the wake of an antisubmarine patrol vessel (PC488) in a quiet sea, as viewed aft from the crow’s nest. The observer in an airplane enjoys ideal condi- tions for the visual study of wakes. Figures 2 to 6 describe better than verbal descriptions what a wake looks like from a great height. The first four were taken from altitudes of 2,500 to 3,000 ft, the plane in level flight overtaking a destroyer, the USS Moale (DD693), which was proceeding on a straight course at constant speeds of 16, 20, 25, and 33 knots. By way of a scale, the ship had an overall length of 376 ft, a beam of 41 ft, and a draft of 13 ft. Figure 6 illustrates what happens to the wake as the ship turns; the foam on the curved section of the path is seen to be visually more dense, especially along the outer edge of the wake. As in turning, acceleration of the ship on a straight course increases the visual den- sity of a wake. Incidentally, the irregular white streaks appearing in Figures 4 to 6 are foam patterns on the water. All the photographs show the delicate bow-wave pattern, fanning out astern with a much greater angle of divergence than the actual wake. Figure 7 is a close-up taken from an altitude of 300 ft, of the bow wave and the wake of another de- stroyer, the USS Ringgold (DD500). The visible structure of the wakes laid by large ships does not differ markedly from that of the wakes of vessels of destroyer size, as may be seen in Figure 8, which gives a view from an altitude of 4,000 ft of a large aircraft carrier, the USS Saratoga (CV3). Beyond the obviously foamy and turbulent nature of wakes, visual inspection does not reveal any of their physical properties. The discovery that “wakes,” using this term in a loose sense, are capable of affecting the propagation of sound energy through the water has suggested a new distinction: an acoustic wake is defined as a volume of the ocean which has acquired, because of the passage of a ship through it, a greater though transitory capacity for absorbing and scattering sound. The expression ‘volume of the ocean” is used advisedly, because acoustic wakes have a definite vertical extension, often rather sharply bounded. Acoustic wakes under the surface, originating from submerged submarines, are of par- ticular interest. During a level run at periscope depth, the upper boundary of the wake, spreading out bodily from the screws, does not reach the ocean sur- face until several hundred yards behind the sub- marine. Several aerial views of submarine wakes, both during surface runs and after a crash dive, are reproduced in Figures 1 to 6 of Chapter 31. The temperature distribution in the top layer of the ocean may be disturbed by the passage of a ship in such a manner as to leave a thermal wake, detect- able by sensitive thermocouples. Evidently, experi- ments must decide to what extent thermal and acoustic wakes coincide with the body of water called a wake by a visual observer. This problem is dis- cussed in Chapter 31, dealing with the geometry of wakes. However, one interesting feature will be men- tioned here: the acoustic and thermal manifestations of a wake may persist over periods of half an hour or more, often long after visible traces of the ship’s passage have disappeared. 26.2 NAVAL IMPORTANCE OF WAKES Wakes can be important in naval warfare in two general ways. In the first place, they may interfere with the successful operation of acoustic devices, by scattering or absorbing sound. In the second place, they may provide a method for detecting, tracking, 441 442 INTRODUCTION ae iy Hi Ficure 1. Wake of submarine chaser (PC 488), seen from crow’s nest. NAVAL IMPORTANCE OF WAKES 443 EERE na Figure 2. Wake of USS Moale (DD 693) at 16 knots from 2,500 feet. or identifying the ship which has produced the wake. Such utilization of wakes in offensive operations com- prises visual detection from the air and thermal de- tection from surface ships or submarines, as well as acoustic detection. However, the present discussion is concerned only with the acoustic properties of wakes and the importance of these acoustic proper- ties in naval warfare. Acoustic interference produced by wakes is fre- quently encountered in the operation of sonar gear. False echoes from submarine wakes may confuse the sonar operator on an antisubmarine vessel and may even lead to an attack on a wake knuckle, a dis- turbance in the water when a submarine suddenly speeds up and turns sharply, while the submarine escapes. During thirty unsuccessful attacks on sub- marines by United States antisubmarine vessels in 1944, where the presence of a submarine was ascer- tained but no damage inflicted, 12 per cent of the failures were attributed to attacking wakes, a larger percentage than assigned to any other single cause. Wakes laid by surface vessels can also be disturb- ing in antisubmarine warfare. After one or more at- tacks in an area, echoes from old wakes from surface ships can be confusing. Moreover, a moderately fresh wake is highly absorbent and may shield a shallow target on one side of the wake from detection by a surface vessel on the other side. In fact a projector surrounded by a fresh wake is almost completely useless, since very little sound can escape through the wake. Thus a surface ship will commonly find that its echo-ranging equipment “goes dead’’ when the ship passes through a fresh wake. Harbor detection equipment can also be seriously hampered by the presence of wakes. When a de- stroyer at moderate speed passes in the neighborhood of bottom-mounted supersonic listening gear, ships passing by subsequently cannot be heard for some time. Similarly, sneak craft in the wake of a large sur- face vessel are very difficult to detect by echo rang- ing. To reduce the seriousness of these effects in combating submarines, or to use them most effec- tively in submarine warfare, accurate information is required on the reflection and absorption of sound by wakes under different conditions. The use of wakes in offensive operations against the wake-laying vessel is a relatively new field. As an example of this utilization of wakes, it was at one 444 INTRODUCTION cr nC. Figure 3. Wake of USS Moale (DD 693) at 20 knots from 2,500 feet. L IMPORTANCE OF WAKES Figure 5. Wake of USS Moale (DD 693) at 33 knots from 2,500 feet. 446 INTRODUCTION Figure 6. Wake of USS Moale (DD 693) as ship turns at 30 knots. time suggested that attacks on submarines could be made by detecting the wake and then following it until the submarine was reached. This suggested pro- cedure turned out to be impractical, owing to the very low scattering power of the wakes behind slow, deep submarines. The wake laid by a surface vessel reflects sound so strongly and so persistently that acoustic methods might possibly be useful for at- tacks on such enemy vessels. Obviously a knowledge of the scattering and absorbing power of wakes at different ranges behind a vessel, and at different depths below the surface, would be very useful in the design of equipment for such methods of attack. 26.3 ACOUSTIC WAKE RESEARCH The aim of current wake studies is twofeld: (1) to explore the overall acoustic properties of wakes with a view to possible tactical applications, and (2) to advance fundamental research on the structure and physical constitution of wakes. The second problem may seem rather academic to those who are prima- rily interested in the first one. But many questions about wakes presented by naval tactics cannot be answered satisfactorily, at present, for lack of. a thorough understanding of the physical constitution of wakes. Thus in the long run, fundamental re- search is indispensable for developing a comprehen- sive doctrine of the use of wakes in naval warfare. The solution of that fundamental problem in itself largely depends on acoustic measurements. Since wake research is still in an early stage, and since only incomplete observations are at hand, it would be im- practical to insist upon strict separation of these two aims. Experimental data frequently are relevant from the point of view either of tactical applications or of fundamental research. Accordingly, a certain shift back and forth between practical and theoretical em- phasis is unavoidable. In order to plan, execute, and interpret acoustic measurements on wakes, some working hypothesis concerning the nature of acoustic wakes must be used as a starting point. Three physical explanations of the causes of scattering and absorption of sound in the sea have been suggested. The scattering and absorbing centers have tentatively been identified ACOUSTIC WAKE RESEARCH Figure 7. Close-up of USS Ringgold (DD 500), from 300 feet. 448 INTRODUCTION Figure 8. Wake of USS Saratoga (CV3) from 4,000 feet. with (1) air bubbles of widely varying size; (2) tur- bulent motion in the sea, on a scale small compared with the dimensions of ships; and (3) thermal in- homogeneities or irregularities in the sea, also on a small scale. Although the bubble theory of acoustic wakes now enjoys general acceptance, it is difficult to put it toa conclusive test; it has been adopted rather by de- fault of the other two explanations. It would seem logical, therefore, to begin by presenting the evi- dence which shows that the turbulent and thermal microstructure of the sea does not provide an ade- quate explanation of the acoustic properties of wakes. However, in order to simplify the exposition, it is pref- erable to discuss first the physical mechanism of the formation and dissolution of bubbles in Chapter 27 and their acoustic properties in Chapter 28, and to defer the necessarily rather cursory treatment of the temperature and velocity structure of the sea until Chapter 29. The theoretical Chapters 27 to 29 com- prise the delineation of the working hypothesis which guides current wake research. Then the bulk of this volume (Chapters 30 to 33) describes the technique and the results of acoustic measurements made on wakes. In Chapter 34, the experimental data are interpreted in terms of the bubble theory; in other words, a test of the working hypothesis is under- taken. In the final Chapter 35, some conclusions which should be relevant in practice are drawn from the previous observations. Incomplete as the experi- mental foundations of some of these conclusions are, it appears useful to formulate some tentative general- izations as to the geometry and acoustic properties of wakes. Pending future research that may fill the con- spicuous gaps in our knowledge of wakes, such gen- eralizations should answer at least some of the ques- tions about wakes raised by the demands of naval tactics. Chapter 27 FORMATION AND DISSOLUTION Ae MAY BE ENTRAPPED mechanically at the ocean surface and dispersed in the form of bubbles; a familiar example is the appearance of white caps on a rough sea. A great dea] of air is also trapped along the waterline of any vessel under way. Proof that such entrained air is capable of producing acoustic wakes comes from experiments on the wakes of sailing ves- sels. Probably the most copious source of bubbles in wakes, however, is propeller cavitation at high speeds. 27.1 FORMATION OF BUBBLES BY CAVITATION When a cavity is created in water containing dis- solved air, gas enters the cavity by diffusion, and when the cavity collapses, this gas remains behind as a bubble. The process of underwater formation of bubbles, therefore, involves two quite different phenomena: (1) the mechanics of cavitation, and (2) the thermodynamics of diffusion and solution of gases in liquids. 27.1.1 Mechanics of Propeller Cavitation The phenomenon of propeller cavitation has long been known to engineers. According to hydrody- namical theory, cavities in liquids originate when certain patterns of flow produce regions of negative pressure near propellers. Such regions are set up in the vortices formed near the propeller tips, provided that the tip speed exceeds a certain critical limit, and also on the back side of the propeller blade. Hence, it is customary to speak of tip vortex cavitation and blade cavitation. These theoretical deductions have been verified experimentally by taking high-speed photographs of propellers running under water, shown in Figures 1, 2, and 3. By driving a propeller in an experimental chamber and observing it through a window, the process of OF AIR BUBBLES cavitation can be followed visually under strobo- scopic illumination. When the speed of the propeller is gradually increased, bubbles are seen first to form at the propeller tips, from which they spiral back- ward in a long stream. Then bubbles begin to cover the part of the blade closest to the tips, forming a sheet on the blade. This phenomenon is sometimes described as laminar cavitation, in order to distin- guish it from the formation of larger bubbles on the blade face nearer to the hub, called burbling cavita- tion, which starts at still higher speeds. Physically, there is no sharp distinction between laminar and burbling cavitation, and it would be more appro- priate to classify them together as blade cavitation. While persistent cavities are particularly likely to be formed in the tip vortices and on the propeller blades, cavitation also may be produced around sharp projections on the ship’s hull, especially during periods of sharp acceleration of the ship. For in- stance, white foamy spots have been observed visu- ally from a launch on the superstructure of a sub- merged submarine that passed at shallow depth. The appearance of the white spots did not suggest the re- lease of a stream of entrapped air; hence, the spots were tentatively attributed to cavitation occurring on the superstructure.! This result cannot be re- garded as general, since the submarine had not been submerged for a long enough time to justify assum- ing that all surface air entrained during the dive had been dislodged by the time of the observation. 27.1.2 Growing and Shrinking of Bubbles After a cavity has been formed in sea water which is saturated with air at an external pressure of 1 atmosphere, gas begins to diffuse into the vacuum from the surrounding liquid. Since the diffusion con- stants for oxygen and nitrogen are nearly equal, the gas collecting in the cavity must have the same com- position as that dissolved in the sea water. This com- 449 450 FORMATION AND DISSOLUTION OF AIR BUBBLES Figure 1. Cavitating model propeller. The picture was made with a 1/30,000-sec flash. Note the heavy tip vortices, considerable laminar cavitation near the blade tips, and the start of burbling cavitation of the blade face near the hub. This is a right-hand propeller and the water is flowing from left to right. position differs markedly from that of atmospheric air because the solubility of nitrogen is twice that of oxygen. Accordingly, the cavitation gas consists of 14 oxygen and 24 nitrogen. The quantity of gas which collects each second in a cavity in moving water is proportional to the surface area of the cavity and to the partial pressure of air dissolved in the surrounding water but is essentially independent of temperature and hydrostatic pressure. The con- stant of proportionality is roughly 4 x 10 mole per sq cm per second per atmosphere.” When the cavity collapses, the gas which has dif- fused into it will be compressed, and a bubble will be formed with a radius such that the gas pressure in- side equals the hydrostatic pressure outside. The cavities formed by blade cavitation collapse so quickly that any air bubbles formed must be very small indeed. However, the cavities originating in the tip vortices last much longer, since the centrifugal force in the whirling vortex remains high for some time. Thus, presumably it is the tip vortex cavita- tion that is primarily responsible for most of the air appearing as bubbles in propeller wakes. It has been observed that sea water at all depths contains dis- solved oxygen and nitrogen in amounts roughly cor- responding to saturation at the surface. For this reason it is undersaturated with respect to a bubble of air or cavitation gas anywhere below the surface, and a bubble of either gas will gradually disappear as the gas reenters the water. The rate of solution agrees with the same simple theory of diffusion as the rate of accumulation of gas in a cavity; indeed, the facts regarding the latter process are largely inferred from a study of the former. The number of moles of air which escape each second from a bubble is ap- proximately proportional to the surface area of the FORMATION OF BUBBLES BY CAVITATION 451 Figure 2. Cavitating model propeller. Picture made with a 1/30,000-sec flash. Shows heavy tip vortices extending down over leading edge and fairly wide area of blade covered by burbling cavitation. This is a right-hand propeller and the water is flowing from left to right. Figure 3. Cavitating model propeller. Short sections of the tip vortices are quite clear and the development, growth, progress, and disappearance of individual bubbles in the cavitation on the back of the upper blade can easily be followed. This is a right-hand propeller and the water is flowing from left to right. F 452 bubble and to the difference between the pressure in the bubble and the partial pressure of air dis- solved in the water. The constant of proportionality is again 4 < 10-° mole per sq cm per second per at- mosphere. An alternative formulation, assuming a spherical bubble, is in terms of the rate of decrease of the bubble diameter per second. In water saturated with air at the surface, this rate increases from 8 xX 10> em per sec at a depth of 5 meters to 18 X 10- cm per sec at a depth of 100 to 200 meters. Beyond these depths there is no further significant increase. RATE OF RISE IN CENTIMETERS PER SECOND a 0.005 0.01 0.02 0.05 Oo. 0.2 0.5 4 2 RADIUS OF BUBBLE IN CENTIMETERS Figure 4. Rate of rise of air bubbles in still water. A. Rectilinear motion, spherical shape. B. Helical and twisting motion, flattened shape. C. Irregular. D. Rectilinear motion, distorted mushroom shape. These theoretical ideas concerning the formation and dissolution of bubbles have been tested in a series of simple experiments; ? their agreement with the theory appears to be satisfactory. However, it remains uncertain to what extent these conclusions reached are applicable to the conditions prevailing in wakes. According to the experiments, a bubble 0.1 em in radius, which is the resonant size for 3 ke sound, should dissolve completely in about 20 min- utes. If the wake originally contains bubbles of all sizes up to 10 em radius, then as the smaller bubbles contract, the larger bubbles also decrease in size; and some bubbles of the smallest size should be found 20 minutes after the formation of the wake. In rough agreement with theoretical expectations, acoustic effects of wakes at supersonic frequencies are ob- served to persist over periods from 15 to 45 min- utes. In a wake, bubbles travel in a field of turbulent motion, rising gradually to the ocean surface where FORMATION AND DISSOLUTION OF AIR BUBBLES they may disintegrate; this process constitutes an- other important factor limiting the lifetime of wakes. The next point to be considered, therefore, is the buoyancy and the rate of ascent of air bubbles in sea water. 27.2 BUOYANCY AND RATE OF ASCENT The unimpeded rise of bubbles through still water has been analyzed in great detail.* From this analy- sis of all available experimental data and from certain theoretical considerations, a curve was constructed which gives the rate of rise of air bubbles in water as a function of the radius of the bubble and is repro- duced in Figure 4. It will be noted that the velocity reaches a maximum at a radius of about 0.1 em and varies only slightly with the radius thereafter. Sev- eral distinct types of motion and shapes of bubbles have been found to be characteristic in various ranges of bubble radii and are shown iv Figure 4. No exact delineation of these radius intervals can, how- ever, be made. All observers agree that tor very small bubbles the motion is linear. For large bubbles the motion is also approximately linear, although some irregularities have been reported. A noteworthy feature of the velocity curve for radii up to 0.04 cm is that it coincides with the empirical curve for the rate of fall through water of spheres of specific gravity 2. In connection with the laboratory experi- ments on bubble screens, which will be described in the next chapter, this relation between bubble radius and rate of rise has been tested empirically, and ex- cellent agreement was found over a range of bubble radii from 0.01 to 0.1 cm. These rates of rise of bub- bles in still water, as predicted from purely gravita- tional theory, would lead to the conclusion that all bubbles of acoustically effective size would reach the ocean surface in a time much shorter than the com- monly observed lifetime of an acoustic wake. However, the motion of the ship’s hull and the action of its propellers continually set up throughout the wake a strongly turbulent internal motion, which interferes with the streaming of bubbles toward the surface resulting from their buoyancy. This phe- nomenon is analogous to the transportation of sus- pended material in rivers. Most suspended material is heavier than water and, therefore, would settle out in nonturbulent flow. But through turbulence this material is maintained in a state of suspension. Similarly, in a wake the bubbles rise toward the sur- face, .while turbulence counteracts this tendency. ANCY AND RATE OF ASCENT Figure 5. Bow wave, hull wake, and stern wake of USS Idaho (BB42). 454 FORMATION AND DISSOLUTION OF AIR BUBBLES Figure 6. Underwater photograph of cavitation spot near bow of a PT boat traveling at 9.5 knots. The analogy with transport in a river is not complete, since the turbulence at any fixed position in a wake dies out gradually and the bubbles, once they have reached the surface, are likely to disintegrate. A semi-theoretical analysis of the lifetime of wakes has been presented which aims at finding precisely how much turbulence is needed in order to account for the observed ages of acoustic wakes.® In this work, the intensity of turbulence is measured by a certain empirical parameter, and it is shown that the theoretical lifetime of the wake passes through a broad but well-defined maximum if the turbulence parameter is increased steadily. This theoretical maximum has a simple qualitative physical explanation. While weak or moderately strong turbulence tends to lengthen the lifetime of a wake, as pointed out before, a very large degree of turbulence will speed the decay of a wake by in- creasing the probability of the bubbles reaching the ocean surface and breaking up, namely, when the average value of the upward components of the tur- bulent motion exceeds the speed of the rise of bub- bles with gravitational force alone. The existence of these opposing effects for very small and very large turbulence accounts for the maximum lifetime reached at some intermediate value of the turbulence parameter. The predicted maximum happens to agree with the average observed lifetime of acoustic wakes, which is from 15 to 45 minutes. Gratifying as this result is, there are not available any measurements of the intensity of turbulence in wakes, and hence the actual value of the turbulence parameter is unknown. Moveover, should the observations necessary to specify the value of the turbulence parameter be made, the analysis® would require some modifica- tion before an exact comparison with the observed lifetime of wakes could be made. In particular, the concentration of bubbles at the ocean surface was assumed to vanish, according to the premise that the bubbles reaching the surface are immediately de- ENTRAINED AIR 455 Figure 7. Underwater photograph of white water under hull of a PT boat traveling at 9.5 knots. stroyed and thus removed from the ocean. Even granting the validity of this physical assumption, the removal of bubbles cannot be expressed mathemati- cally by a vanishing bubble density. Inasmuch as the number of bubbles reaching the surface per unit time and per unit area equals the product of the bubble density and their average velocity upward, a vanish- ing bubble density implies a vanishing number of bubbles reaching the surface and thus does not cor- respond to the physical situation envisaged. In addi- tion, the decay of turbulence as the wake ages may also have to be considered. 27.3 ENTRAINED AIR The fact that sailing ships have a conspicuous wake suggests that a good deal of air is trapped along the waterline of any vessel under way. Such air might materially contribute to the mass of bubbles ap- pearing in the wake of vessels propelled by engines. For instance, if Figure 5 could be relied on, the hull wake on the starboard side of the USS Jdaho (BB42) would be even stronger than the stern wake. Of course, nothing is known about the extension in depth of the respective foam masses. Qualitative tests ® showed that echoes from the wake of a barge towed by a tug alongside could be detected with an NK-1 type shallow depth recorder ranging downward from a launch carried across the wake. However, it was found that this wake was more acoustically transparent than the wakes of vessels propelled by screws and therefore probably had a shorter lifetime. These conclusions were confirmed by experiments in which sailing furnished the motive power. The ship used was a 104-ft yacht; measurements were made as described for other ships in Section 31.3. The wake when using sail was never found to be 456 FORMATION AND DISSOLUTION OF AIR BUBBLES Figure 8. acoustically opaque enough to blank out the bottom of San Diego Bay, where all the experiments were made. The wake thickness did not differ significantly from that observed in runs made with engines only, with the same vessel under comparable weather conditions; the average thickness was 12.4 ft with engines, and 13.0ft under sail. As far as this scanty evidence goes, the geometric form of the wake seems to be determined primarily by the shape of the hull of the vessel and its speed, and it seems to be imma-~ terial whether the bubbles are produced by entering surface air or by propeller cavitation. Underwater view from port quarter of a PT boat traveling at 6 knots showing propeller cavitation. A novel direct approach to the visual study of the subsurface structure of wakes has been made pos- sible by the recent development of underwater mo- tion pictures at the David Taylor Model Basin. This technique should also prove most useful for revealing the distribution of entrained air around the hull. For instance, when a small power boat passed with a speed of 2 to 3 knots over the underwater camera, mounted on the bottom in shallow water, the film shows a strongly foaming, shallow stern wake ex- tending backward from the hull wake over a con- siderable distance. This wake did not reach down to ENTRAINED AIR FIGURE 9. the depth of the screw of the launch. In fact, no stream of bubbles could be detected as emanating from the screw, which was clearly visible since the launch approached the camera as closely as 12 ft; presumably a speed of only 2 or 3 knots was insuf- ficient to reach the cavitation limit. Figures 6 to 11 are selected frames from an under- water motion picture showing a PT boat, outfitted with three screws, and its wake. These pictures were taken in water about 40 ft deep, near the Dry Tor- tugas; the choice of this location was dictated by the need for considerable optical transparency in the ocean. The motion picture camera was mounted, slanting upward, on a steel tower firmly anchored on the ocean bottom, and was operated by a diver. The distance from the camera to the ocean surface was about 15 ft. 457 Underwater view from starboard quarter of a PT boat traveling at £9 knots showing propeller cavitation. At the left in Figure 6, a small amount of entrained air is visible along the water line. Moreover, a sharply outlined cavitation spot is conspicuous; unfortu- nately no attempt was made to ascertain what sort of unevenness on the hull caused this cavitation. In Figure 7 a large amount of entrained air is seen cov- ering the hull. Both Figures 6 and 7 were made as the vessel traveled at a speed of 9.5 knots; there is no explanation of why the amount of entrained air differs so greatly in these two illustrations. Figures 8 to 11 illustrate the progressive develop- ment of propeller cavitation as the speed increases. They furnish an instructive corollary to Figures 1, 2, and 3, and show that tip-vortex cavitation caused by the screws of a vessel under way at high speeds has the same appearance as that behind a laboratory propeller driven by a stream of moving water. 458 FORMATION AND DISSOLUTION OF AIR BUBBLES a ee ae ee ee ee Ficure 10. Underwater view from starboard quarter of a PT boat traveling at 27 knots showing propeller cavitation. ENTRAINED AIR 459 FieurE 11. Underwater view from astern of a PT boat traveling at 36 knots showing propeller cavitation. Chapter 28 ACOUSTIC THEORY OF BUBBLES fi Nee RIGOROUS TREATMENT of the acoustic char- acteristics of bubbles, especially of the cumula- tive effects of a multitude of bubbles, requires a great deal of rather advanced mathematics. For a comprehensive exposition of these theories, reference must be made to several monographs on the sub- ject.'+ In this chapter only the principal features of the problem will be sketched, primarily with a view to the later elementary interpretation of the acoustic properties of wakes in Chapter 34. Actual wakes have such a complicated structure that many physical and mathematical refinements incorporated in the rigorous treatment of certain ideal cases have, at present, only academic interest. The first two sections of this chapter deal with the acoustic properties of individual bubbles. In the third section, the combined acoustic effects of many bubbles are discussed, and the results are applied to the evaluation, from laboratory experiments, of cer- tain physical constants — acoustic cross sections, damping constants, which cannot as yet be pre- dicted from pure theory. SCATTERING BY A SINGLE IDEAL AIR BUBBLE 28.1 For application to wakes, only those air bubbles need be treated whose radius FR is very small com- pared with the wavelength X of sound in water, or 2rR or eee (1) ON R 3yPo From equations (13) and (16) A; can be eliminated and a relation between A and B obtained; if the subscript 0 is omitted from Ro, the bubble radius in equilibrium, then (16) AR 3yPo 2riR- Anf*pR? a r By introducing now the abbreviation f,, defined by B= (17) 1]/ 3yP ie a ||) SUES, af al ; the physical meaning of which will soon become ap- parent, equations (17), and (18) and (1) may be combined to give the result RA (E) — 1+ In order to obtain the scattering cross section o, of the bubble from equation (6), |A|? and |B|? have to be computed from |A|? = AA*, |B|? = BB*, (20) where A* and B* are the complex conjugates of A and B respectively. According to equation (19), RA* ip y if (j Finally, from equations (6), (19), (20), and (21), 4rR? yore For a bubble of a definite radius R, the scattering cross section o, has its peak value if f equals f,; it is then said that the incoming wave is in resonance with the pulsations of the bubble, and hence f, is (18) B= (19) 1a (21) (22) SCATTERING BY A SINGLE IDEAL AIR BUBBLE LOG(o, /wR”) 0,001 0.002 0.004 0.006 00080.01 0.02 0.04 0.06 2mR/d Figure 1. Scattering cross section for an ideal bubble. called the resonance frequency for the bubble of radius R. A plot of o,/7R? as the function of » = 27R/d = 2rRf/c is shown in Figure 1, the outstanding feature of which is a sharp peak. This maximum cor- responds to the resonance value 7, or according to equation (18) 2 R. r 1 Pepi Baki Fy 21s = 136 X10, (23) Cc Cc p if Po is atmospheric pressure and c is the sound velocity in sea water at 60 F. Thus, at resonance, co, is enormously greater than the geometric cross sec- tion of the bubble; specifically Osr 2 \2 = = = 2. 1 6 1 04, awh? (?) s where o;, is the value of o, at resonance. Equation (24) can also be expressed in the form 2 Ug = = ° Tv While equations (24) and (25) must be considerably modified for an actual bubble, as shown in the next section, the phenomenon of resonance is neverthe- less responsible for the great efficiency of bubbles as scattering agents. Moreover, the resonant frequency found from equation (18) is correct for a wide spread (24) (25) 463 of bubble sizes. This equation has been confirmed by observations at low frequencies, between 1,000 and 6,000 ¢ per sec, *? and also at high supersonic fre- quencies, between 20 and 35 ke.’ In each case a single bubble was placed in the sound field, and the sound frequency determined at which the bubble oscillated most violently. The radius of the bubble was then measured either with a microscope, or for the larger bubbles by measurement of the volume of air in the bubble. The values of the resonant frequency f, found in these measurements for bubbles of air, hydrogen, and oxygen in water at different temperatures agreed with equation (18) within the experimental error of about 5 per cent. Thus within the range from 1 to 50 ke equation (18) may safely be used to predict the resonant radius of bubbles in water. Values computed from this equation are given in Table 1. TasLE 1. Resonant radius for air bubbles in water. Frequency in ke 1 5 20 50 Wavelength in centimeters 150 | 30 7.5 3 Pressure Depth of Atmospheres water in feet 1 Surface 0.33 | 0.065 | 0.016 | 0.006 2 35 0.47 | 0.093 | 0.023 | 0.009 5 140 0.73 | 0.15 | 0.037 | 0.015 10 300 1.04 | 0.21 | 0.052 | 0.021 For very small bubbles, with radii less than 10 cm, surface tension becomes important and the com- pressions and expansions of the gas in the bubble be- come isothermal instead of adiabatic. No observa- tions for such small bubbles are available, but a theoretical analysis® shows that equation (22) is still valid provided that f, is defined by the equation 1 2af. = =|/ slay (26) R p where eS Sye7ey (ae) the quantity 7 is the surface tension of the gas- liquid surface, and other quantities have the same meaning as in equation (18). Equations (26) and (27) should not be used for bubbles of radii greater than 10-3 cm. Equation (22), in addition to predicting the im- portance of resonance, also gives correctly the scat- 464 tering coefficient for frequencies considerably greater than the resonant frequency f;. Since 7 is less than one, 72 in equation (22) may be neglected when the ratio f,/f is much greater than one. Consequently, the scattering cross section for low-frequency sound may be written approximately as “or ({)"= we (%) c= tere( J 4nR x This equation is known as Rayleigh’s law of scatter- ing for long-wave radiation. It will be remembered that in optics Rayleigh’s law explained the blue color of the sky, as the resonant frequencies characteristic of the atmospheric gases oxygen and nitrogen are far greater than the frequencies of visible light. Hence, the shorter (blue) waves of sunlight are scattered more strongly than the longer (red) waves and reach our eyes with greater intensity. Equation (28) is also applicable to the high-frequency sound commonly used in echo ranging provided that the bubble radius R is very small; if R is less than 10~* cm, how- ever, f, is given by equation (26) instead of by equa- tion (18). (28) SCATTERING AND ABSORPTION BY AN ACTUAL BUBBLE So far, the attenuation of sound resulting from the absorption of sound energy during the pulsation of the bubble has been neglected. The existence of such an effect is a direct consequence of the second law of thermodynamics, which implies that energy must be extracted from the sound field and dissipated into the surrounding water in the form of heat, in order to maintain the forced pulsation of the bubble against the internal friction of the bubble-water system. In other words, it is thermodynamically madmissible to treat the pulsation of the bubble as if it were a strictly adiabatic process; therefore it becomes neces- sary to amend the analysis given in the preceding section for an ideal bubble. This task is accomplished by adding to equation (13), which expressed the continuity of pressure at the bubble surface, a certain term which takes into account the frictional force modifying the behavior of an actual bubble. Moreover, the exchange of heat between bubble and water by conduction neces- sitates a modification. of-equation (16), which formu- lated the continuity of velocity at the bubble surface. The treatment of the case of the ideal bubble im- plicitly assumed that the pulsations are thermo- dynamically reversible; that is, the work put into the 28.2 ACOUSTIC THEORY OF BUBBLES bubble during compression was supposed to be equal to the work done by the bubble during expansion. Actually, there is heat exchange between bubble and water, but the pulsations are too rapid to permit a complete leveling of temperature at every instant of the cycle. Thus there prevails a continual change of state which is somewhere between the adiabatic and isothermal case. It is not difficult to see that under such circum- stances the pulsation of pressure cannot be in phase with the pulsation of volume. While the bubble is be- ing compressed, the temperature rises steadily; as soon as the rise of temperature becomes appreciable, heat conduction begins to operate and the bubble tends to cool off even before expansion has started. When the minimum volume is reached, the tempera- ture will be decreasing as heat flows from the bubble into the water. Consequently, the temperature maxi- mum will be reached some time before the bubble has been compressed to its minimum volume. Likewise, since the gas pressure is proportional to the tempera- ture, the maximum pressure will not be attained simultaneously with the minimum volume, but some time before. Thus there exists a phase shift between pressure and temperature on one hand, and volume and radial velocity of the bubble on the other hand. For resonant bubbles at frequencies of 100 ke or less, this effect is taken into account’ by inserting a com- plex factor 1 — 67 in the right-hand side of equation (11), where_8 is a positive constant much smaller than one. The two equations of continuity, (13) and (16), must therefore be replaced, for an actual bubble, by the following ones: dR (BO a 1 793 ae (29) t or ; CiB; A B/Ry — 2mikB — A; = ? ame grt afk? and B QrfRoA: ; = il = MB) < ie Gy 8) In equation (29) Ci is a constant measuring the effect of friction, which is assumed to be proportional to the radial velocity dR/dt of the bubble. The term C\dR/dt represents the net pressure on the bubble, which is positive when the bubble is contracting (dR/dt < 0); hence, the correction term appearing on the right side of equation (29) must carry a minus sign. SCATTERING AND ABSORPTION BY AN ACTUAL BUBBLE 465 By proceeding exactly as in Section 28.1, the fol- lowing relation is found instead of equation (19): RA ; 77} Sa GH) a (hee ld ( :) -1+e Sele: +6 eon If one neglects 8° compared to one and defines sf B+ +° se (32) equation (31) becomes RA (33) B = 2 ": (e - 1) + 718( f,Ro) Substituting this expression into equation (6), the cross section for scattering by an actual bubble can readily be evaluated: 4rh? ——] 62 (oy It will be noted that equation (34) is identical with equation (22), which was derived for an ideal bubble, except that 6? has replaced 7? in the denominator. This change affects only the magnitude of the scat- tering cross section near resonance. Thus the fre- quency of resonance and the scattering cross section at frequencies far from resonance are correctly given by equation (22), in agreement with the statements made in the previous section. The knowledge of the scattering cross section does not provide all the information that is wanted in the case of an actual bubble, as the incident flux of energy is reduced both by scattering and absorption of sound. Calling the sum of scattered and absorbed energy the extinguished energy, an extinction cross section o- can be defined by (34) (35) where F, is the total energy extinguished by the bubble per unit interval of time and J; is the intensity of the incident sound energy. The quantity F’. is equal to the work done, per unit interval of time, on the bubble by the incoming sound beam; this extin- guished energy comprises both absorbed and scat- tered energy. Hence, F. is equal to (36) where the bar means the time average; po is the pres- sure of the incident sound wave, and V is the volume of the bubble. To evaluate equation (36) it is simplest to use real quantities. According to equation (2), py = AP, Since the initial phase may be chosen arbitrarily, let A be real, and let the sound pressure and sound velocity be represented by the real parts of the ex- pressions developed above. Then = A cos 2rft. (37) From equations (9) and (15) it follows that dV PIB = = A Bes yan axe GE Reve = fo e Here again only the real part of the entire expression is to be taken. In order to find this real part, split B into its real part B” and its imaginary part 7B’, and express e?””” in terms of its real and imaginary parts: dV 20 — = —— (B+ 7B") (cos 2nft + i sin 2nft) C2 = =a [(B’ cos 2rft + B® sin 2zft) (38) + i (B’ sin 2xft — B" cos 2rft) ]- If equation (37) and the corresponding real part of equation (38) are substituted into equation (36), it is found that F,= — = (eos 2nft) (B’ cos 2rft + B® sin 2rft), (39). p where the bar denotes an average over many cycles Since t = cos? (Qnft) = 5 and (cos 2aft) (sin 27ft) = equation (89) becomes finally AB’ i= —— (40) Jp According to equation (33), RA6 = 7 a (41) (& = 1) + 8 Hence, equation (40) assumes the form RAS 1 R= (42) 466 ACOUSTIC THEORY OF BUBBLES By combining this expression with equations (3) and (32), the cross section for extinction is finally ob- tained: 6 Ark? () 0 The extinguished energy is obviously the sum of the scattered and absorbed energy. Therefore, the absorption cross section oa of the actual bubble can be defined by the relation (43) Te = 03 + Oa (44) and is thus found to be, from equations (34) and (43), ve= 2 2 (45) (Badites Note also the simple relation Os n Te eee 46 Be (46) A word must be said now about the function 6, de- fined in equation (32). If 8 and C, are put equal to zero, for the case of an ideal bubble, 6 reduces to 7, and it is seen that equation (22) is indeed the correct limiting form of equation (34). Numerical values of B and C, can be derived by an analysis of the several physical processes known to contribute to the absorp- tion of sound by the bubble — for instance, heat con- duction, viscosity, surface tension, and other proc- esses. There are also methods for determining 6 empirically from certain observations which will be discussed. Inspection of Figure 2, which shows the damping constant at resonance as a function of fre- quency, will reveal that the predicted values of 6 are much smaller than the observed ones. This discrep- ancy indicates that some relevant physical processes must have been overlooked in the theoretical anal- ysis of the absorption effects. Hence, theoretical evaluation of 8 and C, although carried out else- where,® will be omitted from this review, and the empirical values of 6 will be used for the interpreta- tion of the acoustic properties of wakes to be given in Chapter 34. The physical significance of 6 can best be visualized by plotting o,/4R? against f/f,. A resonance curve, similar to Figure 1, is thus obtained. The peak value of this graph is, according to equation (34), 20 25 30 35 40 fe) 5 10 «(15 FREQUENCY IN KC x Values found from oscillation of a single bubble © Values found from transmission through bubble screen Adopted values of 6r Theoretical curve for air bubbles Theoretical curve for ideal bubbles Figure 2. Damping constant at resonance. Osr 4rR? 8 where 6, is the resonance value of 6 shown in Figure 2. If o.(f) denotes the cross section for any non- resonant frequency, it follows from equations (40) and (47) that 2 olf) _ = v tS) Osr r (£ = 1) + &(7,R) Over a narrow range of frequencies near the peak of the resonance curve, 6(f,R) in the denominator of equation (48) may be replaced by its resonance value 6,, and f,/f is very close to one. Hence, using the abbreviation q = f,/f—1, — (47) f? 2 (EZ a 1) = @@ + 2)? = 49) (49) and equation (48) becomes approximately os(f ) 1 alae Age? (50) 1+ ro ae in other words, for any given small departure q from the resonance frequency, the decline of o, from its peak value is sharper for greater values of 1/6, or for smaller values of 6, itself. The greater the sharpness of SOUND PROPAGATION IN LIQUID CONTAINING MANY BUBBLES 467 a the resonance peak is, the smaller is the damping of the pulsation of the bubble. Therefore, 5, is com- monly called the damping constant. Measurement of Damping Constant 28.2.1 The simplest, most direct way to determine the damping constant 4, is to measure the sharpness of the resonant peak for a bubble in a sound field. Such measurements have been carried out for bubbles in fresh water. In one case,’ the amplitude of oscillation of a single bubble was observed as the sound fre- quency was slowly varied. Since o, is proportional to the square of the amplitude of oscillation, a plot of these observations yields 6, directly. Values of 6, were found by this method for bubbles of hydrogen and bubbles of oxygen, but no systematic difference was found between these two gases. In another case, the transmission loss through a sereen of bubbles all of the same size was observed.’ To produce this screen, six small mzcrodispersers ar- ranged in a line in a laboratory tank were used to produce a stream of bubbles 10 ft below the surface of the water. These bubbles were normally inter- cepted by a hood, which could, however, be swung to one side for about one second to allow a pulse of bubbles to rise to the surface. Since the larger bubbles arrived at the surface first, and the smaller ones at progressively later times, the bubbles near the surface at any one time were of nearly equal radii. The trans- mission loss in decibels of sound at a constant fre- quency crossing this screen was then proportional to c- for a single bubble; from a plot of the transmission loss against bubble radius, a value of 6, could then be determined. A typical set of observed curves ob- tained with this technique is reproduced in Figure 4. In analyzing these data, account was taken of the variation of 6 with bubble radius so that points some distance from resonance could be used as well as those close to resonance. The values of 6, found by these two methods are plotted in Figure 2. The dashed line curve shows the theoretical value of 6, for air bubbles in water, if Cy is set equal to zero, and values of 6B are taken from reference 5. It is evident that at the higher fre- quencies the observed values are much greater than the theoretical values; this discrepancy has already been noted above. The values of 5, found from a single bubble, which are shown as crosses, are somewhat greater than those determined from the transmission loss of sound through a bubble screen, plotted as circles in Figure 2. In the former set of measure- ments, the bubble was not free, but was caught on a small wax sphere fastened to a platinum thread, which oscillated to and fro as the bubble expanded and contracted. Since the damping constant may have been increased in this arrangement over its value for a free bubble, these values cannot be relied upon. Thus, the solid line of best fit shown in Figure 2 is based at high frequencies on the values found with the screen of freely rising bubbles. Confirmation of these observed values of 6, is found in the next sec- tion, where the observed data on scattering and absorption of sound by bubble screens are shown to be in moderately good agreement with the theoretical values based on equations (34) and (43) and on the empirical curve of 6, in Figure 2. For comparison with the observed values, the damping constant 6, com- puted for an ideal bubble resonating in water at at- mospheric pressure is shown as a dashed line in the figure; the value plotted is taken from equation (23). 28.3 SOUND PROPAGATION IN A LIQUID CONTAINING MANY BUBBLES The results derived in the preceding sections for a single bubble are only the first step toward the solu- tion of the general problem, the propagation of sound through a medium containing many bubbles. This problem is complicated because the external pressure affecting each bubble is the sum of the pressure in the incident sound wave and the pressures of the sound waves from all the other bubbles. While the mathematics of the problem is complicated, the gen- eral results to be anticipated can be presented simply. 28.3.1 General Theory First, the presence of the bubbles will affect the nature of the medium through which the sound wave is progressing. If the bubbles are spaced much closer to each other than the wavelength, the sound velocity will be appreciably affected by the presence of the bubbles, which alters the compressibility of the medium. In addition, the sound velocity will have a small imaginary part, resulting from the absorption and scattering of sound, and giving rise to an ex- ponential drop of the sound intensity with increasing distance of travel through the aerated water. Thus a sound wave can be reflected, refracted, and atten- uated as it passes through water containing bubbles. 468 ACOUSTIC THEORY OF BUBBLES On this picture the sound wave behaves as though it were proceeding through a homogeneous medium, in which the sound velocity is a smooth complex function of position. Secondly, this picture must be supplemented to take scattering into account. The sound waves sent out from the different bubbles produce scattered sound, which goes out in all directions. This scattered radiation may be regarded as resulting from the fact that in a random collection of poimt scatterers the number of scattermgs per unit volume is never constant from one region to another, but shows statistical fluctuations. A theory of the scattering of light im air is given along these lines in a well- known text on statistical mechanics.? More simply, the intensity of scattered radiation may be regarded as proportional to the average squared pressure re- sulting from all the individual bubbles. As the bubbles move around, the relative phases of their scattered wavelets will vary widely, and constructive and destructive interference will be equally likely. With this picture, the average squared pressure may be regarded as simply the sum of the squares of the pressures in each of the scattered wavelets. In ship wakes the number of bubbles in a small volume is rarely sufficiently great to produce reflec- tion and refraction of sound waves. The gradual at- tenuation of the incident sound beam and the scat- tering of sound energy in all directions by each bubble individually are therefore the two effects of greatest interest. i The preceding discussion is, of course, not very rigorous. The results stated here have been proved rather generally, however, in an elegant solution to the general problem.** This analysis makes certain assumptions, the most important of which are that the bubbles have diameters much smaller than the wavelength of the incident sound, and that the average distance between bubbles is much larger than their dimensions. The solution, as a result of its physical generality, is of considerable mathematical complexity, and therefore will not be reproduced here. But the mode of approach used in this general theory will be briefly sketched. The chief feature of this theory is its use of con- figurational averages. Different bubbles may be almost anywhere within a certain region. For each distribu- tion of bubbles the sound pressure p at a given time will have some definite value. If now an average value of this pressure is taken for all possible positions of the different bubbles, a configurational average of p, denoted by

, results. Thus is usually not equal to the time average of p, since this time average vanishes because of the oscillations of p between positive and negative values. Similarly, may be defined as the configurational average of p?. The simplified picture presented at the beginning of this section may be given a precise meaning in terms of these configurational averages. The quantity

is found to obey the wave equation in a homogeneous medium in which the complex sound velocity is a function of position. This configurational average acts in general as the pressure from a re- fracted sound wave. Thus

gives rise to a trans- mitted wave; after leaving the scattering region, this transmitted wave bears a definite phase relationship to the incident wave. For any particular configuration, the value of p may differ from

. A measure of this difference is provided by the mean square value of p—

, which is equal to —

?. The analysis shows that this difference is simply the sum of the squares of the pressures in the sound wave sent out from each of the bubbles. These additional terms therefore represent just the scattered sound, includ- ing sound that has been scattered several times. Thus the intensity at any point, which is proportional to p*, is on the average the sum of two terms; the first term

? represents the coherent wave, propagating through a homogeneous medium in which the sound velocity changes in some way with changing position. The second term, —

? represents the sum of the scattered waves from each bubble. At any one time the value of p?, even when averaged over a few cycles, will usually differ from the sum of these two terms, but as the configuration of bubbles changes, the time average of p? should ap- proach the configurational average of p?. In most practical situations a period of several seconds is usually sufficient to bring the time average of p? close to the configurational average. If, then, averages are taken over time intervals of several seconds, the simplified picture presented at the be- ginning of the section may be taken as correct. When the average distance between the bubbles becomes very small, or, in other words, as the average number of bubbles per unit volume becomes very large, this simplified picture becomes inadequate. In this case, another term must be included in , in addition to the two terms representing the re- fracted (coherent) wave and the scattered (incoher- ent) waves. This term is difficult to interpret, but SOUND PROPAGATION contributes to the scattered sound and appears to be due to interference between different scattered wave- lets. It is not easy to determine the precise point at which this term becomes important, but it can be shown to be negligible, for resonant bubbles, pro- vided the attenuation per wavelength is less than a few decibels. This is the same condition that must be satisfied if the change which resonant bubbles pro- duce in the sound velocity of the medium is to be relatively small. Since this condition appears to be satisfied in observed wakes, this additional term will therefore be neglected in the following derivation of practical formulas for the attenuation, scattering, and reflection of sound by water containing bubbles. 28.3.2 Transmission The type of analysis developed in the preceding section will now be applied to find the transmission loss through a region containing bubbles. It will first be assumed that within this region all the bubbles are of the same size. In each cubic centimeter there are assumed to be n bubbles; n may vary from point to point within the region. If J is the intensity in the incident sound beam, the rate at which sound energy is extinguished from the beam by each bubble will be oJ, according to equation (85). Let /(0) be the intensity at the point where the beam enters the region containing bubbles and let /(r) be the in- tensity after the beam has penetrated a distance r through the region; r is measured along a sound ray. The increment of /(r) after passing an infinitesimal distance dr is, of course, negative and has the value dI = —n(r)oI(r)dr. (51) By integration of equation (51) over the path fol- lowed by the sound, it is found that at any distance 7; I(r) = T(0) ene if n(r)dr _ (0) Pe) ‘ (52) where N(r;) is the total number of bubbles in a column of length 7; and unit cross section. If 7 is set equal to w, the total thickness of the region, equa- tion (52) gives the total extinction produced by the bubble screen, or the attenuation as it is usually called in underwater sound work. Expressing the attenua- tion on a decibel scale, equation (52) is equivalent to I(0) I(w) where n is the average bubble density in the screen, defined by 10 log —— = 10 X 0.434 X now = Kaw, (53) IN LIQUID CONTAINING MANY BUBBLES 469 1 fw ] n= | n(r)dr = — N(w)- (54) 0 Ww Ww The quantity K, in equation (53) is usually called coefficient of attenuation, which is conventionally given in units of decibels per yard. Since n and o, are usually expressed in units of em~* and cm, re- spectively, and since there are 91.4 cm to the yard, K, in decibels per yard becomes K, = 396.8nc.. (55) The attenuation coefficient K, is rather easy to determine by acoustic measurements either of a wake (see Chapter 32) or of a bubble screen produced in the laboratory (see Section 28.2). Since o, is known for resonant bubbles from the experimental deter- mination of the damping constant 6, already described in Section 28.2, the bubble density n can be computed from K, and oc, by equation (55), on the assumption that only bubbles of resonant size are present. How- ever, among copious masses of bubbles, as found in wakes, there will usually be a wide dispersion of bubble sizes. It is important, therefore, to evaluate the attenuation produced by such nonhomogeneous bubble populations. Let the number of bubbles per cubic centimeter with radii between R and R + dR be denoted by n(R)dR, and define S, as the total extinction cross section per cubic centimeter. From equations (34) and (48), it is then found, by adding up or integrating the cross sections of all bubbles contained in one cubic centimeter, 4rR?n(R) € *) s= J, arn Bubbles of near-resonant radius will make a large contribution to S.. If n(R) does not change rapidly for radii near resonance, the integral over the reso- nance peak in equation (56) may readily be evaluated. This procedure gives the correct value for S, pro- vided that absorption by bubbles far from resonance can be neglected. Even if the density of bubbles near resonance is comparable with the bubble density at other radii, resonant bubbles will probably make the major contribution to S., since o, is unquestionably much greater for resonant bubbles than for those of other sizes. However, according to what has been said in Section 28.1.2 about the gradual shrinkage of bubbles, a large number of very small bubbles are likely to be present which may contribute apprecia- FR, © (56) 470 bly to the total extinction cross section. Since o, for bubbles of sizes far from resonance size is propor- tional to 6, and since the value of this damping con- stant is unknown for nonresonant bubbles, it is not possible to state the conditions under which non- resonant absorption may become important. In practical applications it is customary to assume that bubbles near resonance provide the dominant source of attenuation in wakes; as shown in Chapter 34, this assumption appears to lead to agreement with experimental results, and is probably correct at supersonic frequencies for the bubble distributions occurring in wakes. Then, in order to compute the value of S, resulting from bubbles near resonant size, n(R), 7(R,f), and 6(R,f) in equation (56) may be taken outside the integral and given their values for R equal to the resonant radius R,. Thus equation (56) is trans- formed into en AnR?n(R,)5, f dR e A ( - (57) 2 2 a le 1) + 8 2 As the radius of the resonant bubbles struck by a sound beam of the frequency f is R,, then according to equation (23) IP, Qrfyp =, (58) pk, and from equations (23) and (58) dip Le Pe (59) and according to equation (49) ie R, —R, =~ -1=—~-1, d= : q="5 ee ape al) By substituting equation (60) into equation (57), S. can be expressed by an integration over the variable q. Only the values near the peak (near to q equals 1) make a considerable contribution to the value of the integral. Therefore, the transformations (49) and (50) may be used, and from equations (57) and (60) it follows that 4rR? ae dq Nr -o 49+ 6 The integral has been extended to infinity. This simplification can be made because on this approxi- mation the contributions which are not very near to the peak can be disregarded. Evaluating the integral in equation (61) gives S. = (61) ACOUSTIC THEORY OF BUBBLES tag, T i 4 + 8 26, (62) and from equations (61) and (62) 2? Rin(R, = 20h n(k) : (63) Nr Let now u(R)dR denote the total volume of air con- tributed by the bubbles with radii between R and R + dR in 1 cu em of the air-water mixture. Hence, 4 u(R) = Ren(R), (64) and from equations (63) and (64) R, S.= 3ru(h,) 3 (65) One The quantity 7,, according to equation (23), has the value 1.36 X 10-? in sea water at 60 F and at atmos- pheric pressure. Hence, S. = 346.5u(R,). (66) In computing the attenuation for a region containing bubbles of many sizes, the equations derived at the beginning of this section may be applied directly. It is necessary only to replace the factor no, in-equa- tion (53) by S., taken from equation (66). If this substitution is made, the coefficient of attenuation is K, = 396.8 X 346.5 X u(R,) K. = 1.4 X 10° u(R,). (67) This expression is the generalization of equation (55) for bubbles with a wide dispersion in size. It will be used in Chapter 34 to compute the amount of air in wakes from the observed attenuation coefficients. 28.3.3 Scattering In accordance with the picture for propagation of sound through a region containing bubbles, as pre- sented in Section 28.3.1, the basic equation for scat- tered sound is very simple. The scattered sound in- tensity from a region is, on the average, simply the sum of the intensities of the waves scattered by each bubble. For a single bubble, the intensity at a dis- tance r is given by the equation Le 4nr? where J) is the intensity of the incident sound at the bubble. This equation may be found from equations T, = Ih, (68) SOUND PROPAGATION IN LIQUID CONTAINING MANY BUBBLES 471 (3), (5), and (6); more simply, it may be written down directly, since by definition oJ is the rate at which sound is scattered by a single bubble, and since the energy is spread out uniformly in all directions, at the distance r it is spread out uniformly over an area 4rr*. In a small region of volume dV, the number of bubbles is ndV, where n is the number of bubbles per cubic centimeter. Equation (68) must be modified to allow for the fact that the scattered sound will be attenuated on its way from the region to a distance r away. Over long distances various sources of attenuation must be considered, such as absorption in the water, scat- tering by temperature irregularities, and so forth. Over short distances, most of these effects may be neglected, and the transmission loss taken from equa- tion (52). The basic equation for the scattered sound measured a distance 7; from the region dV then be- comes ey no,dV = aye where J is the intensity at the region dV. If sound from different directions is incident on the region, I must be averaged over all directions for use in equa- tion (69). Computing the scattered sound intensity from equation (69) is a much more complicated problem than computing the total sound attenuation from equation (51). In the latter case, equation (51) could be integrated along a single sound ray, yielding equa- tion (52) directly. The basic difficulty in solving equa- tion (59) is that the sound intensity J at the volume element dV includes sound scattered in turn from other regions. To consider multiple scattered sound of this type is rather complicated, and leads to inte- gral equations which in general cannot be solved exactly. Methods for treating this problem have been extensively explored in astrophysical literature. 1° 1 The problem of multiple scattering in wakes could probably be studied with success by methods de- -veloped for the corresponding optical problem.” Fortunately, bubbles absorb much more sound than they scatter. From equation (46) and Figure 2 it is evident that the ratio o,/c. for resonant bubbles is less than 1 to 10 for frequencies above 15 ke. For this reason, sound scattered several times from resonant bubbles has usually traveled so far that it is very weak. Multiple scattering will therefore be neglected in all subsequent discussions. In simple cases, the error resulting from this approximation IE n(r)dr e dl, Ie” (69) will be less than half a decibel at frequencies above 15 ke. Even at 5 ke, the error will usually be less than 1 db. For scattering by nonresonant bubbles, multiple scatterings cannot be neglected unless o, is much greater than az. Even with this approximation, the computation of I, from equation (69) is not simple. The quantity I now becomes the sound intensity incident on the region containing bubbles, and attenuated by its passage through part of the region. However, to compute J, at any one point the sound arriving from all parts of the screen must be computed; the total scattered sound must be evaluated by summing up the contributions arriving from all different direc- tions. In any practical situation, the directivity of the receiving hydrophone must also be taken into account in order to find the electrical signal received in the measuring equipment. A detailed considera- tion of these problems in cases of practical impor- tance is given in Chapter 34. To give insight into fundamental features of the scattering problem, it is desirable to eliminate these geometrical complications as far as possible. Equa- tion (69) is here applied to scattering from a bubble screen, that is, from a layer of aerated water bounded by two parallel planes a distance w apart. Instead of integrating over all directions, we shall compute simply the scattered sound reaching the point P from all directions within a small cone of solid angle dQ; the quantity dQ is simply the area of a cross section of the cone divided by the distance r? from P to the cross section. The geometry of this situation is shown in Figure 3. Let [(0) be the intensity of sound incident on the screen; the incident sound is assumed to be a plane wave, whose rays are inclined at an angle 7 with a line perpendicular to the boundary of the screen. Within the screen the intensity falls off exponen- tially; since the path length dr is equal to sec zdz, equation (52) gives for the incident sound at a dis- tance x inside the screen I(z) = IQ) exp | -«. sec i [near |: As Figure 3 shows, the scattered sound which we are considering makes an angle with a line perpendicular to the boundary of the screen. Thus in equation (69) the length dr along the path of the scattered sound is sec edz. Thus we find for the sound scattered from a small element of volume dV, at a distance r; from the point P 472 ACOUSTIC THEORY OF BUBBLES AST INCIDENT SOUND WAVE Figure 3. Scattering from a bubble screen. dl, = n(x)o.dV o,dV 1(0) 4nr? exp| — o-(sec 2 + sec 0) f ncaa |: (70) 0 For the volume element dV within the cone, we have dV = ridQdr; since dr, is simply sec edz as before, equation (70) be- comes dle n(x)o, sec edQdx An exp| — o-(sec 7 + sec of n(a)ar | (71) 0 This equation may be integrated over z from 0 to w, yielding o,dQ oda cost+ cose 1(0) COs 2 dI, = {1 — exp [- o-(sec 2 + sec of nae | \. It is interesting to note that d/, in equation (72) is independent of the distance from the screen to the point P where the scattered sound intensity is measured. This apparent contradiction is resolved when it is realized that with increasing distance a larger area of the screen is intercepted within the solid angle dQ. (72) Equation (72) has two important limiting cases. When the transmission loss across the screen is large, the second term in the brackets is negligibly small, and o;,dQ oAm cosi + cose COs 2 dl, = (73) It may be noted that when e equals, as is the case for backward scattered sound, cos e equals cos 7; if also equation (46) is used for o,/c,, equation (73) yields dQ dat. . 6 8r On the other hand, when the transmission loss across the wake is small, it is possible to use the ap- proximate relationship a (74) e 1 —a, yielding dQ c) dI, = o;— sec € n(x)dx. (75) 4a 0 In terms of the average density n introduced in the previous section, equation (75) becomes he, = (76) dQ = Os, Sec e wn Thus when the transmission loss across the wake is small, d/, is proportional to o, and n. But when the transmission loss is great, the scattered sound reaches a constant value, given by equation (73), and is in- sensitive to changes in n or w. When bubbles of different sizes are present, equa- tion (69) for dZ, may still be used, provided that no, is replaced by S., the total extinction cross section per cubic centimeter, and no, is replaced by S,, the total scattering cross section per cubic centimeter. The quantity S, is discussed in the preceding section; equation (65) gives the relationship between S, and u(R,), the bubble density at resonance. A similar analysis, considering bubbles only of near-resonant size, leads to the following equation for the total scattering cross section per cubic centimeter: 37u(R,) 25, The consideration of only those bubbles near the resonant size is usually legitimate even if absorption by nonresonant bubbles is appreciable. Since o,, the scattering cross section of a single bubble, does not depend on the damping constant 6 for nonresonant bubbles, it is possible to evaluate precisely the con- tribution of bubbles of all sizes. For a single bubble S; — (77) SOUND PROPAGATION IN LIQUID CONTAINING MANY BUBBLES 473 smaller than resonant size, o, falls off as the fourth power of the wavelength; hence such small bubbles are not likely to contribute much to S, unless present in very large numbers. Bubbles larger than resonant size have a scattering cross section about four times their geometrical cross section, but are not likely to be present in greater abundance than smaller bubbles. Thus equation (77) should be valid in a wide range of circumstances. Since the ratio of S,/S, is equal to the ratio of o,/o- at resonance, equation (74) is still valid when the transmission loss across the screen is large; thus the scattered sound in this case is just the same as if all bubbles were of resonant size. When the trans- mission loss across the screen is small, however, equa- tion (76) must be used, with S, substituted in place of nos. 28.3.4 Reflection and Refraction The presence of bubbles changes the velocity of sound. If the bubble density is sufficiently great, this effect may become practically important, leading to reflection and refraction of the sound beam. Since in ship wakes the number of bubbles present per cubic centimeter is usually not sufficiently great to change the sound velocity very greatly, these effects are not discussed in great detail here. The methods of analysis required to deal with this case are briefly sketched, and the results stated. The sound velocity is defined by equations (6), (18), and (26) in Chapter 2 as (78) where p and p are the pressure and density respec- tively of the bubble mixture. If only a volume V of the mixture is considered, equation (78) may be written in the form (79) using the relation pdV + Vdp = 0. The quantities dp/ot and dV /dt may be evaluated from the equations in Sections 28.1 and 28.2 yielding the basic equation NG R)RdR == re __ n(R)RdR 5 (80) y "'(G-1) +8 Sto where ¢y is the sound velocity when no bubbles are present, and n(/) is the number of bubbles per cubic centimeter with radii between R and RdR. The inte- gral in equation (80) extends over all bubble sizes. It is assumed that all bubbles present have a radius much smaller than the wavelength, and that the average distance between bubbles is larger than their radius. If these two assumptions are not fulfilled, the theory on the preceding pages breaks down. The de- tails of the derivation of this equation are given in references 3 and 4. It may be noted that equation (80) is valid only when the density of the liquid-bubble mixture is substantially the same as that of the liquid. Results are given which may be used for any density of bubbles, provided that the bubbles are all much too small to resonate, but much too large for surface tension to become important. For frequencies far from resonance, the imaginary term in equation (80) may be neglected. For fre- quencies below resonance, this leads to the equation a ; 3u ee THE where w is the total volume of air present as bubbles in 1 cu em of the liquid-bubble mixture. Thus wis de- fined by the equation 4 ae ii = R'n(R)dR. (81) (82) The quantity 7, in equation (81) is the ratio of the bubble circumference 27 to the wavelength ) at resonance, as defined in equation (1). Equation (81) is valid only for bubbles which are sufficiently large that surface tension effects can be neglected; more- over, if the expansion and contraction of the bubble are adiabatic rather than isothermal, the last term in equation (81) must be multiplied by the ratio of the specific heats for the gas in the bubble. It is interest- ing to note that, subject to these limitations, equa- tion (81) is independent of the bubble radius. Even if wis as low as 10~ parts of air at atmospheric pres- sure to one part of water, c/cp is 0.62. When the bubbles are all greater than the resonant size, the sound velocity is increased by the presence of the bubbles, and the relation corresponding to equation (81) is wages (83) g fe a M(RAR., where saya eta! in equation (64), is the volume of the bubbles present in 1 cu cm of liquid-bubble 474 ACOUSTIC THEORY OF BUBBLES mixture with radii between R and R + dR. Equation (83) has the surprising implication that when u(R) is sufficiently great, cj/c? becomes negative, the sound velocity becomes purely imaginary on this approxi- mation, and the attenuation becomes very large. Under these circumstances the imaginary term in equation (80) which was neglected in equation (83), determines the wave velocity and the wavelength. For the case of all bubbles with twice the resonant radius R,, the critical value of u at which c becomes infinite is 2 X 10~‘, corresponding to a distance be- tween bubbles of roughly thirty times the bubble radius. When resonant bubbles are present, the imagi- nary part of the sound velocity becomes important. If an integration is carried out only over bubbles close to resonance, and if w(R) is not changing rapidly with R in this region, the real part of the integral in equation (80) is small and may be neglected, yielding (ea 37ik,u(R,) = 1 Of (84) This imaginary part of the sound velocity leads to an exponential decay of sound intensity with distance , since the sound intensity falls off as e@"""/"" If the second term on the right-hand side of equation (84) is small, as it is in most practical cases, the resulting attenuation is exactly the same as was found in equa- tions (53) and (67) in Section 28.3.2. In a region containing bubbles, with any assumed distribution of sizes, and having a sharp boundary, sound incident on this region from bubble-free water will be reflected at the sharp discontinuity. The anal- ysis for this situation is given in Section 2.6.2 where it is shown that the ratio of the amplitude of the reflected and incident waves is given by the equation A” Acme) This is essentially equation (119) of Chapter 2, with pi set equal to p’ and subscripts 0 used for the incident sound wave. The quantity c is the sound velocity in the bubble-free medium, while c; is the corresponding quantity across the boundary, where bubbles are present. The energy reflection coefficient 7. is simply the square of A’’/Ay. The angles z and care the angles which the incident and refracted sound make with a line perpendicular to the boundary. The ratio of cos € to cos e may be found from Snell’s law, yielding cos? Ce c= 1+ tante(1 =. Co COs? L Ci — Co (cos €/cos t) C1 + co (cos €/cos tL) (85) In most cases of practical importance, c is nearly equal to co. Thus, cos ¢ is essentially equal to cos t. By writing equation (81) in the form CF oy 1+0, c the energy reflection coefficient y. found by squaring A’’/A, in equation (85) becomes (86) as long as b is much smaller than 1. This equation may also be used when 6 is complex but less than 1, provided that the absolute value of b is used. When b is comparable to or larger than 1, the formulas be- come considerably more complicated.* 28.3.5 Observed Acoustic Effects of Bubbles The effect of a known distribution of bubbles on the propagation of sound through water has been investigated in the laboratory at frequencies from 10 to 35 ke. The method used for producing a screen of bubbles all of the same size has already been de- scribed in Section 28.2.1. Special measurements were made to determine the number of bubbles per cubic centimeter at various points in the bubble screen. The bubble screen was about 17 in. .ong. Its thick- ness varied with the bubble radius; for bubbles 0.034 cm in radius, corresponding to a resonant frequency of 10 ke, the thickness was about 3 in., while for bubbles 0.020 cm in radius, corresponding to 17 ke, the thickness was more nearly 5 in. In continuous flow, about 1 cu cm of air per second was fed into the screen, resulting in a total density u of about 10~* parts of air per part of water. When a bubble pulse was formed by turning on the stream of bubbles for 1 sec, however, the bubble densities at the level of the acoustic instruments were much less than this, ranging between 10-* and 1077. The acoustic measurements with the bubble pulse consisted in measuring the sound reflected from and transmitted through the screen at a fixed frequency as a function of time since the beginning of the pulse. The transmission loss was measured by reading the sound level in a hydrophone placed on the far side of the bubble screen from the projector. The reflected sound was measured by a hydrophone placed on the same side of the bubble screen as the projector, but separated from the projector by several baffles. The SOUND PROPAGATION IN LIQUID CONTAINING MANY BUBBLES RADII IN CM OF BUBBLES IN SCREEN 20.070 0.040 0.030 0.025 0.020 QOI7 015 0.014 OOS 0.012 0.011 ATTENUATION IN OB TIME IN SECONDS AFTER BEGINNING OF PULSE RADII IN CM OF BUBBLES !N SCREEN 20.070 0,040 0,030 0.025 0.020 0,017 0,015 0,014 0,013 0.012 OI! ie) tl shes |estiee Ona i ae A a TIME IN SECONDS AFTER BEGINNING OF PULSE Figure 4. Acoustic data taken with bubble pulse screen at 20 ke. -40 REFLECTION COEFFICIENT IN 0B hydrophone was placed symmetrically with the pro- jector, so that the sound reflected specularly from the screen could reach the hydrophone. However, scattered sound could also reach the reflection hydro- phone, and presumably contributed to the so-called reflected sound which was measured. As the resonant bubbles passed by the level of the acoustic measuring instruments, the transmitted sound intensity showed a sharp dip. The reflected, or scattered, sound showed a very much broader maxi- mum, in agreement with the constant scattered sound intensity predicted by equation (73) whenever the transmission loss through the region is appreciable. However, the reflected sound showed much greater fluctuations than the transmitted sound. Sample records of transmission through and re- flection from bubble screens are reproduced in Figure 4. The curves show the output of the transmission and reflection hydrophones as a function of time elapsed after a 1-sec pulse of bubbles was formed 6 ft below the acoustic equipment. Also shown are the radii of the bubbles arriving at each time. These radii were measured directly by visual means. The upper diagram in Figure 4 shows three transmission runs at 20 ke, superposed on each other. The radius of the resonance peak in this diagram agrees well with the theoretical value of 0.017 cm found from equa- 475 SOUND FREQUENCY IN KC FOR RESONANCE 4030 20 10 NUMBER OF BUBBLES PER CU CM o e °o o (e) 0.01 0.02 0.03 BUBBLE RADIUS ON ACOUSTIC AXIS IN CM 0.04 SOUND FREQUENCY IN KC FOR RESONANCE 40 30 20 10 ATTENUATION THROUGH SCREEN IN DB BUBBLE RADIUS ON ACOUSTIC AXIS INCM SOUND FREQUENCY IN KC FOR RESONANCE 4030 20 10 REFLECTION COEFFICIENT IN DB (0) 0.01 BUBBLE RADIUS ON ACOUSTIC AXIS IN CM 0.02 0.03 Figure 5. Resonant attenuation and reflection with bubble pulse screen. tion (18). The lower diagram shows a reflection run at 20 kc. The measured reflection coefficient is the difference in level between the incident sound at the bubble screen and the sound measured with the re- flection hydrophone, placed 2.5 ft from the center of the screen. Hach set of observations was repeated at least three times at each of several frequencies. Before and after each group of acoustic measurements, detailed 476 a a lee eee ae AV — Y Be. = Pass = = 7) RESIDUAL > -20 -40 0.015 REFLECTION COEFFICIENT IN 0B 8 0.035 BUBBLE RADIUS IN CM ON ACOUSTIC AXIS 0.020 0.025 0.030 REFLECTION COEFFICIENT IN 0B 0.010 BUBBLE RADIUS IN CM ON ACOUSTIC AXIS 0.015 0.020 0.025 0.030 ——w-—W Upper and lower limits of experimental data Estimated average intensities Ficure 6. Scattering and reflection bubble pulse screen. observations were made of the number of bubbles of different sizes in the screen, since the operation of the microdispersers producing the bubbles tended to be somewhat erratic. If the physical measurements on the number of bubbles of different sizes in the screen did not give the same results before and after the acoustic measurements, the acoustic data were dis- carded. The results of the acoustic measurements on bubble pulses showed moderate agreement with theoretical predictions. Figure 5 illustrates typical results ob- tained for resonant bubbles. The upper diagram shows the total number of bubbles per cubic centi- meter at the level of the transducers at the time when bubbles of each radius reach that level. Since the spread of bubble radii at each time was small com- pared to the width of the resonance peak for a single bubble, all the bubbles at any one time may be as- sumed to be of the same size. In the middle diagram, the continuous curve shows the predicted attenuation through the screen, found by substituting in equation (53) the following quantities: the bubble density taken from the upper diagram; the measured thick- ACOUSTIC THEORY OF BUBBLES ness of the screen; and the value of co, found from equation (43) with f equal to f,, and with values of 6, taken from Figure 2. The average observed trans- mission losses at each frequency are shown by circles, with vertical lines showing the spread of the observa- tions. These experimental points are essentially the maximum difference in sound level produced by the passage of the bubbles; in the middle diagram of Figure 5, for example, the observed resonant trans- mission loss at 20 ke is about 14 db. The lower diagram jn Figure 5 shows the intensity of the reflected or scattered sound for resonant bubbles. To compute the reflection to be expected from resonant bubbles, the specular reflection was first found from equation (86), with b evaluated for bubbles all of resonant size. To this was then added the scattering to be expected; this scattered sound was found from equation (76), since for resonant bubbles the transmission loss through the screen was always great enough to make this equation appli- cable. The value of n, at resonance was taken from equation (23), while values of 6, at resonance were again found from Figure 2. In the computation of this scattered sound account must be taken of the size of the screen and its distance from the sound projector and hydrophone. The solid curve in Figure 5 shows the theoretical predictions; at 10 kc, specular reflec- tion is most important, while at 30 kc, scattered sound is dominant. A similar comparison between theory and observa- tion may be made for nonresonant bubbles. The transmission loss measurements yield nothing further of interest, since the width of the observed resonance curve has already been used to find values of 6,. For bubbles whose size is so far from resonant size that the transmission loss is small, the scattering may be predicted from equation (75), suitably modified to take into account the geometry of the situation. The value of o, to be used may be taken from equation (34). Specular reflection from nonresonant bubbles is negligible. Plots of the observed data are shown in Figure 6, where the crosses represent the computed values for nonresonant scattering. The spread of the observations is indicated by the dashed lines, with the solid line showing the estimated average in- tensities. The circles represent the predicted scatter- ing and reflection from resonant bubbles, already dis- cussed. The dotted linegives the sound levelmeasured at the reflection hydrophone when no bubbles were present. It is evident that the agreement between theory SOUND PROPAGATION IN LIQUID CONTAINING MANY BUBBLES and observation shown in Figures 5 and 6 is not bad. Other runs show about the same agreement, with oc- casional observed transmission losses as low as half or as great as twice the predicted value, and with occasional observed reflection coefficients as much as 6 db outside the spread of the observational data. These discrepancies, which are apparently in random directions, may be the result of irregularities in the bubble-producing devices. It is worth noting that the predicted scattering from nonresonant bubbles should be quite reliable, since the theoretical values are in- dependent of the damping constant. Hence it may be concluded that the agreement of observations with theory is within the observational error, and justifies the practical use of the equations developed in this chapter. Measurements on continuous-flow bubble screens have also been described;’ they showed relatively poor agreement with the theoretical predictions. The observed transmission losses rarely exceeded 25 db, while the predicted transmission losses ranged be- tween 50 and 200 db. It is doubtful whether such great transmission losses could be observed, since sound diffracted around the screen would be expected to become important. In addition, in the continuous- flow screen the smaller bubbles extended over a wider region than the larger ones. At the lower supersonic frequencies this halo of small bubbles would not 477 absorb sound, but would reduce the sound velocity, thus tending to bend the sound rays around the screen. Furthermore, the predicted reflection coefficients for the continuous-flow screen were some 5 to 15 db greater than the observed values. The high specular reflection predicted from theory for these continuous- flow screens would presumably be reduced to a value closer to the observed results if account were taken of the absence of sharp boundaries. In view of the many complexities entering into the explanation of these measurements on the continuous-flow screen, these discrepancies with theory may be disregarded. An important theoretical question which is not answered by these experiments is the absorption pro- duced by bubbles far from resonance. This non- resonant absorption depends on the variations of 6 with bubble radius and sound frequency. Since the values of 6, are unexplained, the predictions of theory as regards values of 6 under other conditions are of little use. The bubble pulse measurements show that the absorption by nonresonant bubbles is usually less than about 5 per cent of the absorption by resonant bubbles. It is not impossible that for some bubble distributions present in wakes nonresonant absorption might be practically important. Further observations under controlled conditions would be required to cast light on this point. Chapter 29 VELOCITY AND TEMPERATURE STRUCTURE Te WATER in the wake of a ship is usually in mo- tion relative to the surrounding water. In addi- tion, the temperature of the water at different points in the wake is sometimes characteristically different from the temperatures found outside the wake. The variations of temperature and velocity are important physical properties of wakes, and might be expected to account at least in part for the acoustic effects ob- served; furthermore, a study of these physical char- acteristics is of independent military interest. Even if air bubbles are responsible for all the observed acoustic effects of wakes, any theory of the origin and persistence of bubbles must be consistent with known facts about the velocity and temperature structure. The present chapter summarizes the fragmentary evidence which is available on these two subjects. Sections 29.1 and 29.2 discuss the available data on the velocity and temperature, respectively. In Section 29.3 the resulting acoustic effects are examined. It is shown that scattering from turbulent but wake-free water is negligible; scattering of sound by water with an irregular temperature distribution may some- times be appreciable, but cannot explain the large acoustic effects observed. Thus, velocity and tem- perature structure alone cannot account for the ob- served acoustic properties of wakes. 29.1 VELOCITY STRUCTURE OF WAKES The simplest wake is that produced by the flow of a fluid past a thin plate parallel to the stream. In this case the plate affects the flow only in a narrow region close to the plate, known as the boundary layer, where the fluid is slowed down. This effect is shown in Figure 1, where the magnitude of the velocity at various points is shown by arrows; for simplicity, only the upper half of the flow pattern is shown. Far behind the plate the velocity distribution still shows the effect of passing by the plate, since the fluid which passed through the boundary layer will be moving less rapidly than the rest of the stream. The arrows in 478 Figure 1 represent the average velocities of the fluid relative to the plate. Thus these results are applicable directly to the reciprocal situation, when the thin plate (or ship’s hull) is moved through still water. In this situation the water in the wake is left moving in the same direction as the plate. It may be noted that in most cases, the flow in the boundary layer becomes turbulent, in which case the flow in the wake will also be turbulent. - — SS (==) -——4 +>} -——1 ———— es t >| Sater ah ies +»! oc +——__>~ —— Pi weil -——+ -———} —_—__ —— -——+4 UNDISTURBED BOUNDARY WAKE FLOW LAYER Figure 1. Velocity structure. In addition to the wake produced in this way by passage of a ship through water, there is also the effect. produced by the screws. To move the ship for- ward, the screws exert a forward force on the ship which is somewhat greater than the frictional foree produced by the flow of water past the hull; the dif- ference is just equal to the retarding force due to wave action and air resistance. For a submerged submarine, however, the propulsive force is just equal to the fric- tional force produced by the flow of the water around the hull. To produce this propulsive force on the sur- face ship or submarine, the screws exert an equal and opposite force on the water, which is forced back- ward. As a result, the water passing through and around the screws moves in a direction opposite to that of the vessel. The flow of water produced by ship screws has already been discussed in detail in Section 27.1.1 in connection with the formation of air bubbles. Thus, close to a ship the wake is made of several component parts: one or more screw wakes, usually called “‘slipstreams,” moving away from the ship as a result of screw action; and the hull wake following the ship as a result of frictional force at the surface of TEMPERATURE STRUCTURE OF WAKES 479 the hull. The backward momentum of the slipstream is nearly canceled‘out by the forward momentum of the hull wake, except at surface ship speeds so high that wave resistance becomes the most im- portant retarding force on the ship. In the wake of a submerged submarine this cancellation is exact. At moderately close distances astern, probably much less than a ship length, these different streams become intermingled and confused, giving rise to a turbulent mass of water in which velocities in almost any direction are equally likely. Over a small distance called the patch size, the velocity at any one time is reasonably constant, but the velocity at any point fluctuates rapidly. Information on turbulent motion is rather incomplete and no velocity measurements are available in surface ship or submarine wakes. As noted already in Section 27.2, not much is known about the magnitude of the turbulent velocities, the average patch size of the turbulent elements, or the rate at which the turbulence gradually dies away. 29.2 TEMPERATURE STRUCTURE OF WAKES ‘ The water temperature at different points in a wake has been the subject of more study than the water velocity. This is partly because small tem- perature differences can be measured much more readily at sea than small fluid velocities. By the use of sensitive thermopiles fastened to a surface vessel, temperature fluctuations as small as 0.01 F may be readily recorded. Data obtained with this technique at the U.S. Navy Radio and Sound Laboratory! and elsewhere show that the presence or absence of ob- servable temperature structure in wakes depends on the presence of vertical temperature gradients in the sea before the passage of the ship. 29.2.1 Constant Temperature in Surface Layer . When a ship is passing through water all of the same temperature, such as is commonly found in the top 50 ft of the ocean, especially during winter months, no thermal structure in the wake can be observed. Repeated wake crossings under these con- ditions have failed to show any trace of temperature structure. In such isothermal water, temperature structure could be produced only by the heating ac- tion resulting from the passage of the ship. Such heating can readily be shown to be negligible. To consider an extreme case, suppose a ship at 30 knots is exerting 30,000 hp, and suppose that all this energy goes into heating a wake with a cross sec- tion 20 ft square. The increase of temperature result- ing in this extreme case is 0.015 F. In most practical cases, the temperature change will be very much smaller. Although small patches of water might be appreciably warmed by water discharged from cool- ing systems, by dissipation of energy in intense vortices, or by similar processes, most of the wake behind a ship in isothermal water will have a tem- perature which is practically the same as that of the surrounding ocean. 29.2.2 ‘Temperature Gradient in Surface Layer When a vertical temperature gradient is observed in the top 20 ft of the ocean, the passage of a ship disturbs the temperature structure and gives rise to a measurable temperature structure in the wake. The thermopiles used in research on this subject have had slow response times, requiring 1 or 2 sec for 80 per cent response; since the surface vessels used in the work were under way at 3 knots or more, changes of temperature over regions less than a few feet in length could not be detected. The most detailed and quantitative work ' was carried out with four thermopiles attached to a long pipe mounted vertically on the bow of a small cabin cruiser; the thermopiles were at depths of 4, 6, 8, and 10 ft below the surface. In each thermopile, one set of junctions was thermally exposed to the sea water; the other set was thermally insulated and remained at the average temperature of the surrounding water, averaged over a period of minutes. The output of each thermopile was measured with a self-balancing potentiometer; since these instruments required some 7 sec to reduce an unbalance to zero, these quanti- tative measurements recorded only the large-scale features of the wake thermal structure. Results obtained with this technique are shown in Figure 2, obtained in successive crossings of a destroyer wake 8 and 15 minutes old. Accompanying bathythermograph records are also shown. It is evident that the fresh wake consists of warmer water at the two sides, with cooler water in the middle. This distribution probably results from descending cur- rents at the sides, and rising currents in the center; such currents could be produced by the rotation of the slipstreams from the two propellers. 480 VELOCITY AND TEMPERATURE STRUCTURE WAKE AGE 8 MINUTES LAUNCH ENTERING FROM WEST T fo) DEPTH IN FEE a fo} we— INCREASING TEMPERATURE—= WAKE AGE 15 MINUTES LAUNCH ENTERING FROM EAST { 2 WwW wW ir z Z td 46 £ = re = 3 B 2 6a Be LA 2 2 __7 NW, ue x CIS: z2 eye FS SM 108 — [ J 10° =| 10 k- BT AT 1316 SEC ELAPSED TIME OF CROSSING EAST SIDE WEST SIDE GF WAKE OF WAKE Figure 2. Horizontal temperature structure of a de- stroyer wake. The thermal structure found for other types of ship wakes is sometimes considerably different from that shown in Figure 2, with single peaks sometimes re- placing the double peaks. In general, however, when- ever the thermopiles were at the depth of a marked negative gradient — 0.5 degree in 10 ft — as shown on a bathythermograph record outside the wake, the wake near the surface was colder than the surround- ing water at the same depth. When the gradient is marked no such general rule may be made. It is interesting to note, however, that thermal wake signals have been readily detected when the gradient outside the wake was almost too weak to be noticed on a bathythermograph record — about 0.2 degree in 20 ft. Measurements have also been made on the thermal properties of the wake behind a submarine at peri- scope depth, with a moderate negative gradient pres- ent in the surface layer. It was found that effects ap- peared even at the surface, where the water behind the submarine was found to be a few tenths of a de- gree cooler than the surrounding water outside the wake. The reason for this rise: of the submarine’s thermal wake to the surface is not known. i The persistence of these thermal effects is some- times quite marked. Identifiable thermal signals have been obtained in crossing wakes an hour or more after these were laid. Not all. wakes exhibit identi- fiable thermal effects for such a long period, even if the gradient is marked. The limiting factors are the decay of the thermal structure of the wake and the background of thermal irregularities present outside the wake. It is sometimes difficult to distinguish the thermal change found in crossing a wake from those frequently found in sailing through wake-free water. The thermal irregularities in wake-free water also tend to increase with increasing temperature gradi- ents; thus a very strong gradient is not necessarily the best for detecting a wake by its thermal proper- ties. As shown in the next section, the acoustic effect of thermal structure is greatest for temperature irregu- larities whose size is about equal to the wavelength of the sound being transmitted through the water. Thus, to compute the scattering of supersonic sound at 24 ke, information on the variation of temperature over regions about 3 in. long would be required. No such information is available, owing to the long time constants of the measuring methods discussed above. Temperature fluctuations over such small regions might be expected in a relatively fresh wake. How- ever, it would be surprising to find such a small-scale temperature structure in a wake more than a few minutes old. 29.3 SCATTERING BY TEMPERATURE AND VELOCITY STRUCTURES Any region in which the velocity of sound varies with position will affect a sound wave passing through it. For example, theory predicts appreciable reflection from a surface separating two large bodies of water differing considerably in temperature.? If variations of the microstructure of the ocean take place over distances not too great compared with the wave- length, an appreciable amount of sound will be scat- tered in various directions. Although the exact anal- ysis of these effects is complicated, certain results may be derived relatively simply. These results, given below, are sufficient to indicate the general magnitude of the scattering of sound by the temperature and velocity structure of wakes. Suppose that in some region S the velocity of sound SCATTERING 481 has some variable value c + Ac, while in the sur- rounding water the sound velocity has a constant value c. Suppose also that a plane sound wave, of in- tensity Jo, and wavelength X, is progressing through the medium in the z direction. The intensity J, of the sound scattered from S may be different in different directions, but at long ranges will fall off as the in- verse square of the radial distance r from the center of the region S. Since J, must be directly proportional to Ih, (1) where & is a constant. A more detailed discussion of this equation is given in Section 19.1 of this volume, describing in general the reflection, or scattering, of sound from objects or scattering regions in the sea. The target strength T as usually defined is simply 10 log k. The quantity k, which depends on the direction of the scattered sound under consideration, must be re- lated to the values of Ac, the sound velocity fluctua- tion, at different points in the region S. Only the energy scattered directly backward need be con- sidered here, since this corresponds to the situation of practical interest. It may also be assumed that the scattering is sufficiently small that the sound level at all points in S is practically equal to its value in the incident sound wave in the absence of scat- tering. This assumption tends to overestimate k; if the scattering is large the sound level will decrease as the wave penetrates the region S, because energy is lost by scattering in the portion of the region S already passed through. Since the scattering is produced by the relative change in sound velocity, it is reasonable to assume (and, in fact, it can be shown) that the pressure dp, of the sound scattered from each volume element dxdydz in S is proportional to the value of Ac/c for each element. In adding up all the sound from differ- ent elements, the differences in phase must be con- sidered. Since sound must travel to the scattering element and then back along the = axis, the difference in phase between two elements separated by a dis- tance x along the x axis will be 47x/\. Thus to find the pressure of the scattered sound, Ac/c must be multiplied by cos (47x%/\ + 2nft), where f is the frequency of the sound, and integrated over the en- tire scattering region S. The scattered sound in- tensity is then proportional to the square of this integral. In this way it may be shown that the quantity k in equation (1) is given by the formula bd E il il il “ aes (4: = a 2nft) aye | "+ (2) By writing 4 cos C= + 2nft) = cos 4a cos 2nft 5 Ce — sin 4r = sin 27ft, (8) the integral in equation (2) becomes the sum of two integrals. Now square this sum, and average over the time t, using the relations cos? 2nft = sin? 2aft = 4 (4) cos 2rft sin 27ft = 0, where the bars denote an average over the time ¢ Then the quantity k, which measures the scattered sound intensity, becomes aEffeaCe\ae) + efffam() cone] As pointed out above, the target strength of the scattering region is 10 log k. When the volume of the scattering region is small compared with the wavelength, the trigonometric functions in equation (5) are constant; since the sum of their squares is unity, -#(([fain) 0 When Ac/c is constant throughout the region, this equation reduces to and where V is the volume of the region. Equation (6) is the so-called Rayleigh scattering law, which predicts only a small amount of scattered sound. On the other hand, when cis constant over a region large compared with the wavelength, k is again small; as a result of the oscillation of the sine and cosine factors in equa- tion (5) each integral adds up to only a small value. 29.3.1 Effect of Temperature Microstructure Equation (5) may be used to compute the sound scattered by a mass of water in which the tempera- 482 ture varies rapidly from point to point. For sim- plicity, suppose that positive and negative values of c are equally likely — that is, that the average tem- perature of the water is just equal to the temperature outside the scattering medium. Although the distri- bution of temperature from point to point is a quantity which fluctuates at random, there is a certain patch size over which the temperature does not usually change appreciably. This is represented mathematically by means of the function p(¢), which is defined by the expression ss Ac(x + 6Y,2) Ac(x,y,2) pS) = ) (7) Ac(x,y,2)? where the averaging is to be carried out in space, over all values of 2, y, and z in the scattering region. While ¢ is a displacement in the x direction in the expression (7) above, the displacement might also be extended in any other direction. The value of the function p(¢) will depend both on the magnitude and on the direc- tion of ¢. If the displacement is zero, then p will equal unity. If the displacement is very large, the values of c at points separated by the distance r show no cor- relation with each other, and their product is alter- nately positive and negative, canceling out on the average; thus for large £, p approaches zero. The patch size is the value of ¢ for which p becomes small, say less than about 14. The function p is called a self- correlation coefficient. The temperature microstructure is described as isotropic if p(¢) is independent of the direction along which ¢ is taken. With some mathematical transformations, equa- tion (5) may be expressed in terms of p(¢). For an isotropic medium, the resulting equation, which is equivalent to that given in a report by Columbia University Division of War Research [CUDWR];3 is 167° Ac\V © sin (4r¢/d) vi (2) vf p(s tee dg, (8) where V is the volume of the scattering region. As one fairly general type of possible correlation coefficient, it may be assumed k= (Shen ae (9) By substituting this expression in equation (8), and integrating, k 1 =) 1 vo ie (1 + 2/16n2A2)2 ©) In actual practice the wavelength \ is usually less VELOCITY AND TEMPERATURE STRUCTURE than the patch size A, and the last term in the de- nominator may be neglected. Correlation coefficients of a form different from equation (10) do not gener- ally give a much greater value of k/V for a given patch size A. Numerical values may be substituted in equation (10). Fluctuations of 0.5 F with a patch size of 6 in. probably represent a rather extreme assumption. For this situation, k/V is about 3 X 10-7 sq yd per cubic yard of volume. The volume scattering coefficient m discussed in Section 12.1 of this volume is related to k by the equation dak a (11) Thus m, in this case, is about 4 X 1077 per yard. If equal energy were scattered in all directions, m would be the fraction of energy scattered per yard of sound travel through the scattering medium. Evidently even these extreme assumptions give a very small scattering coefficient. Even if the scatter- ing volume is 10 yd thick, 30 yd across, and 100 yd long, corresponding to the wake in the path of a sound beam, k is about 10-° yd, corresponding to an effective target strength of —30 db. The transmission loss through such a scattering region would be a very small fraction of a decibel. Temperature microstruc- ture cannot explain the strong echoes or the high transmission losses produced by wakes. 29.3.2 Effect of Velocity Microstructure A separate analysis must be carried out for the case where the velocity of the water varies from place to place in the medium. This is a more complicated situation than the one in which the temperature changes, since the fluid velocity has a direction as well as a magnitude. However, it can be shown that equation (5) is still applicable if the component v, of the fluid velocity in the x direction is used in place of Ac. This seems a reasonable substitution, since it is only the component of the fluid velocity along the direction of the incident sound wave that affects the propagation of this wave. To compute k, then, integrals of the form if sin (4rax\)dx f fi vzdydz must be evaluated. If the integrals over y and z are computed first, it is easy to see that the entire inte- (12) SCATTERING gral vanishes. The integral of v, over the yz plane is simply the net rate at which the fluid is flowing across this plane. At any time, the total amount of fluid passing through the yz plane in one direction must be just equal to the amount of fluid passing through in the other direction, and the net flow vanishes. Thus, a random distribution of velocity does not contribute to backward scattering of sound. However, sound may be scattered in other directions, as indicated in reference 3. Measurements at San Diego* and at Orlando *& are consistent with the result that the sound scat- 483 tered backward from velocity microstructure is very weak. At San Diego attempts were made to obtain echoes from underwater vortex rings, while at Or- lando a mechanical device was used to produce turbulent water in the path of a sound beam and at- tempts were made to measure the reflected sound. In both cases, no reflected sound could be observed. Al- though the data do not exclude the possibility that weak echoes may have been present, the combination of measurements and theory point to the conclusion that backward scattering of sound from velocity microstructure may be practically neglected. Chapter 30 TECHNIQUE OF WAKE MEASUREMENTS Mc OF THE MEASUREMENTS of submarine and surface vessel wakes discussed in Chapters 26 to 35 have been made by University of California Division of War Research [UCDWR] or by Navy observers at the U. S. Navy Radio and Sound Laboratory [USNRSL] in San Diego. The instru- ments and physical principles applied to acoustic observations of wakes do not differ essentially from those employed in other underwater sound measure- ments described in Chapter 4, Chapter 13, and Chapter 21. It is unnecessary, therefore, to introduce here detailed descriptions of instruments and their theory. But before discussing the results, some general features of the experimental work at San Diego on the acoustic properties of wakes will be reviewed. 30.1 LISTENING AND ECHO RANGING Listening through a wake to a ship under way, or to a mechanical noisemaker, constitutes the simplest type of acoustic observation of a wake. The presence of a wake manifests itself by a reduced sound level at the receiving hydrophone, compared with the same level with no wake interposed. Such observations of the acoustic screening effect are the incidental result of numerous measurements of the sound output of ships. But, in order to obtain quantitative results, it is desirable to use as sound source a transducer or mechanical noisemaker of constant power output, instead of the noise from the screws of a ship. To- gether with a hydrophone of constant sensitivity, this equipment makes possible determination of the transmission loss which sound undergoes in passing through a wake. Echoes returned by wakes can be studied by lis- tening or by using objective records of the current generated in the receiving channel of the transducer. While the second method is indispensable for the determination of sound intensities, it does not tell anything about the small changes in frequency that are caused by the relative motion of target and trans- 484 ducer. The acoustic doppler effect is helpful in dis- tinguishing between the echo from a wake, which is nearly stationary, and the echo from the wake-laying vessel. This distinction is occasionally of practical interest, as in the study of the rather weak wakes produced by submarines in submerged level runs. In such cases it may be useful to preserve an audible record, in the form of a phonograph record, of the wake echo. The supersonic echo obtained aboard the experimental vessel is transmitted by short-wave radio to the laboratory ashore, where the phono- graphic recording can be done more conveniently than on a rolling and pitching vessel at sea. 30.1.1 Sound Range Recorder Traces At San Diego it is a standard procedure in all wake work to secure echo records with a sound range recorder of the type in general tactical use. These chemical recorder traces are highly useful for a rapid estimate of the range of the wake and of the decay of its strength. As the chemically treated recording paper is unrolled, with the machine open, the ob- server makes pencil notes on the margin of the record concerning the work in progress, such as the begin- ning and ending of the oscillographie recording, changes of the sound frequency used, and other de- tails. Thus the chemical recorder traces also provide a graphical log of the operations. The general appearance of wake echoes on the sound range recorder paper is illustrated by the photographic reproductions of original records shown in Figures 1 and 2. They are records of wakes laid by the auxiliary yacht E. W. Scripps between the echo-ranging vessel, the USS Jasper (PYc13) and a target sphere buoyed at a center depth of 6 ft below the surface, in the course of experiments described in detail in Sections 31.2 and 32.3.2. The lower part of Figure 1 shows the sphere echo alone. Immediately after passage of the Scripps through the sound beam, there appears a strong wake echo and the strength LISTENING AND ECHO RANGING Ficure 1. from E. W. Scripps. Sound range recorder traces of wake echoes of the sphere echo is markedly diminished by the two-way transmission loss in the wake. Note the 485 en ROJ TRAINED AWAY FROM WAKE, THEN TRAINED — BACK. 4s NING CONSTANT _ Figure 2. Sound range recorder traces of wake echoes from E. W. Scripps. gradual widening of the wake toward the top of the figure, as the wake grows older. The wakes were laid at right angles to the line connecting the transducer on the Jasper with the target sphere. In Figure 1 the projector was kept trained at the sphere in order to study the decay of a fixed part of the wake. Figure 2 shows the effect of gradually changing the training of the projector from its normal training; the range toward the near- est boundary of the wake increases, and since the sound beam now cuts obliquely through the wake, the apparent width of the wake increases propor- tionally to the secant of the angle included between the sound beam and the normal to the wake. On 486 TECHNIQUE OF WAKE MEASUREMENTS Ficure 3. Fathometer record of echoes from ocean surface. training back the transducer, the effect is reversed, thus causing a symmetrical pattern to appear in Figure 2. 30.1.2 Fathometer Records Records of a wake, indicating its thickness and transverse structure, are readily obtained with a fathometer carried across the wake by a survey vessel. Such records may be utilized also to compute the vertical transmission loss, as long as a record from a standard target observable through the wake — for instance, the ocean bottom or surface — is also available (see Section 32.3.3). Early experiments were carried out with the fathom- eter mounted in the orthodox manner on a surface vessel. Records of the ocean bottom are then blanked out in certain cases when the survey boat enters a surface ship wake. This technique suffers from several disadvantages. It does not give an accurate value for the depth of the wake, since the duration of the wake echo is affected by the beam width, the pulse length, and other factors as well as by the depth of the wake. Also, the method is not very suitable for the measure- ment of the transmission loss through the wake, be- cause it requires the echo-ranging vessel to operate in relatively shallow water in order to record the bottom; furthermore, the depth and bottom char- acter may vary considerably while this vessel is moving. If, however, the fathometer is used in the inverted manner, by mounting it on the deck of a submerged submarine, those disadvantages are elimi- nated; clear strong records are obtained both of the highly reflecting ocean surface and of the surface ship’s wake, as illustrated by Figures 3, 4, 5, and 6. Figure 3 shows a record obtained while the sub- marine was diving from the surface. The depth scale marked 5 to 50 applies to this dive, with the time axis running from the right to the left. It can readily be verified from the double record in the center of the illustration that the weaker second reflection corre- sponds to depths that are exactly twice the depth of the stronger first reflection. Thus, the double record LISTENING AND ECHO RANGING 487 Figure 4. Fathometer record of wake echoes from Coast Guard cutter Ewing. is a result of the sound traveling to the ocean surface twice and returning again to the submarine. The dark streaks at the top of this figure result from the acoustically reflecting region formed behind the submarine conning tower, presumably as a result of cavitation originating around the conning tower. The record at the far left is that of the ocean surface after the submarine arrived at a depth corresponding to the scale limit of the recorder and the scale was shifted to bring the record nearer the center of the paper. The small indentations and undulations of the record are produced by the surface swells. Reflection from a surface ship wake under which the submarine is passing produces in these records a shaded area protruding below the ocean surface, as shown in the next three illustrations. Figure 4 repre- sents the record of a wake laid by the USCGC Ewing, proceeding at 13 knots. The submarine in this case passed under the wake at a point 350 yd behind the Ewing. This record was suitable for transmission loss calculations, according to the principles which will be described in Section 32.3.3. The result was a trans- mission loss of 42 db with 21-ke sound traversing the wake twice Note that as a result of the large trans- mission loss, the record of the ocean surface is almost blotted out in the center of the wake, which had a thickness of 15 ft. The same effect is apparent in Figure 5, showing a wake record originating from the destroyer, USS Hopewell (DD681), proceeding at 10 knots. The distance astern is not accurately known, but it is roughly several hundred yards. The transmission loss at 21 ke through the center of this wake was 32 db for the double path. The cause of the extrane- 488 TECHNIQUE OF WAKE MEASUREMENTS Fiaure 5. Fathometer record of wake echoes from USS Hopewell (DD681). ous markings on this record is uncertain; probably they are of instrumental origin. The wake is seen to be 30 ft thick at the maximum point. Figure 6 contains two records of the wake (17 and 11 ft thick, respectively) of the Hwing, proceeding at 13 knots; these records were not suitable for trans- mission loss calculations, since the amplification was increased to record the cross-sectional geometry of the wake. Comparison of Figures 4 and 6 gives an idea of the variations of wake structure occurring in practice; the vessel and speed are the same for both figures. For the proper interpretation of these cross ‘sections, it should be remembered that the sound beam of the customary fathometer is rather broad, including an angle of about 30 degrees, thus causing the fine structure of the cross section to be smoothed out. 30.1.3 Oscillograms In order to obtain permanent sound intensity records suitable for quantitative measurements, the current generated in the hydrophone is amplified and fed into a cathode-ray oscilloscope, the screen of which is photographed continually by a high-speed camera, on standard moving picture film, as described in Section 4.3.3, Section 13.1.1, and Sections 21.2.1 and 21.3.1. The developed negative shows a con- tinuous trace, representing the varying displacement of the luminous spot from its normal position on the oscilloscope screen. Time marks are photographed at suitable intervals as the film moves along steadily. By appropriate design of the electric circuits the dis- placement of the oscillographic trace is made propor- tional to the amplitude of the incident sound wave. The square of the amplitude of the oscillographic trace, therefore, is proportional to the intensity of the sound wave, at the face of the: hydrophone, multi- plied by a factor depending upon the directivity of the hydrophone. If the sensitivity and the directivity pattern of the hydrophone are known, the scale of ordinates on the oscillogram can be calibrated in absolute units to yield the sound pressure in dynes per square centimeter. LISTENING AND ECHO RANGING 489 Figure 6. Fathometer record of wake echoes from Coast Guard cutter Ewing. This type of recording, which has been used widely in other sound studies, has usually been applied only to the analysis of wake echoes rather than to signals transmitted through wakes. The linear distance on the film from mid-signal to mid-echo provides a con- venient record of the range from which the echo was returned, since the distance on the horizontal scale is the product of sound velocity times the time. A number of oscillograms of wake echoes are repro- duced below on the scale of the originals. Figure 7 shows three sets of three successive signals, each 3 msec long, and the corresponding echoes both from a wake, laid by the Z. W. Scripps, and from a target sphere 3 ft in diameter suspended behind the wake at a center depth of 6 ft. The oscillograms were obtained with 24-ke sound during Run 1 of the experiments summarized in Figures 8 and 9 of Chapter 31 and in Table 2 of Chapter 32, which should be consulted for a detailed description of the plan of observations. The numerical evaluation of wake oscillograms has so far been restricted to the visual measurement of peak amplitudes, described in Section 21.3.1, which generally have been held to be sufficiently repre- sentative of the echo as a whole. A more satisfactory though very time-consuming method would be to measure the amplitudes along the entire echo profile, square the amplitudes and integrate them over the time. This integral would be proportional to the total energy contained in the echo. It is possible to design a mechanism which would perform automatically this sequence of procedures. In any event, it would be desirable to supplement and check fundamental wake studies based upon measurement of peak amplitudes by investigating the total energy of echoes. Current procedure is to place the processed film on an illuminated viewer, read the peak amplitude of the echo with the aid of a transparent scale, and cor- rect the measured amplitude, if necessary, for the finite width of the luminous spot on the oscillograph screen. Averages over five successive echoes are 490 P = PING W= WAKE ECHO R= REVERBERATION TECHNIQUE OF WAKE MEASUREMENTS Figure 7. Oscillograms of wake echoes from E. W. Scripps. taken, and the averaged peak amplitude is squared to obtain the echo intensity. The resulting average is different both from the average peak echo intensity and the average of the intensity over the entire echo. Since the spread of peak amplitudes may be as much as 10 db, this difference may be appreciable. The difference between average peak amplitudes and average intensities is discussed in Section 34.3.1. Finally, from the measured peak amplitudes the echo-strength is computed according to the formula: E — § = 20 log A. — 20 logk, where E£ is the echo level in decibels above 1 dyne per sq em, S the source level, defined as the sound level 1 yd from the projector on its axis, also in decibels above 1 dyne per sq cm, and A, is the average peak amplitude of the echo as measured on the oscillo- gram. The constant k on the right side of this equa- tion has to be determined by calibration of the re- ceiving equipment; specifically, k is the amplitude measured on the oscilloscope with an incident wave whose pressure is 1 dyne per sq cm and with the same receiver gain at which A, is recorded. To determine S and k, an auxiliary transducer of known power out- put and of known sensitivity is used. 30.2 OPERATIONS AND MEASUREMENTS Besides the acoustic measurements proper, the study of wakes requires the determination of various auxiliary data. In the first place, the geometric co- ordinates of the part of the wake to which the acoustic data refer have to be known accurately. If the distance from the stern of the ship to the point where the sound beam strikes the wake is known, the age of the wake at the point of measurement may be found by dividing this distance astern by the speed. of the wake-laying vessel and computations will be facilitated by use of Figure 3 in Chapter 35. Since the instrumental characteristics of the sound gear employed may undergo slow changes, it may become necessary to calibrate the gear immediately before or after the observation. Furthermore, there are a number of variable oceanographic factors whose in- stantaneous values have to be taken into account in interpreting the acoustic measurements. In addition to the wake-laying vessel, acoustic measurements on wakes require one vessel for echo ranging and an additional vessel when a transmission run is made in order to measure the horizontal trans- mission loss. For measurements of the transmission OPERATIONS AND MEASUREMENTS loss, the use of a second experimental vessel carrying the receiver might be eliminated by echo ranging through the wake at a target sphere and measuring the intensity of the echo returned to the transducer. From echo ranging at wakes, usually the wake-laying vessel proceeds at constant speed on a straight course past the measuring vessel, which either may run a parallel course with different speed or may be hove to. Maintenance of prescribed speeds and course de- mands accurate seamanship. The relative positions of the two vessels as a function of the time are de- termined by direct triangulation and dead reckoning. During echo-ranging experiments, an incidental check on those geometric data is obtained by the acoustic ranges. During transmission runs, the range from the cruis- ing auxiliary vessel, which carries the projector, to the measuring vessel has been accurately determined by the use of airborne sound; simultaneous radio and sound signals are transmitted from the auxiliary vessel, and the difference between the automatically recorded times of arrival of the two, multiplied by the velocity of sound in air, yields the range. Moreover, for transmission runs, the courses of the operating vessels have to be laid out and maintained with great care in order to avoid interference with the acoustic measurements from the auxiliary vessel’s own wake. In working with wakes which are laid and then allowed to age before the measurements or while the measurements are being conducted, the exact loca- tion of the wake soon becomes difficult to discern. If the wake-laying vessel lies to, it usually soon drifts enough to be useless as a marker for one end of the wake. The following method has proved helpful when working either with surface craft or submerged sub- marines, particularly with very long wakes. A small boat lies to at a point near where the initial end of the wake will be laid. As the wake-laying vessel goes by, the small boat moves into the center of the wake and releases a small amount of fluorescein ;* chrome yel- low or any other nonsoluble dye which floats on the surface is not satisfactory for this purpose, since wind drift can move it away from the wake. In the case of a submarine, it submerges as it passes the small boat or if already at periscope depth, the sub- marine releases the fluorescein. The wake-laying vessel releases fluorescein into the wake at the end of its run. Both this ship and the small boat then keep 8 Before using fluorescein in experiments at sea, it should be ascertained whether special authorization by the Area Com- mander is required. 491 their bows touching the dye spot, thus keeping their net drift the same as that of the wake. More fluores- cein is dropped off the bow of the marker boats at intervals; one marking will not last when working with wakes older than 20 to 30 minutes. If the ob- serving vessel crosses the wake in the course of its measurements, the wake is located by sighting on the marker boats; the use of a simple optical device for lining up the markers is recommended. One of the marker boats takes a stadimeter range on the vessel which crosses the wake; this procedure aids in com- puting the wake age at that point. If successive cross- ings are made, fluorescein is dropped from the ob- serving ship just as it lines up the marker boats. The marker boat closest to this point then moves up to the new dye spot; this insures that the wake between markers remains free of extraneous wakes. Where marker boats are not available, it is helpful to use a mixture of fluorescein and chrome yellow as a marker. The two colors drift apart if any wind is present; the chrome yellow can be seen farther away and is used to locate the fluorescein. 30.2.1 Training Errors In echo ranging on wakes the trainable transducer. is usually operated at a fixed relative bearing. How- ever, in measuring the transmission loss across a wake, it is necessary to keep the trainable projector of the sending vessel aimed at the hydrophone of the receiving vessel; continual changing of the bearing of the transducer is also necessary in echo ranging through the wake at a target sphere. For this purpose, an observer is stationed on the flying bridge to oper- ate a repeating pelorus, to be aimed at the auxiliary vessel or target sphere. A second man stationed at the control stack matches the projector-heading indi- cating “bug” to the target-bearing repeater. Even so, the projector heading does not hold precisely to the true target bearing. The deviation is partly attributa- ble to the lag in the various linkages of the system and partly to the impossibility of holding the pelorus ac- curately on the target at all times. Under practical conditions as prevailing on board the USS Jasper (PYc13), the estimated errors and their sources are as follows: ! (1) pelorus aiming error +2 degrees in fair weather; (2) control stack matching error, projector bearing to target bearing +2 degrees; (8) lag in train- ing system, gears and projector-heading repeater system +2 degrees maximum prior to April 1944, when the system was overhauled and the error re- 492 duced to approximately +1 degree. The maximum error is, therefore, +6 degrees and the probable error +3.5 degrees for data taken prior to April 1944, and +5 degrees and +3 degrees, respectively, for data taken subsequently. At very short ranges there is another correction which may have to be applied because of parallax resulting from horizontal spacing between pelorus position and projector axis. On the Jasper this cor- rection amounts to 2.5 degrees for the aft projector and 3.5 degrees for the forward projector, when the target is 100 yd away and bears either 90 or 270 degrees relative to the sending vessel. Finally there are training errors due to rolling and pitching of the sending vessel. This error can be serious at close range since rolls of 45 degrees have been experienced on the Jasper and rolls of over 20 degrees are common in moderate weather. For the same vessel the pitching angle is of considerably smaller magnitude than that of the roll, rarely ex- ceeding 7 degrees. Installation of a device to record angle of roll, pitch, projector heading, target bearing and ship’s heading for each sound pulse emitted, has been of great help in recognizing and rejecting acoustic observations that have been impaired by serious training errors. Field Calibration The transmitting system’s absolute output has to be checked at the beginning and end of each day’s operation and also during the operation if excessive variations are encountered. For this purpose an aux- iliary transducer, whose performance is known from absolute calibration in the testing laboratory, is lowered into position by means of a special boom which pivots at the rail and swings down to projector depth. To check the actual output of the transducer in use, it is trained on the auxiliary transducer to give a maximum generated voltage; and from the laboratory calibration of the auxiliary transducer, the sound field pressure is determined. For the inverse calibration process, the known power output of the auxiliary transducer is received by the working trans- ducer and the generated current is recorded as in field work. Any appreciable deviation of any of the read- ings from those normally experienced requires an im- mediate investigation to determine the source of the difficulty. According to the experience of the San Diego group with the JK projectors, the standard deviation of the 30.2.2 TECHNIQUE OF WAKE MEASUREMENTS output pressure level was only 1.3 db over a period of 15 months, and 0.4 to 1.0 db for groups of consecutive calibrations within that sequence. However, the standard deviation of the sensitivity of the receiving channel was 3.9 db for the same period and varied from 0.4 to 2.0 db for groups of calibrations within that period. The causes of this variation are un- known. In the course of one day, changes in overall sensitivity, which is the sum of the projector output and the sensitivity of the receiving channel, are negligible; changes in output level did not show any correlation with changes in receiver sensitivity. Also, over the whole period under discussion, changes in output level are not correlated — or at most are weakly correlated — with changes in receiver_sensi- tivity. All these observations refer to 24-ke sound. Incomplete evidence suggests that the performance of 60-ke sound gear is even more variable. The existence of large calibration errors is also suggested by certain discrepancies among the San Diego data on the target strength of spheres, which are discussed in Section 21.4.3. 30.2.3 Oceanographic Factors The weather and state of the sea appears to have some influence on the formation and gradual dis- solution of wakes. It is advisable, therefore, to keep a careful record of the circumstances prevailing at the time of the observations. The momentary oceano- graphic conditions have a profound effect also upon the propagation of underwater sound. Hence, it has become a standard practice to secure bathythermo- grams before and after each set of acoustic observa- tions. The transmission loss in the ocean intervening between sound gear and wake, which must be known in order to correct the measured data, is difficult to determine directly. So far, acoustic observations on wakes have not reached such a high degree of pre- cision as to make it imperative, as in the measure- ment of target strengths, to determine the transmis- sion loss in the ocean simultaneously with the wake observations. For details on the technique of meas- uring the transmission loss consult Chapter 4. Perhaps the most serious disturbances of under- water sound measurements are the rapid and un- predictable changes of the transmission loss, generally referred to as fluctuations and described in Chapter 7, which may amount to many decibels over intervals of only a few seconds. The only way to minimize their OPERATIONS AND MEASUREMENTS influence is to take averages over long series of ob- servations. Even these averages may show:a slow drift with time, sometimes called variation of the transmission loss, but the amplitude of the variations is of a lower order of magnitude than that of the fluctuations. In measuring the transmission loss which sound undergoes while passing across a wake, the fluetuations of the transmission loss in the sur- rounding ocean mask the effect sought after, or even may entirely obscure it for wakes in an ad- vanced stage of decay. In echo ranging, the sound returned by different parts of the wake undergoes destructive and constructive interference, which to- gether with the gradual change of the internal structure of the wake will invariably cause fluctua- tions of the wake echoes that are even more rapid than the fluctuations of the transmission loss. Conse- quently, wake echoes are even more variable in shape than sound signals which have been affected only by 493 fluctuation of the transmission loss in the sea. Figure 7 shows the irregular character of wake echoes re- sulting from changing interference effects. Fluctua- tions of the transmission loss in the ocean are also conspicuous in Figure 7; note the change in strength between the last two echoes from the target sphere, in the lower strip of the illustration. In echo ranging at wakes over short ranges, the reverberation background caused by the scattering of sound in the ocean constitutes an important limit- ing factor. Under conditions giving very high rever- beration, a weak echo may become lost in the back- ground. Strictly speaking, the echo intensities de- rived from measured echo amplitudes, as described above, include the contribution from reverberation and should be corrected for this superimposed inten- sity. In practice, this correction may usually be neglected whenever the wake echo is sufficiently strong to be distinguished from the reverberation. Chapter 31 WAKE GEOMETRY i THIS CHAPTER information of rather heterogene- ous origin, concerning the dimensions of wakes, is brought together. Some types of acoustic observa- tions are in themselves eminently valuable for deter- mining the geometric characteristics of wakes. How- ever, a good deal has to be known about the geometry of wakes in order to plan and execute their investiga- tion by acoustic methods. Such knowledge has been provided by visual and photographic observations. Brief reference also will be made to thermal wakes, although very little is known about them so far. It is undecided whether or not the visual, acoustic, and thermal manifestations of the same wake agree as to the volume of the sea from which they originate; this problem deserves further study. WAKE GEOMETRY FROM AERIAL PHOTOGRAPHS 31.1 The serial views of destroyer wakes shown in Figures 2 to 6 of Chapter 26 were selected from a large series of photographs, made available by the Photographic Interpretation Center, U.S. Naval Air Station, Anacostia. They show wakes of the destroyer USS Moale (DD693) proceeding on a straight course at constant speeds, ranging from 16 to 34.5 knots. For each speed, photographs of three or more differ- ent runs were measured, so that the results represent a fair average. The following conclusions are drawn from measurements made on the original prints. Immediately behind the screws the wake diverges with an included angle of about 50 degrees. Indi- vidual angles measured on different photographs vary between 40 and 60 degrees, but no clear-cut depend- ence on speed is indicated; these variations may well be spurious. It may be mentioned in passing that the wake of a stationary propeller ! showed an angle of divergence of about 20 degrees. At a certain distance astern, the wide divergence of the destroyer wake ceases rather abruptly, and thereafter the wake spreads with a total included angle of about 1 degree. 494 This angle too appears to be independent of the speed with which the wake is laid. However, the distance astern at. which the transition from the 50-degree divergence to the 1-degree divergence occurs in- creases very markedly with the speed of the de- stroyer. At 16 knots, it is about 65 ft, and at full speed about 280 ft; this variation of distance with speed is not linear, as far as present experience indi- cates. The numerical values are given in Section 35.1. Observations of several different types, which will be reported in the rest of this chapter, all seem to indi- cate that the wake spreads out with a large included angle immediately behind the wake-laying vessel, and that at distances astern greater than about 100 yd the wake spreads out with a very small included angle, of the order of 1 degree. However, none of these other observations have the same high intrinsic accuracy as the measurements on aerial photographs. Therefore, the results of these measurements, as in- complete as they are, have been selected for inclusion in Section 35.1. The large initial divergence of a wake is quite conspicuously demonstrated in Figures I and 7 of Chapter 26, showing the wakes of a submarine chaser and destroyer, respectively. Aerial photographs also furnish interesting infor- mation on the cross-sectional structures of wakes. For instance, Figures 2 to 5 of Chapter 26 reveal that at short distances astern and at speeds less than 25 knots the destroyer wake has a dense core and edges that stand out conspicuously; with increasing dis- tance, this internal structure gradually fades out. At speeds above 30 knots, the destroyer wake appears to be so strongly turbulent that the core is largely obliterated. Tranverse structure of a different kind is illustrated by the submarine wakes seen in Figures 1 to 6. The wake of a surfaced submarine shows bifurcation, or twin structure, both at 15 and 20 knots in Figures 3 and 4. The same illustrations clearly differentiate a short wake section immediately behind the sub- marine, which has a large angle of divergence, from RATE OF WIDENING 495 Fieure 1. Wake of surfaced submarine at 6 knots. the long wake proper, whose edges show little diver- gence. Thus, the general wake contour is quite similar for destroyers and surfaced submarines. Figure 7 gives a remarkable aerial view of a PT boat and its wake. The wake proper is narrow and compact, without visible structure, but the bow wave, for a distance astern of several ship lengths, is visually much longer than the wake. 31.2 RATE OF WIDENING The rate of widening of an acoustic wake can be determined by measuring the gradual increase of the duration of the echoes obtained with a horizontal sound beam, as long as the signal length is much shorter than the wake width. In practice, subtracting the signal length from the measured length of the echo will correct for the prolongation of the echo length due to the finite signal length and will make possible a direct determination of the wake width. An analysis along these lines was made for four wakes laid by the H. W. Scripps on November 28, 1944. The Scripps passed between the echo-ranging vessel, the USS Jasper (PYc13), which was hove to, and a target sphere 3 ft in radius buoyed at a center Figure 2. Wake of surfaced submarine at 10 knots. depth of 6.5 ft. The wakes were laid at right angles to the line connecting the sphere with the Jasper, each run being made in a new location of undisturbed water. All echoes were recorded oscillographically and sound range recorder traces were obtained simul- taneously. The signals consisted of pulses of 0.5, 1, and 3 msec long, transmitted in cyclic succession. The duration of the 3-msec echoes was measured both on the oscillograms and on the recorder traces; the results, expressed in yards, are plotted in Figures 8 and 9 as functions of the time elapsed since the Scripps passed. The slope of these curves is the rate of widening. In order to find the width at any time, the plotted values of the echo length should be diminished by 2.4 yd. The‘average rate of widening of the Scripps wake, up to the age of 10 minutes, is 5 yd per minute for the chemical recorder traces, and 6 yd per minute for the oscillograms. This difference of 1 yd per minute can hardly be regarded as significant, in view of the dif- ferences between the several runs. Note that in both illustrations the graph of Run 2 is located between 5 and 10 yd above the graph of Run 1, though both runs were made with 24-ke sound. The origin of this shift remains obscure, as the sea was unusually calm, 496 Figure 3. Wake of surfaced submarine at 15 knots. almost without ripples, during the entire day, and the Scripps maintained the same speed of 9.5 knots dur- ing all four runs. Evidently, the geometric and physical properties of wakes are difficult to reproduce in repeated experiments, even under ideal weather conditions. The wake laid in Run 2 gave distinct 60-ke echoes at an age of 30 to 40 minutes; during this period the wake width measured on the sound range recorder trace increased from 82 to 100 yd, corresponding to a rate of widening of about 2 yd per minute. No similar tests for the persistence of wakes were made during the other runs. Comparison of Figures 8 and 9 reveals that there is no systematic difference between the widths found for different sound frequencies. In particular, the alternating use of 45 and 60 ke during Run 4 gave results which are mutually consistent and agree quite well with the 24-ke graphs. At the stated speed of the Scripps, approximately 300 yd per minute, a rate of widening of 5 to 6 yd per minute means that the total angle of divergence of the wake is about 1 degree. This figure is in excellent agreement with the angle of divergence found for destroyer wakes from aerial photographs. If the WAKE GEOMETRY Figure 4. Wake of surfaced submarine at 20 knots. Scripps wake had the same great initial angle of divergence (about 50 degrees) as the destroyer wakes, it could not have been discovered by acoustic width measurement, because this method lacks the neces- sary “resolving power” along the time axis. At wake ages greater than 10 minutes, the rate of widening appears to decline steadily, and the angle of diver- gence must decrease correspondingly. However, it should be remembered that these observations were made on a calm sea. The rate of widening of thermal wakes can be studied by carrying a sensitive thermocouple across the wake at increasing distances astern. These in- vestigations are still in an exploratory stage, but they are mentioned here because preliminary results have been reported for two wakes laid by the HE. W. Scripps.2 Thus a comparison of the thermal and acoustic dimensions of the wakes laid by the same vessel became possible. Between the ages of 10 and 60 minutes, the thermal wakes of the Scripps showed a linear increase in width from 30 to 50 yd. The rate of widening is about 0.4 yd per minute and the speed of the Scripps was 6 knots, or 200 yd per minute. Hence, the angle of divergence of the thermal wake FATHOMETER STUDIES Figure 5. Wake of submarine during crash dive. is only about 0.1 degree, or one-tenth of the diver- gence of the acoustic wake. In addition, the thermal wake appears to be much narrower than the acoustic one. Since the thermal and acoustic measurements were not made on the same day, it is by no means certain that the thermal and acoustic wakes behave as differently as these observations would seem to sug- gest. The 1-degree divergence found acoustically ap- plied to wakes less than 10 minutes old, while the available thermal data were apparently all for wakes more than 10 minutes old. Furthermore, the rate of widening may possibly depend on oceanographic factors, such as the temperature gradients in the surface layers of the sea. 31.3 FATHOMETER STUDIES At the U. S. Navy Radio and Sound Laboratory [USNRSL], numerous measurements with a record- ing fathometer have been carried out on the wakes of a number of different surface vessels and submarines.* Some of these wakes were investigated systemati- cally, and the width and depth of the wake was de- 497 Figure 6. Swirl behind submarine after crash dive. termined as a function of the distance from the wake- laying vessel and of its speed. 31.3.1 Surface Vessel Wakes With surface ships, two methods were used for measuring the wake width. When ranging on the wakes of ships which happened to be passing, the survey boat, carrying the fathometer, crossed the wake as nearly perpendicularly as could be judged while speed and distance measurements were made. Some inaccuracy arose in the judgment of the angle of crossing when very far behind the wake vessel. However, the error introduced into the measured width by assuming perpendicular crossing was usu- ally negligible. When the survey boat was still farther astern, a greater error was present in determining the onset and disappearance of the wake record. The duration of recording was measured on chart paper (see Figures 3 to 6 in Chapter 30) by @ caliper and rule. From the speed of the chart paper and of the survey boat, the width may be calculated. In the other method, which was suitable at close range when working with an assigned vessel, the 498 Figure 7. Wake of PT boat at 25 knots. survey boat was towed by the vessel laying the wake; the latter is called the wake vessel. Measurements can then be made on a wake whose lifetime is effectively constant, in other words, for a constant boat speed, the survey boat is in a wake of the same age at all times. While the wake vessel maintained a steady course, the survey boat under tow was moved in and out of the wake on either side by using the helm. At the moment the survey boat passed the wake edge, as indicated by the fathometer record, the record was marked and a signal was sent to two observers on the wake vessel. One of these observers was on the stern and followed the transverse movement of the survey boat with a pelorus. At the instant of signaling, the angle of the fathometer mounting on the survey boat relative to the axis of the wake ship was noted. The other observer was on the bridge, and at the signal he instantaneously observed the ship’s compass course, for use in correcting for the angle of yaw of the wake vessel. The record was marked at the time of sig- naling the observers on the wake vessel, so that the data could be discarded if it were found that the survey boat was not exactly at the wake edge. The wake of the Jasper (overall length 127 ft, draft 12 ft, beam 23 ft) was studied by the second WAKE GEOMETRY method over the range from 50 to 500 ft astern. All the measured widths, expressed in feet, agree with the formula W = Wo ar 0.0625vt, (1) where wp is the extrapolated width at the stern of the wake vessel, v the speed of the ship in feet per second, and ¢ the time in seconds since its passage. By dif- ferentiating this formula with respect to the time, — — = 0.0625 = sina, (2) where a, the total angle of divergence of the wake edges, is 3.5 degrees. For similarly small distances be- hind destroyers, the wake edges were found to include an angle of about 50 degrees (see Section 31.1). The conspicuous discrepancy between this value and the corresponding one for the Jasper is doubtless due to the different type of construction of these ships. The extrapolated value wo = 10 ft, in formula (1), is very nearly one-half the ship’s beam. At distances greater than roughly 5 ship lengths, the divergence of the Jasper’s wake ceased at a width of perhaps two and a half times the ship beam; only random measure- ments by the first method were available for this region, however, and thus a small divergence angle of about 1 degree cannot be ruled out as far as large distances behind the Jasper are concerned. The measurement of wake thickness was carried out by proceeding into a wake and either remaining in its center while measuring distances and speeds, or by crisscrossing in order to investigate the thick- ness at points across the wake. Crisscrossing was necessary in order to locate the wake when operating at distances when the wake was not visible. A given wake will frequently have different acoustic trans- parencies at different points along its width. In some cases the thickness is the same along the width, and the greater transparency at the edge is caused by its less effective scattering properties. In other cases the wake is thinner at the edges. Some wakes are quite flat at the bottom, others are rounded at top and bottom, or may have one side which sinks below the other at both top and bottom. A wake cross section asymmetrical in the vertical plane parallel to the beam of the vessel was frequently observed and is apparently correlated with wind di- rection. Such records were first noted when operating with one engine of a twin-screw vessel and were thought to be the result of this asymmetrical source. The wake of a sailing vessel was investigated next, and was found to have an even more pronounced FATHOMETER STUDIES 60 > °o 7) ra) < > as ae = 2 > w x a = AGE OF WAKE IN MINUTES Ficure 8. Increase of wake echo duration for E. W. Scripps at 9.5 knots. Measurements of oscillograms. WAKE WIDTH IN YARDS AGE OF WAKE IN MINUTES FIGURE 9. asymmetry. The sailing vessel heeled over consider- ably during the runs and the varying area of the hull in contact with the water on either side was con- sidered a cause for asymmetry. The wake of a twin- screw vessel when both screws were turning and when the wind was appreciable was then recorded. Again Increase of wake echo duration for E. W. Scripps at 9.5 knots. Measurements on chemical recorder traces. asymmetrical results were found. The effect is inde- pendent of the direction from which the survey boat crosses the wake. A typical value for the slope of the bottom of an asymmetric wake, as found for a 125- ft vessel, is 18 degrees. When the wind shifts from port to starboard, the cross-section geometry of the 500 WAKE GEOMETRY wake should change to a mirror image of its former geometry, but in the majority of cases this expecta- tion is not entirely confirmed. Perhaps some as yet undiscovered parameter is responsible for this puz- zling behavior. results from actual variations of the wake structure. The wake thicknesses were plotted as a function of the distance astern and examined for a possible systematic variation. The slope of the bottom of the wake up to 800 yd astern was found to be 5 minutes TaBLE 1. Wake thicknesses. USS Rathburne | USS Hopewell | USCGC Ewing (ex-DD118) (DD681) Speed in knots 10-12 10 13 Thickness of wake in feet for average distance astern of 400 yd 19.5 + 3.4 23.1 + 3.6 13.7 + 3.3 Range of thickness in feet 12-26 10-32 8-20 Ratio * of wake thickness to ship draft 1.63 1.85 1.52 * These ratios are smaller than those previously found 3 for incidental destroyer wakes, and are believed to be more accurate. Aside from the miscellaneous results just described, an attempt was made to investigate systematically the variation of the thickness h of the wakes of two yachts, the USS Jasper (PYc13) and the LE. W. Scripps, with distance astern up to 3,000 ft and with speed from 3.5 to 11 knots. For either vessel, no systematic changes of h could be noted. A fair average of all measurements of h was 1.70 times the draft, or 2.9 times the screw depth for the Jasper, and 1.11 times the draft, or 3.0 times the screw depth for the Scripps (overall length 104 ft, draft 12 ft, single screw 8 ft above the keel). Scattered measurements made on the wakes of numerous large surface vessels of all types gave an average ratio of thickness to draft of 2.02. The wakes of small craft appear to be relatively thicker, with a thickness to draft ratio of the order of four. The only wake depth shallower than the draft was from a carrier wake 4,000 yd from the ship. For a speed- boat, h appears to increase considerably with speed. All these observations were made with a fathometer ranging downward from a measuring boat in the wake being investigated. Later measurements * were made with a fathometer mounted on the deck of a sub- merged submarine, ranging upward at the surface of the ocean. This method, for several reasons men- tioned in Section 30.1.2, provided more accurate data than was possible with the former. The accuracy of the individual thickness determination is such that the range of wake thicknesses summarized in Table 1 of arc (or 4 ft per 1,000 yd) upward for the USS Rathburne (ex-DD113) and 16 minutes of are (or 14 ft per 1,000 yd) downward for the Ewing. In other words, the differential quotient of the thickness of the wake with respect to the time, which will be required in the later discussion of the decay rate of wake strength, has the following values as upper limits: 1 dh aR —0.08 min“ for the Rathburne at 10 knots, 1 dh ; 5 Rona 0.04 min~ for the Ewing at 13 knots. Additional information on the rate of widening of destroyer wakes is found in a report by UCDWR.* Wakes were laid by three different modern destroyers, running past the H. W. Scripps at 15 knots. The Scripps was hove to and recorded the sound level of a transducer carried repeatedly across the wake by a 50-ft motor launch. The sound level records showed definite breaks whenever the source crossed what may be called the acoustic boundaries of the wake; the time between these breaks was multiplied by the speed of the launch to give the width of the wake, suitable allowance being made for the occasional crossing occurring as much as 30 degrees away from the perpendicular transit. The plot of the entire data collected in this manner (Figure 22 in reference 5) suggested to the experimenters that new wakes widen more rapidly than old ones, with a total included FATHOMETER STUDIES angle of 2 degrees observed as far as 500 yd behind a 15-knot destroyer, and an included angle of 1 degree thereafter. However, a critical examination of the plot reveals such a large quartile deviation that the reality of the differentiation between new and old wakes seems somewhat doubtful. An included angle of 114 degrees for the entire range of observations, with the maximum distance astern of 2,500 yd, gives a fair representation of the plot. The extrapolated initial width of these destroyer wakes is roughly equal to the beam of the vessel, perhaps somewhat smaller. However, the observations do not cast any light on the very large initial divergence of destroyer wakes, revealed by aerial photographs, because the acoustic measurements did not extend to distances less than 100 yd astern. It should be noted that the angle of spread derived from the acoustic measure- ments (114 degrees) is in fair agreement with that derived from aerial photographs (1 degree), as re- ported in Section 31.1. It is possible to attribute the difference of 14 degree between the two figures en- tirely to the inaccuracies inherent in the respective processes of measurement. 31.3.2 Submarine Wakes Information on the geometry of submarine wakes is less detailed. Among the measurements made with the fathometer ranging downward,‘ an investigation TaBLE 2. Wake of submarine at periscope depth. Distance from Wake top Depth of wake periscope in yards in feet bottom in feet 67 39 70 100 0 27 117 0 40 152 0 36 215 0 31 315 0 25 319 0 28 350 0 30 450 0 23 of the wake of a fleet-type submarine 309 ft long is reported. At periscope depth the keel is submerged to a depth of 60 ft, the screws to a depth of 48 ft, and the deck to a depth of 35 ft below the surface; the speed was 5.5. knots. Table 2 contains the observed depths. The same information for a surfaced sub- marine of the same class, moving at 7 knots, is given in Table 3. The maximum distance of 450 yd appearing in Table 2 is not the upper limit of detectability of the wake at a keel depth of 60 ft, as the observers emphasized. The length of the subsurface wake of an S-class submarine,® running at 6 knots, was found to be about 1,000 yd at a depth of 45 ft, 235 yd at a depth of 90 ft, and 100 yd at a depth of 125 ft. These figures give the distances astern of the submarine over which the wake extends before it becomes un- detectable by the gear used in these experiments. The bow-mounted 24-ke transducer was trained at a fixed bearing of 30 degrees relative to the Jasper, TaBie 3. Wake of surfaced submarine. Distance Wake bottom astern depth in yards in feet 100 32 145 24 180 29 300 26 480 18 660 26 800 21 950 22 which was following the submarine on a parallel course and then gradually fell back. At creeping speed (2 to 3 knots) the length of the acoustically effective wake is less than 30 yd for a fleet-type submarine, according to recent San Diego observations.’ It would seem, then, that the subsurface wake is not a good scatterer at greater than periscope depth, particularly at slow speeds. Analogous experiments ° at frequen- cies of 24 and 45 ke were carried out with a fleet-type submarine, running at speeds up to 9 knots and at depths down to 400 ft. According to the observers, during no run was an echo definitely identified as coming from the wake alone. From the data summarized in Table 2, it appears that the wake of a fleet-type submarine, running at 5.5 knots at periscope depth, extends to the surface at distances astern greater than 100 yd; the single record at a shorter distance of 67 yd, which suggested a com- pletely submerged wake, unfortunately was uncer- tain. Later tests, using the same fathometer equip- ment with the fleet-type submarine USS Trepang (SS412), have led to a general confirmation of the previous results.? The Trepang was running at 8 knots at a keel depth of 60 ft, and the wake appeared 502° at the surface at a distance of 600 ft astern. From this figure, the slope of the top of the wake may be computed, assuming it is constant; the ratio of screw depth to distance of emergence of wake, 48/600 or 0.08, corresponds to a total angle of divergence of 9 degrees at the screws. This value, however, is WAKE GEOMETRY based on only one record. No clean-cut wake records were obtained at greater depths (200 and 400 ft), but this may be attmbuted to purely operational diffi- culties since the submarine found it difficult to pass directly under the stationary launch carrying the fathometer. Chapter 32 OBSERVED TRANSMISSION THROUGH WAKES N CROSSING A WAKE, sound undergoes a transmis- I sion loss in addition to that resulting from propa- gation through the ocean at large. Transmission loss in the ocean is primarily geometric — the sound beam spreads over large distances because of the inverse square law and because of refraction condi- tions. At frequencies less than 100 ke, the transmis- sion loss from physical causes, such as scattering and absorption, is not very important at the short ranges —a few hundred yards or so — of interest in wake measurements. The observed transmission loss in wakes, however, is ascribed exclusively to physical causes, scattering and absorption by air bubbles, because the dimen- sions of wakes are much smaller than the distances over which the geometric effects are particularly im- portant. An exception to this rather sweeping state- ment may have to be made in the case of sound originating in the wake, as described in Section 32.3. These phenomena, however, are little understood at present, as they have not been sufficiently studied. 32.1 DEFINITIONS The physics of the transmission of sound through wakes has already been fully discussed in Chapter 28. All that is necessary here is to summarize the conven- tions concerning the expression and presentation of the measurements of the transmission loss through wakes. The total transmission loss undergone by a sound beam on traversing a wake, or the attenuation, as it is usually called in underwater sound work, is defined by the equation 1(0) I(w) where J(0) is the intensity of a parallel beam of underwater sound before entering the wake, and I(w) is its intensity after it has penetrated the entire width w of the wake; the transmission loss in the H. = 10 log (1) wake H., is distinguished by the subscript w from the transmission loss in the ocean at large, which is com- monly denoted by the symbol H. According to equa- tion (53) of Chapter 28, the attenuation by the wake can be represented as a product, namely Hy, = Kw, (2) where w is the geometric width of the wake, usually measured in yards, and K, is the so-called coefficient of attenuation in decibels per yard. Definitions (1) and (2), as they stand, apply to a sound beam im- pinging perpendicularly upon the wake; for oblique incidence w obviously has to be replaced by w sec 8, where f is the angle included between the beam and a line perpendicular to the wake. Note that equation (1) may be written in the form IC) aren =——=s 10. °", 3 1(0) (3) or, by substitution from equation (2), I(w) —Kew/10 Ss = 1 4 7(0) (4) Both H,, and kK, are overall properties of the wake, and it remains to express them as functions of the physical parameters describing the microstructure of the wake, which is known to consist of multitudes of bubbles of all sizes. The acoustic properties of bubbles have been characterized in Chapter 28 by their indi- vidual cross sections os, oa, o- for scattering, absorp- tion, and extinction of sound, respectively. It will be remembered that these quantities vary considerably according to the size of the bubbles, and that, by and large, only bubbles near resonant size make a signif- icant contribution to the average cross section ap- plying to a population of bubbles of all sizes. Should all the bubbles in the wake happen to have exactly the same size, the coefficient of attenuation would be given by [see equation (55) of Chapter 28 ] w H. = K,. = — = 4.34no, db per cm (5) W 503. 504 K, = — = 396.8nc. db per yd , (6) Hy or = w where o, in square centimeters is the extinction cross section of this particular size of bubble and n in em~* is the average number of bubbles of this size per cubic centimeter in the wake, as defined by equa- tion (54) of Chapter 28. In the more realistic case of bubbles of many sizes, the attenuation coefficient is given by equation (67) of Chapter 28, Hy K,= Ww = 1.4 X 10°u(R,) db per yd , (7) where u(R)dR is the total volume of air contributed by bubbles with radii between R and R + dR in 1 cu cm of the air-water mixture, or rather the average of this quantity taken over the entire column in which the sound beam and the wake intersect; R, in equa- tion (7) is the radius of the resonant bubbles cor- responding to the sound frequency used in deter- mining Hy. The total attenuation corresponding to equation (5) is Hy = 4.340.nw = 4.340.N(w) , (8) where N(w) = nw denotes the total number of bubbles in a column of unit cross section. Thus NV (w) is a measure of the total bubble population affecting the sound beam. Differentiating equation (8) logarithmically with respect to the time, the decay of the transmission loss across the wake is obtained. 1 dH, 1 dN(w) H, dt Nw) dt At first, dN(w)/dét perhaps will be positive for suf- ficiently small bubbles, whose number might be in- creased rapidly by the gradual dissolution of bubbles of originally larger size. But ultimately, dN (w)/di must become negative. It is seen then that the decay rate of the transmission loss affords a direct measure of the rate of disintegration of the bubble population. (9) 32.2 EXPERIMENTAL PROCEDURES In principle, the experimental determination of the transmission loss through the wake requires only relative measurements of sound intensities. If over the period of observations the range from transducer to hydrophone, and the transducer output and hydro- phone sensitivity remain constant, then the absolute values of any of these three quantities does not have OBSERVED TRANSMISSION THROUGH WAKES to be known; the difference of sound levels recorded by the hydrophone before and after the wake has been laid across the sound beam is simply the trans- mission loss H». Whenever, during the course of ex- periments, the range changes appreciably a correc- tion must be applied, based on the appropriate value of the transmission loss H in the surrounding ocean. Care should be taken to place both transducer and hydrophone at such a depth that they are completely hidden from each other by the wake. A characteristic feature of transmission measure- ments of this simplest type is that, for sufficiently short wavelengths, only a very narrow cone of the divergent sound beam emitted from the transducer is utilized, namely the solid angle subtended by the face of the hydrophone at the location of the trans- ducer. Thus, the instantaneously recorded transmis- sion loss is for a sharply bounded layer of the wake. The roll and pitch of the vessel carrying the trans- ducer and hydrophone will raise and lower both instruments and will cause that narrow pencil of sound to traverse the wake at different depths below the ocean surface. Since the acoustic thickness of the wake is likely to vary somewhat vertically, corre- sponding variations of the measured transmission loss must be expected. In one respect these variations are even helpful. They afford an automatic smoothing out of the verti- cal variations of the acoustic thickness and thus pro- duce a better representation of the average state of the wake. The directivity of the sound gear is also important, in so far as rolling and pitching of the vessels carrying the transducer and hydrophone, to- gether with possible training errors, may affect their relative orientation and hence may cause fluctuations in the strength of the signals received. In practice, this effect cannot be separated from other fluctua- tions of the signals, resulting from changes of the transmission loss in the- ocean interposed between transducer and hydrophone. By averaging over long series of signals, these disturbing influences may be minimized, though perhaps not fully eliminated. 32.3 TRANSMISSION LOSS ACROSS WAKES 32.3.1 One-Way Horizontal Trans- mission Loss Transmission loss in-wakes has been investigated comprehensively only for five vessels of the destroyer TRANSMISSION LOSS ACROSS WAKES TRANSMISSION LOSS IN 0B AGE OF WAKE IN SECONDS Figure 1. Sound transmission loss due to wake versus age of wake. Ship IV, December 30, 1948, 15 knots. Source beyond wake. type — two old destroyers of the 1916-1917 class, a new destroyer of the Fletcher class, and two destroyer escorts.1 Wakes were laid at speeds of 10, 15, 20, and 25 knots. A 50-ft motor launch repeatedly carried the projecters, mounted at depths of 6 and 7 ft, respec- tively, across the wake, while the hydrophones were suspended from the bow of the EF. W. Scripps at a depth of 10 ft, about half the depth of the wake (see Section 31.3). Sound at frequencies of 3, 8, 20, and 40 ke was recorded both with the launch beyond the wake and with the launch inside the wake; while sound recorded when the launch was on the near side of the wake provided reference values. By applying a correction for the measured average transmission loss in the ocean, all sound levels were reduced to a standard distance of 100 yd, for the three cases of (1) source beyond wake, (2) source in wake, and (3) no wake intervening. The difference between case (3) no wake intervening and case (1) source beyond wake was taken to be the transmission 505 SOURCE IN WAKE EQUATION (10) SOURCE BEYOND WAKE EQUATION (11) TRANSMISSION LOSS IN DB 5 10 1S 20 30 SPEED OF WAKE-LAYING VESSEL IN KNOTS Figure 2. Dependence of transmission loss on speed of wake-laying vessel. loss for the source beyond the wake, or H,, as defined in equation (8) of Section 32.1; similarly, the differ- ence between case (3) no wake intervening and case (2) source in wake was taken to be the transmission loss for the source in the wake. In the original paper, the results are reproduced in separate graphs for each of the several vessels, speeds, frequencies, and locations of the source; one of these is reproduced in Figure 1. However, not all the possi- ble combinations of the different parameters are actually shown. Although not representing the best fit for every single set of observations, the following interpolation formulas are believed to represent ade- quately most of the data. Source in wake Hj, = 2.4(vf)? — (4.8 + 1.6)t, (10) Source beyond wake H,, =1.5(of)?— (3.0 + 1.4)t, (11) where v is the ship’s speed in knots, f is the frequency of the sound in kilocycles and ¢ is the time in minutes which has elapsed since the passage of the screws or age of the wake. No standard errors are assigned in 506 OBSERVED TRANSMISSION THROUGH WAKES the original report to the numerical coefficients of the first terms of equations (10) and (11), but it is stated that the initial values (t = 0) of the transmission loss for individual runs show a scatter of the order of 3 db. Figure 2 gives an idea of the accuracy with which equations (10) and (11) represent the initial trans- mission loss at different speeds and frequencies. A higher transmission loss for case (2) source in wake, than for case (1) source beyond wake, appears to be well established observationally, but the theo- retical explanation for this systematic difference is not at all evident. With the source located inside the aer- ated water of the wake, air bubbles are likely to be held on the face of the transducer by adsorption. There are theoretical reasons for believing that such a layer of adsorbed gas should reduce, or “quench,” the output of the transducer, causing an apparent increase of the transmission loss in case (2). However, it is somewhat surprising that the quenching effect should show a behavior regular enough to follow equation(10). The difference in the decay rate for case (2) the source in wake and case (1) the source beyond wake is dH, ha dH» : —— = 1.8 db per minute. This difference may not be significant in view of the standard errors of these quantities. However, if it is accepted at its face value, the relative rates of decay are equal to each other, and given for fresh wakes by the equation 1 dH, 1 idly Gh Jilin OB GH ee (of) This equality between the two rates is evident from equations (10) and (11) in which corresponding co- efficients have the same ratio of 5/8. According to equation (9) of Section 32.1, the relative rate of decay is a function solely of the rate of disintegration of the bubble population. The physical significance of the observed decay rate will be discussed in Section 34.4. Some incidental information on the transmission loss across wakes has been obtained during measure- ments of the underwater sound output at 5 ke of a destroyer, cruiser, and aircraft carrier? observed at varying speeds. Measurements at higher frequencies were also made, but the results are inconclusive as far as the transmission loss across wakes is concerned. All that can be said about the transmission loss at 25 and 60 ke is that it is distinctly higher than at 5 ke; (12) residual sound intensities, after passage through the wake, in most cases had dropped to the background noise and thus made impossible an evaluation of the transmission loss. Even the 5-ke data, plotted as a function of age of the wake, are rather widely scat- tered. But for each of these vessels the plot is not in- consistent with tentative predictions made from equation (11) above, a fact that is somewhat sur- prising in view of the dimensions of two of these three ships listed in Table 1. The initial transmission loss Taste 1. Ship dimensions. Light Destroyer Cruiser Carrier USS Colhoun | USS Trenton | USS Hancock (DD801) (CL11) (GV19) Length in feet 376 555 874 Beam in feet 39 55 93 Draft in feet 13 13 29 Screw depth in feet 11.25 19.5 21.3 (t = 0) is about 15 db at 5 ke for each vessel, while formula (11) gives 17 db at 25 knots for 5-ke sound. At higher frequencies, the greater absolute value of the transmission loss might facilitate the detection of possible differences between the destroyer and the larger ships. 32.3.2 Two-Way Horizontal Trans- mission Loss Another method of measuring the horizontal trans- mission loss has been tried out in experiments aimed at a simultaneous determination of wake echo strength and transmission loss.*? The EL. W. Scripps, running at 9.5 knots on a straight course, laid a wake between the USS Jasper (PYc13) and a target sphere 3 ft in diameter buoyed at a center depth of 6.5 ft. The drop in the apparent target strength after the wake was introduced thus was taken to represent the two-way transmission loss across the wake. The wake echo intensity could also be measured on each oscillo- gram, giving the effectiveness of the wake as a scatterer of sound. A plot of the sphere and wake echo levels for one of the 24-ke runs is shown in Figure 7 of Chapter 33. The results are summarized in Table 2; further reference to the decay rate of the echo strength will be made in Section 33.4. The 45 and 60-ke data on which the echo strength TRANSMISSION LOSS ACROSS WAKES recovery and decay rates quoted are based were taken quasi-simultaneously by tuning the sonar equipment alternately to the two frequencies for two- minute intervals. The wake and sphere distances for this run are those quoted in the 45-ke row. The 9-db drop in apparent target strength at 60 ke is based on a separate run, with wake and target distances as stated in the third row of the table. 507 32.3.3 Two-Way Vertical Trans- mission Loss A recording fathometer has been used for the meas- urement of sound transmission loss in the vertical direction through surface ship wakes.° The fathom- eter was secured on the deck of the submarine USS 8-18 (SS123) so as to range upward onto the surface TasBLeE 2. Effect of wake on sphere echoes. Maximum drop Rate of Depth at in sphere recovery Rate of which sound echo level of sphere decay of Distance to | Distance beam passed with wake echoes wake echoes Frequency | wake center | tosphere | through wake present in db per in db per in ke in yards in yards in feet in db minute minute 24 270 350 10 6 1.4 1.5 45 97 162 12 No data with fresh wake 0.7 60 58 98 12 9 | 0.8 0.7 Earlier measurements of the Scripps wake * gave a depth of the wake bottom of 13 ft. According to the values quoted in Table 1, the sound beam passed definitely above this bottom depth of 13 ft. However, a short time before the data of Table 2 were obtained, the Scripps had been outfitted with a new engine and propeller, so that the wake dimensions may have been altered to some extent. The present propeller is 3.8 ft in diameter and the shaft is 5.5 ft below water line. Therefore, since the Jasper’s sound projector is 15 ft deep, maximum acoustic shadowing of the sphere by the wake could not be expected immediately after the Scripps had passed. These wakes widened later- ally, as measured by the wake echo elongation, at about 6 yd per minute. The same rate of spreading may also be applicable in the vertical sense without necessarily implying that a strongly absorbent “‘core”’ of the wake ever moves down to an effective screening position in these experiments. This may account for the low magnitude of the observed transmission loss. Similarly the decay rate of the transmission loss dH,,/dté is one-half the rate of recovery of the sphere echoes; hence dH,,/dt is 0.7 and 0.4 db per minute for 24- and 60-ke sound, respectively. These decay rates are much smaller than that of destroyer wakes, which were found to be independent of frequency — 3.0 db per minute. However, the relative rates of decay are in moderate agreement with those computed from equation (10) for a destroyer speed of 10 knots; numerical values are shown in Table 3. of the ocean, the echoes being continuously recorded in the control room. With this arrangement, the effect of a surface ship wake is recorded as the sub- marine passes beneath it. The ocean surface is used as a “standard target.’’ Sample records obtained with this method are shown in Figures 4 to 6 of Chapter 30. TaBLE 3. Relative rates of decay. 24 ke 60 ke 1 dH, = if, Gh. for E. W. Scripps at 9.5 knots (from Table 2) 0.23 min | 0.09 min = oe DD at 10 knots [f: H, at °° a nots [from equation (10) ] 0.13 min | 0.08 min Quantitative transmission loss results are obtained from the fathometer records by a special procedure of operating the instrument in conjunction with calibra- tion records made in the laboratory. As the submarine passes beneath a surface ship wake, an attenuator in the receiver-amplifier is adjusted so that the effect of the wake plus the effect of the attenuator is such that a light gray “‘voltage-sensitive” record of the ocean surface echo is produced on the chart paper. Some practice is required, as very little trial-and-error time is available while the submarine is directly below the wake. 508 The procedure is completed by determining, in effect, the amount of amplifier attenuation required to record the unobscured ocean surface at the same density as that of the record taken below the wake. The difference of attenuator settings in the two cases yields the wake transmission loss directly. The actual procedure was complicated by the lack of a calibrated attenuator; the details of the necessary laboratory calibration of the gain control by matching records for different gain settings and echo levels are de- scribed in reference 5. The fathometer record yields an accurate value of wake thickness in each case so that attenuation coefficients can be computed in decibels per foot of wake thickness. The coefficient of reflection at the ocean surface cancels from the measured transmission loss, because pw affects the sound levels both in and outside the wake in an identical manner, aside from slow varia- tions of the state of the sea. If the ocean surface were a perfect plane, and if the axis of the sound beam impinged upon it perpendicularly, the entire off-axis output of the fathometer would be reflected so as not to return to the transducer. On account of the waves, swells, and other irregularities of the surface, and be- cause of imperfect leveling of the submarine, actually some off-axis sound is reflected back on to the face of the transducer. Hence, it is necessary to keep the submarine at a depth shallow enough to make the central lobe of the sound beam fall entirely inside the wake. This condition was well fulfilled during these experiments, the results of which will now be described. Sound of 21 ke was found to undergo an average attenuation of 18 + 3 db during vertical one-way passage through the wakes about 400 yd behind the USS Rathburne (APD25, ex-DD113), USS Hopewell (DD681) and USCGC Ewing, traveling at speeds of 10 to 13 knots. Combining these total attenuations with the wake depths h for the vessels, accurately determined from the same records and already dis- cussed in Section 31.3, average attenuation coef- ficients in the vertical direction in decibels per yard could be computed and were found to be 3.0 + 0.6 db per yd for the Rathburne and Hopewell and 4.8 + 1.5 db per yd for the Ewing. These are grand averages, disregarding differences in the distance astern, which are unknown in many cases, and disregarding devia- tions of the point of measurements from the center of the wake; moreover, some “knuckles” are included with the straight runs. If only data referring to known distances astern and to the center of straight OBSERVED TRANSMISSION THROUGH WAKES wakes are retained, all observations applying to wakes laid by the Hopewell are eliminated. Plotting as a function of the distance astern, the attenuation coefficients for the wake of the Rathburne, running at a speed of 10 knots (corresponding to a screw-tip speed of 52 ft per sec), the following linear interpola- tion formula is found for the range from 100 to 800 yd: Ay |, = (3-135 + 0.057) + (0.093 + 0.018) x Eee astern 100 yd ) db per yd. (18) Since there is apparently no correlation between the total transmission loss in the wake H, and the dis- tance astern, equation (13) implies that the wake be- comes thinner in the vertical direction as it ages. A similar plot for the Ewing, running at 13 knots, re- veals an enormous variation of the attenuation coefficient ranging from 2.4 to 6.6 db per yd without any clear dependence on the distance astern. The distances astern cannot, however, be very accurately determined in these experiments. There is no obvious explanation why the Ewing data should show a greaterscatter, enough to obliterate any dependence on distance astern. The higher value of the attenuation coefficients for the Ewing has been associated tenta- tively with the greater screw-tip speed (112 ft per sec at 13 knots) of this vessel, compared with the two destroyers. No corroboration for this surmise could be found among the observations of the hori- zontal transmission loss through wakes laid by dif- ferent destroyers, already described in Section 32.3.1, although the screw-tip speeds of these vessels ranged from 80 to 137 ft per sec at 15 knots, and from 53 to 95 ft per sec at 10 knots. It is of interest to compare the attenuation coef- ficient in the vertical direction with that computed from the total transmission loss measured hori- zontally. According to equation (11) in Chapter 32, the one-way horizontal transmission loss through the wake 400 yd behind a destroyer traveling at 10 knots is about 20 db for 21-ke sound. ‘The width of this wake is about 75 ft, according to Figure 22 of refer- ence 1. Hence, the horizontal attenuation coefficient is about 0.9 db per yd, or about one-third of the vertical one, as reported above for the destroyers Rathburne and Hopewell. This discrepancy is probably real, but hardly disturbing. In fact, the average at- tenuation coefficient would be expected to be smaller PROPAGATION ALONG WAKES horizontally than vertically in case the wake has a strong core and weaker fringes, because the vertical measurements refer to the center of the wakes. 32.4 PROPAGATION ALONG WAKES On the whole, the methods employed for the study of sound propagation across wakes, described in Section 32.3, have led to apparently consistent re- sults. For sound propagation along wakes, however, the observations do not fit easily into the general picture; they are a few in number and provide in- sufficient data to permit a complete analysis of all the factors involved. A mechanical noisemaker ° was towed both in and below the wake of a destroyer running at 10 and 14 knots, and sound levels were recorded simultaneously by two hydrophones — one towed in the wake by the destroyer and the other suspended at a depth of 10 ft from a boat which was hove to. The destroyer fol- lowed a straight course past this boat, while the distance between the noisemaker and the towed hydrophone was steadily increased from 50 ft to 1,200 ft by unreeling the hydrophone cable. As the cable lengthened the hydrophone gradually descended, ultimately passing below the bottom of the wake, which was assumed to be 20 ft below the surface.’ The noisemaker was towed 50 ft behind the destroyer, and the hydrophone reached a depth of 20 ft at distances of 400 ft (10 knots) and 1,000 ft (14 knots) behind the noisemaker. Finding the transmission loss along the wake would require comparing the sound levels recorded by the towed hydrophone with levels recorded by a hydro- phone when no wake is present in the direction of the noisemaker. Unfortunately, the levels recorded by the stationary hydrophone, suspended from the boat outside the wake, cannot be used, because the direc- tivity pattern of the noisemaker is unknown. It should be noted that the aspect of the noisemaker, as viewed from the towed hydrophone, is practically constant, while the aspect of the noisemaker relative to the stationary hydrophone changes by about 90 degrees while the destroyer is moving toward, or re- ceding from the point of closest approach. The sound levels obtained by the towed hydro- phone with the noisemaker towed at a depth of 40 ft may serve as an approximate reference level repre- senting the wake-free state, because then most of the path from the noisemaker to the hydrophone runs below the wake. Subtracting these sound levels from 509 the ones applying to the noisemaker towed in a wake, an approximate value for the transmission loss along the wake is found. The numerical values are about 6 db for 3-ke sound and about 13 db for 8-ke sound at a speed of 10 knots; at 14 knots, the values are about 13 db and 30 db for 3-ke and 8-ke sound re- spectively. These transmission losses are of the same order of magnitude as those found in propagation across wakes. The increase of transmission loss with frequency is also in agreement with what has been learned about sound transmission across wakes. However, for the entire range covered (100 to 1,000 ft) the transmission loss along the wake does not show the expected increase with distance from hydrophone to noisemaker. The sound levels used as reference values, with the noisemaker 40 ft below the surface, vary inversely as the square of the distance between hydrophone and noisemaker. But the sound levels recorded with the noisemaker in the wake also follow approximately the same inverse square law. In other words, the measured transmission anomalies fail to show any increase with distance behind the noisemaker, which would readily be interpreted as caused by attenuation inside the wake. There is even a slight decrease, perhaps 3 or 4 db, over a range of 1,000 ft; however, this decrease may result from the presence of bottom-reflected sound. Measurements of the destroyer ship sound, with no noisemaker present, gave results similar to those obtained with the noise- maker. These observations are rather puzzling. The measurements of the sound output of a de- stroyer, cruiser, and aircraft carrier 2 give additional evidence of a very low transmission loss along wakes. During the so-called Z runs, the vessel to be measured passed the measuring vessel, which was hove to, and then made a turn so that, during the receding run, the axis of the wake coincided with the line connecting the stationary vessel with the receding one. The sound levels recorded were corrected for the transmission loss resulting merely from geometrical divergence ac- cording to the inverse square law, and from the cor- rected levels a transmission anomaly was derived. Attenuation coefficients along the wake were found to be 10 to 80 db per kyd; these attenuation coef- ficients were not judged sufficiently accurate to war- rant a discussion of variation with speed (16 to 30 knots), frequency (5, 25, and 60 kc) and ship type. In order to appreciate fully how small those attenua- tion coefficients measured along wakes are, it should be remembered that the measured transmission loss across wakes (see Section 32.3) corresponds to attenu- 510 ATTENUATION IN 0B DEPTH IN FT Figure 3. Ten-inch propeller, 1,600 rpm. ation coefficients of 300 to 6,000 db per kyd. This enormous difference is apparently real but has not yet been explained. 32.5 TRANSMISSION LOSS IN MODEL PROPELLER WAKES Attenuation measurements on wakes of ships under way have been supplemented by experiments with wakes of a stationary model propeller. At the Woods Hole Oceanographic Institution,” an electrically oper- ated device was constructed for driving submarine propellers at speeds ranging from 266 to 1,600 revolu- tions per minute at various depths. This equipment was used in water 70 ft deep. First the relation between sound output and speed of the propellers at constant depth was studied, and the critical speed marking the onset of cavitation was determined. Four propellers ranging from 10 to 20 in. in diameter were employed. The noise level increased sharply whenever the tip speed of the propeller blades exceeded 33 ft per sec. In earlier experiments with 2-in. propellers, mounted in an experimental chamber, a critical speed of 35 ft per sec had been found at the same hydrostatic pressure. The agree- ment between these two figures appears quite satis- factory. Precise measurements of the attenuation were ob- tained by an arrangement in which the transducer and hydrophone were mounted on opposite sides of the wake on a pipe frame attached to the boom carry- ing the propeller and held rigid by wire stays. The instruments were 9 ft behind the hub of the propeller. In this way the axis of the wake was made to pass OBSERVED TRANSMISSION THROUGH WAKES ATTENUATION IN OB DEPTH IN FT Ficure 4. Fourteen-inch propeller, 1,600 rpm. between the transducer and the hydrophone, which were on opposite sides of it at a fixed distance of 6 ft. This arrangement had the advantage that it was easy to handle and could be used in deep water with complete assurance that the position of the instru- ments relative to the propeller would not change. However, it did not allow any. variation of the dis- tance between the instruments and the propeller. Hence, it was impossible to determine the decay rate along thé wake. With this arrangement measurements of sound at- tenuation were made systematically at different depths and with different frequencies. Hach measure- ment of attenuation involved the observation of the response of the hydrophone under three conditions: (1) with oscillator on and the propeller at rest; (2) with the oscillator on and’ the propeller turning; (3) with oscillator off and the propeller turning. By suitable combination of these data it was possible to correct the observations for the noise produced by the pro- peller. Typical results for the different propellers are illustrated in Figures 3 and 4. First, it will be noted that the attenuation in- creases with frequency, being almost absent at 10 ke and rising steadily to 60 ke. This increase with fre- quency, at any fixed depth, is so steep that it defi- nitely exceeds the increase with frequency of the transmission loss through destroyer wakes, which is approximately proportional to the square root of the frequency [see Section 32.3.1, equations (10) and (11) ]. Second, at each frequency, the attenuation diminishes considerably with depth. This effect is more pronounced at the higher frequencies. Since the destroyer wakes have an average depth of 20 ft, and the transmission loss through them is a sort of aver- TRANSMISSION LOSS IN MODEL PROPELLER WAKES age over this entire range of depths, the second effect partially cancels the first one. Moreover, the bubble population found in a destroyer wake may be quite different from that in the wake of the stationary pro- peller (zero slip), and any variation of the relative abundance of bubbles of different sizes is likely to pro- duce frequency-dependent acoustic effects. Hence, the discrepancy noted above is not alarming. In the case of the 10-in. propeller (Figure 3), the attenuation falls to almost zero at a depth of 40 ft for all frequencies. This phenomenon is consistent with the results obtained in the model chamber mentioned. In the model experiments it was found that fewer nonpersistent cavities were formed at higher pres- sures. According to the mechanism of bubble forma- tion described in Section 27.1 higher pressure causes a more rapid collapse of the cavities formed, before there is time for a considerable amount of gas to diffuse into them; moreover, cavities containing a given amount of gas are compressed into bubbles of smaller size at higher pressures. Observations were also made at two frequencies with transducer and hydrophone in the wake, both 511 mounted in the wake axis, but their number is small and no clear-cut conclusions can be drawn from them. There is some indication that for 50-ke sound the out- put of the transducer may be reduced, or “quenched”’ by the wake, but for 10 ke the “‘quenching” effect, if it exists at all, is very much smaller than for 50 ke. In summary, the observations of wakes produced by a stationary propeller are in reasonable agreement with those of destroyer wakes, as far as the depend- ence on frequency of the transmission loss is con- cerned. By dividing the attenuations plotted in Figures 3 and 4 by the distance between transducer and hydrophone (6 ft), attenuation coefficients can be computed. For instance, at a depth of 15 ft attenu- ation coefficients of 3.6 and 2.3 db per yd are found for 25-ke sound, which is the same order of magnitude as found for destroyer wakes at speeds of 10 to 15 knots. Since the diameter of the wake probably was smaller than the distance from transducer to hydro- phone, the values quoted for the attenuation coef- ficient are actually lower limits; the true attenuation coefficient may have been greater by 50 to 100 per cent. Chapter 33 OBSERVATIONS OF WAKE ECHOES hoe FROM WAKES, like those obtained from other targets, vary considerably with the type of sound gear employed, the prevailing oceanographic conditions, and the physical constitution of the wake. Before the observations are reviewed, it is necessary to outline the theoretical concepts entering into the reduction of the crude data obtained by measure- ment. As regards the physical mechanism by which sound is returned from a wake to an echo-ranging trans- ducer, two limiting cases can be imagined. On one hand, the multitude of microscopic scatterers may be spread out so thinly that the phases of the scattered sound waves are distributed at random — that is, so that constructive and destructive interference are equally probable. Then the average power returned to the transducer is obtained by summing up the contributions from the individual scatterers. On the other hand, a wake might reflect sound specularly. This alternative would occur only if the concentra- tion of scatterers near the wake surface increased in- wardly very rapidly. It is undecided as yet whether or not specular reflection from wakes does occur; in- conclusive evidence on this point will be discussed in Chapter 34. In the present chapter, wake echoes will be treated on the first assumption. Experience has shown that this approach is usually quite satisfactory. 33.1 CONCEPT OF WAKE STRENGTH 33.1.1 Target Strength of a Wake and Wake Strength Kchoes from wakes differ in two important respects from echoes from ships and other small targets. The concept of target strength has been analyzed in Sec- tion 19.1 where it was shown that for a target of finite size the target strength becomes independent of range at very long ranges and may be computed from the equation T=EH-—S+2H, (1) 512 where £ is the echo level in decibels above 1 dyne per sq cm, S the source level or pressure level 1 yd from the projector, in decibels above 1 dyne per sq cm, and H the one-way transmission loss from the source to the target in decibels. If equation (1) were used to compute the target strength T., for a wake from the echo level at long ranges, T., would increase with the range because, for practical purposes, the wake extends infinitely in the horizontal direction; as the range increases, more of the wake becomes ex- posed to the sound beam, more scattering occurs, and the target strength increases.. For the same reasons. a transducer with a broad horizontal beam would yield a higher echo level than a transducer with a narrow pattern beaming sound at the same wake, other things being equal. It is desirable, therefore, to introduce in place of the target strength of a wake another characteristic, which is essentially the target strength of a l-yd length of the wake. This quantity is principally a function of the geometric dimensions and of the physical properties of the wake alone and, therefore, will be called wake strength and denoted by the sym- bol W. The wake strength will here be defined in a simple manner for an ideal wake, without regard to the physical structure of actual wakes. In Section 33.1.2, an analysis of this wake strength in terms of the physical constitution of the wake will be given, including the effects originating from the finite length of the sound pulses used in practice. The wake echo will now be treated as if it were the echo from a plane strip having infinite horizontal ex- tension (— ~ < y <+) and a constant vertical height h (depth of the wake) which is supposed to be much smaller than the distance to the transducer. The hypothetical strip is assumed to have a rough surface, so that the reflection of sound by it is non- specular and perfectly diffuse, with a dimensionless coefficient of reflection s which is the fraction of sound energy returned into a unit solid angle. The fraction of sound energy reflected back, regardless of direc- CONCEPT OF WAKE STRENGTH tion, is then 27s. By comparison with Section 19.1 it is readily verified that the target strength of one square yard of this wake surface, placed perpendicu- larly to the sound beam, is 10 log s. Since the depth of the wake is h yards, the target strength of a l-yd length of wake is 10 log hs. In order to relate this wake strength to the observed echo intensity, con- sidering the directivity of the transducer and the scattering of sound from elements of the wake surface not perpendicular to the sound beam, a more detailed exposition of this simple case is required. The geometry of this experimental situation is illustrated in Figure 1, from which it is apparent that NORMAL TO THE WAKE TRANSDUGER AXIS i ee FicgurE 1. Horizontal plane through transducer and wake. WAKE the surface element of the strip having the area hdy receives from the transducer the power — 0.lar Ip — (9) c0s (8 + @)hdy, 2) where J, is the output of the transducer on the axis; b(@) measures the angular variation of the output around the horizontal plane; r is the distance from the transducer to the surface element hdy; ¢ is the angle between this ray and the axis of the transducer, which subtends the angle 6 with the normal to the wake so that cos (8 + ¢) measures the geometric foreshortening of the insonified area; and a is the coefficient of attenuation in the ocean. If the trans- mission loss on the return path is taken into account, the equivalent echo intensity I, is yar 1 le ate f y=— 0 o — 0.2ar b()b’() cos (8 + )hdy, (3) r4 where b(¢)b’(¢) is the composite pattern function of the echo-ranging transducer. The factor b’(¢) is the ratio between the response of the hydrophone to a signal incident at an angle ¢ to the axis and the response to a signal of equal strength incident on the axis. Thus the equivalent echo intensity J, is propor- 513 tional to the output voltage of the transducer acting as hydrophone. By substitution of the perpendicular range D from transducer to wake, D = rcos (6 + ¢) y = Dtan (6+ ¢), equation (3) is transformed into g=+5-8 ee-aih ffi b(¢)b’(g) 10°77” °° °F) cos3(B + ¢) de, Se, OS or, to a very good approximation o=+5-6 Io D = 10°22? se°6 Ds = hs | b(¢)b'(g) cos*(B + o)db- (A) 0 x Sin B This approximation neglects the variation of the transmission anomaly along the wake, which is in- significant because of the narrow beam pattern of the transducers used in practice. By writing cos (8 + ¢) = cos B (cos ¢ — tan B sin 4) and D cos B =7, where 7 is the range to the wake measured along the transducer axis, equation (4) becomes : ¢=+5-6 10°77 = hs J (6)6'(#)(cos # — tan 8 sin 4)*dd. : o--2-8 (5) By collecting the terms representing the transmission loss, ar + 20 log 7 = H, (6) and adopting the abbreviation $=+5-8 10 log | b(¢)b’(¢)(cos ¢ — tan Bsin¢)'dp =, (7) S2=3 = equation (5) can be expressed, in decibels, as E—S+ 2H — 10log7 —W = 10loghs = W. (8) The quantity W defined in equation (7) will be called the wake index, analogous to the reverberation index defined in Chapters 11 through 17. The product hs in equation (8) has the dimension of a length. Since the ranges appearing in (8) are customarily measured in yards, the wake strength W is the ratio of hs to one yard, expressed in decibels. Then, by comparing equa- tions (8) and (11), the relation between the wake 514 strength W and the target strength of a wake T, becomes T, =W+ 10log7r+ Y¥, (9) where 7 is the range to the wake, measured along the transducer axis, and W is the wake index defined by equation (7). The physical meaning of equation (9) has already been noted in the first paragraph of this section. 33.1.2 Dependence of Wake Strength on Physical Parameters The fundamental definition of wake strength, given in the preceding section, was facilitated by treating the wake as if it were a plane strip with the coefficient of reflection s. In effect, this approach neglected the wake structure along the transverse x axis. Moreover, that analysis tacitly assumed the use of a continuous signal for measuring the wake strength. But if short sound pulses are beamed at a diffusely reflecting plane, the echo profile on the oscillogram, in general, will not reproduce the shape (usually square-topped) of the signal, and the de- pendence of echo intensity on signal length must be investigated. For a brief theoretical demonstration of this fact see Section 19.3. On inspection of the sample oscillograms repro- duced in Figure 7 of Chapter 30, it will be observed that reflection from a wake also alters the shape of square-topped sound pulses. However, the explana- tion of this effect is more complicated than that of the variation with pulse length of the echo intensity re- turned by a plane target. In fact, the echo profile de- pends on the transverse structure of the wake, which therefore must form an integral part of a compre- hensive theory of wake echoes. According to the working hypothesis adopted in Section 26.3 wake echoes are composed of a multitude of reflections originating throughout the entire wake. Superposition of these scattered waves leads to con- structive and destructive interference, because their phases are distributed at random. Consequently the echo intensity measured at any instant will not equal the average value. The difference between the in- stantaneous and average echo intensity is a rapidly fluctuating quantity, evidently beyond the reach of theoretical analysis, because it depends on the micro- structure of the entire wake. Physical significance can be attributed only to the average of many echo pro- files recorded in rapid succession. Such averaging is also necessary in order to minimize the effect upon OBSERVATIONS OF WAKE ECHOES the echoes of the rapid fluctuations of the transmis- sion loss in the ocean at large, which were discussed in Chapter 7. Accordingly, the theoretical analysis about to be presented refers to “average” echo in- tensities throughout. The problem now is to evaluate the relation be- tween the total number, arrangement and physical parameters of the bubbles and the overall reflectivity s of the wake, filling a volume of constant depth h and width wand of infinite length (—- 1 ] the exponential in the bracket under the integral is very much less than 1. Hence, in the case of infinite acoustic thickness of the wake, there follows the rigorous formula +26 ae b ve (¢)[cos ¢ — tan B sin ¢ d¢- (17) By comparison ita eo (5), the value of the wake index W.,, where the subscript has been added to emphasize that this index refers to infinite acoustic thickness, can be identified: ra G b(¢)b’(¢) [cos ¢ — tan B sin ¢ |'d¢. 2 I e =G 73] (90-247 = jibe WV.. = 10 log (18) OBSERVATIONS OF WAKE ECHOES Conversely, for highly transparent wakes [o.N (w) < 1] the second bracket under the integral in equa- tion (16) may be developed into a series and the quad- ratic and higher terms neglected, yielding 1 — ¢ 2eNiu) seo (B+ 8) — 2g.N(w) sec (8 + 4)- The formula for the wake strength of highly trans- parent wakes then reads +i_p ue PAHs = = (w) sec af oc (o) - 0 ‘ns 5 = /3 [cos ¢ — tan B sin ¢ Pd¢, he D = 10 log ae B b(6)b" (¢) [eos —=-8 — tan B sin atao - (19) Numerically, the difference between VY... in equation (18) and W% in equation (19) is negligible for direc- tional transducers as long as 8 is small. To a very good approximation the general equation (16) can, therefore, be written as ¥2) _ ol — @ r7eN(w) sec B 1 ’ (20) I be — © -34 (\0.2arT— ine 10 Expressed on a decibel ited equation (20) becomes E—S+2H + 10 logr—wy hes 2 = 10 log {a fil = GOs a Sib) 810. Hence, the reflectivity of the wake per unit solid angle is os We Ce [1 c By this equation, the problem proposed at the outset of this section is solved for sufficiently long pulses (ro > w). However, it should be remembered that equation (21) does not represent the entire echo pro- file, but applies merely to its central part which has a constant average intensity because the pulse over- laps the entire wake. The rise and fall of the average echo profile, when only part of the pulse intersects the wake, cannot be represented by a simple formula, be- cause of mathematical difficulties of the same nature as will become apparent presently in the discussion of short-pulse echoes. —2ceN(w) sec F] i (22) SHortT PULSES For pulse lengths smaller than the wake width (vm) < w), equation (11) has to be integrated over the CONCEPT OF WAKE STRENGTH volume in which the wake and the cylindrical shell (thickness ro, inner radius 7) of the pulse intersect. Hence, bb Pics 5; = My, (jy b(p)b’(o) - eee sec (B+ drdp- (23) The limits ¢. and ¢, of the integral over dd, unspeci- fied for the time being, are determined by the relative position of pulse and wake and, therefore, vary with time; their explicit form will be evaluated after the discussion of the integral over dr has been finished. Equation (28) is valid only for rectangular pulses — that is, for echo-ranging signals whose intensity is constant for their duration. The variation of n(x) and N(a) across the wake prevents integration of (23) in closed form. In order to gain any insight at all into the behavior of short- pulse echoes, a drastic simplification becomes neces- sary. For this reason the discussion is confined to a wake of constant bubble density in the transverse direction. Putting thus n(x) = constant = n = N(w)/w and N(a) = nx the integral reads go m+ ao ‘ I, _A ‘3 10— ar afte nae 2 b(b)b’(d) me 27 °° +H rd. iia ga Th (24) While in the general case ¢. and ¢} vary asr increases from 7; to 71 + 7, for short pulses this change is quite negligible. The integration over dr may then be carried out before the integration over d@ without difficulty. On account of the geometric relation, from Figure 2, ~ xsec (8 + ¢) =r — Dsec (6 + 9), the integral over dr in equation (24) becomes m+ 190702" ru f :; ne *°"[r — Dsec (8 + ¢) ]dr - 1) (25) After applying the theorem of the mean value of an integral with respect to the factor 10°”? the inte- gral (25) is transformed into 7#-319-020* 1 [1 a pac es 2cen[ri—D sec (8 +¢)1 Since r* differs very little from 7; (because 7 < r* < ™-+ 1, by definition), r* can be replaced by 71 517 without any appreciable loss of accuracy. Thus the echo integral, equation (24), reads I, 3 0.2ar; ee lea e7 2eenre iff Ip ett ie Gall z : b’ ()e — 2cen[ri — Bee (8+¢)] do. (26) The exponential under the integral measures the transmission loss resulting from absorption and scat- tering inside the wake, since 7 — D sec (8 + ¢) is the distance, along any ray (¢ = constant), from the inner boundary of the pulse to the front of the wake. Now by making the substitution — Dsec (8 + $) D sec B — D [sec (6 + ¢) — sec B], equation (26) assumes the form =7T1— pb Ho ho. - sae a rsa = acl eet I a: — Dsec » fos) 5 0 TO e be b' (per? [sec (6+ ¢)— sec B] dd. (27) Here the factor (1 — e~”"”) comprises the effect of the pulse length on the echo strength. The transmis- sion loss inside the wake has been split into two factors. The first one, namely e27"™~ Ps?) de- pends only on the range 7; of the pulse, which in- creases with time, but not on the directivity of the transducer; in fact, this first factor is simply the transmission loss, measured along the sound beam axis, from the boundary of the wake to the pulse. The second factor is independent of time and appears as an exponential under the integral over d¢. Using the abbreviation WV = 10loe} [ooo @) 1Q27e"Plsee (6+ ¢)— sec maa ; (28) ga which defines another wake index applying to short- pulse echoes, equation (27) may be written on a decibel scale as follows. E—S+2H + 10logn — wv’ h = = = 10 log = 7 Gel oie ee —D sec ot. (29) The quantity (r1 — Dsec B) is the distance, meas- ured along the transducer axis, which the rear boundary of the pulse has penetrated into the wake. Hence, for a directional transducer no appreciable echo intensity will be obtained outside the range of penetration which is given by DsecB 1, and long signals are used, or 7) > w (for example, 7) > 2w), according to equation (18) which turned out to be identical with equation (7): bei Wo= 10 log f b($)b’(¢) [cos @ — tan B sin ¢ }'dg. (40) aie) b All the numerical values of W reported in this chapter have been computed according to the definition given by equation (38). However, when the original publications are consulted, care should be taken in ascertaining what particular definition of wake strength was used by the author. For in- stance, in one paper,} u is set equal to 2 in correcting echo levels for transmission loss; moreover, a term 10 log (47) is also added to equation (37). The net result is that the values of the wake strength in reference 1 are 5.0 db larger than those computed from equation (38). Since the reflectivity of a target has been defined, in Section 19.1 of this volume, in terms of the sound reflected into a unit solid angle, it seems desirable to maintain the same convention for the reflectivity s of a wake. Accordingly, the term —10 log (4) here appears in equations (45) and (46), instead of in equation (37). 520 However, if the wake is highly transparent, or N(w) <1, the echo level EL, and also W as computed from W,, will be found to increase with the oblique- ness 8 of the impinging sound beam. In this case, the replacement of V.. by WY, according to equation (19), may be expected to give a wake strength independent of B: VY, = 10 log {sce a| b(¢)b’(¢) [cos¢ — tangsing Fah. a i (41) For short pulses (77 K w, say mo w W = 10 log hs = 10 toe AL = FA \ (46) T Oe It will be noted that the first exponential in equation (45) becomes equal to that in equation (46) for 7 =w, because of equation (54) of Chapter 28, reading nw = Nw), which is the definition of 7. Equation (45) is an ap- proximate formula, because in its derivation the as- sumption n(x) = constant = n had to be made. However, while equation (46) applies to the constant average echo intensity constituting the central part of long-pulse echoes, equation (45) represents the entire average profile of short-pulse echoes. So far the entire discussion has been restricted to average echo intensities. But, as described in Section 30.1.3, peak echo intensities are customarily measured by the San Diego observers. The measurements re- ported in Chapters 11 through 17 of this volume on the “band” or ‘“‘point’”’ method of reading reverbera- tion records imply that about 6 db must be sub- tracted from the wake strengths computed by equa- tion (38) from peak intensities, in order to express them on the scale of average intensities envisaged in equatioris (45) and (46). This correction will be ap- plied only in Section 34.3.1, where the interpretation of the observed wake strengths by the acoustic theory of bubbles is discussed. In that context, the wake strength computed from the measured peak ampli- tudes of short-pulse echoes, and then corrected by subtracting 6 db, will be regarded as corresponding to the maximum of the profile (45), or rm — DsecB = 0. 33.1.4 Decay Rate of Wake Strength The decay rate of wake strength, in terms of the physical properties of the wake, is found by differen- tiating equations (45) and (46) with respect to the time. Before doing so, it is advantageous to make the substitutions es Hy n= anal ae MOD) eae 6 CONCEPT OF WAKE STRENGTH where H, is the one-way transmission loss for hori- zontal passage of sound through the wake, as defined in equation (8) of Chapter 32. Equations (45) and (46) then read h W = 10 log eel - Eien (m Kw) (47) h os W = 10 toe | ge Et — oN, (r) >> w). (48) The result of the differentiation is: Short pulses, 7 K w Wane. OMGerce oe” dW _ gayi dh , 0.4Ger Otel? ty dt Gh Sa Oae aD == (49) dt w dt Long pulses, 75 > w dw chen AGeme ard Fie, mre Tf ape =046Hw ? (50) dt hdt 1-—e dt The first term in these equations is the same for long and short signals. It represents the effect of the change in depth of the wake and is known to be quite small; for two destroyer wakes, according to data re- ported in Section 31.3.1, (1/h) (dh/dt) was found to be —0.08 and +0.04 db per minute, respectively. The second term seems to be the dominant one. While the factor in front of it, containing the exponential, is ex- ceedingly small for fresh wakes, it grows rapidly and approaches infinity for very old wakes; numerical values of this factor can be read from the graph in Figure 4. The differential quotient dH.,,/dt is equalto3 db per minute, for destroyer wakes, and (H.,/w) (dw/dt) can be estimated from the same data to be of the order of 1 to 2 db.per minute. It will be noted that equation (49) would be transformed into equation (50) by setting 7>/w equal to one — except for the term pro- portional to (H,./w)(dw/dt) which does not appear in equation (50). The physical meaning of this term is interesting: as the wake ages, it spreads laterally, causing dw/dt to be a positive quantity; conse- quently, the factor H.,/w is bound to decrease, even if H.,., the total attenuation across the wake, remains constant. According to equations (46) and (48), for long pulses the wake strength is a function of N(w), which is directly proportional to H,, or the total atten- uation, and which is not affected by a mere spreading laterally of the wake without simultaneous disinte- gration of the bubble population. But for short pulses FACTOR H, IN DB Ficure 4. Factor appearing in formula (50) for the de- cay rate of wake strength. [see equations (45) and (47) | the wake strength is a function.of the product of signal length rp times the average bubble density 7 which is proportional to the attenuation coefficient. Hence, the decay rate of the wake strength for short pulses is a function of the decay rate d(H./w)/dt of the attenuation coef- ficient which gives origin to a term proportional to (dw/dt) or the lateral spreading, even if dH.,/dt is negligibly small, corresponding to an extremely small physical disintegration of the bubble population. Summing up, for short pulses, whose volumes do not intersect the entire wake, there exists a progressive decay of wake strength having a purely geometric origin — namely the lateral spreading of the wake. Naturally, for long pulses, which overlap the entire wake, such an effect cannot arise. If the decay of a wake is followed over a very long period of time, and a constant pulse length is employed, it may well happen that the pulse which was chosen, at zero age of the wake, so as to be long will finally become short with respect to the steadily growing width of the wake. At the moment the critical point w = 7% is passed, the term (H,,/w) (dw/dt) suddenly begins to operate, causing an accelerated decay. In order to avoid all unnecessary complications, it may be ad- visable, therefore, to choose very long signals for the study of the decay rate of wake strength. The general significance of equations (49) and (50) is that they establish a relation between the decay rates of the transmission loss and the wake strength which can be tested by observation. 522 OBSERVATIONS OF WAKE ECHOES 33.2 EXPERIMENTAL PARAMETERS The formulas derived 1n the preceding section are applied in later sections to the interpretation of echoes from actual wakes. The theoretical results may also be applied to indicate what type of echo-ranging ex- periment is most suited for fundamental studies of wakes. Certain considerations along this line, espe- cially concerning the choice of transducer directivity, pulse length, and frequency are presented in this section. 33.2.1 Transducer Directivity The mathematical intricacies of the analysis given in Section 33.1.2 are essentially a consequence of the imperfect directivity of the transducers and the finite range over which the wake is observed. The chief re- sult is a variety of wake indices pertaining to specific experimental situations. Fortunately, the picture is greatly simplified in practice, because of the proper- ties of the transducers customarily employed in echo ranging. The numerical differences between the different wake indices, for the same directivity pattern, are quite insignificant in proportion to the accuracy at- tainable in acoustic measurements. As an example, Table 2 gives the wake indices computed from the composite directivity pattern of a particular transducer. The integral V taken over the composite directivity pattern alone, as defined by equation (43), is given for comparison. It would seem that Y = J, + 8 db TaBLE 2. Typical wake indices —UCDWER trans- ducer No. 1917 at 45 ke. f B = 0° B = 60° Yoo —9.75 db —9.87 db Yo —9.75 ~6.85 v'} 2o.nr = 20 —9.58 2aenr = 40 —9.52 v -9.74 —6.74 may be used in place of any of the rigorous values of the wake indices, except for YW with obliquely imping- ing sound beam (6 = 60 degrees). In this particular case the wake index includes the factor sec £, as is physically evident for reflection from a semi-trans- parent layer of finite thickness. It is concluded, then, that for all practical purposes Y may be substituted for the other wake indices, if the correction factor sec B is applied for oblique incidence of sound on semi- transparent wakes. Since the wake index W’ applying to short-pulse echoes depends on the range and on the attenuation coefficient inside the wake, Table 2 gives a more de- tailed illustration of the influence of the variable parameters. The effect is seen to be quite small; there- fore it does not influence the interpretation of the experiments on the E. W. Scripps wake carried out on November 28, 1944, during which the two trans- ducers referred to in Table 3 were employed. As far as the range is concerned, there is a distinc- tion between short and long pulses. In order to obtain short-pulse echoes of a kind that can be treated by a simple acoustic theory, the sound beam must be trained perpendicularly at the wake and the range must be shorter than 200 times the signal length; equation (35) of Section 33.1.2, which formulates this condition in an exact manner, shows that the exact factor is a function of the directivity pattern of the transducer. Short-pulse echoes produced with a sound beam trained obliquely at the wake defy any simple mathematical analysis and, therefore, are of little use in the study of wakes. As regards long- pulse echoes, however, the range is of minor im- portance, and the aspect of the wake is of no conse- quence whatever because the dependence of the wake index on the aspect angle @ is fully taken into account by equations (40) and (41). In practice, it should suffice to keep the ratio of wake width w to range r less than about 0.1; this value of w/r makes it possible to neglect the inverse-cube correction factor appear- ing in equation (15), which was omitted from there on. Barone Pulse Length Pulses varying in length from 0.3 to séveral hun- dred milliseconds have been employed in echo rang- ing at wakes. There are some general considerations concerning signal length that apply primarily to the tactical use of wake echoes. In practice, the design of the keying circuits and the build-up time of the transducer set a lower limit to the pulse length. While so far no special study has been made of the optimum conditions for recognition of wake echoes, it may be surmised, from experience with echoes from finite targets,> that signals shorter than 10 msec are not suitable for satisfactory recognition by ear. But wake echoes obtained with 1-msec signals, and even with shorter ones, are readily recognized on sound range ECHOES FROM SUBMARINE WAKES recorder traces and oscillograms. Reverberation, par- ticularly at long ranges, imposes an upper limit to the practicable pulse length. Rather different considerations govern the choice of pulse length for fundamental research into the physical constitution of wakes. The aim of such work 523 which the internal density distribution is undergoing all the time, aside from echo fluctuations due to ran- dom interference and variable transmission loss. Only by averaging numerous instantaneous profiles could a, truly representative picture of the n(x) distribution be obtained. TABLE 3. Wake indices. JK GD 1143 Transducer 24 ke 40 ke 50 ke 60 ke Qcenr Wo (8 = 0°) | —7.48 db —6.35 db —7.36 db —7.39 db —7Al —6.22 —7.28 —7.30 5 —7.36 —6.14 —7.22 —7.25 10 wy’ —7.31 —6.05 —7.17 —7.19 15 —7.26 —5.96 —7.12 —7.13 20 (6B = 0°) —7.20 —5.87 —7.06 —7.08 25 —7.15 —5.77 —7.00 —7.02 30 —7.03 —5.57 —6.88 —6.89 40 —6.90 —5.35 —6.75 —6.76 50 may be either to establish the overall properties of a wake, or to resolve its microstructure. In the first case the use of long signals, overlapping the entire wake, is indicated, whereas in the second case maxi- mum resolving power is achieved by extremely short pulses. According to equation (40) the wake strength determined with long pulses is a function of (1) the depth of the wake, (2) the average cross section for scattering and extinction by the bubble population, and (3) the acoustic thickness o.N(w) of the entire wake, which may be determined quite independently by measurement of the horizontal transmission loss. Therefore, a simultaneous observation of the echoes returned by the wake and of the horizontal transmis- sion loss through it offers the greatest promise for testing the adequacy of equation (40) for long signals. The corresponding equation (89) for short pulses has been derived by neglecting the microstructure of the wake, by putting n(z) = constant = 7. However, on inspection of the rigorous equation (24), it will be seen that the echo profile on the oscillogram is es- sentially proportional to the function n(x)e~27-N@ for the case of extremely short signals, and of an ideal sharp sound beam which could be realized approxi- mately by placing the transducer very close to the wake. Such an analysis of the microstructure of wakes by short-pulse echoes would be of rather limited practical value, because of the rapid changes As to signals of intermediate length, it may be pre- sumed that equation (89) will represent the variation of W with 7 reasonably well. 33.2.3 Frequency Most echo ranging at wakes has been carried out with frequencies between 20 and 60 ke. The available observations suggest a conspicuous variation of wake strength with frequency, but no such dependence can be anticipated theoretically. The dominant factor o;/oe in the formula for the wake strength does not change much with frequency, for bubbles of resonant size. At present little is known about the relative proportion of resonant bubbles in the total popula- tion and how this proportion changes with time; but there is no definite reason to believe that bubbles of nonresonant size predominate, in which case o;/o. would vary more markedly with frequency. In any event, the influence of o,/c, on the wake strength would not be expected to account for more than a few decibels. However, some frequency effect may result from the factor (1 — e *””’), provided that the wake is highly transparent; otherwise the exponential would be small compared with 1. 33.3 ECHOES FROM SUBMARINE WAKES Quantitative data on the strength of submarine wakes have recently been computed from the original measurements of echo levels. Some of these have been 524 published before;* ° others were obtained from the files of the San Diego laboratory. During these ex- periments, the echo-ranging vessel overtook the sub- marine while proceeding on a parallel course; the observations comprise surface runs, dives, and sub- merged level runs. In arder to-reduce the uncertain- OBSERVATIONS OF WAKE ECHOES ular (8 = 0 degrees) and oblique (6 = 60 degrees) incidence of the sound beam on the wake. With a few exceptions illustrated in Figures 5 and 6, the obser- vations did not extend over sufficiently long periods of time ta reveal the gradual decay of the wakes. Hence only average values of the wake strength W TasB_e 4. Submarine wake strengths. Ping Wake strength W Wake strength W in db Average Bin Wo | length | Frequency | in db 9.5 knots 6 knots submerged aletanee ; i faced astern Run | degrees | in db | in msec in ke surface Depth 45 ft | Depth 90 ft angel USS S-23 a 0 —9.4 30 60 —18 —28 400 (SS128) 2 0 —9.4 30 60 —16 —25 400 0 —9.4 30 60 —19 —26 400 Avg —18 Avg —26 USS 8-34 1 0 —8.6 30 45 —12 300 (SS139) 2 60 —6.5 30 45 —14 300 Avg —13 1 0 —8.6 30 45 —22 200 2 60 —6.5 30 45 —24 200 Avg —23 USS Tilefish | 1 0 —8.6 30 45 =15 —22 600 (SS307) 2 0 —8.6 30 45 —13 —20 600 3 60 —6.5 30 45 —11 —19 600 Avg —13 Avg —20 USS 8-18 1 60 —6.5 10 45 —34.6 200 to 250 (SS123) 30 45 —32.6 200 to 250 100 45 —30.6 200 to 250 Avg —32.6 2* 60 —6.5 10 20 —21.8 10 to 500 30 20 —18.7 10 to 500 100 20 —16.7 10 to 500 Avg —19.0 2* 60 —6.5 10 20 —23.6 650 to 850 30 20 —21.0 650 to 850 100 20 —19.3 650 to 850 Avg —21.3 25 60 —6.5 10 20 —1.8 Decay of wake 30 20 —2.3 Decay of wake 100 20 —2.6 Decay of wake Avg —2.2 * This run is illustrated in Figure 1. ties of the relative position during the submerged portions of the runs, the submarine towed a marker buoy. Pelorus bearings on this buoy were logged from the echo-ranging vessel. With the aid of the original logs, a diagram was constructed for each run, giving the relative position of submarine and measuring vessel. The distances behind the submarine to which wake echoes belonged were then read from these dia- grams. Measurements were made both for perpendic- Absolute values of the wake strength W are uncertain because of lack of adequate calibration. are given in Table 4, together with the approximate distances astern to which they refer. The transducers used had a narrow directivity pattern horizontally and a very wide pattern vertically, so that even dur- ing the deepest dives — to 400 feet — there was no significant loss of sensitivity. Although the 0 point of the W scale in Figure 5 is rather uncertain because an adequate calibration of the sound gear is lacking for that particular day, the ECHOES FROM SUBMARINE WAKES 525 AGE OF WAKE IN MINUTES WAKE STRENGTH (W) IN OB DISTANCE ASTERN IN YARDS Ficure 5. Dependence of wake strength on distance astern. Plot for USS S-18, submerged to a depth of 45 ft, for run 2 of Table 1. Echo-ranging vessel and submarine were proceeding on parallel courses at constant speeds of 8 and 6 knots respectively. plot illustrates some significant features of the obser- vations. The individual points of the diagram are computed from the average of five successive echo levels, and the scattering of these averages gives a good idea of the magnitude of echo fluctuations en- countered in practice. Signals 10, 30, and 100 msec long were sent out in cyclic succession, so that the three curves for the different pulse lengths refer to the same wake. Despite the large echo fluctuations, there is good evidence for an increase of W with the signal length 7. The steep rise of the curves at zero distance astern probably is due to the stern of the submarine. Up to 500 yd astern — corresponding to a wake age of 2 minutes — the wake strength changes very little, if any. But when the observations were resumed at 670 yd astern, the decay of the wake had definitely set in. The values given in Table 4 suggest that for this wake the decay rate increased with increasing pulse length. All reliable numerical values of W have been col- lected in Table 4, together with the values of the wake index used in the individual computations. The latter will permit the computation from W of the corre- sponding target strength of the wake, if desired. The 526 outstanding feature of the table is the greater strength of the wake laid by surfaced submarines, compared with those from submerged runs. However, during a dive the wake strength does not decline steadily. In- stead, repeated peaks occur. Some of the peak values even equal the strength of the surface wakes, as il- lustrated in Figure 6. These peaks are undoubtedly connected with the diving operations, movement of diving planes, blowing of tanks, and other operations. While the surface values of W are surprisingly con- sistent — about —15 db — only the order of magni- tude of the subsurface strength can be regarded as established, perhaps —25 db to —30 db. The relative acoustic weakness of wakes behind submerged sub- marines probably results from several causes, such as lack of entrained air and the reduction of cavitation and bubble production at the higher pressure. A small but definiteincrease of W with pulse lengthas the pulse length changes from 10 to 100 msec is found for both submerged runs of the USS S-18 (SS123). The in- crease is small and results largely from the extension of the wake along the axis of the sound beam. Even for a wake whose thickness is less than the signal length, the echo will vary with signal length when the transducer is pointed obliquely at the wake. Only for normal incidence of the sound beam is the change of target strength with pulse length a simple, readily predictable effect. ECHOES FROM SURFACE VESSEL WAKES 33.4 The San Diego group has studied echoes from wakes laid by numerous surface craft. Early experi- ments were carried out in San Diego harbor. During 1944, the group carried out a large program of record- ing echoes from the wakes of a number of surface vessels, including aircraft carriers, destroyers, and some small craft. Wakes for this program were laid on the open sea off San Diego. Echo Ranging at Wakes in San Diego Harbor, 1943 For these experiments an echo-ranging transducer was mounted on a barge moored to one side of the harbor channel.! Most of the measurements were made on wakes laid by a motor launch (length 40 ft, beam 11 ft, draft 214 ft) traveling at 4 to 6 knots. Incidental results were also obtained by echo ranging at wakes produced by other vessels which happened 33.4.1 OBSERVATIONS OF WAKE ECHOES to pass; these vessels probably did not travel at full speed in the harbor. The chief interest of these experiments, which have been reported in detail in reference 1, lies in the fact that short signals — only 9 msec long — were trans- mitted alternately at 15, 24, and 30 ke. Thus it is possible to analyze the results for a possible depend- ence on frequency both of the wake strength and its decay rate. The absolute values of the wake strength appear to be less reliable, for two reasons. First, difficulties with the calibration of transducers seem to have been experienced during the early phases of the San Diego wake studies; such would affect the absolute values of W, without impairing the results concerning the dependence of W on frequency. Sec- ond, the measurements in the shallow waters of San Diego harbor are likely to have been disturbed by bottom-reflected sound; indeed, an apparent de- pendence of W on the range was explainable only as caused by some peculiarity in the bottom contour. The results of these early measurements are sum- marized in Tables5 and 6. Values of the wake strength, obtained with 9-msec signals at three different fre- quencies, are collected in Table 5, which also contains the attenuation coefficients a and the wake indices Yo used in these computations. There is no information available on the wake age at which the observations on the larger vessels were made; probably the age did not exceed a few minutes, and the initial wake strength had decayed only slightly. The decay of the wakes laid by the launch was studied systematically over a period of 12 min- utes, after which time the echo intensity had dropped to the reverberation level. While the decay of the 15-ke echoes appeared to start immediately after pas- sage of the launch, the echoes at 24 and 30 ke main- tained their initial strength for about 2 minutes before they began to decay. The decay of the echo intensity follows a simple exponential law, to a good approxi- mation; thus the strength of the echo expressed on a decibel scale decreases linearly with time. The decay rates found are listed in Table 6. Within the errors of observation, there is no dependence of the decay rate on frequency. But the wake strength W seems to in- crease with frequency. From the average W for each frequency of the vessels contained in Table 5, exclud- ing the launch and the three fishing boats, the follow- ing differences are found: Wa — Wis = +1.0 db Wa — Wu = +4.6 db. ECHOES FROM SURFACE VESSEL WAKES 527 DIVE BEGINS VENTS OPEN WAKE AT SOFT TIME IN SECONDS Figure 6. Wake strength and distance astern. TasLeE 5. Surface vessel wake strengths. W in db for 9-msec pulses Range Vessel in ilo Ike 2 Se 30 ke d a = 3.0 db per kyd a = 5.0 db per kyd a = 7.0 db per kyd y W, = —5.8 db WY) = —8.0 db i es GA al 40-ft motor launch 60-150 — 2.9 + 3.1 + 84 Tanker 330 + 8.1 Stats + 9.0 Fishing boat 298 + 0.7 +10.7 ate Fishing boat 150 0.0 — 0.6 Fishing boat 152 Ofc — 0.4 ee Kelp barge 140 + 7.3 + 5.5 +12.2 Kelp barge 190 + 4.5 + 9.1 +14.3 50-ft boat 170 + 2.7 + 7.9 +12.9 Transport 450 +18.9 +17.5 +22.9 Tank boat 580 +10.9 +10. +14.7 Avg (excluding launch and fishing boats) + 8.7 + 9.7 +14.3 The same trend is definitely established by the values of W for the 40-ft launch in Table 5, which are the averages resulting from 16 wakes. 33.4.2 Deep Water By 1944 considerable progress had been made in standardizing the sound gear at frequencies in the neighborhood of 24 ke. It is believed that the relia- bility of the absolute calibration of the transducers used for these later measurements is much greater at these frequencies than during the earlier measure- ments. Moreover, this program was executed in deep water off San Diego, so that interference from bottom- reflected sound was avoided. Pending a comprehen- 528 OBSERVATIONS OF WAKE ECHOES sive report on these investigations, a summary of pre- liminary results has been made available for the purposes of this volume. The experiments with craft other than the USS Jasper (PYc18) itself followed a single pattern. The recording vessel, the Jasper, was lying to in the open sea, and the wake vessel approached and passed her while maintaining a straight course. As the wake vessel approached, the Jasper echo-ranged on her, training the sonar projector with the aid of a pelorus manned on the flying bridge. When the wake vessel came abreast of the Jasper at the time of closest ap- proach, the training of the sonar projector was halted, and its true bearing was held fixed and approximately TaBLE 6. Decay rate of wake of 40-ft launch. Mean decay rate and its prob- Standard Number Frequency able error in deviation in of in ke db per minute db per minute wakes 15 6.8 + 0.6 4.0 20 24 7.0 + 0.4 2.5 20 30 7.0 + 0.5 3.0 20 perpendicular to the wake until the end of the run. When possible, recording was continued, either con- tinuously or intermittently, until no more echoes from the wake were detectable above the background of reverberation. When the Jasper was studying her own wake, a different technique was necessarily adopted. In this case, the Jasper ran on a straight course at 12 knots for 10 or 15 minutes; then she turned around and ran back parallel to her original course with her sonar projector trained abeam so that the sound beam was directed normal to the original course. Recording was continued until a wake echo was no longer discernible above the reverberation. A sample graph of the peak echo level received from a wake against time is shown in Figure 7, which also contains the echo levels received from a sphere 3 ft in diameter buoyed behind the wake at a depth of 6 ft. On this run three different signal lengths, re- peated in cyclical succession, were used to give wake echoes. This system of interchanging signal lengths was used on a number of wakes. On some occasions the gear used for cycling the pulses was not in order, and five or six echoes were recorded at one pulse length before the pulse length was changed manually. On a few occasions only one pulse length was used throughout the run. The initial wake strength is determined by the initial echo level — the level of the wake echo at the time (zero time) when the stern of the wake vessel has just passed out of the sound beam. This time can only be estimated, and sometimes echoes were not recorded until some time after the wake was laid. In such cases the values to be used for initial echo level are obtained by extrapolating the observations available back to the zero time. The decay of the wake is measured as the slope in decibels per minute of the echo level-time curve. Some thought was given to the possibility of two decay rates in the wake, the dividing line between them being rather sharp in time, but the data were not sufficiently well defined to allow such a distinction to be made. Therefore, only one decay rate was obtained for each wake. This is the rate at which the echoes seemed to decay steadily for several minutes before they became indistin- guishable. 20 SPHERE ECHO LEVEL *40 LOG R(R=350 YD) 6 TIME IN MIN AFTER CROSSING SOUND BEAM Figure 7. Decay of wake from EZ. W. Scripps. Run 1 of Table 8, 24 ke. It seems fairly certain that there is a systematic increase in W with the size of the wake vessel, but Table 7 shows that the magnitude of the effect is not very large. All that can be said about the speed of the wake vessels during these experiments is that they seemed to be running within their normal range of operating speeds. These averages include echoes obtained both with 10-msec and 30-msec pulses, as the number of avail- able data was small and the increase of W20 msec OVer Ww msec is moderate (see Table 8). Presumably, the standard deviation would not be reduced much by separating the results according to signal length. The wake strength appears to increase with fre- ECHOES FROM SURFACE VESSEL WAKES 529 TABLE 7. Dependence of wake strength on wake-laying vessel. 24 ke 60 ke Type of wake vessel aRe = Pa Average W | Standard deviation | Number of | Average W | Standard deviation | Number of in db of W in db wakes in db of W in db wakes CVE’s and AP’s — 7.7 4.1 5 abe wert DD’s and DE’s — 9.6 6.3 5 +7.9 1.1 2 Laboratory yachts —13.6 2.6 5 +1.6 3.0 8 (Scripps & Jasper) Small boats —18.2 2.0 2 —3.7 2.1 2 TaBLe 8. Dependence of wake strength on pulse length. Average difference of wake strength in db Total Type of Frequency number of wake vessel in ke Wio msec — W1 msec | W30msec — Wid msec | wakes DD's 24 6.5 3.5 3 CVE’s . 24 8.5 3.0 3 E. W. Scripps 24 9.0 1.5 1 E. W. Scripps 60 8.0 4.0 2 USS Jasper 24 9.5 3.0 3 (PYce13) USS Jasper 60 7.5 3.5 3 (PYc13) Avg of all vessels for all frequencies 8.2 3.7 quency. From Table 7, it can be seen that the average difference between wake strength at 60 ke and wake strength at 24 ke is 16 db. But the reality of this phenomenon is doubtful, because use of this same underwater sound equipment in measurements of tar- get strengths of submarines has yielded results at 60 ke which are also 10 to 20 db above the 24 kc results, thus contradicting theoretical expectations (see Sec- tion 23.6.2). It is also important that measurements on submarine wakes made with different equipment and discussed before (see Table 4) show a decrease of W with increasing frequency rather than an increase. In a separate set of careful experiments on surface- vessel wakes, which are summarized in Table 9, a single instance was found where there was a marked difference of opposite sign between wake strength at 60 ke and wake strength at 24 ke. The existence of an isolated but well documented instance like this where an apparent trend is contradicted must be given con- siderable weight when conclusions are drawn about the frequency dependence of the wake effect. The decay rate of surface wakes shows very little dependence on frequency between 24 and 60 ke. The average decay rates of the wakes described above are 1.36 db per minute at 24 ke, and 1.18 db per minute ' rn 24 YO PING LENGTH O = a €CHO LEVEL IN DB ABOVE ARBITRARY REFERENCE AGE OF WAKE IN MINUTES Figure 8. Echo level as function of age of wake, for various ping lengths at 24 ke. Wake vessel: Small carrier at about 15 knots. at 60 ke. The standard deviations of these measure- ments are0.59 db per minute at 24 ke and 0.69 db per minute at 60 ke. This difference in averages is so much 530 OBSERVATIONS OF WAKE ECHOES smaller than the spread of the data that it cannot be said that there is any significant difference between the decay rate at 60 ke and the decay rate at 24 ke. Figure 8 shows a typical example of the variation of the echo level, or of the wake strength, with signal length. Numerical values of the variation of the echo level, or of the wake strength, with signal length are summarized in Table 8. This dependence of W on the signal length was pre- dicted from general theoretical considerations (see Section 33.1.2), and the observed magnitude of the effect permits an estimate of the average concentra- tion of bubbles in a wake (see Chapter 34). The num- ber of observed data is too small to warrant any con- clusions as to the influence of frequency and size of the wake vessel on the pulse length effect. TaBLE9. Wake strength and decay rate, EH. W. Scripps. Wake strength | Decay rate of Wake | W in db at age | wake strength Frequency | index | 0-2 minutes aW /dt Run in ke in db'| 3-msec pings | in db per minute 1 24 —7.0 — 5 —1.5 2 24 —7.0 —11 3 60 —7.0 —21 —0.7 4 60 —7.0 —16 4 45 —6.5 —22 —0.7 Table 9 contains some additional values of W for several wakes laid by EZ. W. Scripps on a day when the sea was unusually calm. Experimental details concerning these observations have already been given in Section 32.3.2, and the transducers used are listed in Table 3; the echo level-time curve of Run 1 of Table 9 is reproduced in Figure 7. The average W at 24 ke (mean of Runs 1 and 2) is —8 db for 3-msec pulses. In order to make this value comparable with the average value of W at 24 ke for the wakes of the Scripps and Jasper in Table 7, which is —13.6 db, a correction for the difference in signal lengths used must be made; from Table 8 it may be estimated that Wiomsee— Wimsec is of the order of +6 db. The corrected Wes xc of Table 9 is then —2 db, or about 12 db greater than Wes xe in Table 7. After the corresponding correction of the 60-ke dataof Run 3 has been made, We xe in Table 9 is still 17 db smaller than its counterpart in Table 7; the origin of this serious discrepancy remains unexplained. As for the minor discrepancy at 24 ke it seems worth men- tioning that the H. W. Scripps had been outfitted with a new propeller and engine in the fall of 1944, so that the data in Tables 7 and 9 are not strictly comparable. The decay rates in Table 9 do not differ significantly from the averages for all surface vessels quoted before. ECHOES FROM MODEL PROPELLER WAKES 33.5 At the Woods Hole Oceanographic Institution,® a number of experiments were made on the scattering of sound by the wakes of stationary model propellers. Although the published data do not yield absolute values of the wake strength, they give some interest- ing information on the relative echo intensity as a function of the frequency of sound and of the depth of the propeller. In order to measure the scattering, the hydrophone and transducer were mounted on the same side of the wake in a horizontal plane including the wake axis. The axis of the hydrophone was vertical and the transducer was directed toward the wake. Both instruments were secured to a pipe frame and were separated by a baffle, in order to reduce the passage of the direct signal from the transducer to the hydro- phone. The baffle consisted of a sheet of Celotex 32 in. square and 1¢ in. thick, sheathed with copper; the plane of this sheet was perpendicular to the axis of the wake. This single baffle was found to be preferable to a wedge-shaped baffle composed of two sheets of Celotex making an angle with one another. In order to reduce the direct signal still further, the hydro- phone was partially enclosed in a box lined with Celotex and open on the side toward the wake. The perpendicular distance from the instruments to the wake axis was 5 ft; the plane of the baffle, midway between the instruments, was 10 ft from the plane parallel to it through the propeller. With this arrange- ment, scattering measurements were made in the deep spot 200 ft off the wharf at depths varying from 5 to 60 ft and at frequencies from 30 to 60 ke. At lower frequencies the reflection was too small to measure. Each determination of the scattering involved the measurement of the signal at the hydrophone under three conditions: (1) with the propeller at rest and the transducer on; (2) with the propeller running and the transducer on; (3) with the propeller running and the transducer off. The results of these three measure- ments, in decibels, will be referred to by 21, 22, and 23, respectively, with z, representing the direct signal from the transducer in the absence of scattering — ECHOES FROM MODEL PROPELLER WAKES SCATTERED SOUND IN DB fo) 10 20 30 40 50 60 DEPTH IN FT Figure 9. Dependence of sound scattered from 10- inch propeller at 1,600 rpm on depth below surface. Direct signal constant for each frequency. the sound which travels around and through the baffle, and z3; representing the cavitation and pro- peller noise. The true value of the scattered sound in decibels, which we call z,, is in general different from 22 but may be obtained from it by correction for the direct signal (z:) and for cavitation and propeller noise (23). It is given by the equation 1077" = 102" —_ 102” Le 102”. The results presented below were calculated in this way. It should be pointed out that the effect of z3 was in all cases negligible. An interval of a minute or a minute and a half was always allowed between successive determinations to make sure that there should be no residual wake from the previous determination to interfere with the fol- lowing one. Only the 10-in. and 14-in. propellers were used, and only at the highest speed, 1,600 rpm. Under other conditions the scattered sound was too small to measure satisfactorily. The results of the study are shown in Figures 9 and 10. Although the scatter of the observations is large, particularly with the 14-in. propeller, there can be no question of the general effect. It is evident that there SCATTERED SOUND IN DB OEPTH IN FT Ficure 10. Dependence of sound scattered from 14- inch propeller at 1,600 rpm on depth below surface. Direct signal constant for each frequency. is a marked decrease in sound scattering with depth. At a frequency of 60 ke the scattered sound is less than 49 as much at 60 ft as at 5 ft. In this respect the situation is similar to that observed in the case of attenuation (see Section 32.5). The data plotted in Figures 9 and 10 give simply the total reflected sound in decibels at the hydro- phone. They take no account of the strength of the direct signal from the transducer. Since the oscillator was always set to give the same output, this signal may be regarded as constant for each frequency. Consequently at each frequency the change in the decibel level of the reflected signal with depth gives the change in the scattering coefficient. Nevertheless, in order to obtain absolute values of the scattering coefficient and to discover its dependence on fre- quency it is necessary to take into account the strength of the direct signal which would be received by the hydrophone in the absence of a wake at the position of what may be called the “virtual image”’ of the hydrophone with respect to the wake. This is a point at the same distance from the wake as the hydrophone, but on the opposite side of it. It was estimated to be 6 ft away from the transducer. With this in mind, throughout the study, daily determina- tions were made of the response of the hydrophone 532 OBSERVATIONS OF WAKE ECHOES 6 ft in front of the transducer and in the same orien ta- tion as in the actual measurements. Such determina- tions were made for each frequency used in the meas- urements. The results were found to be independent Taste 10. Direct signal as function of frequency. Frequency inke 30 40 50 60 Direct signal zo 22.5 26.2 34.0 36.8 of depth, as would be expected, and were reasonably constant from day to day. Relative minor variations are probably attributable to small differences in the spacing of the two instruments. Values of the direct signal, which will be called zo, measured in this way are given in Table 10. On the basis of these results, it is a simple task to calculate the relative intensity of the reflected sound. This, in terms of decibels, is simply z, — zo. Table 11 gives the values of z, — Zo obtained with each of the two propellers at a depth of 10 ft. In arriving at these results values of z, were read off the smooth curves of Figures 9 and 10; values of Z) were taken from Table 10. The intensities of the scattered sound at other depths are, of course, less than these, in accordance with the way in which the curves of Figures 9 and 10 drop off. It is evident that there is no considerable TaBLE 11. Reflected sound as function of frequency. 10-in. propeller at 1,600 rpm and depth of 10 ft Frequency in ke 30 40 50 60 Zrii 20 —24.5 —25.2 —24.0 —20.8 14-in. propeller at 1,600 rpm and depth of 10 ft Frequency in ke 30 40 50 60 = A — 22.0 — 22.2 —24.0 — 22.8 effect of frequency between 30 ke and 60 ke. The de- crease of the echo intensity with depth is again a manifestation of the influence of increased pressure on the formation and dissolution of bubbles, as in the decrease of the attenuation with depth described in Section 32.5. Chapter 34 ROLE OF BUBBLES IN ACOUSTIC WAKES bare PREVIOUS CHAPTERS have developed a general theoretical background for the study of wakes and have presented the results of acoustic measure- ments on wakes. In this chapter, a review is first given of the evidence that bubbles are the chief source of the acoustic properties of wakes. Next, the quanti- tative acoustic measurements are compared with the theoretical formulas derived in Chapter 28. From this comparison, conclusions are drawn as to the amount of air present in wakes. Finally, the rate of decay of acoustic wakes is discussed, and shown to be roughly similar to the rate at which air bubbles disappear in sea water. EVIDENCE FOR AIR BUBBLES IN WAKES 34.1 At the present time it seems almost certain that small air bubbles are responsible for the observed re- flection and absorption of sound by surface ship and submarine wakes. The evidence for this is of two general types, qualitative and quantitative. From a qualitative standpoint, air bubbles provide the only mechanism yet proposed which can explain the general behavior of wake echoes. In particular, no other explanation seems capable of explaining the very marked dependence of scattering and absorbing power on the depth of the wake. The measurements with the model propeller, described in Sections 32.5 and 33.5, show unmistakably a pronounced weaken- ing of both attenuation and scattering when the pro- peller is below about 30 ft. Measurements of echoes from submarine wakes show a similar decrease of about 5 to 10 db in wake strength when the sub- marine dives from the surface to periscope depth. Practical echo-ranging trials confirm the disappear- ance of wake echoes when the submarine dives below 200 or 300 ft. These observations cannot be explained on the assumption that turbulence or temperature effects are responsible for the acoustic properties of wakes, but they follow naturally from the assumption that bubbles are the important agents. From a quantitative standpoint, the magnitude of the observed effects is enormously greater than can apparently be explained by any assumed mechanism besides the presence of small bubbles in the wake. It has already been noted, in Chapter 29, that on the basis of present acoustic theory, neither turbulence nor temperature irregularities could account for any appreciable scattering or attenuation by wakes. The absorbing and scattering power of a single resonant bubble, analyzed in Section 28.1, is so great, how- ever, that a relatively small number of bubbles is required to explain the observed acoustic effects. Any theory of the acoustic properties of wakes can- not be regarded as completely confirmed until reliable quantitative data are shown to be in close numerical agreement with the theoretical predictions. Until in- dependent nonacoustic measurements are made of the bubble density in wakes, or until accurate and reproducible acoustic data can be obtained on wakes under a variety of conditions, it is not possible to verify the “bubble hypothesis” explaining the origin of the acoustic wake. Nevertheless, the general evi- dence seems sufficiently strong to make this hy- pothesis highly probable. 34.2 TRANSMISSION THROUGH WAKES The attenuation of sound by air bubbles in water has been discussed in Section 28.2. The conclusion reached was that probably most of the attenuation is produced by bubbles whose radii are close to the radius R, of a resonant bubble. Integrating the con- tributions to the attenuation from all bubbles near resonant size leads to equation (67) of Chapter 28 for K., the attenuation coefficient in decibels per yard: K, = 1.4 X 10°u(R,) , () where u(R,)dR is the volume of air per cu cm in bubbles whose radii lie between R, and KR, + dk. If K, is known at all frequencies, equation (1) gives u(R,) for bubbles of any radius. The total volume u 533 534 ROLE OF BUBBLES IN ACOUSTIC WAKES of air in one cu cm of water is then given by the integral Rmax u = f waoar,, 0 (2) where Rmax, the maximum radius of any bubble present, is assumed to be much less than 1 cm. Since the attenuation coefficient K, is directly pro- portional to the bubble density u(R,), and since also the damping constant 6 discussed in Section 28.2 does not affect K., measurements of acoustic attenuation provide a sensitive determination of the amount of air present in wakes. The actual attenuations ob- served, however, are somewhat complicated by the geometry, since the wake is never sufficiently deep to ensure that no sound reaches the measuring hydro- To find the absorption in decibels per yard, the re- sulting transmission losses have been divided by the wake widths for the destroyers given in Section 31.3.1. The values of K, for a destroyer speed of 15 knots are listed in Table 1, together with the cor- responding values of u(R,). The values of u(R,) for different ages of the wakes were plotted against R, for destroyer speeds of 10, 15, 20, and 25 knots, re- spectively, and the areas under these curves were determined by graphical integration. The resulting values of u, the relative amount of air present, in bubbles of all sizes for different destroyer speeds and wake ages are given in Table 2. Starred values are uncertain, since they are based primarily on the extrapolated parts of the graphs. Apparently no direct estimates have been made of air present as bubbles in destroyer wakes. The only Taste 1. Attenuation coefficient and density of resonant bubbles—destroyer at 15 knots. Age of wake and distance astern 1 minute 3 minutes 5 minutes 500 yd astern 1,500 yd astern 2,500 yd astern Frequency R, in ke in cm K. u(R,) K. u(R,) K. u(R,) 3 0.35 2.5 * 10% | 0.03 | 2.1 X 107 BEKO aks 0.107 8 0.67 4.8 X 10* | 0.21 1.5 X 10* | 0.03 | 2.1 x 107 0.040 20 1.11 7.9 X10* | 0.48 | 3.4 x 10% | 0.22 1.6 X 10* 0.016 40 1.65 | 1.18 x 10 | 0.79 | 5.6 xX 10 | 0.48 | 3.1 x 10° 0.008 phone below the wake. As a result of this uncertainty, the bubble densities found by use of equations (1) and (2) are somewhat indefinite, though they are probably not in error by a factor of more than two. Bubble densities may be computed from acoustic measurements for destroyers at different speeds and for different wake ages. They may also be computed for a small high-speed propeller with no forward motion. Wakes of Destroyers and Destroyer Escorts 34.2.1 The computations for destroyers and similar ves- sels are based on the extensive transmission measure- ments across wakes reported in Section 32.3.1. The smoothed curves represented by equations (10) and (11) of Chapter 32 have been used, and an average taken for source outside the wake and source inside the wake, since these represent lower and upper limits to the absorption in the top 10 ft of the wake. Tas_Le 2. Fraction of air present as bubbles in de- stroyer wakes. u Destroyer speed in knots Age of wake 1 minute 3 minutes 5 minutes 10 5.2 X 1077* 1.4 x 107 6.5 X 10-8 15 7.4 X 107™* 2.0 x 107 6.9 < 108 20 7.0 X 1077* 2.3 X 107 8.5 X 10° 25 9.1 X 1077* 2.1 X 107 8.7 < 1078 * Uncertain. attempts to collect bubbles in ship wakes are ap- parently the attempts made with a 78-ft yacht." About 1 cu em per minute of air was collected through a ring 8 in. in diameter, 6 ft behind a propeller 38 in. in diameter rotating at tip speeds between 50 and 60 ft per second. Cavitation bubbles could be seen in the water, but the bubble density computed for a slip- stream speed of 5 ft per second is only 5 X 10~ parts of air by volume in 1 part water. This value is in ECHOES moderately good agreement with the values shown in Table 2. 34.2.2 Wakes of Model Propellers A similar computation may be carried out for the wakes of small propellers. Measured values of the absorption across a wake are reported in Section 32.5. The cross section of the wake was about 1.5 yd wide at the point where the measurements were made. The values of u(R,) were computed by use of equation (1) for the 10-in. propeller at 1,600 rpm and for depths of 10 ft, 20 ft, and 30 ft. Somewhat smaller values are found for the 14-in. propeller at the same rpm, possibly as a result of the narrower blades and lower pitch of this propeller. The corresponding values of u — the relative amount of air present in bubbles of all sizes, found directly from these curves — are given in Table 3. TaBLE 3. Fraction of air present as bubbles in wake of 10-in. model propeller. Depth in feet w 10 3 X 10% 20 2X 10% 30 9 X 107 It is perhaps unexpected that the bubble density in the wake of a 10-in. propeller be from five to ten times as great as the corresponding density in a destroyer wake. Further analysis shows this is not too surpris- ing. The propeller developed 11 hp during operation, with a tip speed of 70 ft per second. When a destroyer is making 15 knots, its two propellers with diameters between 9 and 11 ft, have a comparable tip speed, about 80 ft per second. Moreover the destroyer is moving rapidly, and it is well known that a propeller which is held stationary in the water tends to produce stronger tip vortices than one at the same rpm which pushes itself through the water. Thus the small pro- peller may be expected to cavitate more vigorously than the propeller of a destroyer at 15 knots. The volumes over which the bubbles produced in one second are spread in these two cases are proportional to the total propeller areas. Thus it would not be surprising to find that the bubble density measured behind the small propeller is greater than the cor- responding density in the destroyer wake. 34.3 ECHOES FROM WAKES The wake strength W is related to the bubble den- sity m a more complicated way than is the attenua- FROM WAKES 535 tion coefficient K,. In addition, W depends both on the detailed geometrical properties of the wake, and on the physical properties of bubbles of different sizes, and cannot therefore be predicted with any exactness for a known distribution of bubbles. Thus, at most, a rather general agreement can be expected between observed and predicted wake strengths. The formulas are simplest for long pulses; when bubbles of a single size are present, the wake strength W for long pulses is given by the equation W = 10 log f ho. [1 — aa |: (3) (81, taken from equation (48) of Chapter 33. The quanti- ties o, and o, are the scattering and absorption cross section defined by equations (34) and (43) in Chap- ter 28, while h is the depth of the wake, measured in yards. N(w) is the total number of bubbles in a col- umn one sq em in cross section extending through the wake in a direction parallel to the sound beam [see equation (54) of Chapter 28], and the product oN (w) is 0.23 times H.,,, which is the total transmis- sion loss across the wake measured in decibels. Thus when this transmission loss is high, the exponential term is very small, and W approaches the limiting value Co == |}, 4 816. ) (@) Equation (46) of Chapter 3 gives the ratio of c, to a. in terms of 6, the so-called damping constant, and n, the ratio of bubble circumference to the wave length of the sound which represents the contribution of radiation damping to the damping constant. Values W = 10logh + 10 oe ( Taste 4. Observed frequency dependence of ratio of scattering to extinction cross sections. Frequency in ke 1 5 8 138 19 26 36 45 10 log (=) —21 —22 —93 —24 25 —26 —27 —28 of these two quantities have been taken from Figure 2 and equation (23) of Chapter 28 and the resulting values of 10 log (¢,/8mc.) shown in Figure 1 and Table 4 of this chapter. At 24 ke, this quantity is —26 db, and the maximum value of W is equal to W = 10logh — 26. (5) For a typical wake 10 yd deep this gives a maximum wake strength of —16 db. 536 ROLE OF BUBBLES IN ACOUSTIC WAKES 2s). 10 106 (Sra) k FREQUENCY IN KC Figure 1. Frequency dependence of ratio of scattering to extinction cross sections. Considering the systematic difference between the observed and theoretical values of 5, as evident in Figure 2 of Chapter 28, it appears highly probable that at 60 kc the damping constant will not be smaller than its theoretically predicted value, since the actual damping by dissipative effects should not be less than o:/o-. However, the scattering and absorption cross sections of a resonant bubble are so much greater than those of other sizes that it seems unlikely that bubbles other than those near resonance can contrib- ute appreciably to either the scattering or the ab- sorption. Thus equation (5) may be used for actual wakes, provided that a value of appropriate to a resonant bubble is taken. On the other hand, when both the product oN (w) and the transmission across the wake are negligible, equation (3) gives for bubbles all of the same size the equation W = 10 log (Bee a ollor (‘evies) 19.6, (6) Ar An where w is the width of the wake in yards and n is the average number of bubbles per cubic centimeter. Since N (w) is the number of bubbles per square centi- meter appearing in projection on a plane perpendicu- lar to the sound beam, the equivalent product nw in equation (6) must have the same units — that is, square centimeters. It is customary to measure the wake width w in yards, or units of 91.5 cm. Hence, in order to keep equation (6) dimensionally correct, a Tas.e 5. Frequency dependence of damping constant. Frequency in ke 5 10 15 20 25 30 35 40 45 10 log (3/85) —-64 —52 —42 —3.4 —2.7 —2.1 —1.6 that resulting from the flow of heat in and out of the oscillating bubble. This predicted value, derived from the theory given in Section 29.2, happens to be about one-third of the observed value of 6 at 24 kc. Hence the true damping constant for 60-kec sound very likely is greater than one-third of the observed damp- ing constant at 24 ke. This surmise implies that the theoretically predicted maximum wake strength for 60-ke sound should not exceed the observed value of ‘W at 24 kc by more than 5 db — because 7 is inde- pendent of frequency in this range — unless the ef- fective value of the wake depth h is quite different at the two frequencies. For the general case of a bubble population com- prising all sizes from the largest to the smallest, the analysis is more complicated. If many bubbles of very large radii are present, they will scatter without much absorbing, and o,/c. will be increased. On the other hand if many bubbles of very small radii are present, these will absorb without much scattering, decreasing term 10 log 91.5, which is equal to +19.6, has been added to the right-hand side of equation (6) since w is measured in yards. When bubbles of varying sizes near resonance are considered, equation (6) is modi- fied by the substitution of S, for 7ic,; S; is a weighter mean of o, for bubbles near resonance, according to equation (77) of Chapter 28, and is equal to ug 37u(R,) ; 26, Equation (6) then may be written W = 10logh + 10 log w + 10 log u(R,) + 10 log (2) + 19.6, (8) where h and w are the depth and width of the wake, respectively, both measured in yards. Values of 10 log 3/86 are shown in Table 5 for resonant bubbles at different frequencies. In principle, equation (8) can be used to determine u(R,) from the observed value (7) s ECHOES FROM WAKES of W for any wake across which the transmission loss is less than 1 db. In practice, if the wake strength is less than about —30 db, the echo is difficult to dis- tinguish from the background. Since the theoretical maximum value of W is only —16 db for a wake 10 yd deep, there is a relatively narrow spread of values over which u(R,) can be varied to give meas- urable variations in W. 34.3.1 Surface Vessels For surface ships the transmission loss across the wake is usually large. Thus in theory all surface wakes should exhibit a wake strength W given by equation (5). All wake strengths should be nearly constant and equal to — 16 db, except for small varia- tions in 10 log h, presumably not exceeding 3 db at most. An examination of the surface vessel wake strengths tabulated in Table 5 of Chapter 33 shows that the wake strengths are highly variable. The variability of transmission loss, which could not readily be taken into account in the measurements, probably accounts at least in part for this failure of the wake strengths to remain at a constant level. Even the average observed values of W, however, cannot be compared directly with the theoretical predictions. In the first place, the measured wake strengths all refer to peak amplitudes. Extensive measurements of reverberation records ? show that the average peak amplitude is about 7 db higher than the average amplitude; these measurements refer toa segment of reverberation three to six times as long as the signal length. Moreover, since the rms amplitude is about 1 db above the average amplitude, it follows that —6 db should be applied as a net correction. According to the observations, this correction does not change rapidly in magnitude when the length of the reverberation segment analyzed is changed. Since echoes from wakes are structurally similar to rever- beration, it is concluded that a correction of —6 db applied to the observed values of W listed in Chapter 33 presumably will suffice to express them on the in- tensity scale envisaged in equations (3) to (8). In addition, if surface-reflected sound reaching the wake is of the same intensity as the direct sound, the actual transmission anomaly is 3 db less than as- sumed; another 6 db should then be subtracted from the wake strengths reported in Chapter 33 to give the correct values. If the correction for surface-reflected sound is neg- 537 lected, values of the observed wake strengths on an intensity scale may be found by subtracting 6 db from the values of W listed in Table 7 of Chapter 33. The resulting values are shown in Table 6, together with TaBLE 6. Observed and predicted wake strengths. Maximum theo- Observed wake retical wake strengths at h strength at 24 ke in db in yds 24 ke in db CVE’s and AP’s —14 15 —14 DD’s and DE’s —16 8 —17 E. W. Scripps —20 4.4 —20 USS Jasper —20 6.7 —18 (PYe13) Small boats —24 2(?) —23(?) the wake depths h taken from Chapter 31 and the theoretical limiting values of W found from equation (5). The close agreement between theory and obser- vation for the larger vessels suggests that no large correction is required for the presence of surface-re- flected sound. This same conclusion is supported by agreement between direct and indirect determina- tions of submarine target strength at beam aspect, reported in Sections 21.5.4 and 23.8.1 of this volume. There are a few cases of anomalously high wake strengths, discussed in Section 33.4. These are diffi- cult to explain on the basis of scattering by bubbles. One possible effect worth considering, that could in principle give rise to very high wake strengths, is the specular reflection of sound from wakes. As pointed out in Section 28.3.4, bubbles not only scatter sound, but also affect the sound velocity. If the boundary of the wake is sufficiently sharp, some sound will be re- flected backward. Since the reflected sound rays will go predominantly in the backward direction, rather than out in all directions, the resulting wake echo can be quite high even though the coefficient of re- flection is not very great. For the bubble densities found in destroyer wakes, and summarized in Table 2, the reflection coefficient found from equation (85) of Chapter 28 is less than 0.4 X 10~* and therefore quite negligible. It is possible that higher bubble densities might be present in the highly reflecting wakes of the vessels discussed in Section 33.4, but this seems unlikely. These high values (see Table 5 of Chapter 33) were found in early measurements in shallow harbor waters and have not been reproduced in later, more accurate determinations on wakes of the same vessels. For example, early measurements 538 on a 40-ft motor launch gave a value of 2 db for W; later measurements on the same ship, with more standard equipment, gave a value of —21 db. In view of the failure of the later measurements to reproduce the early high values, these early values can probably be neglected. Until more detailed information is avail- able it may therefore be assumed that on the average the wake strength of large moving surface vessels, measured with long pulses, are all close to the theo- retical maximum values found from equation (5); that is, about — 16 db for rms amplitudes and —10 db for average peak amplitudes, at 24 ke. The high values of W found at 60 ke are not easily explained. These values are believed to be less ac- curate than those at 24 ke, since the equipment had not yet been wholly standardized. It is perhaps sig- nificant that in one of the most careful tests — the measurements on the wake of the Scripps discussed in Section 33.4 — the value of W at 60 ke was actu- ally less than that at 24 ke (see Table 9 of Chapter 33). Moreover, use of this same underwater sound equipment in measurements of target strengths of submarines has yielded results at 60 ke which are also 10 to 20 db above the 24 ke results, in contradiction to theoretical expectations (see Sections 21.4.3 and 23.6.2). It is also important that measurements on submarine wakes, made with different equipment and discussed below, show a decrease of W with increas- ing frequency rather than an increase. It is possible that the bubble density at 60 ke is sufficiently high and the wake boundary sufficiently sharp that specu- lar reflection of sound at the wake boundary is suf- ficient to account for the high wake echoes observed; this possibility has not been investigated theoreti- cally. Until the high wake strengths found at 60ke can be either explained or shown to be the result of ob- servational error, they will remain a serious discrep- ancy in the study of wakes. 34.3.2 Submarines Values of W for submarines both submerged and surfaced are presented in Table 4 of Chapter 33. For surfaced submarines no estimates are available at 20 kc; but at 45 ke, the value of W found for two sub- marines is —13 db. When 6 db is subtracted to give the wake strength in terms of the average intensity, this value is in close agreement with the maximum wake strength of about —16 db found at 24 ke. While no experimental data are available on the value of the damping constant 6 at 45 ke, Figure 1 suggests that ROLE OF BUBBLES IN ACOUSTIC WAKES the value of 10 log (¢,/8ze,) at 45 ke does not differ by more than a few decibels from its value at 24 ke. Thus it may be inferred that the wake strength for a surfaced submarine is quite comparable with that for any large moving surface vessel. The decrease of W shown at 60 ke in Table 4 of Chapter 33 is probably not significant. This same Table 4 shows that the wake of a sub- merged submarine is a much poorer reflector than the wake of a surfaced submarine. Since the wake strength is less than its maximum value, W should vary with the bubble density, and therefore with submarine depth and speed. While the measurements are not very conclusive, they indicate that for a sub- marine at 6 knots and a depth of 45 to 90 feet, W is about — 25 db at 45 kc; this estimate may well be in error by as much as 5 db. As before, an additional 6 db must be subtracted to convert to an intensity scale, giving —31 db for W. If 10 log (3/865) at 45 ke is taken from Table 5, equation (8) gives 10 log u(R,) = —31 — 0.8 — 10 logh — 10 log w — 19.6. (9) If the wake is 10 yd deep and 30 yd across, the bubble density u(R,) is about 3 X 10-8, less than a hundredth of the values for destroyer wakes 1 minute old at 15 knots found in Table 1; the assumed wake dimensions are somewhat uncertain, but any reasonable varia- tion of these figures would not change the order of magnitude of u(R,). If the curve of u(R,) against R, were the same as the typical curves for destroyers, the total fraction u of the wake volume occupied by air bubbles would be only about 1 X 10-*. While no other quantitative measurements are available, prac- tical echo-ranging tests indicate that the bubble density decreases with increasing submarine depth as would be expected at the greater pressure. In the top 60 ft this decrease is about as rapid as would be expected from the experiments with model propellers. Both the transmission measurements dlis- cussed above and the reflection measurements dis- cussed below show that with a 10-in. propeller all acoustic effects are much reduced at depths below 30 ft. However, the reflection measurements indicate that the acoustic effects have largely disappeared at depths below 60 ft, while echoes from submarine wakes have been reported at greater depths. This difference may be due to lack of sensitivity of the acoustical equipment used for the model propeller experiments. Alternatively, the greater size of the full-scale propellers may enable the formation of DECAY OF WAKES larger bubbles, which would persist longer at great depths. More complete measurements would be re- quired on submarine wakes of different depths before any detailed conclusions can be drawn as to the varia- tion of wake strength with depth. Model Propellers The studies of echoes from wakes of model pro- pellers, reported in Section 33.5, are not sufficiently detailed to compare with theoretical predictions. While echo levels were quantitatively determined, neither the geometry of the experiment nor the trans- ducer and hydrophone directivities are well enough known to make possible a prediction of the echo levels from the known properties of air bubbles. These measurements are of theoretical interest, however, because they provide information on the change of wake echoes with depth. This information has al- ready been discussed before. The data also provide an interesting qualitative confirmation of scattering theory. As is evident from Figures 9 and 10 of Chapter 33, the echo level re- mains relatively constant in the first 30 ft of increas- ing depth. In this same depth interval the attenua- tion in the wake, shown by Figures 3 and 4 of Chap- ter 32, decreases from a high value near the surface to less than 5 db at 30 ft. This behavior is in accord with equation (8); this equation predicts that as long as the transmission loss is more than a few decibels, the amount of sound scattered from a collection of air bubbles in water will be independent of the density of bubbles in the water. 34.3.3 34.4 DECAY OF WAKES The observations on the decay rate of a wake’s acoustic properties should be consistent with the rate of disappearance of bubbles, if bubbles are actually responsible for scattering and attenuation of sound by wakes. Although optical measurements of the bubble density concentration in wakes have been contemplated, at present there are not available any nonacoustic observations of the rate of decay of bubbles. Neither does physical theory permit pre- dicting the rate of wake decay. As set forth in Section 27.2, the turbulent internal motion may be the factor which determines the “Tife-time’’ of wakes. However, an adequate theoretical analysis of the effect of tur- bulence on the rate of rise of bubbles in wakes is still lacking. 539 Pending the solution of this fundamental problem, the equations (49) and (50) of Chapter 33 suggest a partial test of the theory of decay of acoustic wakes. These equations established a quantitative relation between the decay rate of the wake strength and that of the total transmission loss across the wake; more- over, they do not involve any quantities which are unknown or difficult to determine. With this test in mind, simultaneous observations of the decay rates dH_,,/dt and dW /dt were made for a number of wakes laid by the H. W. Scripps, on November 28, 1944; these experiments have already been described, and the results were summarized in Table 2 of Chapter 32 and Table 9 of Chapter 33. The results, though in- sufficient to verify the relationship predicted theo- retically, do not seem to be inconsistent with it. According to the discussion in Section 33.1.4, the following relation should hold for short pulses and fresh wakes: OY FOLGE Nn alel, Ty ele 7 | ae |e ew ae, OY e w dt w dt The factor F in brackets can be read from Figure 4 in Chapter 33, using H.r)/w as argument; 7) was equal to 2.4 yd, as 3-msec signals were used. The width of the Scripps wake is 45 yd at the age of 5 minutes; hence 7/w is about 0.05. The results of the numerical test of equation (10) are presented in Table 7. The observed and computed ratios (dW/dt)/(dHw/dt) TasLe 7. Observed and predicted decay rates. Frequency in ke 24 60 dH. ane in db per minute 0.7 0.4 ay in db per minute 1.5 0.7 dW /dHw aa Gna observed ~2 ~2 H,, at age of 5 minutes in db 3.0 4.5 Hyro/w at age of 5 minutes in db 0.15 0.22 F 7 4 Fro/w 1.05 0.88 dW /dH. Tap a computed ~1 ~1 seem to agree as to order of magnitude. Little more can be expected, considering the high sensitivity of the test following from the rapid variation of the function F with H,. 540 ROLE OF BUBBLES IN ACOUSTIC WAKES At any rate, equation (10) and the corresponding one for long pulses, resulting from putting 7)/w equal to 1, seems to account qualitatively for the shape of curves obtained by plotting wake strength against wake age, as illustrated by Figure 8 in Chapter 33. Generally, W remains constant during the first 5 minutes after the wake has been laid, or it may even increase slightly. Thereafter W decreases linearly with time. However, the transmission loss Hy of the wake appears to decrease linearly with time right from the beginning of the wake. The explanation is that for young wakes the factor F in equation (10) is so much smaller than 1 that dW/dt equals 0. After about five minutes H,, seems to have decreased to such an extent that F becomes of the order of one, or dW /dt and dH.,,/dt have reached the same order of magnitude. The observed rates of decay of wake echoes, noted in Chapter 33, are mostly between 1 and 2 db per minute; the much higher values recorded in Table 6 of Chapter 33 may be caused by the rather shallow depth of the wake of the launch, as distinguished from the much deeper wakes of the larger surface vessels. In the interpolation formula for H» — equation (10) in Chapter 32 — H, was assumed to decrease linearly with increasing time. An exponential decay would be more consistent with the observations of wake echoes, if equation (10) of this section is fulfilled; the meas- urements are not sufficiently accurate, however, to indicate which type of decay is actually followed. Thus it may be concluded that the observed decay rates for scattering and attenuation are mutually con- sistent, as far as the rather scanty evidence goes. Even if future wake observations would establish beyond doubt that equation (10) is satisfied, these results would by no means suffice to confirm the bubble hypothesis. It should be realized that equa- tion (10) represents a relationship of a quite formal nature and physically does not imply more than the plausible proposition that the acoustic effects of wakes are proportional to the volume density of some unspecified agent. The total time required for wakes to decay, how- ever, is consistent with the time required for small bubbles to disappear by resolution in sea water. The experiments discussed in Section 27.2.2 indicate that a bubble whose initial radius is 0.10 cm will disappear in about 30 minutes by gradual resolution of air back into the water. Turbulent motion is needed to keep air bubbles from reaching the surface but cannot pro- long the life of a wake beyond the time limit set by the resolution process. Thus 30 minutes is an upper limit for the life of an acoustic wake if the greatest air bubbles present are initially 0.10 cm in radius. Since bubbles of this size resonate to sound of 3 ke, the transmission loss observations described in Sec- tion 32.3.1 indicate that bubbles of this size are present initially. The observed length of time during which echoes are observed from a surface ship wake averages in the neighborhood of 30 minutes. Thus the observed rate of decay of acoustic wakes is at least generally consistent with the hypothesis that bubbles are responsible for the wake’s acoustic properties. In- formation on turbulence in wakes would be necessary for more detailed comparison. However, this general consistency lends added support to the “bubble hypothesis,” especially when added to the data al- ready discussed on (1) the variation with depth of the cross section for scattering and extinction, and (2) the value of the ratio o,/o., and its variation with frequency, which affects the absolute values of the wake strength. Chapter 35 SUMMARY fl Rs WAKE of a moving ship scatters and attenu- ates sound. The following sections summarize existing data in the form of rules for predicting the geometry of acoustic wakes—their depths and widths, the attenuation of sound crossing wakes, and the scattering of sound from wakes. In some cases, these rules are based on few ob- servations. Moreover, the degree of reliability of most of the rules is difficult to assess, and an adequate ap- praisal of it in most cases can be reached only by study of the detailed expositions given in the pre- ceding chapters. 35.1 WAKE GEOMETRY For surface ships, the depth h of an acoustic wake is approximately twice the draft of the wake-laying vessel, and is practically constant up to distances of at least 1,000 yd behind the ship (see Section 31.3.1). The depth of the wake laid by a surfaced submarine decreases from about 30 ft at a distance 100 yd be- hind the screws to about 20 ft at a distance astern of 1,000 yd. The wake of a submerged submarine, run- ning at a periscope depth with a speed of 6 knots, reaches the ocean surface at distances astern greater than 100 yd, corresponding to a half-angle of diver- gence at the screws of about 5 degrees in the vertical direction (see Section 31.3.2). The width w of a wake increases with the range r behind the wake-laying vessel. For destroyer and destroyer escort wakes at distances astern greater than 100 yd, the wake fans out laterally in a regular manner, with the wake edges including a total angle of 1 degree (see Section 31.2). At distances less than 100 yd astern, the wake geometry is less regular and depends upon the speed of the destroyer in a complicated manner. This de- pendence may be represented by the following equa- tion: 2k w = —r = 0.86r, (1) r which isvalid at distances astern r less than r*. For r* the values in Table 1, which were deduced from aerial TasiEe 1. Dependence of r* and w* on ship speed. Ship speed Pe w* in knots in yards in yards 16 21 18 20 39 33 25 75 64 33 93 80 photographs (see Section 31.1) of destroyer wakes, should be used. At distances astern greater than 7%, one can compute the wake width by the equation w = w* + 0.017(r — r*), (2) using the same values of r* and w* as before. Acoustic determinations of the width of destroyer wakes (see Section 31.3.1) are much less accurate than the photographic measurements, and seem to be in moderate agreement with the predictions made on the basis of equation (2). The acoustic properties of the wake apparently vary with position inside the wake, although no defi- nite predictions can yet be made for a particular wake. Outside the boundaries established by the above relationships, the acoustic effects produced by the water are no greater than those typical of the ocean with no wakes present. 35.2 ABSORPTION BY WAKES When sound from a shallow projector is received on a shallow hydrophone, the transmission loss is in- creased by an amount H,, if a wake is present between the projector and the hydrophone. This attenuation by the wake H,, may be expressed as Hy = K.« (3) where K, is the attenuation coefficient in decibels per yard, and z is the length in yards of the sound path within the wake [see equation (2) of Chapter 32]. 541 542 SUMMARY @s=\NITIAL TRANSMISSION LOSS IN DB 15 20 25 30 VELOCITY IN KNOTS Initial transmission loss across destroyer IN DB TRANSMISSION LOSS the wake, as bubbles with radii between R, and R, + dk, the attenuation coefficient A, in decibels per yard may be written [see equation (7) of Chapter 32] K, = 1.4 X 10°u(R,) . (4) In the wake less than 500 yd behind a destroyer or destroyer escort, the attenuation coefficient in the horizontal direction K, = H,,/w is about 1 db per yd at 20 ke (see Section 32.3). If attenuation at other frequencies by the same wake is taken into account, the total amount of air is about 0.7 X 10-* cu em per cu cm of water in the wake of a destroyer at 15 knots, one minute after the passage of the vessel (see Tables 1 and 2 of Chapter 28). For sound transmitted vertically upward and reflected back by the surface, thus traveling twice through the center of a destroyer wake about 20 ft thick, the attenuation coefficient is found from the equation TIME IN MINUTES Decay of transmission loss across destroyer wakes. a = initial transmission loss in db. FIGURE 2. The attenuation coefficient A. is determined by the density of air in resonant bubbles of radius R,. If u(R,)dR is the fraction of air present, in 1 cu cm of value of K, observed in this case is about 3 db per yd at 20 ke [see equation (13) of Chapter 32]. For sound transmitted along a horizontal path per- ECHOES FROM WAKES VELOCITY IN KNOTS 543 TIME IN MINUTES Figure 3. Distance astern in yards as function of wake age and speed of wake-laying vessel. pendicular to the wake axis, and within 10 feet of the surface, the transmission loss in destroyer wakes is given by the equation (see Section 32.3.1) Hw = 1.5(of)? — 3.0t = a — 3.08 (5) where f is the frequency of the sound in ke, v the ship’s speed in knots, and ¢ the age of the wake in minutes. When the projector is in the wake, the factor 1.5 in equation (5) should be replaced by 2.4; however, the value of H,, in this case may be different for different projectors, since the sound output of the projector may be affected by the presence of the wake. Numeri- cal values of a and H, resulting from equation (5) can be read from Figures 1 and 2, respectively; Figure 3 may be used to find the distances behind the wake-laying destroyer corresponding to different wake ages and ship speeds. For the wakes of large surface vessels at speeds be- tween 10 and 25 knots, the value of K, and H are probably much the same as those given by equation (5) applying to destroyers and destroyer escorts. These values of AK. and H, are averages over the cross section of the wake and do not take into account possible large changes in these quantities with posi- tion in the wake. For transmission along wakes, equation (3) cannot be used for distances large compared to the depth of the wake, since scattered sound traveling other than straight paths through the wake may become im- portant. In particular, the transmission loss H,, for propeller sounds observed directly behind a ship with a hydrophone at a depth of 10 to 20 ft is of the order of 10 to 100 db per kyd, for frequencies between 5 and 60 ke. This low value may also be due in part to re- duction of the absorption coefficient K, at depths greater than 10 ft in the wake (see Section 32.4). 35.3 ECHOES FROM WAKES The level EF of the echo received from a wake can be determined from the so-called wake strength W using the equation H=S+W — 2H + l0logr+yv (6) where S is the level of the rms pressure on the axis of the projector, measured one yard from the projector in decibels above 1 dyne per sq cm; H is the rms pres- sure level of the echo, again in decibels above 1 dyne per sq em; 7 is the range in yards from the projector to the wake, measured along the projector axis, il- lustrated in Figure 4; H is the transmission loss from the projector to the wake, defined as ten times the 544 PROJECTOR AXIS PROJECTOR Ficure 4. Range from transducer to wake. logarithm of the ratio of rms pressures at a point one yard from the projector and at a point r yards away; and W is a wake index based on the transducer pattern which differs for different conditions. Equation (6) may be written If = SJ = Wel sp Wp (7) where 7’ is the target strength of the wake. Then T., is related to W, the target strength of a one-yard length of wake, by the equation T» =W + 10logr+ Vv. (8) While in some ways it is convenient to picture the quantity W, called wake strength, as representing the target strength of a one-yard length of wake, Chapter i} ic} (Ty 7) IN OB 399 20 50 100 200 500 RANGE IN YARDS 1000 Figure 5. Wake target strength as function of wake strength and range. 33 — especially Sections 33.1.1 and 33.1.3 — should be studied for a full understanding of the physical meaning of wake strength. In particular, it should be noted that, according to equation (8), the target strength of the wake T,, depends on the transducer pattern and on the range over which the echoes are SUMMARY received. Values for 7’, for different values of W and r may be found from Figure 5 if W is known. These re- lationships all assume that both the top and bottom of the wake are in the sound beam. Since the echo fluctuates, the rms pressure will not be constant within one echo. In this summary, the rms pressure is averaged within each echo and then over several echoes. If in each echo the peak rms pressure recorded is taken and then averaged over several echoes, the wake strength W and the echo level E will be about 6 db higher than the values given here. 35.3.1 Long Pulses For pulses of duration 7 sufficiently long so that cr/2 exceeds the extension of the wake along the pro- jector axis, reasonably good predictions of the wake strengths of surface vessels and submarines can be made. If the attenuation H, across the wake exceeds a few decibels, the wake strength W is given by W = 10 logs + 10 logh (9) where h is the depth of the wake in yards, and s is a function of frequency only, with the values indicated in Table 2. Thus the wake strength of an opaque wake TaBLE 2. Dependence of reflection coefficient s on frequency. Frequency 10 log s in ke in db 1 —21 5 —22 8 —23 13 —24 19 —25 26 —26 36 —27 45 —28 10 yd deep is —16 db at 24 ke. As shown in Figure 1 of Chapter 34 (see curve marked OBSERVATIONS), wake strengths at 60 ke are uncertain. Values at fre- quencies between 10 and 30 ke are probably correct to within about 3 db. A correction of 6 db must be added to the wake strengths computed from equa- tion (9) in order to make them apply to the peak amplitude of the average echo. For a moderately directional transducer, the value of W in the case of long pings is given by Wes dade Sy (10) ECHOES FROM WAKES 545 where J, is the surface reverberation index, defined J. = 10 log E f veo" eae | (11) Qa where 6(¢)b’(¢) is the composite pattern function of the echo-ranging transducer. For typical transducers, the surface reverberation index can be computed from the equation J, = 10 logy — 23.8, (12) where 2y is the horizontal angular width, measured in degrees, of the sound beam between points down 3 db from the axis. Thus in this simple case, the target strength of an opaque wake at 24 kc is given by T» = —26+ 10logh + 10logr +10logy—24+8 (13) This equation may be used to predict the initial strength of echoes received from the wakes behind surface vessels. If the total attenuation H. across the wake is less than 1 db, the wake strength W may be less than the value found from equation (10). The wake strengths of observed surface wakes are constant for about 2 to 5 minutes, and thereafter decay at a rate of 1 to 2 db per minute; wake echoes can be observed, under good conditions, for 20 to 40 minutes after the passage of a vessel. These times are not inconsistent with what is known of the times required for air bubbles initially 0.1 cm in radius to disappear by diffusion back into sea water. In this situation, WV for a directional trans- ducer is wv =J,+ 8 + 10 log sec 8B (14) where B is the angle between the projector axis and a line perpendicular to the wake axis. The increase of echo strength with increasing 6 predicted by equation (14) holds only so long as the ping length is greater than the extension AB of the wake along the pro- jector axis in Figure 4, and so long as the absorption loss along the path AB is less than 1 db. The wake strengths of submerged submarines at 45 and 90 ft at speeds of 6 knots are about —30 db. Surfaced submarines appear to have about the same wake strength as that predicted for large surface vessels from equation (10) (see Section 33.1.2). Short Pulses When the pulse length cr/2 is less than the exten- sion AB of the wake along the projector axis in Figure 35.3.2 4, the preceding equations are less useful. Although it is possible to predict wake strengths by adding to equation (10) a correction term depending on the signal length, the resulting values of W cannot simply be transformed into echo levels, using equation (7), or into target strengths, using equation (9), unless the echo ranging transducer is beamed perpendicu- larly at the wake. For short pulses, the wake strength W decreases with the.decreasing ratio of the geometric pulse length 7) measured in yards to the geometric width w of the wake, and can be predicted from the following equation W = 10logs + 10logh SET tore [il SO Oy) SL (15) where h is the depth of the wake, measured in yards, and 10 log s = kis the same function of the frequency only as in equation (9), with the values indicated in Table 2. Numerical values of the third term on the right side of equation (15) can be read from Figure 6 Figure 6. Wake strength term as function of attenu- ation and ratio of ping length to wake width. as a function of H_,, the total attenuation across the wake, and of 7/w, the ratio of signal length in yards to wake width. The wake strengths and echo levels computed from equation (15) refer to the peak of the average echo. Echo levels # and target strengths T,, predicted for values of W computed on the basis of equations (15) and (11) should be quite satisfactory, provided that the sound is beamed at the wake nearly perpen- dicularly. For lack of anything better, the same pre- dictions may be used in case the sound beam strikes the wake obliquely. The expected discrepancies be- tween observations and predictions, for that case, are believed to be smaller than +5 db. 546 35.3.3 Angular Variation of the Echo Level When a wake is insonified by a stationary trans- ducer and the echo is recorded by a different hydro- phone at several positions, the average echo level thus determined may show moderate variations with posi- tion of the hydrophone even after corrections for range to the wake, measured along the hydrophone axis, have been applied. This angular variation of the echo level has not been investigated experimentally ; however, a simple theoretical estimate of the order of magnitude of this effect can be made for long pulses and may be useful [see equations (72), (73), and (76) of Chapter 28 ]. For pulses longer than the width of the wake meas- SUMMARY ured along the sound beam, the echo level should be proportional to cos B ~ (16) cos a + cos B if the wake is highly opaque (total attenuation across the wake more than a few decibels) ; and proportional to sec @ (17) if the wake is acoustically transparent (total attenua- tion less than 1 db). In equations (16) and (17), B denotes the angle between the transducer axis and a line perpendicular to the wake, as illustrated in Figure 4; a is the corresponding angle between the axis of the hydrophone and a line perpendicular to the wake. BIBLIOGRAPHY Numbers such as Div. 6-510-M1 indicate that the document listed has been microfilmed and that its title appears in the microfilm index printed in a separate volume. For access to the index volume and to the microfilm, consult the Army and Navy agency listed on the reverse side of the half-title page. Chapter 2 . Sound Transmission in Sea Water, Report G1/1184, WHOI, Feb. 1, 1941. Div. 6-510-M1 . An Acoustic Interferometer for the Measurement of Sound Velocity in the Ocean, Robert J. Urick, Report S-18, USNRSL, Sept. 18, 1944. Div. 6-510.22-M6 . Theory of Sound, Lord Rayleigh, 2, The Macmillan Co., 1940. 4. The Propagation of Underwater Sound at Low Frequencies as a Funetion of the Acoustic Properties of the Bottom, J. M. Ide, R. F. Post, and W. J. Fry, Report S-2113, Naval Research Laboratory, Aug. 15, 1943. Div. 6-510.5-M1 Chapter 3 . Higher Mathematics for Engineers and Physicists, 1. S. and E. S. Sokolnikoff, McGraw-Hill Book Co., New York, 1941, p. 146. . Calculation of Sound Ray Paths Using the Refraction Slide Rule, BuShips-NDRC, NavShips-943 D, May 1943. . Calculation of Sound Ray Paths in Sea Water, R. H. Fleming, UCDWR, and R. Revelle, USNRSL, Jan. 16, 6.1-sr30-1741, Project NS-140, Report U-246, Aug. 8, 1944. Div. 6-510.11-M8 Sound Beam Patterns in Sea Water, NDRC 6.1-sr31-1730, WHOI, Oct. 10, 1944. Div. 6-510.11-M9 Some Characteristics of the Sound Field in the Sea, OSRD 546, C4-sr30-083, San Diego Laboratory, UCDWR, Mar. 13, 1942. Div. 6-510.11-M4 Theory of Diffraction of Sound in the Shadow Zone, C. L. 1942. Div. 6-510.11-M3 Pekeris, NDRC 6.1-sr20-846, CUDWR, PAG (A), May 5, . The Sonic Ray Plotter, L. I. Schiff, UCDWR, NDRC 1943. Div. 6-510.11-M6 Chapter 4 . Measurement of Projector and Hydrophone Performance — Definition and Terms, E. Dietze, NDRC 6.1-sr1130-1833, USRL, Sept. 19, 1944. Div. 6-551-M12 . Apparatus for Recording Reverberation in the Sea, L. N. Liebermann, OEMsr-31, WHOI, Feb. 23, 1945. Div. 6-520.2-M3 . Operational Procedure and Equipment Used in Sonar Sound Field Studies, Report U-295, Project NS-140, NDRC 6.1-sr30-2024, UCDWR, Feb. 15, 1945. Div. 6-510.2-M8 . Sonar and Submarine Diving, Monthly Progress Report for June 1945, WHOI, July 11, 1945, p. 1. Div. 6-530.22-M21 7. 10. Supplement to the Triplane, D. E. Ross and F. N. D. Kurie, OSRD 1689, NDRC 6.1-sr30-961, Report U-4a, UCDWR, June 29, 1943. Div. 6-643.21-M3 Experimental Surface Model Echo Repeater, E. M. MeMil- lin and W. A. Myers, June 20, 1942. Div. 6-643.27-M1 Experimental Underwater Towed Model Echo Repeater, E. M. McMillin, W. A. Myers, and D. J. Evans, NDRC C4-sr30-515, UCDWR, Sept. 1, 1942. Asdic Area Trials, G. E. R. Deacon and H. Wood, British Internal Report 127, HMA/SEE, Fairlie Laboratory, Great Britain, OSRD Liaison Office WA-669-14, May 10, 1943. Div. 6-570.21-M3 . The Sound Field of Echo-Ranging Gear, OSRD 2011, 11. Some Characteristics of the Sound Field in the Sea, OSRD NDRC 6.1-sr30-1206, Interim Report U-113, UCDWR, 546, NDRC C4-sr30-083, Oceanographic Section, Oct. 1, 1948. Div. 6-510.22-M3 UCDWR, Mar. 13, 1942. Div. 6-510.11-M4 . The Triplane, D. E. Ross and F. N. D. Kurie, OSRD 12. Sound-Ranging Experiments at Key West, July 23-30, 1098, NDRC C4-sr30-402, Report U-4, UCDWR, Nov. 1941, M. Ewing, OSRD 725, NDRC C4-sr31-130, WHOL, 23, 1942. Div. 6-643.21-M2 May 23, 1942. Div. 6-570.21-M1 Chapier 5 . Prediction of Sound Ranges from Bathythermograph Obser- 3. The Oceans, H. V. Sverdrup, M. W. Johnson, and R. H. vations — Rules for Preparing Sonar Messages, BuShips, Fleming, Prentice-Hall, Inc., 1942. NDRC, NavShips 943-C2, March 1944. 4. Sound-Ranging Charts of the Oceans, U.S. Hydrographic Office. . Use of Submarine Bathythermograph Observations, BuShips, 5. Submarine Supplements to the Sailing Directions, US. NawvsShips 900,069, Apr. 25, 1945. Hydrographic Office. 547 548 BIBLIOGRAPHY 6. 10. 11. 12. 13. 14. 16. 17. 18. 19. 20. 21. 22. 23. Measurements of the Horizontal Thermal Structure of the Ocean, N. J. Holter, Report S-17, USNRSL, Aug. 18, 1944. Div. 6-540.4-M1 . Fluctuation of Transmitted Sound in the Ocean, NDRC 6.1-sr1131-1883, Technical Memo 6, Sonar Analysis Sec- tion, CUDWR-SSG, Jan. 17, 1945. Div. 6-510.3-M4 . Some Experiments on the Transmission of Continuous Sound in 100-fathom to 600-fathom Water, NDRC 6.1- sr30-1842, Report M-193, Project NS-140, Listening Sec- tion UCDWR, Mar. 15, 1944. Div. 6-510.2-M4 . Transmission Measurement in the Vicinity of San Diego, California, W. B. Snow, H. N. Hoff, J. J. Markham, NDRC 6.1-sr1128-1574, Report D12F/R822, Project NO-163, NLL, Sept. 20, 1944. Div. 6-510.2-M7 Sound Transmission Measurements Made Off San Diego from August 31 to September 10, 1943, Projects NS-140, NS-164, and MIT Research Project DIC-6187 [The Theory of Propagation of Low Frequency Sound in Deep Water], NDRC 6.1-sr1046-1047, MIT, Feb. 10, 1944. Div. 6-510.21-M1 Transmission of Underwater Sound at Lower Frequencies, Interim Report U-362, Nobs-2074, UCDWR, Nov. 1, 1945. Div. 6-510.21-M2 Sonic Listening Aboard Submarines, NDRC 6.1-sr1131- 1885, Sonar Analysis Section, Sec. 2.2, CUDWR-SSG, February 1945. Div. 6-623.1-M8 The Attenuation of Sound in the Sea, C. F. Eckart, NDRC 6.1-sr30-1532, Report U-236, Project NS-140, UCDWR, July 6, 1944. Div. 6-510.22-M4 The Influence of Thermal Conditions on Transmission of 24-Kc Sound, Report U-307, Nobs-2074, Sonar Data Division, CUDWR, Mar. 16, 1945. Div. 6-510.4-M5 . Transmission of Sound in Sea Water. Absorption and Re- flection Coefficients and Temperature Gradients, EK. B. Stephenson, Report S-1204, NRL, Oct. 16, 1935. Div. 6-510.22-M1 Absorption Coefficients of Sound in Sea Water, EK. B. Stephenson, Report S-1466, NRL, Aug. 12, 1938. Div. 6-510.222-M1 Absorption Coefficients of Supersonic Sound in Open Sea Water, E. B. Stephenson, Report S-1549, NRL, Aug. 2, 1939. Div. 6-510.222-M2 Attenuation of Underwater Sound, F. A. Everest and H. T. O’Neil, NDRC C4-sr30-494, UCDWR, Revised July 30, 1942. Div. 6-510.2-M1 Attenuation of Sound in Sea Water, G. J. Thiessen, OSRD Liaison Office ITI-I-830, Report PS-162, CNRC, June 10, 1943. Div. 6-510.22-M2 “Ultrasonic Absorption in Water,” F. EH. Fox and G. D. Rack, Journal of the Acoustical Society of America, 12, 1941, p. 505. “Ultrasonic Interferometry for Liquid Media,” F. E. Fox, Physical Review, 52, 1937, p. 973. “Ultrasonic Absorption and Velocity Measurements in Numerous Liquids,” G. W. Willard, Journal of the Acous- tical Society of America, 12, 1940, p. 938. Acoustique-absorption des ondes ultra-sonares par l'eau, Note (1) De M. B. Biquard, Comptes Rendus, 1931, 25. 26. 27. 28. 29b. 29c. 29d. 29e. 29f. 30. 31. 32. 33. 34. 35. 36. pp. 198, 226. (The Diffusion and Absorption of Ultra Sonics in Liquids), B. Biquard and R. Lucas. . “Absorption of Supersonic Waves in Water and in Aqueous Suspensions,” G. K. Hartmann and H. Facke, Physical Review, 57, 1940, p. 221. “Absorptions Geschwindigkeits und Entgasungsmessun- genism Ultraschallge-beit,” C. Sorenson, Ann. d. Phys., 26 [5], 1936, p. 121. “Absorptions of Ultra Sonic Waves in Liquids,” J. Claeys, J. Errera, H. Sack, Faraday Soc. Trans., 33, 1936, p. 136. The Extinction of Sound in Water, C. F. Eckart, NDRC C4-sr30-621, UCDWR, Aug. 31, 1941. Div. 6-510.11-M1 Asdic Area Trials, G. E. R. Deacon and H. Wood, OSRD Liaison Office WA-669-14, British Internal Report 127, HMA/SEE, Fairlie Laboratory, Great Britain, May 10, 1943. Div. 6-570.21-M3 . Biweekly Report covering period July 25—Aug. 7, 1943, NDRC 6.1-sr31-753, Project NO-140, WHOI, Aug. 11, 1943, pp. 1-2. Div. 6-510.41-M1 Biweekly Report covering period Sept. 5-18, 1943, NDRC 6.1-sr31-757, Project NO-140, WHOI, Sept. 22, 1943, p. 2. Div. 6-510.41-M2 Biweekly Report covering period Sept. 19—-Oct. 2, 1943, NDRC 6.1-sr31-758, Project NO-140, WHOI, Oct. 6, 1943, pp. 1-2. Div. 6-510.41-M3 Biweekly Report covering period Oct. 3-16, 1943, NDRC 6.1-sr31-759, Project NO-140, WHOI, Oct. 20, 1943, p. 3. Div. 6-510.41-M4 Biweekly Report covering period Oct. 17-30, 1943, NDRC 6.1-sr31-1060, Project NO-140, WHOI, Nov. 3, 1943, p. 2. Div. 6-510.41-M5 Biweekly Report covering period Nov. 14-27, 1943, NDRC 6.1-sr31-1062, Project NO-140, WHOI, Dec. 1, 1943, p. 2. Div. 6-510.41-M6 A Comparison of Calculated and Observed Intensities for Some Split Beam Sound Field Runs, R. R. Carhart and L. A. Thacker, Internal Report A-26, Oceanographic Sec- tion, UCDWR, Aug. 2, 1944. Div. 6-510.22-M5 Sound Beam Patterns in Sea Water, NDRC 6.1-sr31-1730, WHOL, Oct. 10, 1944. Div. 6-510.11-M9 Layer Effect, Echo-Ranging Section, R. W. Raitt and M. J. Sheehy, Internal Report A-35, UCDWR, Sept. 9, 1944. Div. 6-510.41-M7 Layer Effect at 24 Kc and 60 Ke, M. J. Sheehy, Internal Report A-51, UCDWR, Dec. 27, 1944. Div. 6-510.41-M8 The Sound Field of Echo-Ranging Gear, NDRC 6.1- sr30-1206, Report U-113, UCDWR, Oct. 1, 1943. Div. 6-510.22-M3 Sound-Ranging Experiments at Key West, July 23-30, 1941, M. Ewing, OSRD 725, NDRC C4-sr31-130, WHOI, May 23, 1942. Div. 6-570.21-M1 Asdic Area Trials, G. E. R. Deacon and H. Wood, OSRD Liaison Office WA-669-14, British Internal Report 127, HMA/SEE, Fairlie Laboratories, Great Britain, May 10, 1943. Div. 6-570.21-M3 BIBLIOGRAPHY 549 Chapter 6 1. Attenuation of Sound in the Sea, C. F. Eckart, NDRC 10. Some Sound Propagation Measurements in the Four- 6.1-sr30-1532, Report U-236, Project NS-140, UCDWR, teenth Naval District, NDRC 6.1-sr30-1691, Report M-226, July 6, 1944. Div. 6-510.22-M4 Project NS-140, Listening Section, UCDWR, June 19, 2. Some Evidence for Specular Bottom Reflections of 24-Ke 1944. Div. 6-510.2-M6 Sound, R. R. Carhart, Report A-17, San Diego Labora- 11. Some Shallow Water Sound Propagation Measurements tory, UCDWR, June 9, 1944. Div. 6-510.5-M2 in the Thirteenth Naval District, NDRC 6.1-sr30-1317, 3. Bottom Sediment Charts [for the guidance of submarines ], Report M-126, Projects NS-140, NS-163, Listening Sec- The Hydrographic Office, July 1944. tion, UCDWR, Oct. 26, 1943. Div. 6-510.2-M2 4. Bottom Reverberation. Dependence on Frequency, NDRC 12. Transmission of Continuous Sound, Biweekly Report 6.1-sr30-677, Report U-79, UCDWR, June 16, 1943. Covering Period January 23 to February 5, 1944, NDRC Div. 6-520.21-M1 6.1-sr30-1233, Project NS-140, Report U-176, CUDWR, 5. Reverberation Studies at 24 Kc, OSRD 1098, NDRC 6.1- Feb. 11, 1944, pp. 9-12. Div. 6-510.2-M3 sr30-401, Report U-7, UCDWR, Nov. 23, 1942. 13. Transmission Survey Block Island Sound, W. B. Snow, Div. 6-520-M2 H. B. Hoff, and J. J. Markham, NDRC 6.1-sr1128-1027, 6. Sonar and Submarine Diving: Monthly Progress Report Report D12/R616, CUDWR-NLL, Mar. 16, 1944. for June 1945, Nobs-2083, WHO, July 11, 1945, pp. 2-4. Div. 6-510.2-M5 Div. 6-530.22-M21 14. Sonic Listening Aboard Submarines, NDRC 6.1-sr1131- 7. Transmission of 24-Ke and 60-Ke Sound in Very Shallow 1885, Sonar Analysis Section, CUDWR-SSG, February Waiter, M. J. Sheehy, Internal Reports A-31 and A-3la, 1945. Div. 6-623.1-M8 UCDWR, Aug. 26 and Oct. 23, 1944. 15. Transmission of Underwater Sound over a Sloping Bottom, Div. 6-510.221-M1 R. R. Carhart and K. O. Emery, Internal Report A-39, Div. 6-510.221-M2 UCDWR, Oct. 1, 1944. Div. 6-510.5-M3 8. Acoustic Properties of Mud Botioms, G. P. Woollard, 16. Transmission of Continuous Sound, Biweekly Report WHOI, Dec. 6, 1944. Div. 6-510.5-M4 Covering Period from January 23 to February 5, 1944 9. Long Range Sound Transmission, M. Ewing and J. L. NDRC 6.1-sr30-1233, Project NS-140, Report U-176, Worzel, Interim Report 1, Nobs 2083, WHOI, Aug. 25, UCDWR, Feb. 11, 1944. Div. 6-510.2-M3 1945. (See also Chapter 9 of this volume.) 17. Sonic Listening Aboard Submarines, NDRC 6.1-sr1131- Div. 6-510.1-M4 1885, CUDWE-SSG, February 1945. Div. 6-623.1-M8 Chapter 7 1. The Sound Field of Echo-Ranging Gear, OSRD 2011, C. F. Eckart, OSRD 173, NDRC C4sr30-175, UCDWR, NDRC 6.1-sr30-1206, Report U-113, UCDWR, Oct. 1, May 12, 1942. Div. 6-560.32-M1 1943. Div. 6-510.22-M3 6. Lloyd Mirror Effect in a Variable Velocity Medium, 2. Amplitude Fluctuations of Transmitted and Reflected R. R. Carhart, Memorandum for File 01.92, Report Sound Signals in the Ocean, M. J. Sheehy, Internal Re- M-140, UCDWR, Oct. 23, 1948. Div. 6-510.111-M1 port A-29, UCDWR, Aug. 17, 1944. Div. 6-510.3-M3 7. Measurements of the Horizontal Thermal Structure of the 3. Correlation of Simultaneous Transmission in Deep Water Ocean, N. J. Holter, Report S-17, USNRSL, Aug. 18, at Different Frequencies, M. J. Sheehy, Internal Report 1944. Div. 6-540.4-M1 A-44, UCDWR, Oct. 28, 1944. Div. 6-510.222-M3 8. Fluctuation of Transmitted Sound in the Ocean, Technical 4. Variation of Signal Amplitude after Transmission in the Memorandum 6, NDRC 6.1-sr1131-1883, Sonar Analysis the Sea, M. H. Hebb and N. M. Blachman, HUSL, Dec. Section, CUDWR, Jan. 17, 1945. Div. 6-510.3-M4 19, 1944. Div. 6-510.11-M10 9. Theoretical Discussion of Reverberation, C. L. Pekeris, 5. Detection of an Echo in the Presence of Reverberation, OSRD 684, NDRC C4sr20-097, CUDWR, May 29, 1942. Div. 6-520.1-M7 Chapter 8 1. Underwater Explosives and Explosions, August 15 to Sep- 5. A Study of the Transmission of Explosive Impulses in Sea tember 15, 1942, Section B1, NDRC. Div. 2-130-M1 Water, T. F. Johnston, OF Msr-30, UCDWR, June 25, 2. Relate Pressure Measurements in Shock Wave from Small 1942. Div. 6-510.23-M4 Underwater Explosions, M. F. M. Osborne and A. H. 6. Underwater Explosives and Explosions, February 15 to Taylor, Report-S-2305, NRL, June 10, 1944. March 15, 1944, Division 8, NDRC, Report UE-19. Div. 6-551-M11 ; Div. 2-130-M1 3. Underwater Explosives and Explosions, April 15 to May 16, 7. Supersonic Flow and Shock Waves, AMP Report 38.2R, 1944, Report UE-21, Division 8, NDRC. Div. 2-130-M1 OEMsr-945, AMG-New York University, August 1944. 4. Transmission of Explosive Impulses in the Sea, T. F. Div. AMP-101.1-M9 Johnston and R. W. Raitt, NDRC C4sr30-403, Report 8. Hydrodynamics, H. Lamb, Cambridge University Press, U-8, UCDWR, Dec. 2, 1942. Div. 6-510.23-M6 Sixth Edition, 1932, pp. 481-489. 550 10. 11. BIBLIOGRAPHY Chapter 9 . Relative Pressure Measurements in Shock Waves from Small 12. Depth Charge Range Meter Tests, H. B. Hoff, G. R. Perry, Underwater Explosions, M. F. M. Osborne and A. H. et al., Memorandum D50/R1222, Project NS-238, Taylor, Report S-2305, NRL, June 10, 1944. CUDWR-NLL, Nov. 24, 1944. Div. 6-642.31-M1 Div. 6-551-M11 13. The experimental points for Figure 5 are taken from Development of Single Sweep Equipment for Impulse Work, reference 7, but the theoretical curves for Figures 5 and 6 T. F. Johnston, OSRD 766, NDRC C4-sr30-189, are taken from a recent unpublished calculation made by UCDWR, Apr. 29, 1942. Div. 6-510.23-M3 UCDWR. The Use of Electrical Cables with Piezoelectric Gauges, 14. Theory of Diffraction of Sound in the Shadow Zone, C. L. R. H. Cole, Report A-306, OSRD 4561, OEMsr-596, Pekeris, NDRC 6.1-sr20-846, CUDWR, May 5, 1943. Projects OD-03, NO-144, Division 2, NDRC, WHOI, Div. 6-510.11-M6 January 1944. Div. 2-111.11-M4 15. The Sound Field of Echo-Ranging Gear, OSRD 2011, Nature of the Pressure Impulse Produced by the Detonation NDRC 6.1-sr30-1206, Report U-113, UCDWR, Oct. 1, of Explosives Under Water. An Investigation by the Piezo- 1943. Div. 6-510.2-M3 Electric Cathode-Ray Oscillograph Method, Report CB- 16. Propagation of Sound in a Medium of Variable Velocity, 01670-12, OSRD Liaison Office W-201-1E, Admiralty Re- C. L. Pekeris, NDRC C4-sr20-001, NLL, Sept. 29, 1941. search Laboratory, Teddington, England, November Div. 6-510.11-M2 1942. Div. 6-510.23-M1 17. Hydrophone Calibration by Explosion Waves, J. L. Carter Propagation of Steep-Fronted Sonic Pulses Through the and M. F. M. Osborne, Report S-2179, NRL, Apr. 19, 1944. Div. 6-510.23-M11 Sea, OSRD Liaison Office W-215-5, Internal Report 66, 7 ane 18. Factors Affecting Long Distance Sound Transmission in HMA/SEE, Fairlie Laboratory, England, Mar. 17, 1942. : PA) RM ke aos Sea Water, G. P. Woollard, NDRC 6.1-sr31-426, OSRD The Error in the Measurement of Pressure in an Explosion wae, WRC, Mies 30, ne ; PN SOO eee ee ; 19. Bibliography and Brief Review of Published Material on Pressure Wave Due to Finite Gauge Size and to Inadequate iho Dingien! Exaneilies af Gobaeains (Dates M. F Frequency Response of the Recording Amplifier, Report ang were EES ie CG oy tek aoe ana ADM/219/ARB, OSRD Liaison Office WA-4243-2C, Mlevormayy, NIRID.S Cky tapos Ue, aaa 20. Long Range and Sound Transmission, Interim Report 1, Road Research Laboratory, Great Britain, February Mar. 1. 1944—Jan. 20. 1945. M. Ewi GUL i. Viorel 1945. Div. 6-510.23-M13 eee! Be ey 12 Getto eee OAR F ; Nobs-2083, WHOI, Aug. 25, 1945. Div. 6-510.1-M4 Uigatenitier 1B losis Gg Mgulosiies, Pemrdery 19 te 21. Deep Water Sound Transmissions from Shallow Explosions, March 15, 1944, Report UE-19, Division 8, NDRC. 5 J. L. Worzel and M. Ewing, WHOI, (n.d.). DIN EUR! Div. 6-510.23-M14 A Study of the Transmission of Explosive Impulses in 99. Explosion Sounds in Shallow Water, M. Ewing and J. L. the Sea Water, T. F. Johnston, OEMsr-30, NDRC Worzel, N111s-38137, NOL and WHOI, Oct. 11, 1944. UCDWR, June 25, 1942. Div. 6-510.23-M4 Div. 6-510.23-M12 Transmission of Explosive Impulses in the Sea, T. F. 93. Theory of Propagation of Explosive Sound in Shallow Johnston and R. W. Raitt, NDRC C4-sr30-403, Report Water, C. L. Pekeris, OSRD 6545, NDRC 6.1-sr1131- U-8, UCDWR, Dee. 2, 1942. Div. 6-510.23-M6 1891, January 1945. Div. 6-510.12-M5 Solution of Acoustic Boundary Problems, Parts I to III, 24, The Propagation of Underwater Sound at Low Frequencies L. I. Schiff, University of Pennsylvania, Sept. 4, Oct. 7, as a Function of the Acoustic Properties of the Bottom, and Nov. 2, 1943. Div. 6-510.1-M3 J. M. Ide, R. F. Post, and W. J. Fry, Report $-2113, Explosive Sound Waves in the Sea. Observations with a NRL, Aug. 15, 1948. Div. 6-510.5-M1 2500-cycle Moving-Coil Oscillograph, T. F. Johnston and 25. Theory of Characteristic Functions in Problems of Anom- R. W. Raitt, Memorandum M-10, OEMsr-30, UCDWR, alous Propagation, W. H. Furry, Report 680, MIT-RL, Sept. 16, 1942. Div. 6-510.23-M5 Feb. 28, 1945. Chapter 11 . Reverberation in Echo-Ranging: Part I, General Prin- 2. Reverberation in Echo-Ranging: Part II, Reverberation ciples, T. H. Osgood, OSRD 807, NDRC C4-sr20-149, Found in Practice, T. H. Osgood, OSRD 1422, NDRC CUDWR, July 28, 1942. Div. 6-520-M1 6.1-sr20-84C, Project NS-140, CUDWR, Apr. 14, 1943. Div. 6-520-M3 Chapter 12 . Measurements of the Horizontal Thermal Structures of the 3. The Discrimination of Transducers Against Reverberation, Ocean, N. J. Holter, Report S-17, USNRSL, Aug. 18, 1944. OSRD 1761, NDRC 6.1-sr30-968, Report U-75, UCDWR, Div. 6-540.4-M1 May 31, 1943. Div. 6-520.1-M8 . Theory of Sound, Lord Rayleigh, The Macmillan Com- 4. Bottom Reverberation. Dependence on Frequency, NDRC pany, New York, 2, 1926, p. 126. 6.1-sr30-677, Report U-79, UCDWR, June 16, 1943. Div. 6-520.21-M1 10. BIBLIOGRAPHY . Bottom Reverberation at 24 Ke. E. W. Scripps Data, R. R. Carhart, Internal Report A-7, UCDWR, May 18, 1944. Scattering of Sound by the Surface of the Sea, L. I. Schiff, Project NS-140. Memorandum for file M-217, UCDWR, May 15, 1944. Div. 6-520.11-M4 551 C. F. Eckart, Memorandum for File No. 01.40, UCDWR, Apr. 18, 1942. Div. 6-520.11-M2 Div. 6-520.21-M3 8. Relation between Scattering and Absorption of Sound, . Mathematics of Physics and Chemistry, H. Margenau and Memorandum for File No. 01.40 x 01.72, Report SAS-8, G. Murphy, D. Van Nostrand and Company, New York, CUDWR-SSG, Dee. 11, 1944. Div. 6-520.11-M5 1943, p. 246. 9. Theory of Sound, Lord Rayleigh, The Macmillan Com- . Multiple Scattering, C. F. Eyring, R. J. Christiensen, and pany, New York, 2, 1926, p. 145. Chapter 13 . Reverberation Studies at 24 Kc, OSRD 1098, NDRC 6.1- 7. Bottom Reverberation at 24 Kc. E. W. Scripps Data, R. R. sr30-401, Report U-7, UCDWR, Nov. 23, 1942. Carhart, Internal Report A-7, UCDWR, May 18, 1944. Div. 6-520-M2 Div. 6-520.21-M3 . Ibid., p. 23. 8. Operational Procedures and Equipment Used in Sonar . A System for Recording Reverberation as it Occurs in the Sound Field Studies, NDRC 6.1-sr30-2024, Report U-295, Ocean, NDRC 6.1-sr30-1202, Report M111, UCDWR, Project NS-140, UCDWR, Feb. 15, 1945, p. 18. Aug. 28, 1943. Div. 6-520.2-M1 Div. 6-510.2-M8 . Operational Procedures and Equipment Used in Sonar 9. Apparatus for Recording Reverberation in the Sea, L. N. Sound Field Studies, NDRC 6.1-sr30-2024, Report U-295, Liebermann, OEMsr-31, WHOI, Feb. 23, 1945. Project NS-140, UCDWR, Feb. 15, 1945, p. 8. Div. 6-520.2-M3 Div. 6-510.2-M8 10. Volwme Reverberation. Scattering and Attenuation versus Limitation of Echo Ranges by Reverberation in Deep Frequency, OSRD 1555, NDRC 6.1-sr30-670, Report Water, Report M-361, Nobs-2074, Sept. 20, 1945. U-50, UCDWR, Apr. 13, 1943. Div. 6-520.3-M1 Div. 6-520.22-M2 11. Characteristics of Some Transducers Used by UCDWR, Summary of the Calibration of the Reverberation Equip- Report U-23, UCDWR, May 6, 1943. ment, November 24, 1943 to February 23, 1945, T. H. 12. A Practical Dictionary of Underwater Acoustical Devices Schaefer, UCDWR, Apr. 18, 1945. Div. 6-520.2-M4 NDRC 6.1-sr20-889, CUDWR-USRL, July 27, 1943. Chapter 14 Reverberation Studies at 24 Kc, OSRD 1098, NDRC 6.1- 11. Solution of Acoustic Boundary Problems: Part I, L. 1. st30-401, Report U-7, UCDWR, Nov. 23, 1942. Schiff, University of Pennsylvania, Sept. 4, 1943. Div. 6-520-M2 Div. 6-510.1-M3 Volume Reverberation. Scattering and Attenuation versus 12. Solution of Acoustic Boundary Problems: Part II, L. I. Frequency, OSRD 1555, NDRC 6.1-sr30-670, Report Schiff, University of Pennsylvania, Oct. 7, 1943. U-50, UCDWR, Apr. 13, 1943. Div. 6-520.3-M1 Div. 6-510.1-M3 Theory of Sound, Lord Rayleigh, The Macmillan Com- 13. Solution of Acoustic Boundary Problems: Part IIT, L. 1. pany, New York, 2, 1926. Schiff, University of Pennsylvania, Nov. 2, 1943. Limitation of Echo Ranges by Reverberation in Deep Water, Div. 6-510.1-M3 Report M-361, Nobs-2074, Sept. 20, 1945. 14. Echoes from Swells, G. E. Duvall, Report A-43, UCDWR, hr it, Geen Oct. 27, 1944. Div. 6-540-M1 Peso Aisi of teed ee cho Ranges, Sarid 15. Multiple Scattering, C. F. Eyring, R. J. Christiensen, and 2 Bey NAVY, epartment, NavShips 900,055 (labeled C. F. Eckart, Memorandum for File 01.40, UCDWR, Apr. NavShips 900,050), December 1944. 18. 1942 Div. 6-520.11-M2 . Survey of Underwater Sound. Ambient Noise, V. O. 16 Th Sh 7 R nee aa : ; A th inudsenteRaS-vAlfordsrand'-.We) EmlingOSRD#4333) (yy een ce ey EGR, aeer Tu Oe Casremenis Or ine NDRC 6.1-1848, Report No. 3, Sept. 26, 1944. JK-SK4926 Transducer at 24 Ke, N. Most, Internal Re- Div. 6-580.33-M2 port No. A-52, UCDWR, Jan. 5, 1945. . The Influence of Thermal Conditions on Transmission of Div. 6-510.221-M3 24-Ke Sound, Report U-307, Nobs-2074, UCDWR, Mar. 17. The Effect of the Ship’s Roll on Echo Ranging, J. S. 16, 1945. Div. 6-510.4-M5 MeNown and C. F. Eckart, NDRC 6.1-sr30-1205, Re- . Theoretical Physics, G. Joos, G. HE. Stechert and Com- port M-114, UCDWR, Oct. 8, 1943. Div. 6-510.3-M2 pany, New York, 1932, p. 581. 18. The Discrimination of Transducers Against Reverberation, . The Sea Surface and its Effect on the Reflection of Sound and OSRD 1761, NDRC 6.1-sr30-968, Report U-75, UCDWR, Light, C. F. Eckart, Report M-407, Nobs-2074 UCDWR, May 31, 1943. Div. 6-520.1-M8 Mar. 20, 1946. Div. 6-520.11-M6 19. Computed Maximum Echo and Detection Ranges for Sub- marine Echo-Ranging Gear, W. B. Snow and E. Gerjuoy, NDRC 6.1-sr1131, 1128-1688, CUDWR, July 1944. Div. 6-570-M2 552 BIBLIOGRAPHY Tate 12. 13. 14. Chapter 15 Water, Part II, L. I. Schiff, University of Pennsyl- vania, June 5, 1943. Div. 6-520.11-M3 Probability and its Engineering Uses, Fry. Fluctuation of Transmitted Sound in the Ocean, Technical Memorandum 6, NDRC 6.1-sr1131-1883, CUDWR, Jan. 17, 1945. Div. 6-510.3-M4 The Effect of the Ship’s Roll on Echo Ranging, J. S. MeNown and C. F. Eckart, Report M-114, NDRC 6.1- sr30-1205, UCDWR, Oct. 8, 1943. Div. 6-510.3-M2 Theory of Random Processes, H. Uhlenbeck, Report 454, MIT-RL, Oct. 15, 1943. Div. 14-125-M7 Coherence of CW Reverberation, Memorandum for File No. 01.40, Report SAS-11, CUDWR, Dec. 20, 1944. Div. 6-520.1-M9 The Fluctuations in Signals Returned by Many Inde- pendently Moving Scatterers, A. J. F. Siegert, MIT-RL, Report No. 465, Nov. 12, 1943. Div. 14-122.113-M7 The Appearance of the A Scope When the Pulse Travels Through a Homogeneous Distribution of Scatterers, A. J. F. Siegert, Report 466, MIT-RL, Nov. 9, 1943. Div. 14-124.2-M2 “Stochastic Problems in Physics and Astronomy,” S. Chandrasekhar, Rev. of Mod. Phys., 15, January 1943. 20. 21. 22. 23. 24. 25. . Range Limitation in Shallow Water as Controlled by Bot- 6. Bottom Reverberation. Dependence on Frequency, NDRC tom Character, State of Sea, and Thermal Structure, F. P. 6.1-sr30-677, Report U-79, UCDWR, June 16, 1943. Shepard, Report A-10, UCDWR, May 22, 1944. Div. 6-520.21-M1 Div. 6-520.21-M4 . . ae . Bottom Reverberation at 24 Kc. E. W. Scripps Data, R. R. i a Aa aes eee Pectuiey Carhart, Report A-7, UCDWR, May 18, 1944. ; ani tS 3 Div. 6-520.21-M3 Div Gozo MS . Bottom Reverberation, R. J. Christiensen, Internal Report 8. Comp uted Maximum Echo and Detection Ranges for Sub- A-5, UCDWR, May 16, 1944. Div. 6-520.21-M2 ern Echo-Ranging Gear, W. B. Snow and E. Gerjuoy, . Calculation of Sound Ray Paths Using the Refraction Slide NDRC 6.1-srl1131, 1128-1688, CUDWR, July 1943. Rule, NavShips 943, BuShips-NDRC, May 1943. Div. 6-570-M2 The Short Range Spatial Pattern Measurements on the 9- Bottom Reverberation in Very Shallow Water, NDRC 6.1- JK-SK4926 Transducer at 24 Kc, N. Most, Internal sr30-1845, Report SM-249, Projects NS-140, NS-297, Report A-52, UCDWR, Jan. 5, 1945. Div. 6-501.221-M3 Aug. 18, 1944. Div. 6-520.21-M5 Chapter 16 . Theory of Sound, Lord Rayleigh, The Macmillan Com- 15. ‘‘Mathematical Analysis of Random Noise,” S. O. Rice, pany, New York, 1, 1926. . Bell System Technical Journal, 23, July 1944, p. 289. . Reverberation Studies at 24 Kc, OSRD 1098, NDRC 6.1- 15a. “Mathematical Analysis of Random Noise,” S. O. Rice, sr30-401, Report U-7, UCDWR, Nov. 23, 1942. Bell System Technical Journal, January 1945. Div. 6-520-M2 16. The Extrapolatory Interpolation and Smoothing of Station- The Detection of an Echo in the Presence of Reverberation, ary Time Series, N. Wiener, NDRC Progress Report No. C. F. Eekart, OSRD 173, NDRC C4sr30-175, UCDWR, 19 to the Services, MIT, Feb. 1, 1942. May 12, 1942. Div. 6-560.32-M1 17, Frequency Characteristics of Echoes and Reverberation, Reverberation and Scattering, Series I, Sonar Data, Report W. M. Rayton and R. C. Fisher, OSRD 4159, Project MR-345-I, Nobs-2074, UCDWR, July 1945, pp. 4-6. NS-140, NDRC 6.1-sr30-1740, Report U-244, UCDWR, Div. 6-520.22-M1 Aug. 9, 1944. Div. 6-520.3-M2 Reverberation and Scattering, Series I, Sonar Data, Report 1g The Theory of Reverberation and Echo, C. F. Eckart, MR-365-I, Nobs-2074, UCDWR, September 1945, NDRC C4sr30-005, UCDWR, July 7, 1941. j pp. 4-6. Div. 6-510.22-M7 Div. 6-520.1-M1 Fluctuations in Reverberation Due to Scattering Centers in 19. Theoretical Discussion of Reverberation, C. L. Pekeris, OSRD 684, NDRC C4-sr20-097, CUDWR, May 29, 1942. Div. 6-520.1-M7 Frequency Spread of Reverberation as Measured with the Periodmeter, Memorandum for File No. 01.40, Report SAS-15, Sonar Analysis Section, CUDWR-SSG, Jan. 17, 1945. Div. 6-520.3-M5 Frequency Characteristics of Reverberation, Memorandum for File No. 01.40, Report SAS-16, Sonar Analysis Sec- tion, CUDWR-SSG, Nov. 23, 1944. Div. 6-520.3-M4 The Dependence of the Operational Efficacy of Echo-Ranging Gear on its Physical Characteristics, H. Primakoff and M. J. Klein, NDRC 6.1-sr1130-2141, Project NS-182, CUDWR-USRL, March 15, 1945. Div. 6-551-M14 Frequency Modulation in Echo Ranging, C. F. Eckart, NDRC C4-sr30-236, UCDWR, July 21, 1942. Div. 6-635.1-M3 Observations of Echo Signals Obtained Using Variable Frequency Transmission, E. M. Macmillan, NDRC C4 sr30-208, UCDWR, July 4, 1942. Div. 6-510.3-M1 Coherence and Fluctuation of FM Reverberation, M. J. Sheehy, Report A-37, UCDWR, Sept. 19, 1944. Div. 6-520.3-M3 BIBLIOGRAPHY 553 Chapter 20 1. The Theory of Sound, Lord Rayleigh, London, 1896. 6. Reflection and Scattering of Sound, H. F. Willis, OSRD 2. “On the Absorption of Sound Waves in Suspensions and Liaison Office WA-92 10f, NDRC C4-brts-501, British Emulsions,” P. S. Epstein, Theodor von Karman Anni- Internal Report 50, HMA/SEE, Fairlie Laboratory, versary Volume, California Institute of Technology, Great Britain, Dec. 20, 1941. Div. 6-530.1-M1 May 11, 1941, p. 162. 7. Reflections from Submarines, M. J. Klein and J. B. Kellar, 3. H. Stenzel, Ann. d. Physik, Series 5, 41, 1942, p. 245. NDRC 6.1-sr1130-1376, Project No. 222, USRL, Apr. 15, 4. H. Reissner, Helvetia Physica Acta, 11, 1935, p. 140. 1945. Div. 6-530.1-M3 5. The Acoustic Properties of Domes: Part II, H. Primakoff, 8. General Information and Sketch Book for the Engine Room NDRC 6.1-sr1130-1366, USRL, Feb. 18, 1944. Personnel of German Submarines, Type VII C, U.S. Navy, Div. 6-555-M17 DTMB, May 1942. Div. 6-530.22-M1 Chapter 21 1. An Analysis of Reflections from Submarines, NDRC 10. Measurements made with 26-Ke DSS on USS Cythera 6.1-sr1131-1846, File 01.80, Technical Memorandum 4, (Memorandum), C. A. Ewaskio, HUSL, Feb. 21, 1945. Sonar Analysis Section, CUDWR-SSG, Sept. 9, 1944. Div. 6-632.422-M3 Div. 6-530.22-M9 11. Submarine Runs with Directional and Nondirectional 2. Reverberation Studies at 24 Ke, OSRD 1098, NDRC C4- Transmitting Beams, 26-Ke DSS on-USS Cythera (Memo- sr30-401, File 01.40, Report U-7, Reverberation Group, randum), C. M. Clay, HUSL, June 18, 1945. UCDWR, Nov. 23, 1942. Div. 6-520-M2 Div. 6-632.422-M13 3. Listening Techniques, Biweekly Report Covering Period 12. Sound Ranges Under the Sea— 1944, OSRD 4400, October 4 to October 17, 1942, NDRC C4-sr30-396, NDRC _ 6.1-sr1131-1880, Sonar Analysis Section, UCDWR, Nov. 7, 1942, p. 5. Div. 6-530.22-M2 CUDWE-SSG, November 1944. Div. 6-500-M2 4. Target Strength of a Submarine at 24 Ke, G. E. Duvall, 13. Small Object Detection, Sonar Data: Monthly Progress Re- File 01.80, Internal Report A-4, Echo-Ranging Section, port, Series I, Report MR-323-I, Project NS-140, Nobs- UCDWR, May 10, 1944. ____ Div. 6-530.22-M6 2074, UCDWR, May 1945, pp. 9-10. Div. 6-530.22-M18 5. Data at 45 Ke on Echoes from a Diving Submarine and its 14. Reflection of Sound from Targets, Sonar Data: Monthly Wake, [W. M. Rayton], Report M-172a, Project NS-141, Ean Reve Sonics ER t MR-334-I. Nobs-2074 : gress Report, Series I, Repor , Nobs 5 NDRC 6.1-sr30-1475, Sonar Section, UCDWR, Mar. 3, UCDWR, June 1945, pp. 10-12. Div. 6-530.22-M20 1944. Div. 6-530.22-M4 : ; : P ; 15. Reverberation and Scattering, Sonar Data: Monthly 6. Internal Waves, Biweekly Report Covering Period Jan- P. Report, Series I, Report MR-345-I, Nobs-2074 uary 21 to February 3, 1945, NDRC 6.1-sr30-2025, asibit CF PAL al oe Dee perce ane CDWR, July 1945, pp. 4-6. Diy. 6-520.22-M1 Report U-297, UCDWR, Feb. 10, 1945, p. 6. : 5 Div. 6-501.4-M3 16. The Attenuation of Sound tn the Sea, C. F. Eckart, 7. Relative Echo Intensity versus Aspect, F. E. Gilbert, Jr., NDRC 6.1-sr30-1532, Report U-236, File 01.70, Project and J. K. Nunan, Report P29/R789, CUDWR-NLL, NS-140, UCDWR, July 6, 1944. Div. 6-510-22-M4 Mar. 10, 1944. Div. 6-530.22-M5 17. Echoes of Very Short Pings from Submarines, W. M. 8. Sonar and Submarine Diving: Monthly Progress Report Rayton, Report M-301, Project NS-140, Nobs-2074, for May 1945, Report 3, Nobs-2083, WHOI, May 10, UCDWR, Mar. 1, 1945. Div. 6-530.22-M16 1945, p. 3. Div. 6-530.22-M19 18. Surface Reflected Submarine Echoes, Report M-306, File 9. Sonar and Submarine Diving: Monthly Progress Report for 01.80, Project NS-140, Nobs-2074, Echo-Ranging Sec- June 1945, Report 4, Nobs-2083, WHOI, July 11, 1945, tion, UCDWR, Mar. 15, 1945. Div. 6-530.22-M17 pp. 2-4. Div. 6-530.22-M21 Chapter 22 1. Reflection of Light from a Submarine Model, R. B. Tibby, NS-222 and MIT Research Project DIC-6187, MIT- Memorandum for File 02.30, Report M-61, UCDWR, USL, Apr. 12, 1944. Div. 6-530.23-M3 May 12, 1943. Div. 6-530.23-M1 4. Studies of Optical Reflections from Submarine Models: 2. Reflections from Submarines at Close Ranges. Model Ex- Part II, OSRD 3706, NDRC 6.1-sr1046-1668, File 07.10, periments Using Optical Method, Project NO-222 and Navy Project NS-222 and MIT Research Project DIC- MIT Research Project DIC-6187, MIT-USL, Apr. 8, 6187, MIT-USL, Aug. 15, 1944. Div. 6-530.23-M4 1944. Div. 6-530.23-M2 5. Measurement of Reflections from Submarines Using Models 3. Studies of Opticul Reflections from Submarine Models: and High-Frequency Sound, J. B. Kellar, OSRD 4439, Part II, OSRD 3706, NDRC 6.1-sr1046-1053, Project NDRC 6.1-sr1130-1834, Navy Project NS-140, USRL, Sept. 27, 1944. Div. 6-530.23-M5 504 BIBLIOGRAPHY 10. 11. 12. 14. 15. Chapter 23 An Analysis of Reflections from Submarines, NDRC 6.1- sr1131-1846, File 01.80, Technical Memorandum 4, Sonar Analysis Section, CUDWR-SSG, Sept. 9, 1944. Div. 6-530.22-M9 Target Strength of a Submarine at 24 Ke, G. E. Duvall, File 01.80, Internal Report A-4, cho-Ranging Section, UCDWR, May 10, 1944. Div. 6-530.22-M6 Sonar Sound Field, Biweekly Report Covering Period September 17 to September 30, 1944, NDRC 6.1-sr30- 1862, Report U-262, UCDWR, Oct. 5, 1944, pp. 4-5. Div. 6-530.22-M10 Pillenwerfer Design, OSRD Liaison Office WA-328-16, British Internal Report 100, HMA/SEE, Fairlie Labora- tory, Great Britain, Sept. 15, 1942. Div. 6-651-M1 Studies of Optical Reflections from Submarine Models: Part IT, NDRC 6.1-sr1046-1668, Navy Project NS-222 and MIT Project DIC-6187, File 07.10, MIT-USL, Aug. 15, 1944. Div. 6-530.23-M4 Reflections from Submarines, M. J. Klein and J. B. Kellar, NDRC 6.1-sr1130-1376, Navy Project NO-222, USRL, Apr. 15, 1944. Div. 6-530.1-M3 Listening Techniques, Biweekly Report Covering Period October 4 to October 17, 1942, NDRC C4-sr30-396, UCDWR, Nov. 7, 1942. Div. 6-530.22-M2 Reverberation Studies at 24 Kc, OSRD 1098, NDRC C4- sr30-401, File 01.40, Report U-7, Reverberation Group, UCDWR, Nov. 23, 1942. Div. 6-520-M2 Data at 45 Ke on Echoes from a Diving Submarine and its Wake, [W. M. Rayton], NDRC 6.1-sr30-1475, Memo- randum for File 01.50, Service Project NS-141, Report M-172-A, Sonar Section, UCDWR, Mar. 3, 1944. Div. 6-530.22-M4 Sonar und Submarine Diving. Monthly Progress Report for June 1945, Report 4, Nobs-2083, WHOI, July 11, 1945, pp. 2-4. Div. 6-530.22-M21 Measurements Made with 26-Kc DSS on USS Cythera (Memorandum), C. A. Ewaskio, HUSL, Feb. 21, 1945. Div. 6-632.422-M3 Measurement of Reflections from Submarines Using Models and High Frequency Sound, J. B. Kellar, OSRD 4439, NDRC 6.1-sr1130-1834, Navy Project NS-140, USRL, Sept. 27, 1944. Div. 6-530.23-M5 . Relative Echo Intensity versus Aspect, F. E. Gilbert, Jr., and J. K. Nunan, Report P29/R789, CUDWR-NLL, Mar. 10, 1944. Div. 6-530.22-M5 Submarine Runs with Directional and Nondirectional Transmitting Beams, 26-Kc DSS on USS Cythera (Memo- randum) C. M. Clay, HUSL, June 18, 1945. Div. 6-632.422-M13 Sonar Submerged Submarine Wakes, [P. H. Hammond], BuShips Problem U2-9CD, Serial S-RS-96, Report ND11/NP22/S68, USNRSL, Aug. 9, 1944. Div. 6-540.31-M3 16." 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. General Information and Sketch Book for the Engine Room Personnel of German Submarines, Type VII C, U.S. Navy, DTMB, May 1942. Div. 6-530.22-M1 Studies of Optical Reflections from Submarine Models, Part I, OSRD 3706, NDRC 6.1-sr1046-1053, Navy Project NS-222 and MIT Project DIC-6187, File 07.10, MIT-USL, Apr. 12, 1944. Div. 6-530.23-M3 Change of Average Peak Echo Intensity with Changing Ping Length, Lyman Spitzer, Jr., Memorandum for File 01.80, Report SAS-30, Sonar Analysis Section, CUDWR- SSG, Mar. 22, 1945. Div. 6-530.1-M4 Preparation of Charts of Average Echo-Ranging Condi- tions, Biweekly Report Covering Period July 23 to August 5, 1944, NDRC 6.1-sr30-1745, Report U-248, Project NO-140, UCDWR, Aug. 10, 1944, p. 5. Div. 6-530.22-M7 Preparation of Charts of Average Echo-Ranging Condi- tions, Biweekly Report Covering Period August 6 to August 19, 1944, NDRC 6.1-sr30-1750,. Report U-253, Project NO-140, UCDWR, Aug. 23, 1944, p. 4. Div. 6-530.22-M8 Reflectivity of Targets, Biweekly Report Covering Period October 1 to October 14, 1944, NDRC 6.1-sr30-1865, Report U-264, Project NS-140, UCDWR, Oct. 31, 1944, pp. 3-5. Div. 6-530.22-M 11 The Attenuation of Sound in the Sea, C. F. Eckart, NDRC 6.1-sr80-1532, Project NS-140, Report U-236, File 01.70, UCDWR, July 6, 1944. Div. 6-510.22-M4 Reflectivity of Targets, Biweekly Report Covering Period October 29 to November 11, 1944, NDRC 6.1-sr30-1874, Report U-274, Project NS-140, UCDWR, Nov. 16, 1944, pp. 5-6. Div. 6-530.22-M13 The Influence of Thermal Conditions on Transmission of 24-Kc Sound, Sonar Data Division, Problem 2A, Report U-307, Nobs-2074, UCDWR, Mar. 16, 1945. Div. 6-510.4-M5 Internal Waves, Biweekly Report Covering Period Jan- uary 21 to February 3, 1945, NDRC 6.1-sr30-2025, Report U-297, UCDWR, Feb. 10, 1945, p. 6. Div. 6-501.4-M3 Echoes of Very Short Pings from Submarines, W. M. Rayton, Problem 2C, Report M-301, File 01.80, Project NS-140, Nobs-2074, CUDWR, Mar. 1, 1945. Div. 6-530.22-M16 Reflectivity of Targets, Biweekly Report Covering Period January 7 to January 20, 1945, NDRC 6.1-sr30-2021, Report U-292, Project. NS-140, UCDWR, Jan. 26, 1945. Div. 6-530.22-M14 Origin of Nearest Echo, W. E. Benton, G. M. Johnson, W. A. Jones, and R. J. W. Morrison, British Internal Report 209, OSRD Liaison Office WA-4297-1 HMA/ SEE, Fairlie Laboratory, Great Britain, Feb. 15, 1945. Div. 6-530.22-M15 BIBLIOGRAPHY 555: Chapter 24 . Thermal Wake Detection, D. H. Garber, R. J. Urick, and J. Cryden, Report S-20, USNRSL, Jan. 12, 1945. Div. 6-540.4-M2 . Reflection of Sound in the Ocean from Temperature Changes, R. R. Carhart, NDRC 6.1-sr30-960, Project NS-140, Report U-74, UCDWR, May 17, 1943. Div. 6-510.4-M3 . Theoretical Discussion of Reverberation, C. L. Pekeris, OSRD 684, NDRC C4-sr20-097, CUDWR-PAG, May 29, 1942. Div. 6-520.1-M7 . Surface Vessel Target Strengths, Memorandum for File 3. Status Report on Task No. 5, Effect of Short Pulse Lengths 01.80, SAG-38, Sonar Analysis Group, CUDWR-SSG, and Receiver Bandwidth on Echo Ranging, R. W. Kirkland, July 5, 1945. Div. 6-530.21-M3 Report 3510-RWK-HP, BTL, July 15, 1944. Div. 6-632.03-M5 . Oscillograms of 23-Kce Echoes from a Destroyer and its 4. Underwater Sound Reflecting Characteristics of Surface Wake, [C. F. Eckart], Memorandum for File 01.50, Re- Ships, C. Shafer, Jr., Report 2320-CS-PD, BTL, Oct. 6, port M-141, UCDWR, Jan. 3, 1944. Div. 6-530.21-M1 1944. Div. 6-530.21-M2 Chapter 27 . Sonar Submerged Submarine Wakes, P. H. Hammond, Part II, Experimental Results and Theoretical Interpreta- BuShips, Problem U2-9CD, Serial S-RS-96, Report tion, BE. L. Carstensen and L. L. Foldy, OSRD 3872, ND11/NP22/S68, USNRSL, Aug. 9, 1944, modified as of NDRC 6.1-sr1130-1629, Project NS-141, USRL, June 23, Nov. 1, 1944. Div. 6-540.31-M3 1944. Div. 6-540.3-M4 . Laboratory Studies of the Acoustic Properties of Wakes, . . J. Wyman, W. Lehmann, and D. Barnes, NDRC 6.1- a Bees : ae ee ae Hees Hate ol, Hise tof sr31-1069, Project. NS-141, WHOI, March 1944. BUGS Ci © WER Jo Be MUON OI, IDR @.1eseile, ; Div. 6-540.3-M3 File 01.50, Report U-25, UCDWR, Feb. 19, 1943. . The Rate of Rise and Diffusion of Air Bubbles in Water, Div. 6-540.21-M3 C. L. Pekeris, OSRD 976, NDRC C4-sr20-326, CUDWR- 8. Geometry on Surface Wakes and Experiments on Artificial PAG, Oct. 22, 1942. Div. 6-540.21-M2 Wakes, N. J. Holter, BuShips Problem U2-9CD, Report . Propagation of Sound through a Liquid Containing Bubbles, S-10, USNRSL, May 22, 1943. Div. 6-540.1-M1 Chapter 28 . “On the Absorption of Sound Waves in Suspensions and 7. E. Meyer and K. Tamm, Akustische Zeitschrift, 4, 1939, Emulsions,” Paul S. Epstein, Theodor von Karman Anni- p. 145. versary Volume, CIT, May 11, 1941, pp. 162-168. 8. Propagation of Sound through a Liquid Containing Bub- . The Stability of Air Bubbles in the Sea and the Effect of bles, Part II, Experimental Results and Theoretical Inter- Bubbles and Particles on the Extinction of Sound and pretation, E. L. Carstensen and L. L. Foldy, OSRD 3872, Light in Sea Waiter, P. S. Epstein, NDRC C4-sr30-027, NDRC 6.1-sr1130-1629, Project NS-141, USRL, June UCDWR, Sept. 1, 1941. Div. 6-540.21-M1 23, 1944. Div. 6-540.3-M4 . Propagation of Sound through a Liquid Containing Bub- 9. Statistical Mechanics, R. H. Fowler, Cambridge Univer- bles: Part I, General Theory, L. L. Foldy, OSRD 3601, sity Press, 1929, p. 154. NDRC 6.1-sr1130-1378, Project NS-141, USRL, Apr. 25, 10. The Internal Constitution of the Stars, A. S. Eddington, 1944. Div. 6-540.22-M2 Cambridge University Press, 1929. . Leslie L. Foldy, The Physical Review, 67, 1945, p. 107. 11. Handbuch der Astrophysik, Julius Springer, Berlin, 1930. . Acoustic Properties of Gas Bubbles in a Liquid, Lyman 12. “On the Illumination of a Planet Covered with a Thick Spitzer, Jr.. OSRD 1705, NDRC 6.1-sr20-918, CUDWR, Atmosphere,”’ B. P. Gerasimovic, Bulletin de l’Observa- July 15, 1943. Div. 6-540.22-M1 toire Central & Poulkovo (Russia), 15, No. 127, 1937, p- 4. . M. Minnaert, The London, Edinburgh and Dublin Philo- 13. A Textbook of Sound, A. B. Wood, The Macmillan Com- sophical Magazine and Journal of Science, 16, 1933, p. pany, 1941, p. 362. 235. Chapter 29 4. The Geometry of Surface Wakes and Experiments on Artificial Wakes, N. J. Holter, Report S-10, USNRSL, May 22, 1943. Div. 6-540.1-M1 Preliminary Measurements on the Acoustic Properties of Disturbed Water, E. Dietze, NDRC C4-sr20-205, USRL, Sept. 7, 1942. Div. 6-540.3-M1 Propagation of Sound Through a Liquid Containing Bub- bles, Part II, Experimental Results and Theoretical Inter- pretation, E. L. Carstensen and L. L. Foldy, OSRD 3872, NDRC 6.1-sr1130-1629, Service Project NS-141, USRL, June 23, 1944. Div. 6-540.3-M4 556 BIBLIOGRAPHY Chapter 30 Operational Procedure and Equipment Used in Sonar Sound Field Studies, NDRC 6.1-sr30-2024, Service Project NS-140, Report U-295, UCDWR, Feb. 15, 1945. Div. 6-510.2-M8 Chapter 31 Laboratory Studies of the Acoustic Properties of Wakes (Parts I and II), J. Wyman, W. Lehmann, and D. Barnes, NDRC 6.1-sr31-1069, Service Project NS-141, WHOI, March 1944. Div. 6-540.3-M3 Thermal Wake Detection, D. H. Garber, R. J. Urick, and Joseph Cryden, Report 8-20, USNRSL, Jan. 12, 1945. Div. 6-540.4-M2 . The Geometry of Surface Wakes and Experiments on Artificial Wakes, N. J. Holter, Report 5-10, USNRSL, May 22, 1943. Div. 6-540.1-M1 . Sound Transmission Loss Through and Thickness of the Wakes of Antisubmarine Vessels, N. J. Holter, Report 8-13, USNRSL, Nov. 22, 1943. Div. 6-540.32-M2 Sound Transmission Through Destroyer Wakes, OEMsr-30, Project NS-141, Report M-189, UCDWR, Mar. 8, 1944. Div. 6-540.32-M3 6. Chemical Recorder Traces of Submarine Wakes at 24 Ke, Internal Report A-23, G. E. Duvall, UCDWR, July 18, 1944. Div. 6-540.31-M2 7. Reflectivity of Targets, Biweekly Report Covering Period October 15 to October 28, 1944, NDRC 6.1-sr30-1871, Project NS-140, Report U-271, UCDWR, Oct. 31, 1944, pp. 5-6. Div. 6-530.22-M12 8. Wake of a Fleet-Type Submarine, W. M. Rayton and G. E. Duvall, Internal Report A-34, Echo-Ranging Section, UCDWR, Sept. 5, 1944. Div. 6-540.31-M4 9. Sonar Submerged Submarine Wakes, P. H. Hammond, BuShips Problem U2-9CD, Code 940, Serial S-RS-96, Report ND11/NP22/S68, USNRSL, Aug. 9, 1944. Div. 6-540.31-M3 Chapter 32 Sound Transmission through Destroyer Wakes, OK Msr-30, Project NS-141, Report M-189, Listening Section, UCDWR, Mar. 8, 1944. Div. 6-540.32-M3 Underwater Sound Output of Cruiser, Destroyer, and Aircraft Carrier, Report SM-268, UCDWR and MIT- USL, Oct. 28, 1944. Div. 6-580.2-M4 Reflectivity of Targets, Biweekly Report Covering Period January 7 to January 20, 1945, NDRC 6.1-sr30-2021, Project NS-140, Report U-292, UCDWR, Jan. 26, 1945, Artificial Wakes, N. J. Holter, Report S-10, USNRSL, May 22, 1943. Div. 6-540.1-M1 5. Sound Transmission Loss Through and Thickness of the Wakes of Antisubmarine Vessels, N. J. Holter, Report 8-13, USNRSL, Nov. 22, 1943. Div. 6-540.32-M2 6. Transmission of Sound Along Wakes, NDRC 6.1-sr1046- 1054, Project NS-141 and MIT Research Project DIC- 6187, MIT-USL, July 26, 1944. Div. 6-540.32-M4 7. Laboratory Studies of the Acoustic Properties of Wakes, (Parts I and II), J. Wyman, W. Lehmann, and D. pp. 5-6. Div. 6-580.22-M14 Barnes, NDRC 6.1-sr31-1069, Project NS-141, WHOI, The Geometry of Surface Wakes and Experiments on March 1944. Div. 6-540.3-M3 Chapter 33 . Acoustic Measurements on Surface Wakes in San Diego Harbor, R. R. Carhart and G. E. Duvall, OSRD 1628, NDRC 6.1-sr30-961, Report U-62, UCDWR, May 8, 1943. Div. 6-540.32-M1 The Discrimination of Transducers Against Reverberation, OSRD 1761, NDRC 6.1-sr30-968, Report U-75, UCDWR, May 31, 1943. Div. 6-520.1-M8 Status Report on Task No. 5. Effect of Short Pulse Lengths and Receiver Bandwidth on Echo Ranging, Robert W. Kirkland, Report 3510-RWK-HP, BTL, July 15, 1944. Div. 6-632.03-M5 4. Preliminary Report on Echoes from a Diving Submarine and Its Wake, Project M-172, Report M-172, Sonar Sec- tion, UCDWR, Jan. 22, 1944. Div. 6-530.22-M3 5. Data at 45 Ke on Echoes from a Diving Submarine and its Wake, W. M. Rayton, NDRC 6.1-sr30-1475, Project NS-141, Report M-172a, UCDWR, Mar. 3, 1944. Div. 6-530.22-M4 6. Laboratory Studies of the Acoustic Properties of Wakes, (Parts I and II), J. Wyman, W. Lehmann, and D. Barnes, NDRC 6.1-sr31-1069, Project NS-141, WHOI, March 1944. Div. 6-540.3-M3 Chapter 34. . Laboratory Studies of the Acoustic Properties of Wakes, J. Wyman, W. Lehmann, and David Barnes, NDRC 6.1-¢r31-1069, Project NS-141, WHOI, March 1944. ; Div. 6-540.3-M3 2. Reverberation Studies at 24 Kc, OSRD 1098, NDRC 6.1- sr30-401, Report U-7, UCDWR, Nov. 23, 1942. Div. 6-520-M2 CONTRACT NUMBERS, CONTRACTORS, AND SUBJECT OF CONTRACTS Contract No. NDCre-40 OEMsr-20 OEMsr-30 OEMsr-31 OEMsr-287 OEMsr-346 OEMsr-1046 OEMsr-1128 OEMsr-1130 OEMsr-1131 Name and Address of Contractor Subject Woods Hole Oceanographic Institution Woods Hole, Massachusetts The Trustees of Columbia University in the City of New York New York, New York The Regents of the University of California Berkeley, California Woods Hole Oceanographic Institution Woods Hole, Massachusetts President and Fellows of Harvard College Cambridge, Massachusetts Western Electric Company, Inc. 120 Broadway New York, New York Massachusetts Institute of Technology Cambridge, Massachusetts The Trustees of Columbia University in the City of New York New York, New York The Trustees of Columbia University in the City of New York New York, New York The Trustees of Columbia University in the City of New York New York, New York Studies and experimental investigations in connection with the structure of the super- ficial layer of the ocean and its effect on the transmission of sonic and supersonic vibra- tions. Studies and investigations in connection with the oceanographic factors influencing the transmission of sound in sea water. Studies and experimental investigations in connection with and for the development of equipment and methods pertaining to submarine warfare. Maintain and operate certain laboratories and conduct studies and experimental in- vestigations in connection with submarine and subsurface warfare. Studies and experimental investigations in connection with the structure of the super- ficial layer of the ocean and its effects on the transmission of sonic and supersonic vibrations. Studies and experimental investigations in connection with the development of equip- ment and devices relating to subsurface warfare. Studies and experimental investigations in connection with submarine and subsurface warfare. Studies and experimental investigations in connection with (1) underwater sound transmission and boundary impedance measurements; (2) ship sound surveys at high frequencies; (3) development of de- vices for the control of underwater sounds; and (4) development of intense underwater sound sources for special purposes. Conduct studies and experimental investiga- tions in connection with and for the de- velopment of equipment and methods in- volved in submarine and subsurface war- fare. Conduct studies and experimental investi- gations in connection with the testing and calibrating of acoustic devices. Conduct studies and investigations in con- nection with the evaluation of the applica- bility of data, methods, devices, and systems pertaining to submarine’ and sub- surface warfare. 557 558 SERVICE PROJECT NUMBERS The projects listed below were transmitted to the Executive Secretary, National Defense Research Committee [ NDRC], from the Navy Depart- ment through the Office of Research and Inventions (formerly the Coor- dinator of Research and Development), Navy Department. These are the principal Navy projects relating to the physics of sound in the sea. Service Project Number Subject NO-163 Cooperation with the Navy in harbor surveys and surveys of ambient underwater noise conditions in various areas. NO-222 Acoustic reflection fields of submarines. NS-140 Acoustic properties of the sea bottom. NS-140 Range as a function of oceanographic factors. (Ext. ) NS-141 Acoustic properties of wakes. INDEX The subject indexes of all STR volumes are combined in a master index printed in a separate volume. For access to the index volume consult the Army or Navy Agency listed on the reverse of the half-title page. “Absorption cross section’ of bubble, 466 Absorption effect in underwater sound transmission absorption coefficient, 97-100 attenuation measurements, 102-105 bubble formation, 465-467 coefficient of attenuation, 100 frequency ranges, 105-107 thermal structure, 102 transmission anomaly, 100-101 wakes, 541-543 Acoustic interference echoes, 377 intensity, 168-170 target strength measurements, 410 Acoustic interferometer for sound ve- locity measurements, 17 Acoustic measurements in underwater transmission, 243-244, 474-477 Acoustic wakes see Bubbles in acoustic wakes; Wakes, acoustic Acoustical axis of sound projector, 26— 27 Adiabatic pressure changes bubble formation, 461 Aerial photographs in acoustic wake geometry, 494-495 Air bubbles in acoustic wakes see Bubbles in acoustic wakes Airey phase of water waves, 232 Anchored ships, target strength meas- urements, 424—425, 437 Angular variation of echo level, 546 Antinodes of stationary sound waves, 33 Aspect angle, target strength measure- ments, 388-398, 424 Asymmetry effects on target strength measurements, 400—402 Attenuation coefficient in sonic trans- mission bottom scattering, 320-321 bubble formation, 469-470 isothermal water, 100, 104-107 shadow boundary, 124-125 target strength measurements, 370, 373, 411-413 transmission anomaly, 129-131 wake thickness, 503-504, 508-509 Attenuation of sound bubble theory, 583-534 explosions, 193-197 during frequency effects, 209-211 long range transmission, 216-219 propeller wakes, 510-511 scattering layer, 299-301 shadow zone, 67-68 transmission anomaly, 100, 105-107 wake theory, 503-504 wave theory, 27-28 Average layer effect in underwater sound transmission, 112 Averaging methods for reverberation data, 278-280 B-19 H magnetostrictive hydrophone, 74 Backward scattering coefficient of sound, 252, 266, 306, 335 Backward scattering of sound, 254, 483 Band method of averaging reverber- ation data, 279-280 Bathythermograph classification, 92-95 description, 76 ray tracing, 60-63 velocity-depth variations, 197-200 Beam target strengths in echo ranging, 415-417, 435-436 Bell Telephone Laboratories (BTL), surface vessel target strengths, 423-424 ; Blade cavitation in acoustic wakes, 449 “Blobs’”’ in reverberation of sound, 335 Bottom reverberation of sound, 264 average intensities, 321-323 data analysis, 319-321 deep-water transmission, 86-87 definition, 264 description, 308-312 frequency, 318-319 grazing angle, 314-318 refraction, 312-313 scattering coefficients, 314, 319-321, 338 summary, 338-339 Bottom scattering coefficients of sound, 314, 319-321, 338 Bottom-reflected sound attenuation coefficient, 103-104 dispersion phenomena, 228-229 normal modes theory, 222-224 predictions of ray theory, 224 ray intensity, 55-56 reflection coefficient, 219-221 shallow-water transmission, 137-138 simple harmonic propagation, 224— 227 summary, 243 supersonic frequencies, 140-141 times of arrival, 221-222 wave equation, 33-34 Boundary conditions in sound propa- gation point source far from surface, 33-34 point source near surface, 31-33 reflection and refraction of plane waves, 30-31 reflection from sea bottom, 33-34 target strengths, 353 transition conditions, 28-31 wake theory, 478 wave equation, 13-14 BTL (Bell Telephone Laboratories), surface vessel target strengths, 423-424 Bubbles in acoustic wakes absorption during bubble pulsation, 464167 acoustic effects, 474-477 attenuation, 469-470 “bubble hypothesis”, 533 buoyancy, 452-455 damping constant, 467 decay of wakes, 539-540 echo intensities, 514-515 entrained air, 455-457 long pulses, 515-516 multiple scattering, 470-473 oscillograms, 186-190 propeller cavitation, 449-452, 539 reflection, 473-474 scattering by an ideal bubble, 460- 464 scattering coefficient, 306-307 short pulses, 516-519 submarine wake strengths, 538-539 surface vessel wake strengths, 537— 538 theory, 448, 467-469 transmission loss, 503-504, 533-535 wake echoes, 535-537 Bulk modulus of a disturbed fluid, 12 Buoyancy of bubbles in underwater sound, 452-455 Burbling cavitation for bubble forma- tion, 449 “Burning” process in underwater ex- plosions, 173-174 559 560 INDEX Cable hydrophones, 74-75 Calibration techniques for sound meas- urements, 492 reverberation intensities, 277 target strengths, 368-369 transmission loss, 76-78 Canadian National Research Council, attenuation measurements, 105 Cathode-ray oscilloscope for acoustic wake measurements, 488-490 Cavitation in bubble formation, 191, 449-450 CHARLIE bathythermograms, 93 Chemical recorder traces in acoustic wake measurements, 484 “Chirp” signal in echo-ranging gear, 23 CN-8 crystal hydrophone, 74 Coherence in sound reverberation, 335 amplitude, 327-329 intensity, 339 transmission, 71 Compression viscosity in attenuation of sound, 28 Configurational averages for acoustic theory of bubbles, 468 Conservation of energy law for second- ary sound pressure waves, 186— 188 Continuity law in sound wave propaga- tion, 8-10 Continuous-flow bubble screens for acoustic measurements, 477 Convex surface, target strength meas- urements, 359, 434 “Cross-section” of bubble, 461 CW pings, frequency analysis of rever- beration, 329-331 Cylinder surfaces, target strength meas- urements, 360, 435 Damped vibration, 28 Damping constant, 467, 535-536 Decay rate in sound transmission acoustic wakes, 520-521, 539-540 bottom reverberation, 322 echo intensity, 526 shock waves, 184-186 surface reverberation, 337 Deep-water reverberation of sound average reverberation levels, 304-306 deep scattering layers, 282-284 definition, 86-89 echo ranging, 527-530 frequency effects, 284-288 multiple scattering effects, 303-304 oceanographic conditions, 289 ping length, 302 range dependence, 289-302 scattering coefficient, 306-307 transducer directed downward, 281-— 288 transducer horizontal, 288-299 volume reverberation, 281-282, 284— 288, 335-337 Deep-water transmission of sound see Transmission of sound, deep- water Density-pressure properties of a dis- turbed fluid, 11-12 Depth effects in sound transmission bathythermograms, 92-95 bottom reverberation level, 338 corrections, 49-51 ray diagrams, 89-90 temperature gradients, vertical, 90— 92 thermocline transmission, 115-117 volume reverberation, 282-284 Destroyer wakes, air bubble hypothesis, 534 Detonation process in underwater ex- plosions, 173-175 Diffraction of sound waves hypothesis, 201 nonspecular reflection, 361 pressure-time records, 204-206 ray theory, 41 shadow zones, 65-66, 200-201 wave equation, 66-68 Direction of sound propagation definition, 5-6 directivity index, 72 double source, 24-26 pattern functions, 26-27 point source, 24 transducer patterns, 429-430, 522 Doppler effect reverberation, 329-331 wake measurements, 484 Double layer effect in sound propaga- tion, 200 Double sources of sound, 24—26 Drift effect in echo variability, 376 EBI-1 crystal transducer, 276 Echo intensity measurements angular variation, 546 definition of echo level, 434 long pulses, 515-516 short pulses, 516-519 target strength, 347-348, 351, 377 variability, 374-378 Echo ranging equipment, 85 frequency, 523 projectors, 241 pulse length, 522-523 shallow-water, 321-323 submarine wakes, 523-526 surface vessel wakes, 526-530 target strengths, 343-344, 376 temperature gradients, 3-4 thermocline, 109-110 transducer directivity, 522 wake measurements, 484, 490-493 Echoes, wake beam echoes, 415-417, 435-436 decay, 539-540 off-beam echoes, 417-420, 436-437 propellers, 539 repeater, target training, 85 source, 420—421 submerged submarines, 437 surface vessels, 437 target strengths, 377, 435 wake theory, 535-537, 543-546 Eckart, self-correlation coefficient for sound intensity fluctuations, 166 Eikonal wave equation in ray acoustics, 44-45, 64-65 Electromagnetic sources for sonic fre- quencies, 72 Elongation phenomena of off-beam echoes, 418-420 Entrained air in acoustic wakes, 455-457 Equations for target strengths definition, 347 derivations, 348-350 reflected pressure, 355 Equations of wave propagation, 8-14, 43-45 - boundary conditions, 13-14 continuity, 8-10 differential equations of rays, 45-46 differential equations of wave fronts, 43-45 forces in a perfect fluid, 10-11 initial conditions, 13-14 motion, 10 ray paths, 46-47 state of fluid, 11-12 wave equation, 12-13 Equipment for reverberation measure- ments, 272-277 Explosions, underwater, 173-235 attenuation, 193-197 bottom reflection, 219 cavitation, 191 deep sound channels, 213-216 diffraction, 200-206 Fourier analysis, 206-211 long-range propagation, 216-219 normal mode theory, 224—229 predictions of ray theory, 222-224 pressure waves, secondary, 186-190 reflection coefficients, 220-222 refraction, 197-200 shallow-water experiments, 229-235 shock fronts, 175-177, 182-184 shock waves, 184-186 summary, 173-175 surface reflection, 190-191 211-213, transmission, 192-193 variations, 211 wave theory, 178-182 “Extinction cross section’”’ of bubbles, 465-466 Fathometer records for acoustic wakes submarines, 501-502 surface vessels, 497-501 thickness and structure, 486-488 two-way vertical transmission loss, 507-509 Fermat’s theorem of reverberation in- tensity, 253, 269 Fluctuations in sound transmission beam echoes, 436 echo intensity, 377 interference, 167-170 magnitude, 158-160 microstructure, lens action, 170-171 off-beam echoes, 437 probability distributions, 160-164 reverberation, 324-327, 335, 339 roll and pitch effects, 167-168 sound pulses, 211 space patterns, 167 supersonic frequencies, 241 time patterns, 164-167 Fluid velocity of sound waves see Velocity of sound in water Fluorescein for acoustic measurements of wake-laying vessel, 491 FM sonar, reverberation from wide- band pings, 75, 332-333 Forced vibrations of bubbles, 461 Forces in a perfect fluid) sound wave equation, 10-11 Fort Lauderdale, Florida, target- strength measurements, 366, 368 Forward reverberation of sound, 80 Fourier theory in sound propagation, 23, 36, 206-211, 329 “Free vibrations” of bubbles, 461 Frequency of sound attenuation, 138 bottom reverberation, 338 ekaracteristics, 23-24 deep-water reverberation, 284-288 echo ranging, 408-410, 523 narrow-band pings, 329-331 periodmeter, 330 shallow-water reverberation, 240, 318-319 sonic, 238-239 supersonic, 238-239 surface reverberation, 337 target strengths, 433 volume reverberation, 336 wide-band pings, 332-333 INDEX Fresnel zone theory of target strengths, 356-360 applications, 358 convex surface, 359 cylinder, 360 method, 356-357 sphere, 358-359 Gaussian distiibution of sound in- tensity fluctuations, 161-162, 326 Geometry of acoustic wakes see Wake geometry in sound trans- mission Grazing angle variation in reverber- ation of sound bottom scattering coefficients, 314— 318 transducer horizontal, 299-301 Ground wave in sound transmission, 230-232 “Group velocity” of a wave train, 227 Harbor detection equipment for sub- marine wakes, 443 Harmonic waves in sound propagation, 17-18, 22-23 Heterodyned reverberation of sound, 339 “Hidden periodicities’ of sound in- tensity fluctuations, 166-167 “Highlight” in Fresnel zone theory of target strengths, 357, 358 Horizontal transmission of sound beam echoes, 317-318 bottom reverberation, 321-323, 339 deep-water reverberation, 337-338 transmission loss, 504-507 transmission run, 79 Hugoniot equation for shock fronts, 180, 184 Hull reflections of underwater sound, 415 Hull wake in sound transmission, 478 Huyghen’s principle for reflected sound pressure, 356 Hydrodynamic theory of bubble forma- tion, 449 Hydrographic conditions for sound transmission anomalies, 119 Hydrophone depth in sound transmis- sion, 72-74, 148-150 Image effect in sound transmission, 95— 97, 190 Image interference in sound field in- tensity, 32-33, 163, 301 Index of refraction, wave front equa- tions, 44 “Instantaneous frequency” of rever- beration, 329-330, 339 561 Intensity of echoes see Echo intensity measurements Intensity of sound, 6 see also Fluctuations in sound trans- mission contours, 62-63 experiments, 114-117 formulas, 51-53 interference effects, 168-170 linear gradients, 57-58 phase distribution, 37-38 plane waves, 21 rays, 65-66 reverberation, 265-266, 334 scattering, 532 shadow zone, 65-68 spherical waves, 21—22 thermocline, 112-114 transmission anomaly, 53-54, 58-59 velocity-depth variation, 54-57 wake measurements, 488-490, 504 wave equation, 22 Interference effects in sonic transmis- sion echoes, 377 intensity, 168-170 target strength measurements, 410 Interferometer for sound velocity meas- urements, 17 Inverse square law for underwater sound, 6-7, 237, 345-347 Isothermal water, sound transmission absorption, 97-104 attenuation coefficient, 104-107 bottom reverberation, 313 deep-water transmission, 238-239 echo ranging, 109-110 image effect, 95-97 layer effect at 24 ke, 112-114 layer effect at 60 ke, 117 ray theory, 61 short range transmission, 108-109 temperature-depth pattern, 93 thermocline depth, 114-117 transmission loss, 107-108 transmission runs, 110-111 Isovelocity layer effect, ray acoustics, 56-57 Kennard’s theory of propagation of cavitation fronts, 191 Khintchine’s theorem, self-correlation coefficient for sound intensity fluctuations, 167 Lambert’s law for surface reverberation of sound, 300, 314 Laminar cavitation in bubble forma- tion, 449 562 INDEX Lee nnn Law ot conservation of energy for secondary sound pressure waves, 186-188 Law of motion for sound wave equa- tion, 10 Law of similarity for shock waves, 182 Layer effect at 24 ke, underwater sound transmission ray acoustics, 56--57 theory, 112-114 thermocline depth, 115-117, 238 University of California studies, 114— 115 Layer effect at 60 kc, underwater sound transmission, 117 Lens action of microstructure, sound in- tensity fluctuations, 170-171 Listening equipment for wake measure- ments, 484 Lloyd Mirror effect in wave acoustics, 32-33, 299, 301 Long Island area survey in sonic trans- mission, 154-156 Long-range sound channel propagation, 211-219 deep channels, 213-216 experimental results, 216-219 introduction, 211-213 Loops of stationary sound waves, 33 Magnetostrictive effect in sound trans- mission, 5, 72 Mean echo intensity for target strength measurements, 377-378 Microdispersers for measuring damping constant in sound field, 467 MIKE bathythermograms, 93-95 Motion law for sound wave equation, 10 Motion pictures of subsurface structure of wakes, 456 Moving vessels, target strengths, 425— 426, 437 Multiple scattering of sound, 268-269, 2303-304 NAN bathythermograms, 93-95 Narrow-band pings, frequency analysis of reverberation, 329-331 Naval warfare, acoustic wakes, 443-448 Navy echo ranging see Echo ranging Newton’s second law of motion for sound wave equation, 10 NK-1 type shallow depth recorder for acoustic wakes, 455 Nodes of stationary sound waves, 33 Noises, sinusoidal sound vibrations, 23 Nonisothermal water, bottom rever- beration of sound, 321-323 Nonresonant bubbles, acoustic meas- urements, 476, 477 Nonspecular reflections of sound, 361— 362, 410 Normal mode theory of sound, 34-38, 222-229 bottom reflection, 222-224 characteristic functions, 35-36 dispersion phenomena, 225-228 general waves, 36-37 intensity of sound, 37-38 plane waves, 34-36 prediction of rays, 224-225 pressure-time records, 228 OAX transducers, 78 Ocean bottoms, acoustics properties, 139-141 Oceanographic conditions for sound transmission bathythermographs, 76 measurements, 243 target strengths, 411-413 wakes, 492-493 Off-beam target strengths, 417-420, 436-437 One-way horizontal transmission loss, acoustic wakes, 504-506 Optical experiments for target strength measurements, 379-381, 386, 410 Oscillograms for underwater sound data beam echoes, 415 dispersion phenomena, 233-235 echo intensities, 377 explosive sound, 229-231 ground wave phase of disturbance, 230-233 hydrophone output, 74-76 pressure-time records, 204-206 reverberation data, 278 wake measurements, 488-490 “Overtaking effect’? in shock wave theory, 177, 183 “Patch size” of acoustic wake, 479 Pattern function for intensity of back- ward scattered sound, 254 Peak echo intensity in target strength measurements, 373-374, 377— 378 Perfect fluid, law of forces, 10-11 Periodmeter for frequency analysis of reverberation, 330-331, 339 PETER bathythermograms, 93 Phase constant in ray acoustics, 41 Phase distribution in wave acoustics, 37-38, 266 Photographic Interpretation Center, Anacostia, wake acoustics, 494 Physical parameters of acoustic wake strength, 514-519 echo intensity, 514-515 long pulses, 515-516 short pulses, 516-519 Piezoelectric effect in sound transmis- sion, 5, 72 “niling-up” effect in long-range sound transmission, 218 Pings in reverberation theory coherence, 327-329 duration, 334 narrow-band, 329-331 short pulses, 326 surface reverberation, 302 volume reverberation, 336 wide-band, 332-333 Plane waves, sound propagation intensity of sound, 21 normal mode theory, 34-36 pressure versus fluid velocity, 19-20 reflection and refraction, 30-31 velocity of sound, 15-17 wave equation, 14-15 Point method of averaging reverber- ation data, 279-280 Point source of sound boundary conditions, 33-34 equation, 22, 31-32 image interference effect, 32-33 surface reflection, 32 target strength, 353 Poisson distribution of fluctuations of sound intensity, 326 Power level recorders for underwater sound transmission measure- ments, 75 Pressure of reflected sound wave, 352- 355 boundary conditions, 353 mathematical analysis, 353-355 physical analsyis, 355 “Pressure pattern function” of sound receiver, 265 Pressure versus fluid velocity of sound waves, 19-20 Pressure waves (secondary) in sound propagation, 186-190 oscillatory motion, 186-188 spherical symmetry, 188-190 Pressure waves (nonlinear) Riemann’s theory, 178-179 Pressure-density properties of a dis- turbed fluid, 11-12 Pressure-time curves of shock waves, 184-186, 204-206, 228 Probability coefficients for reverber- ation levels, 328 Probability distributions of intensity fluctuations of sound field, 160— 164 distribution functions, 160-162 Gaussian, 161 image interference, 163 Rayleigh, 161-163 Propagation of progressive waves, 14— 15, 17-18 Propeller wakes, sound transmission bubble density, 535 bubble formation, 449 scattering measurements, 530-532, 539 transmission loss, 510-511 underwater explosions, 173-175 Pulse length, sound measurements Fresnel zone theory, 362 long pulses, 515-516 short pulses, 516-519 target strengths, 350-351, 404408, 432 wakes, 522-523, 544-546 QB crystal transducer, 275-276 QCH-3 crystal transducers, 273-275, 290-292 Rankine-Hugoniot theory of shock fronts, 179-181 Rarefractional shock waves in under- water explosions, 180-181 Ray acoustics, 41-68 curvature of ray, 46-47 depth correction, 49-51 diagrams, 59-60, 89-90 eikonal wave fronts, 64-65 general waves, 42-43 intensity along a ray, 51-54 long-range transmission, 216-219 plotter for ray-tracing, 59 ray patterns, equations, 45-46 refraction, 197-200 shadow zones, 65-68 “Sound channel” propagation, 211— 216 spherical waves, 41-42 temperature-depth patterns, 60-63 transmission anomalies, 58-59 velocity-depth variation, 54-58 vertical velocity gradients, 46-49 wave front equations, 43-45 Ray acoustics, theory of normal modes, 222-229 computations, 224-228 dispersion phenomena, 225, 228-229 predictability, 222-224 Rayleigh’s sound scattering law deep-water reverberation, 288 equation, 325-327, 481 intensity fluctuations, 161—163, 169 nonspecular reflection, 362 radiation, long-wave, 464 INDEX Receivers for underwater sounds, 73-76 Reciprocity principle in sound propaga- tion, 38-39, 269-270 Recommendations for sonar research, 241-244, 339-340 acoustic measurements, 244 bottom reflection, 243 oceanographic measurements, 243 reverberation, 339-340 surface reflection, 243 velocity of sound, 242 volume scattering, 242-243 Reflected beam in linear gradient, ray acoustics, 55-56 Reflected wave, 29 Reflection and refraction of plane waves, 30-31 Reflection coefficients for underwater sound long range transmission, 218-219 ocean bottoms, 220-222 sonic, 137-138 supersonic, 140-141 Reflection of sound bubble pulses, 473-474 close ranges, 360 submarines, 361, 386 surface of water, 190-191, 373-374 surface vessels, 437 underwater targets, 352-355 Refraction of sound bottom reflection, 138 bottom reverberation, 312-313, 338 bubbles, 473-474 explosions, 197-200 fluctuations, 170-171 “Resolving time’ for short-range sound propagation, 193 Resonant frequency of an air bubble, 462-464, 536-537 Reverberation of sound see also Bottom reverberation of sound; Deep-water reverberation of sound; Surface reverberation of sound; Volume reverberation of sound analytical procedures, 278-280 backward scattering coefficients: 266, 335 bottom levels, 310, 338-339 coherence, 327-329, 335, 339 deep-water levels, 335-338 definition, 247, 334 duration, 309 equipment for measuring intensity, 272-278 Fermat’s principle, 269 fluctuation, 158-160, 324-327, 335, 339 forward, 80 frequency, 258-259, 329-333, 339 563 intensity, 252-258, 265-266, 304-306, 334 level, 258-259, 334 peak, 321-323 ping length, 258-259 properties, 247-249 reciprocity theorem, 269-270 recommendations for future research, 338-339 seattering, 250-252, 266-269 strength, 259 surface reflection, 270-271 wakes, 492-493 Riemann’s theory for sound waves of finite amplitude, 178-179 Rigorous intensity in ray acoustics, 65-66 Roll and pitch effects on sound in- tensity fluctuations, 167-168, 377 Rough surface effects on reflection of sound, 361 Salinity effect on sound velocity, 17 Scattered sound see also Bubbles in acoustic wakes absorption, 242-243 average levels, 304-306 backward, 266, 335, 483 bubble theory, 306-307, 470-473 deep-water reverberation, 286-288 duration, 266-268 multiple, 268-269, 303-304 nonspecular reflection, 361 propeller wakes, 530-532 shadow zone, 125-129 shallow-water reverberation, 316-317 surface reverberation, 299-302 temperature and velocity of wakes, 480-483 theory, 250-252 Screw wakes, sound transmission, 478 Sea bottoms, acoustic properties, 139— 141 “Secondary sources” of sound, 356 Self-correlation coefficient for sound in- tensity fluctuations, 164-166, 482 Shadow boundary of sound attenuation coefficient, 124-125 ray theory, 65-68, 89-90 seattered sound, 125-129 zones, 120-122, 200-206 Shadowing effect in surface reverber- ation, 301 Shallow-water reverberation see Bottom reverberation of sound Shallow-water transmission of sound see Transmission of sound, shallow- water 564 INDEX Ship draft and tonnage, effect on target strengths, 432 Shock wave fronts, sound transmission, 174-186 law of similarity, 182 Rankine-Hugoniot theory, 177, 179- 181 Riemann’s theory of waves of finite amplitude, 176-179 structure and decay, 184-186 thickness of pressure region, 182-184 Short-range sound propagation, 108- 109, 193-211 diffraction hypothesis, 201-206 Fourier analysis, 206-211 pulse measurements, 193-197 refraction effects, 197-200 shadow zones, 200-201 transmission variations, 211 Similarity law for shock waves, 182 Sinusoidal sound experiments, 192 Slide rule for sound ray tracing, 59 “Slipstreams” in sound transmission, 478 Snell’s law of refraction for bottom re- verberation of sound, 318 Sonic transmission analysis of records, 83-84 deep-water, 238-239 frequency effects, 138-139 listening gear, 87 Long Island area survey, 155-156 Pacific Ocean measurements, 156 shallow-water, 240 summary, 156-157 “Sound channel” propagation deep sound channels, 213-216, 240 experiments, 216-219 long range transmission, 211-213 surface sound channels, 239-240 temperature gradients, 133-135 Sound field measurements see Transmission loss measurements Sound propagation in liquid containing many bubbles, 467-477 acoustical observations, 474-477 reflection, 473-474 scattering, 470-473 theory, 467-469 transmission, 469-470 Sound range recorders for wake meas- urements, 484 Sound transmission, underwater see Fluctuations in sound transmis- sion; Transmission of sound, deep-water; Transmission of sound, shallow-water Sources of sound see also Explosions, underwater directivity, 24-27 108-109, echoes, 420-421 frequency, 23-24 levels, 347-348, 434 transmission runs, 72-74 Space pattern of fluctuation of sound intensities, 167 Spectrum level in Fourier analysis of explosive sound, 208-209 Specular reflection of sound beam echoes, 415-417 convex surface, 434 frequency factors, 410 Fresnel zones, 356 surface vessel, 430 target strengths, 373-374 Speed of ship, effects on strengths, 402, 431 Sphere target strengths definition, 434 derivation, 348-350 Fresnel zone theory, 358-359 Spherical sound waves, 21-22, 41-42 intensity, 21-22 pressure versus fluid velocity, 20 ray acoustics, 41-42 wave equation, 18-19 “Spines” of echoes in surface-reflected sound, 373-374 Split-beam patterns in ray acoustics, 61-62 Standard reverberation level of sound, 259 Stationary waves in underwater sound see Normal mode theory of sound Still vessels, target strengths, 424-425, 437 Stoke’s hypothesis for attenuation of sound, 28 Submarine reflectivity, 379-381, 386 Submarine tactics in sound transmis- sion, 4 Submarine target strengths, 388-421 altitude angle, 395-397 aspect angle, 388-393 asymmetry, 400-402 beam echoes, 413-417 frequency, 408-410 measurements, 397-400 oceanography, 410-413 off-beam echoes, 417-420 orientation, 388 pulse length, 404-408 range, 402-404 source, 420-421 speed, 402 Submarine wakes, acoustic measure- ments echoes, 523-526 experiments, 501-502, 538-539 Supersonic transmission data-analysis, 80-84 target deep-water, 238-239 frequency effects, 138 listening gear, 87 sea bottoms, 139-141 summary, 153 transmission runs, 141-153 velocity gradients, 142-150 wind force, 152 Surface reverberation of sound average levels, 304-306 definition, 259 elimination, 281 grazing angle, 300-302 index, 262 intensity, 259-263 “Jevel” concept, 263-264 multiple scattering, 303-304 ping length, 302 range, 289-299 reflection effects, 270-271 scattering coefficient, 306-307, 337 summary, 336-338 wind speed, 298-299 Surface vessel target strengths aspect angle, 424-428 deep-water transmission, 527-530 frequency, 433 introduction, 422 measurements, 422424 pulse length, 432 range, 426-431 reflection, 437 ship type, 4382 speed, 431 wake echoes, 497-501 Surface-reflected sound fluctuations, 377 reverberation, 301 short-range propagation, 196 summary, 243 transmission loss, 373-374 Target strength measurement approximations, 352-353 calibration errors, 368-369 comparison of methods, 387 computation, 358-360 convex surface, 434 cylinder, 435 definition, 347-348 echo variability, 374-378 experiments, 363-366 Fresnel zones, 356-358 introduction, 343 Massachusetts Institute of Tech- nology, 379-381 mathematical theory, 353-355 Mountain Lakes, N. J., 381 nonspecular reflection, 361 principles, 363-364 pulse length, 350-351, 362 INDEX 565 reflectivity, 353-355, 361, 386 San Diego, 379 scattering, 481-482 spherical, 348-350, 434 summary, 434-435 surface vessels, 422-424 transmission loss, 345-347, 369-374 uses, 343-344 wakes, 512-513 wavelength effects, 386-387 Targets, echo-ranging, 84 Taylor Model Basin, sonic transmission experiments, 188, 456 Temperature gradients in the ocean introduction, 3 microstructure, 90-92, 482-483 ray diagrams, 89-90 refraction, 312-313 60 ke transmission, 135 surface effects, 239, 296-297 velocity, 15-17 wake structure, 441, 479-480 Temperature gradients (negative) in the ocean attenuation coefficient, 124-125 ray theory, 61-62 shadow zones, 120-122, 125-129 sharp gradients, 120-129 60 ke transmission, 135 sound channels, 131-133 transmission anomalies, 122-123 weak gradients, 129-133 Temperature-depth patterns, ray dia- grams, 60-63 Thermal microstructure for sound in- tensity fluctuations, 169-171 Thermal wakes, sound transmission, 441, 479-480, 496 Thermocline, sound transmission see also Isothermal water, sound transmission below isothermal layer, 238 ray theory, 61 submarine target strengths, 411-413 temperature gradients, 89 Thermocouple recorder for sound ve- locity measurements, 17 Thermodynamic law for absorption of sound, 464 Thickness of acoustic wakes, 498-500 Thickness of shock wave fronts, 177, 182-184 dissipation of energy, 183-184 Hugoniot equation, 184 Riemann overtaking effect, 183 summary, 177 “Time of arrival’? of bottom-reflected sound pulses, 221—222 “Time of rise’ data for short-range sound propagation, 193 118-120, Time patterns of sound intensity fluctuations, 164-167 “hidden periodicities”, 166-167 self-correlation coefficient, 164-166 Training errors in echo-ranging on wakes, 491-492 Transducers for acoustic measurements calibration, 78 directivity, 522 EBI-1; 276 JK, 276 QB-crystal, 275-276 QCH-3; 273-275, 290-292 reverberation intensities, 272-273, 277 target strengths, 429-430 wakes, 492 Transmission anomaly in underwater sound see also Attenuation of sound average, 122-123, 131-133 bottom scattering, 319-321 definition, 70-71, 237 image effect, 95-97 isothermal water, 100—104 ray theory, 53-54, 58-59, 67-68 supersonic, 147-150 target strength, 369-371 temperature gradients, 118-120 Transmission loss measurements attenuation, 373, 503-504 background, 3-4, 69-71 bubble theory, 469-470 echo runs, 84-85 equipment, 76-78 inadequacy, 372 methods, 78-80 observed echo ranges, 85 oceanographic factors, 492-493 one-way horizontal transmission, 504— 506 propagation along wakes, 509-510 propeller wakes, 510-511 receivers, 73-76 sources, 72-74 summary, 71-72 supersonic frequencies, 80-84 surface reflections, 373-374 target strengths, 369-372, 411-413, 430-431 two-way horizontal transmission, 506-507 two-way vertical transmission, 507— 509 variation, 107-108 wakes, 345-347, 504 Transmission of sound, deep-water absorption, 97-104 attenuation coefficient, 103-107 bathythermograms, 92-95 characteristics, 86-89 echo-ranging trials, 109-110 image effect, 95-97 introduction, 86 isothermal water, 95, 238-239 layer effect, 112-117 long-range experiments, 216-219 negative temperature gradients, 118- 120 scattered sound in shadow zone, 125- 129 sharp temperature gradients, 120- 125, 239 short-range, 108-109 60 ke transmission, 135 sound channels, 133-135, 239-240 thermocline, 110-111, 238 transmission loss, 107-108 variability of vertical temperature gradients, 90-92 vertical temperature structure, 89-90 weak temperature gradients, 129— 133, 239 Transmission of sound, shallow-water dispersion phenomena, 228-229 experiments, 229-235 reflection coefficient, 140-141 sea bottoms, 137-140 sonic, 154-157 summary, 240-241 supersonic, 139-140 24 ke transmission, 141-143 velocity gradients, 142-150 wind force, 152-154 Transmitted wave, 30 Triangulation in long-range sound transmission, 219 Triplane in echo ranging, 84 Turbulence parameter for acoustic wakes, 452-455 Two-way horizontal transmission loss in acoustic wakes, 506-507 Underwater sound transmission, 236— 244 see also Fluctuations in sound trans- mission; Transmission of sound, deep-water; Transmission of sound, shallow-water recommendations for future research, 241-244 summary of definitions, 236-238 Variability of echo intensity, 374-378 “Variance of amplitudes” for sound in- tensity measurements, 237 Variations in sound transmission short-range propagation, 211 summary, 241 transmission loss, 71, 107—108 Velocity of sound in water bubble theory, 473-474 microstructure, 482 566 pressure effects, 19-20 ray equations, 46-49 refraction effects, 197-200 shallow-water transmission, 138 summary, 242 supersonic transmission, 142-150 target depth correction, 49-51 wake theory, 478-479, 480-483 wave equations, 15-17 Velocity-depth variation in ray acous- ties, 54-59 beams in linear gradients, 54-58 layer effect, 56-57 transmission anomalies, 58-59 Vertical temperature gradients in the ocean see Temperature gradients in the ocean Vertical transmission of underwater sound, 79, 507-509 Viscosity (fluid) effects on sound in- tensity, 27-28 Volume reverberation of sound average intensity, 255-256 definition, 253-254 depth, 282-284 frequency, 284-288 index, 259 intensity, 256-258 level, 258, 335-337 range, 281 scattering coefficient, 243, 286, 336— 337 Wake geometry in sound transmission aerial photographs, 494-495 submarines, 501-502 INDEX summary, 541-542 surface vessels, 497-501 target strength, 513-514 widening measurements, 495-497 Wake-laying vessel, acoustic measure- ments, 491 Wakes, acoustic absorption, 541-543 decay rate, 520-521, 539-540 definition, 441 echo ranging, 484, 543-546 evaluation of research, 443-448 fathometer records, 486-488, 497-501 frequency, 523 geometry, 494-495, 513-514, 541 index, 513, 519-520, 543 listening gear, 484 long pulses, 515-516 measurements, 490-492 oceanographic effects, 492-493 oscillograms, 488-490 physical properties, 514-515 propellers, 530-532 pulse length, 522-523 scattered sound, 480-483 short pulses, 516-519 sound range recorder, 484 submarine, 501-502, 523-526, 538- 539 surface vessel, 526-530, 537-538 target strength, 512-513 temperature structure, 479-480 theory, 541-546 thickness, 498-501 training errors, 491-492 transducer directivity, 522 velocity structure, 478-479 widening rate, 495-497 Water wave, sound transmission, 233- 235 Wave acoustics boundary conditions, 138-14, 28-34 equation, 10-18, 43-45, 242 equation of continuity, 8-10 equations of motions, 10 fluid viscosity effects, 27-28 general waves, 36-37 harmonic waves, 17-18, 22-23 intensity, 20-22, 37-38 mathematics, 39-40 normal mode theory, 34-38 plane waves, 14-17, 34-36 pressure versus velocity, 19-20 reciprocity principle, 38-39 sources, 23-27 spherical waves, 18-19 Wave equation for shadow boundary, ray acoustics, 66-67 Wave fronts, ray acoustics eikonal equation versus general equa- tion, 64-65 equations, differential, 43-46 general waves, 42-43 spherical waves, 41-42 Wave length effects on target strength measurements, 386-387 Wavelets for reflected sound pressure, 356 Wide-band pings, frequency analysis of reverberation, 332-333 Widening rate of acoustic wakes, 495- 497 Wind effects in sound transmission force, 337 speed, 295-299 supersonic, 150 *% U.S. GOVERNMENT PRINTING OFFICE : 1969 O—357-418 j 7 4 vt ‘ t She pf f VW DATE DUE SRE Saeeies Sen