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,-dvmtJ. n oJ^- A* INTERNATIONAL SERIES IN PURE AND APPLIED PHYSICS Cat. P. HARNWELL, Consulting Editor Advisory Editorial Committee: E. U. Condon, Goorge R. Harrison, Elmer HutcMasoo, K. EL Darrow VIBRATION AND SOUND The quality of Uie materials used in ike manufacture of this book is governed hy continued postwar shortage*. INTERNATIONAL SERIES IN PURE AND APPLIED PHYSICS G. P. HarkwbI/L, Consulting Editor Rachhb and QoTOtMn— ATOMIC ENERGY STATES Bktbb— INTRODUCTION TO FERROMAGNETISM Brillouin— WAVE PROPAGATION IN PERIODIC STRUCTURES Cauy— piezoelectricity Cla-rk— APPLIED X-RAYS Curtis ELECTEICAX MEASUREMENTS Davey-CRYSTAL STRUCTURE AND ITS APPLICATIONS Edwards— ANALYTIC AND VECTOR MECHANICS Haujjv and PaaWH— TH3B PRINCIPLES OF OPTICS IIauxwell-ELECTRICITY AND ELECTROMAGNETISM HABNWBU. and Lmxcoon— EXPERIMENTAL ATOMIC PHYSICS Houston— PRINCIPLES OF MATHEMATICAL PHYSICS Hughes and DuBridgk— PHOTOELECTRIC PHENOMENA HTOO>-- HIGH-FREQUENCY MEASUREMENTS PHENOMENA IN HIGH-FREQUENCY SYSTEMS Keuble-PRIXCIPLES OF QUANTUM MECHANICS Keknabd— KINETIC THEORY OF GASES Kollbb— THE PHYSICS OF ELECTRON TUBES Morse— VIBRATION AND SOUND Paulino and Goudsiot- THE STRUCTURE OF LINE SPECTRA Richtmyerand Kshhab©— INTBODtTCTlONTO MODERN PHYSICS Ruark and Urey— ATOMS, MOLECULES AND QUANTA Seitz— THE MODERN THEORY OF SOLIDS Slatuib— INTRODUCTION TO CHEMICAL PHYSICS MICROWAVE TRANSMISSION Slater and Frank— ELECTROMAGNETISM INTRODUCTION TO THEORETICAL PHYSICS MECHANICS Smythe-STATIC AND DYNAMIC ELECTRICITY Stratton— ELECTROMAGNETIC THEORY White— INTRODUCTION TO ATOMIC SPECTRA Williams— MAGNETIC PHENOMENA Dr. Lrso A. Du Bridge was consulting editor of the series from H139 to 1046. VIBRATION AND SOUND By PHILIP M, MORSE Director, Broukhaven National Laboratory SECOND EDITION NEW YORK TORONTO LONDON McGRAW-HILL BOOK COMPANY, INC, 1948 VIBRATION AND SOUND Copyright, 1936, 1048, by the McCiuAW-HiLL Book Company, Inc. PRINTED IN THE UNITED STATES OF AMERICA All riykts reserved. This book, or parts thereof, may not be reproduced in any form without per mission of (he publishers, THE MAPLE PRESS COMPANY, YORK, PA. TO PROFESSOR DAYTON C. MILLER PREFACE TO THE SECOND EDITION The recent war induced a considerable development in the science of acoustics and in the mathematical techniques that are particularly useful in theoretical acoustics. Studies in ultrasonics have quickened interest in problems of radiation and scattering and in transient phenomena; and the rapid development of microwave techniques, which parallel acoustic techniques, has stimulated interest in the general theory of wave motion. Reflecting these developments, the present edition of this volume includes more detail than the first edition on radiation problems and introduces the important subject of transient phenomena and the technique of the operational calculus. Both of these subjects arc usually shunned because of their reputed difficulty. The writer is convinced that they are not particularly difficult conceptually, although they usually necessitate calculations of rather wearisome extent. They are here discussed with the intent to bring out funda- mental ideas, rather than to ensure mathematical rigor of treatment. In keeping with the plan of the first edition, the more difficult sub- jects are segregated at the terminations of each chapter, so that they need not be assigned if the book is to be used for a beginning course. The writer is pleased to acknowledge his indebtedness to a large number of friends for valuable suggestions as to improvements in exposition, grammar, arithmetic, and aJgebra. Some of the most obvious errors in the first edition have been corrected. Especial thanks are owing J. R. Pel lam, who has expended much time and effort in checking the manuscript and mathematics, and to Dr. Cyril Harris for his many helpful suggestions concerning subject matter. Philip M. Mohsk Upton, n. Y. January, 1948 rii PREFACE TO THE FIRST EDITION The following book on the theory of vibrations and sound is intended primarily as a textbook for students of physics and of com- munications engineering. After teaching the introductory course in this subject at the Massachusetts Institute of Technology for several years, the author has become persuaded that there is need for a new textbook in the field. There are, of course, many other boobs on the theory of sound. The author's excuse for adding another to the list is that in the past ten years the rapid growth of atomic physics has induced a complete reorganization of the science of acoustics. The vacuum tube and the other applications of electronics have provided immensely powerful tools for the measurement, recording, and reproduction of sound; tools which have revolutionized acoustic technique. Another useful tool, perhaps not so obvious, is the new mathematical technique which has been developed for the working out of quantum mechanics, and which is capable of throwing light on all problems of wave theory. The last chapter of this book is an example of the utility of these methods. In it the mathematical methods developed for the study of the radiation of light from an atom are applied to the theory of the acoustic properties of rooms. During the recent rapid change in the science of sound, certain parts of the subject have gained and other parts have lost importance. The present book attempts to follow this change in emphasis and to discuss the new development as well as those portions of the older theory which are still import-ant. The book has been planned as a textbook with a twofold aim in view. The first aim, of course, is to give the student a general introduction to the theory of vibration and sound. An introduc- tory course in this subject must of necessity be more theoretical than practical. In no other branch of physics are the fundamental measure- ments so hard to perform, and the theory relatively so simple; and in few other branches are the .experimental methods so dependent on a thorough knowledge of theory. Since this is so, the student must first be given a physical picture of the fundamental theory of the ix x PREFACE TO THE FIRST EDITION vibration of solid bodies and the propagation of sound waves before he can appreciate the techniques used in the measurements of sound, and before he can begin to design acoustical apparatus. The second aim is to give the student a series of examples of the method of theoretical physics; the way a theoretical physicist attacks a problem and how he finds its solution. This subject is too often neglected, especially in engineering courses. The student is usually given a series of formulas to use in standardized cases, the formulas sometimes introduced by a cursory derivation and sometimes with no derivation at all. After such a course the student is capable of using the formulas on standard problems, but he is unable to devise a new formula to use in unusual cases. In this book the author has tried to derive every formula from the fundamental laws of physics {there arc a few exceptions to this procedure) and to show in some detail tin: steps in these derivations and their logical necessity. This docs not mean I. hut the mathematical machinery is given in excessive detail, but that the steps in the physical reasoning are brought out. Often generality and mathematical rigor have been sacrificed to make the chain of logic more distinct. It has been the author's experience that once the student can grasp the physical picture behind a mathematical derivation, he can himself add what extra generality and rigor he may need. Often, too, the author has supplemented or replaced the rigid and esoteric technical vocab- ulary by more colloquial phrases, in order to make vivid a concept, or to suggest a new point of view. It is assumed that the student has a thorough knowledge of calculus, and some acquaintance with the fundamental laws of mechanics. A knowledge 6f differential equations is helpful but is not necessary, for the solutions of the various differential equations encountered are worked out in the text. Tables of the functions used are given in the back of the book. Although the book is designed primarily as a textbook, a certain amount of material of an advanced nature has been introduced. In this way, it is hoped, the volume will be useful as a fairly complete reference work for those parts of the theory of sound which seem at. present to be most important for the acoustical scientist. The advanced material has been included in the form of extra sections placed at the end of various chapters. The instructor may assign the first few sections of these chapters for the introductory course, and the student may refer to the other sections for further details when he needs them. PREFACE TO THE FIRST EDITION » The author wishes to express his gratitude to Professor R. D. Fay and to Dr. W. M. Hall, whose help in choosing subject matter and methods of presentation has been invaluable. Ho is also indebted to Dr. J. B. Fisk for his willing and painstaking aid in correcting proof, and to many other colleagues in the Department of Physics at the Massachusetts Institute of Technology, for their many helpful suggestions. Philip M. Morse Cambridge, Mass. August, 1936 CONTENTS Preface to the Second Edition vii Preface to the First Edition. . -, ix Chapter I INTRODUCTORY 1. Definitions and Methods. 1 Units. Energy 2. A Little Mathematics 3 Tho Trigonometric Functions. Beasel Functions. The Exponential, Conventions as to Sign. Other Solutions. Contour Integrals. Infinite Integrals. Fourier Trans- forms Problems . • 1? Chapter II THE SIMPLE OSCILLATOR 3. Free Oscillations 20 The General Solution. Initial Conditions. Energy of Vibration 4. Damped Oscillations 23 The General Solution. Energy Relations 5. Forced Oscillations 27 The General Solution. Transient and Steady State. Impedance and Phase Angle. Energy Relations. Electro- mechanical Driving Force. Motional Impedance. Piezo- electric Crystals xiii xiv CONTENTS Section 6. Response to Transient Forces 42 Representation by Contour Integrals, Transients in a Simple System. Complex Frequencies. Calculating the Transients. Examples of the Method, The Unit Func- tion. General Transient, Some Generalizations. La- place Transforms 7. Coupled Oscillations . 52 The General Equation. Simple Harmonic Motion. Nor- mal Modes of Vibration, Energy Relations. The Case of Small Coupling. The Case of Resonance. Transfer of Energy. Forced Vibrations. Resonance and Normal Modes. Transient Response Problems . . 66 CfiAl'TEK III THE FLEXIBLE STRING 8. Wares on a Siring 71 The Wave Velocity, The General Solution for Wave Motion. Initial Conditions. Boundary Conditions. Re- flection at a Boundary. Strings of Finite Length 9. Simple Harmonic Oscillations 80 The Wave Equation. Standing Waves. Normal Modes. Fourier Series. Initial Conditions. The Scries Coeffi- cients, Plucked String, Struck String. Energy of Vibration 10. Forced Vibrations. . . , 91 Wave Impedance and Admittance. General Driving Force. String of Finite Length. Driving Force Applied Anywhere. Alternative Scries Form. Distributed Driv- ing Force, Transient Driving Force. The Piano String. The Effect of Friction. Characteristic Impedances and Admittances 11. Strings of Variable Density and Tension , . . , 107 General Equation of Motion. Orthogonality of Character- istic Functions. Driven Motion. Nonuniform Mass. The Sequence of Characteristic Functions. The Allowed CONTENTS XV Section Frequencies. Vibrations of a Whirling String, The Al- lowed Frequencies. The Shape of the String. Driven Motion of the Whirling String 12. Perturbation Calculations. 122 The Equation of Motion. First-order Corrections. Ex- amples of the Method. Characteristic Impedances. Forced Oscillation. Transient Motion 13. Effect of Motion of the End Supports. . . 133 Impedance of the Support. Reflection of Waves. Hyper- bolic Functions. String Driven from One End. Shape of the String. Standing Wave and Position of Minima. Characteristic Functions. Transient Response. Reca- pitulation Problems M7 Chafer TV THE V JURATION OF BARS 14. The Equation of Motion 151 Stresses in a Bar. Bending Moments and Shearing Forces. Properties of the Motion of I he Bar. Wave Motion in an Infinite Bai- ls. Simple Harmonic Motion 156 Bar Clamped at One End. The Allowed Frequencies. The Characteristic Functions. Plucked and Struck Bar. Clamped-clamped and Free-free Bars. Energy of Vibra- tion. Nonuniform Bar. Forced Motion 16. Vibrations of a Stiff String 1G6 Wave Motion on a Wire. The Boundary Conditions. The Allowed Frequencies Problems 170 ClI.Vl'TKlt V MEMBRANES AND PLATES 17. The Equation of Motion ', 172 Forces on a Membraue. The Laplacian Operator. Bound- xvi CONTENTS Section ary Conditions and Coordinate Systems. Reaction to a Concentrated Applied Force IS. Thz Rectangular Membrane. 177 Combinations of Parallel Waves. Separating the Wave Equation. The Normal Modes. The Allowed Fre- quencies. The Degenerate Case. The Characteristic Functions 19, The Circular Membrane . , 183 Wave Motion on an Infinite Membrane. Impcrmanence of the Waves, Simple Harmonic Waves. Bessel Func- tions. The Allowed Frequencies. The Characteristic Functions. Relation between Parallel and Circular Waves. The Kettledrum. The Allowed Frequencies 2U. Forced Motion, The Condenser Microphone . 195 Neumann Functions. Unloaded Membrane, Any Force. Localized Loading, Any Force. Uniform Loading, Uni- form Force. The Condenser Microphone. Electrical Con- nections. Transient Response of Microphone 21. The Vibration of Plates 208 The Equation of Motion. Simple Harmonic Vibrations. The Normal Modes. Forced Motion Problems , 213 Chapter VI PLANE WAVES OF SOUND 22. The Equation of Motion. 217 Waves along a Tube. The Equation of Continuity. Compressibility of the Gas, The Wave Equation. Energy in a Plane Wave. Intensity. The Decibel Scale. Inten- sity and Pressure Level. Sound Power. Frequency Dis- tribution of Sounds. The Vowel Sounds 23. The Propagation of Sounds in Tubes . ........... 233 Analogous Circuit Elements. Constriction. Tank. Ex- amples. Characteristic Acoustic Resistance. Incident and Reflected Waves, Specific Acoustic Impedance. CONTENTS xvii Snrnont Standing Waves. Measurement of Acoustic Impedance, Damped Waves. Closed Tube. Open Tube. Small- diameter Open Tube. Reed Instruments. Motion of the Reed, Pressure and Velocity at the Reed. Even Har- monics. Other Wind Instruments. Tube as an Analo- gous Transmission Line. Open Tube, Any Diameter. Cavity Resonance, Transient Effects, Flutter Echo 24. Propagation of Sound in Horns , . 265 One-parameter Waves. An Approximate Wave Equation. Possible Horn Shapes. The Conical Horn. Transmission Coefficient- A Horn Loud-speaker. The Exponential Horn. The Catenoidal Horn, Reflection from the Open End, Resonance. Wood-wind Instruments. Transient Effects Problems 288 CnAI'TKlt VII TBS RADIATION AND SCATTERING OF SOUND 25. The Wave Equation, . 294 The Equation for the Pressure Wave. Curvilinear Coordi- nates 26. Radiation from Cy folders , 297 The General Solution. Uniform Radiation. Radiation from a Vibrating Wire. Radiation from an Element of a Cylinder. Long- and Short-wave Limits. Radiation from a Cylindrical Source of General Type, Transmission inside Cylinders. Wave Velocities and Characteristic Imped- ances. Generation of Wave by Piston 27. Radiation from Spheres 311 Uniform Radiation. The Simple Source. Spherical Waves of General Form. Legendre Functions. Bessel Functions for Spherical Coordinates, The Dipole Source. Radiation from a General Spherical Source. Radiation from a Point Source on a Sphere. Radiation from a Piston Set in a Sphere xviii CONTENTS Section 28. Radiation from a Piston in a Plane Wall 326 Calculation of the Pressure Wave. Distribution of In- tensity. Effect of Piston Flexure on Directionality, Radi- ation Impedance, Rigid Piston. Distribution of Pressure over the Piston, Nonuniform Motion of the Piston. Radiation out of a Circular Tube. Transmission Coeffi- cient ■ for a Dynamic Speaker. Design Problems for Dynamic Speakers. Behavior of the Loud-speaker, Transient Radiation from a Piston 29. The Scattering of Sound 346 Scattering from a Cylinder. Short Wavelength Limit. Total Scattered Power. The Force on the Cylinder. Scattering from a Sphere. The Force on the Sphere. Design of a Condenser Microphone. Behavior of the Microphone 30. The Absorption of Sound at a Surface 360 Surface Impedance, Unsupported Panel. Supported Panel. Porous Material. Equivalent Circuits for Thin Structures. Formulas for Thick Panels. Reflection of Plane Wave from Absorbing Wall 31. Sound Transmission through Ducts 308 Boundary Conditions. Approximate Solutions. Principal Wave, Transient Waves. The Exact Solution. An Ex- ample Problems , , . . 37(3 Chapter VIII STANDING WAVES OF SOUND 32. Normal Modes of Vibration 381 Room Resonance, Statistical Analysis for High Fre- quencies. Limiting Case of Uniform Distribution. Ab- sorption Coerneinnt. Reverberation. Reverberation Time. Absorption Coefficient and Acoustic Impedance. Standing Waves in a Rectangular Room. Distribution in Frequency of the Normal Modes. Axial, Tangential, and Oblique Waves. Average Formulas for Numbers of Allowed Fre- quencies. Average Number of Frequencies in Band. CONTENTS xix SECTION The Effect of Room Symmetry. Nonrect angular Rooms. Frequency Distribution for Cylindrical Room 33. Damped Vibrations, Reverberation 401 Rcct tmgular Room, Approximate Solution. Wall Coeffi- cients and Wall Absorption. Reverberation Times for Oblique, Tangential, and Axial Waves. Decay Curve for Rectangular Room. Cylindrical Room. Second -order Approximation. Scattering Effect of Absorbing Patches 34. Forced Vibrations. 413 Simple Analysis for High Frequencies. Intensity and Mean-square Pressure. Solution in Series of Characteristic Functions. Steady-state Response of a Room, Rec- tangular Room. Transmission Response. The Limiting Case of High Frequencies. Approximate Formula for Response. Exact Solution. The Wall Coefficients, Transient Calculations, Impulse Excitation. Exact Solu- tion for Reverberation Problems 429 bibliography 433 Glossary of Symbols .,,,,.,.,,......... 435 Tables of Functions 438 I and II, Trigonometric and Hyperbolic Functions. Ill and IV, Hyperbolic Tangent of Complex Quantity. V, VI and YU, Bessel Functions. VTII, Impedance Functions for Piston. IX, Legendre Functions. XII, General Im- pedance Functions for Piston. XIII, Absorption Coeffi- cients Plates 453 I and II, Hyperbolic Tangent Transformation.. Ill, Magnitude and Phase Angles of sinh and cosh. IV, Standing Wave Ratio and Phase vs. Acoustic Impedance. V, Hxact Solutions for Wave Modes in Rectangular Ducts and Rooms. VI, Absorption Coefficient vs, Acoustic Impedance Index 459 VIBRATION AND SOUND CHAPTER I INTRODUCTORY 1. DEFINITIONS AND METHODS The discussion of any problem in science or engineering has two aspects: the physical side, the statement of the facts of the case in everyday language and of the results in a manner that can be checked by experiment; and the mathematical side, the working out of the intermediate steps by means of the symbolized logic of calculus. These two aspects are equally important and are used side by side in every problem, one checking the other. The solution of the problems that we shall meet in this book will, in general, involve three steps: the posing of the problem, the inter- mediate symbolic calculations, and the statement of the answer. The stating of the problem to be solved is not always the easiest part of an investigation. One must decide which properties of the system to be studied are important and which can be neglected, what facts must be given in a quantitative manner and what others need only a qualitative statement. When all these decisions are made for problems of the sort discussed in this book, we can write down a statement some- what as follows: Such and such a system of bodies is acted on by such and such a set of forces. We next translate this statement in words into a set of equations and solve the equations (if we can). The mathematical solution must then be translated back into the physical statement of the answer: If we do such and such to the system in question, it will behave in such and such a manner. It is important to realize that the mathematical solution of a set of equa- tions is not the answer to a physical problem; we must translate the solution into physical statements before the problem is finished. Units.— The physical concept that force causes a change in the motion of a body has its mathematical counterpart in the equation F = Jt (mv) (1.1) 2 INTRODUCTORY [LI In order to link the two aspects of this fact, we must define the physical quantities concerned in a quantitative manner; we must tell how each physical quantity is to be measured and what standard units of measure are to be used. The fundamental quantities, distance, mass, and time, can be measured in any arbitrary units, but for convenience we use those arbitrary units which most of the scientific world is using: the centimeter, the gram, and the second. (The units of the few other quantities needed, electrical, thermal, etc., will be given when we encounter them.) The units of measure of the other mechanical quantities are denned in terms of these fundamental ones. The equation F = d(mv)/dt is not only the mathematical statement of a physical law, it is also the definition of the unit of measure of a force. It states that the amount of force, measured in dynes, equals the rate of change of momentum in gram centimeters per second per second. If force were measured in other units than dynes, this equation would not be true; an extra numerical factor would have to be placed on one side or the other of the equality sign. Energy. — Another physical concept which we shall often use is that of work, or energy. The wound-up clock spring can exert a force on a gear train for an indefinite length of time if the gears do not move. It is only by motion that the energy inherent in the spring can be expended. The work done by a force on a body equals the distance through which the body is moved by the force times the component of the force in the direction of the motion; and if the force is in dynes and the distance is in centimeters, then the work is given in ergs. The mathematical statement of this is W = fP-ds (1.2) where both the force F and the element of distance traveled ds are vectors and their scalar product is integrated. If the force is used to increase the velocity of the body on which it acts, then the work that it does is stored up in energy of motion of the body and can be given up later when the body slows down again. This energy of motion is called kinetic energy, and when measured in ergs it is equal to mv 2 /2. If the force is used to overcome the forces inherent in the system — the "springiness" of a spring, the weight of a body, the pull between two unlike charges, etc. — then the work can be done without increasing the body's velocity. If we call the inherent force F (a vector), then 1.2] A LITTLE MATHEMATICS 3 the work done opposing it is W = — JF • ds. In general, this amount depends not only on the initial and final position of the body but also on the particular way in which the body travels between these posi- tions. For instance, owing to friction, it takes more work to run an elevator from the basement to the third floor, back to the first, and then to the fourth than it does to run it directly to the fourth floor. In certain ideal cases where we can neglect friction, the integral W depends only on the position of the end points of the motion, and in this case it is called the potential energy V of the body at its final position with respect to its starting point. If the body moves only in one dimension, its position being given by the coordinate x, then F--j>* or ,--g) (1.3) In such a case we can utilize Newton's equation F = m(dv/dt) to obtain a relation between the body's position and its velocity. V = — I Fdx = — ra I (-rrjdx = — ra I vdv = — %mv 2 + constant or imv 2 + V = constant (1.4) This is a mathematical statement of the physical fact that when friction can be neglected, the sum of the kinetic and potential energies of an isolated system is a constant. This proof can be generalized for motion in three dimensions. 2. A LITTLE MATHEMATICS With these definitions given we can go back to our discussion of the twofold aspect of a physical problem. Let us take an example. Suppose we have a mass m on the end of a spring. We find that to keep this mass displaced a distance x from its equilibrium position requires a force of +Kx dynes. The farther we push it away from its equilibrium position, the harder the spring tries to bring it back. These are the physical facts about the system. The problem is to discuss its motion. We see physically that the motion must be of an oscillatory nature, but to obtain any quantitative predictions about the motion we must have recourse to the second aspect, the mathematical method. We set up our equation of motion F — m(d 2 x/dt 2 ), using our physical knowledge of the force of the spring F = — Kx. 4 INTRODUCTORY [1.2 ^ + n 2 x = (2.1) where n 2 stands for (K/m). The physical statement corresponding to this equation is that the body's acceleration is always opposite in sign and proportional to its displacement, i.e., is always toward the equilibrium point x = 0. As soon as the body goes past this point in one of its swings, it begins to slow down; eventually it stops and then returns to the origin again. It cannot stop at the origin, however, for it cannot begin to slow down until it gets past the origin on each swing. The Trigonometric Functions. — Equation (2.1) is a differential equation. Its solution is well-known, but since we shall meet more difficult ones later, it is well to examine our method of obtaining the solution, to see what we should do in more complicated cases. We usually just state that the solution to (2.1) is x = a cos (nt) + fli sin (nt) (2.2) but if we had no table of cos or sin, this statement would be of very little help. In fact, the statement that (2.2) is a solution of (2.1) is really only a definition of the symbols cos and sin. We must have more than just symbols in order to compute x for any value of t. What we do — what is done in solving every differential equation — is to guess the answer and then see if it checks. In most cases the guess was made long ago, and the solution is familiar to us, but the guess had nevertheless to be made. So we shall guess the solution to (2.1). We shall make a pretty general sort of guess, that re is a power series in t, and then see if some choice of coefficients of the series will satisfy the equation: x = a + ait + a 2 t 2 + a s t s + a^ A + • • • For this to be a solution of (2.1), we must have d 2 x/dt 2 , which equals 2a 2 + Qa 3 t + I2a4 2 + 20a 5 t 3 + • ■ • plus n 2 x, which equals n 2 a + n 2 a x t + n 2 arf, 2 + n 2 a z t z + • • • equal zero for every value of t. The only way that this can be true is to have the sums of the coefficients of each power of t each zero by themselves; i.e., a 2 = —n 2 a /2, a 3 = — n 2 ai/6, a 4 = — ra 2 a 2 /12 = n 4 a /24:, etc. Therefore the series that satisfies (2.1) is ( i _ VlIl a- Vl!l _ Vl!l 4- ^ V 2 + 24 720 + ' ' ' / V 6 + 120 ' ' / X = Go + aAt- 1.2] A LITTLE MATHEMATICS 5 By comparing this with (2.2) we see that what we actually mean by the symbols cos, sin is »2 «4 «*> cos(*) = l-- 2 - + 2l-^+ t sm(z) = 2 _._ + _ _ ^q + (in the problem 2 = nt) and that when we wish to compute values of cos or sin we use the series expansion to obtain them. For instance, cos (0) = 1 - + • • • = 1 cos(i) = 1 - 0.125 + 0.003 • • • = 0.878 cos (1) = 1 - 0.5 + 0.042 - 0.002 • • • = 0.540, etc. We see that the mathematical solution involves certain arbitrary constants «o and a\. These must be fixed by the physical conditions of the particular experiment we have made and will be discussed later (Sec. 3). Of course, we usually mean by the symbols cos (z), sin (z) certain ratios between the sides of a right triangle whose oblique angle is z. To make our discussion complete, we must show that the trigonometric definitions correspond to the series given above. In any book on ele- mentary calculus it is shown that the trigonometric functions obey the following relations: ^ cos (z) = -sin (z), -£ sin (z) = cos (z) cos(0) = 1, sin(0) = By combining the first two of these relations, we see that both the sine and cosine of trigonometry obey the equation d 2 y/dz 2 = — y, which is equivalent to (2.1). Taking into account the third and fourth relations, we see that the series solutions of this equation which correspond to the two trigonometric functions must be the ones given in (2.3). Once this is known, we can utilize the trigonometric properties of the sine and cosine to simplify our discussion of the solution (2.2) [although, when it comes to computing values, the series (2.3) are always used]. For instance, we can say that a; is a periodic function of time, repeating its motion every time that nt increases by 2ir or every time t increases by (2w/ri). The value of {2k /n) is called the period of oscillation of the mass and is denoted by the symbol T. The number of periods per second {n/2ir) is called the frequency of 6 INTRODUCTORY [1.2 oscillation and is denoted by v. Remembering the definition for n, we see that t =**4> -(1)4 Bessel Functions. — More complicated differential equations can be solved by means of the same sort of guess that we used above. For instance, we can solve the equation s+iS+0-?)'- ™ by again guessing that y = a + a x x + a 2 x 2 + a 3 x z + • • • and set- ting this guess in the equation. Thus: a + dix + a 2 x 2 + a 3 x s + • • • ai — a 3 x — aix 2 — a&x s — • • • x — + 2a 2 + 3a 3 x + 4a 4 x 2 + ba h x z + • • • x 2a 2 + Qa 3 x + 12a 4 ;c 2 + 20a 5 £ 3 + ■ • ■ must equal zero. Equating coefficients of powers of x to zero as before, we have a = 0, o 2 = — 7^ = 0, a 3 = — g- rt a 3 ai , a 4 = 0, a 5 = ~ 24 = g . 24 * etc - Therefore, 2/ = 1 minus ^ y = X 2 , 1 rfw plus - ~r = x ax 1 d 2 y P lus ^i = / z 3 ^"2^4 2/ = a x [ x - ^ + 2 . 4 . 4 . 6 - 2-4-6-4-6-8 + We shall call this series a Bessel function, just as we call each of the series in (2.3), a trigonometric function. To save having to write the series out every time, we shall give it a symbol. We let 1/ x s 2\ X 2-4 Ji(x) = 2V X " 2^1 + 2-4-4-6 ' ) (2 ' 5) just as we represent series (2.3) by the symbols cos and sin. Essentially the series (2.5) is no more complicated than series (2.3). We can compute values of J\(x) in the same manner as we computed values of cos (2). 1.2] A LITTLE MATHEMATICS 7 J x (0) = - • • • = JiG) = 0.250 - 0.008 ■ • • = 0.242 Ji(l) = 0.5 - 0.063 + 0.003 • • • = 0.440, etc. Once someone has obtained series (2.5), given us the symbol Ji(x) and computed its values, we can use the symbol with as great freedom as we use the symbol cos (z). We can say that the solution of (2.4) is y = AJi(x), where the arbitrary constant A is to be determined by the physical conditions of the problem, as we shall discuss later. We notice that the solution of (2.4) involves only one function and one constant A, whereas solution (2.2) involved two constants and two different functions cos (nt) and sin (nt). Actually, there is another solution of (2.4), designated by the symbol Ni(x) and called a Neumann function; but Ni(x) becomes infinite at x = 0, and so we shall not be able to use it for many of the problems discussed in this book. We can say that the complete solution of (2.4) is y = AJi(x) + BNx(x) but that, since a solution representing physical facts must not become infinite, the constant B must be made zero in all cases where we let x become zero. These functions will be dealt with more fully in Chap. V. The Exponential. — Another very useful way of dealing with the solution of (2.1) can be obtained by the following chain of reasoning. We utilize the series method to show that a solution of the equation (dy/dz) = y is the series called the exponential: z 2 z 3 z 4 z 5 «,= !+* + _ + _ + _ + __+... (2.6) By repeated differentiation we see that x = Ce at is a solution of the equation (d 2 x/dt 2 ) — a 2 x = 0. If a 2 were to equal — n 2 , this equation would be the same as (2.1). Therefore we can say that a solution of (2.1) is Ce~ int , where i = \/—l (we could also use e +int ). The function e~ iz is a complex number, with real and imaginary parts, and can be represented in the usual manner by a point on the "complex plane" whose abscissa is the real part of the function and whose ordinate is the imaginary part. It is also represented by the vector drawn from the origin to this point in the complex plane. If we expand e~ iz in its series form '*-( 1 -5 + H----)- < (*-£ + i35---') we see immediately that the relation between the imaginary expo- nential and the trigonometric functions is INTRODUCTORY e -i z _ cog (^ _ £ gm ^ cos (2) = i(e i!S + e~ iz ) sin (2) = — %i (e iz — e~ iz ) [L2 (2.7) This shows that the number e~ iz can be represented on the complex plane (see Fig. 1) by a vector of unit length inclined at an angle — z Imag. Axis C=cio+icv ContouiG. Direction of Rotation Direction of Rotation Fig. 1. — Representation of complex numbers on the complex plane. Integration around a contour G in the complex plane. Relation between i and j. with respect to the real axis. It also shows th at the co mplex number a + ib can be expressed as Ae'*, where A = Va 2 + b 2 and tan (<3?) = (b/a). For Ae™ = A cos($) + iA sin(<i>); and since cos(3>) = (a/A) 1.2] A LITTLE MATHEMATICS 9 and sin ($) = (b/A), we have Ae i# = a + i&. The factor e** signifies a rotation of the vector A through an angle $ in the complex plane. Thus we see that Cer int is a solution of (2.1) and can be expressed as a combination of cos (nt) and sin (nt), as is demanded by (2.2). But this new solution is a complex number, and the results of physics are, in general, real numbers, so the new solution would seem to have little value to us. What can be done, however, is to write down the solution as being Ce~ int and make the convention that we use only the real part of this solution when we use it to check physical measure- ments. All through this book we shall be writing down complex solutions of differential equations, with the convention that we use only their real parts. It is possible to do this with the solution of any linear differential equation (i.e., equations containing only the first power of the unknown function and its derivatives) ; for if a com- plex function is a solution of a linear differential equation, then both its real and imaginary parts by themselves are also solutions. (Why is this not true for nonlinear differential equations?) We could, of course, make a convention that we use only the imaginary part of the function, for the imaginary part is also a solution of the equation, but the usual convention is to 'take the real part. We can therefore express solution (2.2) as the real part of x = Ce~ int , C = a + iax (2.8) This can be checked by the use of (2.7), for Ce~ int = a cos (nt) + ia x cos (nt) — ia sin (nt) + ai sin (nt) and the real part of this corresponds to (2.2). The advantages of this method will become apparent as we use it. For instance, since C = Ae { *, A 2 = al + a\, tan(f>) = (ai/o ), we see that the real part of Ce~ int = 4 e -*'c»*-*) is A cos (nt + <£>). Therefore we can express the solution of (2.1) in any of four ways: x = a cos (nt) + ai sin (nt) = A cos (nt — $) (2.9) or x = Ce~ int = Aer-t^-v where our convention requires that we take only the real part of the last expressions. The constant A is called the amplitude of oscillation of the mass, since it is the value of the maximum displacement of the mass from equilibrium. Our conventions as to the use of complex quantities must be used with tact when we come to compute powers and energies, where squares 10 , INTRODUCTORY [1.2 Of quantities enter. For instance, in a given problem the power radi- ated may turn out to be Rv 2 , where v is the velocity of a diaphragm and R is a real number. The velocity may be represented by the expression v= Ce~ 2 * ivt , C = a + ia h just as x is in Eq. (2.9), But to compute the average power radiated, we must take the real part of the expression before squaring and averaging, Rv 2 = RA 2 cos 2 (2irvt - $) according to Eq. (2.9) . Since the average value of cos 2 is £, the average power radiated will be iRA 2 = iR(a 2 + a 2 ) = i#|C| 2 = %R\v\ 2 \v\ 2 = (real part of v) 2 + (imaginary part of v) 2 If z — x + iy is any complex number, the quantity \z\ is called the magnitude of z. It is the distance from the origin to the point in the complex plane corresponding to z, and it equals the square root of the sum of the squares of the real and imaginary parts of z. There- fore another important rule concerning the use of complex quantities in physical problems is the following: the average value of the square of a quantity represented by the complex function Ce~ iat is equal to one-half of the square of the magnitude of C. The angle <i> in Eq. (2.9) is called the phase angle of the complex quantity C, since it measures the angle of lag of the quantity Ce~ int behind the simple exponential e~ int . We notice that the phase angle of i is 90 deg, that of (-1) is 180 deg, and that of (-i) is 270 deg or 90deglead. If a; = Ce' int , the velocity v = (dx/dt) = - inCer*" leads the displacement x by 90 deg, as is indicated by the fact that v = ( — i)nx. Conventions as to Sign. — In the present volume we use the letter i to stand for V^-l and the symbol e~ int to express simple harmonic dependence on time. Many books on electrical engineering use the symbol j instead of i and e jnt with the positive sign in the exponential. Since we intend to take the real part of the result, the choice of symbol and sign is but a convention; either convention is satisfactory if we are only consistent about it. As long as we are studying simple systems, with displacements a function only of time, there is little to choose between the two con- ventions; as a matter of fact the positive exponential e int would be slightly preferable. However, as soon as we come to study wave motion, with displacement a function of position as well as of time, it turns out that the form involving the negative exponential e~ int is 1.2] A LITTLE MATHEMATICS 11 rather more satisfactory than the positive. This will become appar- ent later in this book. Since we are going to deal with problems of wave motion in this book we shall consistently use the negative exponential e~ int . And since this convention differs from the positive exponential used in most electrical engineering books, we shall use the symbol i instead of j, to emphasize the difference. It will, however, always be possible to transform any of the formulas developed in this book into the usual electrical engineering notation by replacing every iin the formulas by —j. We might, if we wish, consider i and j as the two roots of ( — 1), so that i = -j; (iy = (j)2 = _i In this notation, the impedance of a circuit with resistance and induct- ance in series will be R — iuL. Other Solutions. — Power series are not always the best guesses for the solutions of differential equations. Now that we have denned the exponential function, we can sometimes express solutions of other equations in terms of exponentials. Consider the equation -^ + n 2 x = ae~ ipt (2.10) where, to represent a physical problem, we must use our convention on the right-hand side of the equation. This could be solved by expanding the exponential in a series and solving for the series for x as before. But since an exponential is in the equation, and since we know that Ce~ int is the solution of (2.10) when a is zero, it will be simpler to guess that x = Ce~ int + Be~ lpt Setting this in the equation, we have that n 2 x = n 2 Ce~ int + n 2 Be~ ipt plus d 2 x/dt 2 = -n 2 Ce~ int - p 2 Be~ ipt must equal ae~ ipt . This means that B = a/(n 2 — p 2 ). ,If we have used our convention on the term ae -i P t m (2.10), we must use it on the answer. The real part of x = Ce~ int + .i 2 — p 2 is x = a cos (nt) + a x sin (nt) + -^ — ^ cos (pt) (2.11) if a is a real quantity. If a is complex and equals De**, then we can 12 INTRODUCTORY [1.2 write x = A cos (nt - 4> ) + n2 _ 2 cos (p£ - tfO We notice again that in (2.11) our solution has two arbitrary con- stants a and a x . The arbitrariness corresponds to the fact that this solution must represent all the possible sorts of motion which the system can have when it is acted on by the forces implied by Eq. (2.10). A mass on a spring can have different motions depending on how it is started into motion at time t = 0. Therefore the par- ticular values of the arbitrary constants a and a x in (2.2) are deter- mined entirely by the physical statements as to how the system was started into motion. These physical statements are called initial conditions and are usually stated by giving the position and velocity of the system at t = 0. More will be said on this point in the next section. Contour Integrals. 1 — In a number of cases discussed in this volume it will be necessary to use integrals of complex quantities of the general form $F(z) dz, where F = U + iV is a function of the complex varia- ble z = x + iy. Since z can vary over the x,?/-plane, it will in general be necessary to specify the path over which the integration is carried out. A natural extension of our usual definition of integration indi- cates that the integral is the limit of a series of terms, a typical term being the product of the value of F{z) for a value of z on the path, and a quantity dz = \dz\e i& , with length \dz\ and with phase angle & deter- mined by the direction of the tangent to the path at the point z. The integral is the limit of such a sum as [dz\ goes to zero. This is shown in Fig. 1. Another definition is that jF dz = j[U + iV][dx -f- i dy] where x and y are related in such a manner that the integral follows the chosen path. When the path chosen is a closed one, the integral is called a contour integral (labeled f) and the chosen path is called a contour. Contour integrals have certain remarkable properties, proved in standard texts on theory of functions of a complex variable. We have space here to outline only some of these properties. For sim- plicity it will be assumed that the function F{z) to be dealt with is a smoothly varying, reasonable sort of function over most of the z- plane. The value of a contour integral depends largely on whether the integrand F(z) becomes zero or infinity for some value or values of z 1 This section need not be studied unless the later sections on transient phe- nomena are to be studied. 1.2] A LITTLE MATHEMATICS 1,3 inside the contour. Near such points F(z) would take on the form (z — z ) n R(z) where R(z ) is neither zero or infinity and R(z) is not discontinuous near z . When n is not an integer, positive or negative, the point z is called a branch point and the problem becomes more complicated than is necessary to discuss here. When n is a negative integer, the point z is called a pole of F(z), the pole f or n = — 1 being called a simple pole and that for n = —2,-3 • • • being called a poZe of second, third, etc., order. The statement at the beginning of this paragraph can now be made more specific: the value of the contour integral $F(z) dz depends on the behavior of F at the poles and branch points that happen to be inside the contour. When F has no branch points or poles inside the contour, the value of the contour integral is zero. This can be verified by a tedious bit of algebraic manipulation for those cases where F can be expressed as a convergent power series in z. When F has a simple pole at z = z , but has no other pole or branch point inside the contour (see Fig. 1), then the contour integral can be shown to be equal to that around a vanishingly small circle drawn about z . Since z is specified as being a simple pole, F(z) has the form R(z)/(z — z ) near z where R(z) is continuous and finite in value near z . We can write the equation for the resulting circle as z — z = ee iv and dz = ue iv d<p, where e is vanishingly small. The contour integral then reduces to £F(z) dz = R(z ) f ' l * e% \ d<P = iR(z ) I d<p = 2iriR(z ) Jo &* Jd where R(z ) — lim [(z — z )F(z)) is called the residue of F{z) at its z—*zo simple pole zo. When F(z) has N simple poles, at z , z x • • • zn, and no other poles or branch points, then the contour integral j>F{z) dz equals (27ri) times the sum of the residues of F at the poles that are inside the chosen contour. We note that the direction of integration around the con- tour is counterclockwise; changing the direction will change the sign of the result. This is a remarkably simple result; in fact, it seems at first too simple to be true, until we realize that the function F(z) we are dealing with is a very specialized form of function. F is not just any complex function of x and y; its dependence on these variables is severely limited by the requirement that it be a function of z = x + iy. It can have the form bz 2 + (C/z) or z sin(z), but not the form x sin(z) -f iy or \z\. For such a specialized function F(z), which is called an 14 INTRODUCTORY (1.2 analytic function, the positions of its branch points and poles, and its behavior near these branch points and poles, completely determine its behavior everywhere else on the complex plane. When this is under- stood, the result we have quoted for the value of a contour integral does not seem quite so surprising. As an example of the rule of residues, we can take the integral <f sin (2) dz/(z 2 — a 2 ), where the contour is a circle of radius greater than a, with center at the origin 2 = 0. The integrand has two simple poles, at 2 = a and z = —a, with residues (l/2a) sin (a) and — (l/2a) sin ( — a). The result is therefore *^*-OW>[*f] A special case of the residue theorem can be stated in equation form § z dz = 2tiR(z ) (2.12) where the contour of integration does not enclose a branch point or pole of R(z). By differentiating once with respect to 2 on both sides of the equation, we obtain the equation § , R(Z \ 2 dz = 2riR'(z ) (2.13) \Z Zoj where R'(z) = dR(z)/dz. This indicates how we can evaluate con- tour integrals around poles of second order. Cases for poles of higher order may be dealt with by further differentiation of Eq. (2.13). Infinite Integrals. — An oft-encountered class of integral may be evaluated in terms of contour integrals. Consider the integral /-". w dz where the integral is taken along the real axis. If F(z) has no branch points or poles of higher order, if the simple poles of F are not on the real axis, and if F(z) goes to zero as \z\ goes to infinity, then the integral can be made into a contour integral by returning from + 00 to — ».." along a semicircle of infinite radius. When t is positive (to make it more general we should say when the real part of t is positive, but let us assume that t is real), the exponential factor ensures that the integral along this semicircle is zero if the semicircle encloses the lower half of the 2 plane. For along this path, 2 = pe iv = p cos <p + ip sin <p, 1.2] A LITTLE MATHEMATICS 15 with p infinitely large and with ^ going from to — -ny so that sm<p is negative. Therefore along the semicircle pr-izt ^ g— ipt cos <P + fit sin (P As long as sin<p is negative this vanishes, owing to the extremely large value of p. Therefore the addition of the integral around the semi- circle turns it into a contour integral but does not change its value. The final result is that the integral equals (— 2xi) times the sum of the residues of F(z)e~ izt at every one of its poles located below the real axis. The negative sign is due to the fact that the contour in this case is in the clockwise direction. When t is negative, we complete the contour along a semicircle enclosing the upper half of the z-plane, which now has the vanishing integral, and the resulting value is (27rc) times the sum of the residues at each of the poles above the real axis. The limitations on F have already been given. As an example, we can write down the result « 2 f " e ~** _ | (a/2)e at (t < 0) %r J- . z 2 + a 2 dZ \ (a/2)e~ at (t > 0) practically by inspection. Another result of interest later is ia f °° 2^ J_ ^ *-{•?_ <!<2 (2-14) z + ia [ ae~ at (t > 0) In some cases of interest, F has poles on the real axis. These cases can also be treated by making the convention that the integration is not exactly along the real axis but is along a line an infinitesimal amount above the real axis. With this convention, and for functions F(z) that have only simple poles and that vanish at \z\ — * « , we have the general formula sum of residues of \ (2ti) I ^'^ at aU itS I it > 0) \ poles on and below J / + » „ , , . ± , I \the real axis / F{z)e->»dz=( pi«~~» (2l5) /sum of residues ot\ + (2wi)( Fe ~ Ut at aU itS | (t < 0) \ poles above the I \real axis / ./^ f^^^eU 16 INTRODUCTORY [1.2 As an example, we can write down the very useful formulas _^p e -i, (t<0) 2tJ-„z* - a*™ \ sin (<rf) (< > 0) C ^ 17; _J_p^T 0>O) Fourier Transforms. — When the quantities / and F are related by the formula /(0 = j^ a) F{z)e- i " t dz (2.19) then / is said to be the Fourier transform of F. A great deal of abstruse mathematical reasoning, which we shall not go into here, is needed to prove that if / is the Fourier transform of F, according to Eq. (2.19), then F{z) = ± J ^ f(f)e** dt (2.20) so that, contrariwise, 2tF is the Fourier transform of /. Naturally Eqs. (2.19) and (2.20) are valid only when F and / are reasonable sorts of functions, which approach zero in a proper manner as \z\ approaches infinity. When f(t) is zero for t < 0, as it is in Eqs. (2.14), (2.16), (2.17), and (2.18), this reciprocal integral may be written 2irJo FHV) =2^J d f( l ) e ~ Pt dt (2.21) where p = — iz. This integral converges satisfactorily as long as the real part of p is positive, i.e., as long as the imaginary part of z is positive. Formula (2.21) provides a simple means of checking Eqs. (2.14) to (2.18). The usefulness of these integral formulas will become apparent in the succeeding chapters, as we deal with problems of the response of vibrating systems to transient {i.e., nonperiodic) forees. If, instead of Eq. (2.10), we had put d 2 x C " ^f + n 2 x = I F(p)e-^ dp Ml' A LITTLE MATHEMATICS 17 then the solution for x could have been expressed in terms of a con- tour integral. This problem will be considered in the next chapter. Problems 1. A mass m. slides without friction on a horizontal table. It is attached to a light string which runs, without friction, through a hole in the table. The other end of the string is pulled downward by a constant force F. The mass (which is too large to drop through the hole) is held at rest a distance D from the hole and then let go Set up the equation of motion of the mass, and solve it. Is the motion periodic? If so, what is its frequency, and how does the frequency depend onZ)? 2. A bead of mass m slides without friction on a straight wire that is whirling in a horizontal plane, about one end, with a constant angular velocity w radians per sec. It is found that the centrifugal "force" on the bead is mw 2 r, where r is the distance of the bead from the center of rotation. The direction of the force is away from the center. Set up the equation of motion of the bead, and solve it. Is the motion periodic, and, if so, what is the frequency? 3. Show that a solution of the equation g+*g+-»-° can be represented by the series _ c A _ n 2 x 2 ntx* n 6 x" \ \ 2 2 ~*~ 2 2 • 4 2 2 2 -4 2 -6 2 ' " / The series inside the bracket i3 called the Bessel function J (nx), where J (z) = 1 - (2 2 /2 2 ) + • • • . Compute values of /,(0), Jo(l), and J (l) to three places of decimals. 4. Show that the solution of the equation g + (1 - kx 2 ) y = is y = a D e (k, x) + aiD„(k, x) where D e and D are symbols for the following series: nri.~\ . * s 7k . , Ilk + 15k 2 , D e (k,x) -ooB* + i5 **-3g 6 *« + ^j )Mr - ** n ti. \ • , * « 13* , , 17* + 63* 2 „ D (k,x) - sinx + 25 *• - 2 52 q *' + -^720" * Compute values of D e and D , to three places of decimals, for * = 1 and for x =» 0, i, and 1. 6. Show that the solution of the equation Tx [ (1 ~ x2) 2] + Xy = ° is y = a <>P e (\x) + ai P (\,x) 18 INTRODUCTORY 0L2 where P, and Pt, are symbols for the following series: P.(X,x) = 1 - ±x> - ^^ x« - M6-X)(20-A ) ;c< , P,(M)-» + V' , + <a "" X) 5 ( ? 2 "" X) "' | (2-X)(12-A)(30-\) x , { Compute values of P e and P , to three places of decimals, for X = 0, 1 and 2, for x = 0, £, and 1. 6. Show that the solution of the equation J^(* 2 |0 + ( 1 -§) y = O is »-**<*>+«»*»<*> where ji and ni are symbols for the following functions: . , . sin a; cosx . N cosx sin a; Ji(x) = — ; , n x (x) = 5 x % x ' x 2 x What are the values of j t and rti at x = 0, £ and 1 ? What is the solution of the equation ?s(-£)+»- M 7. What is the length of the line drawn from the origin to the point on the complex plane represented by the quantity (a + ib)~*? What is the angle that this line makes with the real axis? What are the amplitudes and phase angles of the following quantities: (a - to)-*; (a + t&) -1 + (c + id)- 1 ; (a + ib) n e- iat 8. What are the real and imaginary parts, amplitudes, and phase angles of Va + ib, log (a + ib), e - 2Ti( - v+i ^ t , e~ ix (l + e^ 5 )" 1 9. The hyperbolic functions are defined in a manner analogous to the trigo- nometric functions [see Eq. (2.7)]. cosh (z) = §(e* + e - *); sinh (z) = §(e* — e~ g ) tanh (z) = , ,{ = — ,. . . (see Tables I and II) cosh (z) coth (z) v ' Show that cosh (iy) = cos(y); cos (iy) =cosh(y); sinh (iy) =i sin(y); sin (iy) = i sin (y). Find the real and imaginary parts of cosh (x + iy), cos (x + iy), sinh (x + iy), sin (x + iy). What are the magnitudes and phase angles of these quantities? 10. Find the real part R and imaginary part X of tanh (x + iy). Show that B. + ^ + ootCW-j^ 1.2] A LITTLE MATHEMATICS 19 Plot the curves on the complex plane for tanh (x + iy) when y is allowed to vary but x = 0, (ir/10), oo ; when x is allowed to vary but y = 0, (tt/4), (x/2). 11. Where on the complex plane are the poles of the functions 1 /cosh (z) ; tan (z) ; e «/ z 2( z 2 _ a 2). tan(«)/z(z 2 +a 2 )? Are they all simple poles? Calculate the residues at all the simple poles of these functions. 12. Compute the values of the integrals /» g—izt /• 00 , , , — 5T dz; I tan (z)e - «. z(z 2 + a 2 ) J- * rfz z X2x (1 — 2p cos + p 2 ) -1 d0, to a contour integral over z, by setting e i9 = z. What is the shape of the contour? Where are the poles of the integrand, when p is real and smaller than unity? Calculate the value of the integral when < p < 1. 14. Calculate the value of the integral 16. Show that _ il f°° -■ f- x 2 dx -» (1 + x 2 )(l - 2x cos + x 2 ) l — (_l)n e i2(nir/u)-| I (^ < 0) g2 _ m2 \dz = ^ sin {at) [0 < t < (nw/o>)] ( [(n7r/a>) < t] CHAPTER II THE SIMPLE OSCILLATOR 3. FREE OSCILLATIONS Now that we have discussed, to some extent, the mathematical methods that we shall need in our work, we shall come back to the physics. The whole study of sound is a study of vibrations. Some part of a system has stiffness; when it is pulled away from its position of equilibrium and then released, the system oscillates. We shall first study the simplest possible sort of vibrations for the simplest sort of system, a mass m fastened to some sort of spring, so that it can oscil- late back and forth in just one direction. A very large number of vibrators with which we deal in physics and engineering are of this type or are approximately like it. All pendulums (the "spring" here is the force of gravity) are like this, and all watch balance wheels. Loud-speaker diaphragms which are loaded so that their mass is con- centrated near their center are approximately like this (at least at low frequencies), as are loaded tuning forks, etc. Even when an oscillating system is more complex than the simple oscillator, many of its properties are like it. Later, when we study these more compli- cated systems, we can simplify our discussion considerably by point- ing out first the properties wherein the system behaves like the simple case and then showing where it differs. The forces exerted by the various springs in the examples of the simple oscillator mentioned above have one property in common: The restoring force opposing the displacement of the mass from its position of equilibrium is proportional to this displacement if it is small enough (we remember the discussion of Hooke's law for a spring and the discussion of the force on a single pendulum given in the ele- mentary physics course). In some problems it is not very accurate to assume that the displacements are small enough, but in our work it is a good assumption, for the vibrations with which we deal in sound have displacements that are very small indeed. For instance, an air mole- cule needs to vibrate with an amplitude of motion of only about a tenth of a millimeter to do its full part in transmitting away the racket 20 H.3] FREE OSCILLATIONS 21 generated at Times Square on New Year's Eve. Seldom does the amp- litude of oscillation of a loud-speaker diaphragm exceed a millimeter. The General Solution. — We shall assume, then, that the "springi- ness" force acting on the simple oscillator can be expressed by the equation F = -Kx (3.1) where x is the displacement of the mass from equilibrium, K is called the stiffness constant (its value depends on the sort of spring we use), and the minus sign indicates that the force opposes the displacement. This will be a very good assumption to make for most of the cases dealt with in this book. The reciprocal of the stiffness constant, C m = (1/K), is called the compliance of the spring. To start with, we shall consider that no other forces act on the oscillator. This is, of course, not a good assumption in many cases; usually there are frictional forces acting, and sometimes external forces come in. There are many cases, however, where the frictional force is negligible compared with the springiness force, and we shall treat these first, bringing in friction and external forces later. This brings us to the equation of motion (2.1) discussed in Sec. 2: ^ = -co 2 z, w 2 = (K/m) = (l/mC m ) (3.2) We have already seen that the solution of this equation can be ex- pressed, in terms of our convention, as x = Ce-™ 1 , C = a + ia x or as x = a ccs {u)t) + «i sin (ut) (3.3) We must now discuss the physical implications of this solution. It has been shown in Sec. 2 that the moti on of the body is periodic, having a frequency v = (o)/2ir) = (l/27r) s/K./m. This frequency is larger for a stiffer spring (for a larger K) and is smaller for a heavier mass. Initial Conditions. — We have seen that the specific values of the constants a and ai are determined by the way that the mass is started into motion. Ordinarily, we start an oscillator into motion by giving it a push or by pulling it aside and letting it go, i.e., by giving it, at t = 0, some specified initial displacement and initial velocity. Once we have fixed these two initial conditions, the motion is completely determined from then on, unless we choose to interfere with it again. It is not hard to see that a is the value of the initial displacement 22 THE SIMPLE OSCILLATOR [IL3 and ojai that of the initial velocity. Solution (3.3) can then be rewrit- ten in the following forms: x = x cos (2rvot) + ( ~- J sin (2rvot) = 4 cos (2™ * -<*>); v = Q^^;} (3.4) *- * + &)'.■ tan * = fek) - where ar is the initial displacement and v the initial velocity of the mass. These equations reemphasize the fact that only the initial value of x and of (dx/dt) need be given to determine the subsequent motion of an oscillator completely. Once x and v are specified, then, even if we did not have a completely worked-out solution of (3.2), we could find the initial value of the body's acceleration by inserting x on the right-hand side of (3.2). The initial value of the third deriva- tive can be obtained by differentiating (3.2) with respect to t and placing the value of v on the right-hand side in place of (dx/dt); and so on. If it is recalled that the behavior of a function is com- pletely specified by its Taylor's-series expansion (dj\ ,?(d?f\ ,t2(<pf\ W*-o 2 W/«-o "*" 6 \dt*/ t m = /co) + 1 y; (=o + - 2 ^ + ff ^ + - . . (within a certain range of t whose limits are of interest in specific problems, but which need not bother us here), we see that once x and v are given, thus fixing the values of all the higher derivatives at t — 0, the future motion is determined. It is not hard to generalize this reasoning, to see that a body, acted on by any sort of force that depends on x and on v, will have its motion specified completely just by assigning definite values to its initial position and velocity. The mathematical counterpart to this state- ment is the rule that the solution of any second-order differential equation (one having a second derivative term in it but no higher derivative) has two arbitrary constants in it. Another very important physical fact which can be deduced from (3.4) is that the frequency of the oscillation depends only on K and m and not at all on xo or v . This means that for a given mass and a given spring, as long as the law of force of the spring is F = —Kx, then the frequency of oscillation will be the same no matter how we U.4] DAMPED OSCILLATIONS 23 start the system to oscillate, whether it oscillates with an amplitude of motion of 1 cm or 0.001 mm. This is a very important fact in its practical applications, for if the. law of force of actual springs were not nearly F — —Kx, or if this property did not hold for solutions of Eq. (3.2), then no musical instrument could be played in tune. Imag- ine trying to play a piano when the frequency of each note depended on how hard one struck the keys! Compare this with Prob. 1, page 17. We might have found this fact for a number of cases by a long series of experimental observations, but our mathematical analysis tells us immediately that every mass acted on by a force F = —Kx has this property. Oscillations of this type are called simple harmonic oscillations. Energy of Vibration. — We shall need an expression for the energy of a mass oscillating with simple harmonic motion of amplitude A and frequency v . The energy is the sum of the potential and kinetic energies W = \mv 2 + JT* Kx dx = %mv 2 + iKx 2 = 2-K 2 mv\A 2 sin 2 (2W - $) + \KA 2 cos 2 (2W - *) But 4t 2 v% = (K/m) so that W = i£A 2 [sin 2 (2W - $) + cos 2 (2W - $)] or W = \KA 2 = 2-K 2 mv\A 2 = \mU 2 (3.5) where U = 2ttv A is the velocity amplitude of the motion. The total energy is thus equal to the potential energy at the body's greatest displacement (iKA 2 ) or is equal to the kinetic energy at the body's greatest speed (kmll 2 ). Expressed in terms of v and A, we see that W depends on the square of these two quantities. 4. DAMPED OSCILLATIONS So far, we have not considered the effects of friction on oscillating systems. In general, friction does not play a very important role in the problems that we shall consider in the first part of the book. If we show its effects on the simplest system with which we deal, we can deduce its effect, by analogy, on more complicated systems; therefore, we shall mention friction only occasionally, later on in this book. The friction that is most important in vibrational problems is the resistance to motion which the air surrounding the body manifests. 24 THE SIMPLE OSCILLATOR [TLA Energy in the form of sound waves is sent out into the air. From the point of view of the vibrating system, this can be looked on as friction, for the energy of the system diminishes, being drained away in the form of sound. This resisting force depends on the velocity of the vibrator, and unless the velocity is large (much larger than those with which we shall deal), it is proportional to the velocity. It can be expressed mathematically as —R m (dx/dt), where the constant R m is called the resistance constant. The total force on a simple oscillator acted on by both friction and springiness is therefore — R m (dx/dt) — Kx, and the equation of motion becomes dt 2 + dt + 47r VoX " °' 4tt 2 . 2 = (K/m) (4>1; The value of v is the frequency that the oscillator would have if the friction were removed (R m were zero) and is called the natural frequency of the oscillator. It is interesting to notice that the equation for the free oscillation of charge in a circuit containing inductance, resistance, and capacitance has the same form as (4.1). The inductance is analogous to the mass m, the resistance to the resistance factor R m , and the inverse of the capacitance to the stiffness constant K. The General Solution. — To solve Eq. (4.1) we make use of the exponential function again. We guess that the solution is Ce bt and solve for b. Substitution in (4.1) shows that (6 2 + 2kb + 4r 2 v$)e bt must be zero for all values of the time. Therefore 6 2 + 2kb + 4tt 2 ^ = 0, or b = -k ± Vk 2 - 4tt 2 ^ In all the problems that we shall consider, the stiffness constant K is much larger than the resistance constant R m {i.e., friction will not be big enough to make the motion much different from that discussed in the last section). Since 4r 2 vl is supposed to be larger than k 2 , then the square root in the expression for b is an imaginary quantity, and we had better write b = -k ± %riv f , v f = vo Vl ~ (k/2rv ) 2 (4.2) This means that, following our convention, we can write x = Ce- fc <- 2 ™"< (4.3) or x = e- ki [a cob (2rv ft) + «i sm(2irv f t)] = A e- kt cos(2ttj7J - $) The values of a and a x are again fixed by the initial conditions for H.4] DAMPED OSCILLATIONS 25 the oscillator; a must equal the initial displacement x , and the initial velocity in this case is equal to %cv } a x — ka , so that eti ( t»o + kx<\ 2rv f . J The solution is not periodic, since the motion never repeats itself, each swing being of somewhat smaller amplitude than the one before it. However, if A- is quite small compared with v f , we can say that it is very nearly periodic. In any case, the frequency of the oscilla- tions is v f , which is very nearly equal to v if k is small. It again turns out that the frequency is independent of the amplitude of the motion. Of course, strictly speaking, we should not use the word frequency in connection with nonperiodic motion. But when the damping is small, the motion is almost periodic, and the word will have some meaning, although a rather vague one. There are several respects in which the motion of the damped oscil- lator differs from that of the simple oscillator. The most important difference is that the amplitude of motion of the damped oscillator decreases exponentially with time; it is A e~ kt , instead of being just A (A is the initial value of the amplitude). The amplitude decreases by a factor (1/e) in a time (1/k) sec (e = 2.718). This length of time is a measure of how rapidly the motion is damped out by the friction and is called the modulus of decay r of the oscillations. The fraction of this decrease in amplitude which occurs in one cycle, i.e., the ratio between the period of vibration and the modulus of decay, is called the decrement 8 of the oscillations. Another method of expressing this is in terms of the "Q of the system," where Q = (co m/# m ) is the num- ber of cycles required for the amplitude of motion to reduce to (1/e*) of its original value. If these quantities are expressed in terms of the constants of the system (the small difference between v f and v being neglected), it turns out that «-fe)-(?> -**-$ *-£) r441 r=l = -^; 5 = l = l = J- ( ( } k irvn' Q v Q tvq J The smaller R m is, the larger Q and r are, indicating that it takes a longer time for the oscillations to damp out, and the smaller 5 is, indicating that the reduction in amplitude per cycle is smaller. These properties, of course, are independent of the way the oscillator is started into motion. Another difference between the damped and the undamped oscil- 26 THE SIMPLE OSCILLATOR [II.4 lator is the difference in frequency. When (k/2rv ) is small, the expression for v f can be expanded by means of the binomial theorem, and all but the first two terms can be neglected. Vf =-v^(sy = "-(§^) + -'- (4 - 5) In most of the cases with which we shall be dealing, k and v have such values that even the second term in the series is exceedingly small, so that the change in frequency is usually too small to notice. Energy Relations. — The subject of damped oscillations can be considered from a quite different point of view — that of energy loss. We must first develop an expression for the average energy of the system at any instant. We cannot use the formulas (3.5) because the amplitude of oscillation in the present case is not constant. The sum of the kinetic and potential energies of a body of mass m acted on by a springiness force Kx, whose displacement is given by the formula A(t) cos{2irvt — <£), is W(t) = \mv 2 + iKx 2 = 2-K 2 mv 2 A 2 - 2mmv (^7 J A sm(2rvt - <3>) cos(2tt^ - <S>) + im i^£\ cos 2 (2rvt - *) When averaging this value over a single oscillation, the second term on the right drops out. If A is very slowly varying, so that (dA/dt) is small compared with vA, then the third term can be neglected, and we have the approximate formula for the energy of motion and position {i.e., the energy that can be recovered, that is not yet irre- vocably lost in heat) W(t) ~%R[A(t)] 2 = 2T 2 mv 2 [A(t)Y = im[U(t)] 2 (4.6) where the symbol ~ means "is approximately equal to." In the case of the damped oscillator, this "free energy" is 2-K 2 mv%A\e~ 2U \ which diminishes exponentially with time. The rate of loss of energy due to friction is equal to the frictional force opposing the motion R m v multiplied by the velocity v (since force times distance is energy, rate of change of energy is force times velocity). The rate of loss of energy is P = R mV 2 = [Air 2 R m v f 2 sin 2 (2jrvft - 4>) + 4;irR m Vfk sin (2irv f t — 3>) cos (2irv f t — 3>) + Rmk 2 cos 2 (2rv ft - $)]A§e~ 2fct D.6] FORCED OSCILLATIONS 27 Using the same approximations as before, we have for the average loss of energy per second • --&) Wv\R m A\e-™ = %R m [U(t)]* (4.7) If we had started out without formula (4.3) for the details of the motion but had simply said that the free energy at any instant was given by (4.6) and that the energy loss was given by (4.7), where R m is small, we could have found the dependence of W on the time by means of (4.7). Eliminating U from Eqs. (4.6) and (4.7) results in P ~ (R m /m)W = 2kW, so that we have p = ~ (t) - 2kW > or w - +2kW ( 4 - 8 ) The solution of this is W~ W e~ 2kt = imU%e- 2kt = %c i v\mA\e-' lu which checks with Eq. (4.6). We see from this that the damping out of the motion is required by the fact that the energy is being lost by friction. We might point out here that the fraction of free energy lost per cycle is just (2k/v f ) ~ (2k/v ) = (2w/Q), where Q is given in Eq. (4.4). In nearly every more complicated case of vibrations, the effect of friction is the same as in this simple case. The amplitude of vibration slowly decreases, and the frequencies of natural oscillation are very slightly diminished. Usually, the change of frequency is too small to be of interest. 5. FORCED OSCILLATIONS It often happens that a system is set into vibration because it is linked in some way with another oscillating system (which we shall call the driving system). For instance, the diaphragm of a microphone vibrates because it is linked, by means of sound waves, to the vibra- tions of a violin string; and a loud-speaker diaphragm vibrates because it is linked to the current oscillations in the output circuit of an ampli- fier. The system picks up energy from the driving system and oscil- lates. In the two instances mentioned, and in many others, the driven system does not feed back any appreciable amount of energy to the driving system, either because the linkage between the two is very weak (as is the case with the violin and microphone) or else because the driving system has so much reserve energy that the amount fed 28 THE SIMPLE OSCILLATOR [II.6 back is comparatively negligible (as is the case with the amplifier and loud-speaker). In these cases the only property of the driving system that we need to know is that it supplies a periodic force which acts on the driven system. The more complicated case, where the feedback of energy cannot be neglected, will be considered in the next section. The General Solution. — For the present, we ask what happens to a simple oscillator when it is acted on by a periodic force F cos (2-irvt), or Fe~ 2lcivt , according to our convention. We wish to know what its motion is just after the force has been applied and, more important, what its motion is after the force has been acting for a long time. We are also interested in how this behavior depends on the frequency v of the driving force (which does not have to be the same as the natural frequency v of the oscillator). The total force on the oscillator is a combination of the springiness and the frictional and the driving force -R m v - Kx + Fe~ 2 * ivt . The equation of motion is ^§ + 2k d 4i + 4ttV .t = ae- 2 * M , a = (F/m), k = (R m /2m), at 2 at co = 2rv, 4tt 2 ^ = (K/m) (5.1) We discussed a similar equation in Sec. 2, where we showed that a choice of two exponentials, one corresponding to the free vibration of the oscillator and the other to the forced motion, was a good guess for the solution. Substituting x = Ce~ u '^ iv t t + Der 2 * M in (5.1), we find that it is a solution if n _ a = v(vl - v 2 ) + ivk = R itf U ~ 4tT 2 (^ - V 2 ) - ^ivk ±TC\V% ~ V 2 ) 2 + \KV 2 k 2 where B = (<*/4ir) _ (F/2tp) \Ar 2 ("o ~ v 2 ) 2 + v 2 ¥ 4 RI.+ o K ZTV and tan# = Rr *{v\ - v 2 ) (K/2-kv) - 2Tvvm The solution can be thrown into a semblance of simplicity, and the analogy with electric circuits can be made more apparent, if we define a mechanical impedance for the system II.5] FORCED OSCILLATIONS 29 Z m = R m - dam - — J = R m - iX m = \Z m \e~ i -» (5.2) co = 2wv t \Z m \ = ^Jr* + U m _ ±) „ = * - 90° = tan- [ " m ~^ /c ° ] We recall that the usual electrical-engineering notation is obtained by substituting —j for i, so we can see that the equation for Z m is exactly analogous to the equation for the complex electrical impedance of a series circuit, with the mechanical resistance R m analogous to the electrical resistance, the mass m analogous to the electrical inductance and the mechanical compliance C m = (l/K) analogous to the electrical capacitance. The quantity X m = urn — (K/co) can be called the mechanical reactance of the system. The units in which this mechanical impedance are expressed are not ohms, for the quantity is a ratio between force and velocity rather than between voltage and current. The symbolic analogy is close enough, however, to warrant the use of the same symbol Z with the subscript m to indicate "mechanical." The units of mechanical impedance are dyne-seconds per centimeter, or grams per second. The solution of Eq. (5.1) can therefore be written in either of the two alternate forms F X = Q e -hi-1riv t t _ -«-"•-' lirivZn or \2Tv\Z m \J x = e~ kt [a cos (2irv f t) + a t sin (2irv f t)] + I - — r T? — x ) cos (2irvt — &) \ZTV\Z/ m \/ The constants a and ai are determined, as before, by the initial posi- tion and velocity of the mass. Transient and Steady State. — When the force is first applied, the motion is very complicated, being a combination of two harmonic motions of (in general) different frequencies. But even if the friction is small, the first term, representing the free, or "transient," vibra- tions, damps out soon, leaving only the second term, which represents simple harmonic motion of frequency equal to that of the driving force (see examples of this in Fig. 2). 30 TEE SIMPLE OSCILLATOR IIL6 x --*■ — F 2irivZ r , e - *"', or x — » i? 2tv\Z„ sin (%irvt — ■ <p) where y ~ * \ 7" ) e ~ iat ' 0r U — * \ l7~T ) C0S ^^ ~ ^ (com) - (X/co) (5.3) tan co = — cot# = Rt! ) co = 2^^ This part of the motion is called the steady state. We see that it is completely independent of the way in which the oscillator is started V=" 2 A> f V=Vf \=2v* Time-*- Fig. 2. — Forced motion of a damped harmonic oscillator (A; = weff/10)- Curve a shows free oscillations, and curves b, c, and d show forced oscillations due to sudden application of force at t = 0. Dotted curves give force as function of time; solid curves give displacement. Effect of transient is apparent at the left side of the curves; at the right side the steady state is nearly reached. into motion, its amplitude, phase, and frequency depending only on the constants F and v of the force and on the oscillator constants m, R m , and K. No matter how we start the oscillator, its motion will eventually settle down into that represented by (5.3). II.5] FORCED OSCILLATIONS 31 Steady-state motion is motion of a system that has forgotten how it started. Impedance and Phase Angle. — The amplitude and velocity ampli- tude are proportional to the amplitude of the driving force and are inversely proportional to the magnitude of the mechanical impedance Z m . The analogy with electric circuits is therefore complete. The velocity corresponds to the current, the mass to the inductance, the frictional constant to the resistance, and the stiffness constant to the reciprocal of the capacitance. The impedance is large except when v = v , but at this frequency, if the friction is. small, it has a sharply defined minimum. Therefore, the amplitude of motion in the steady state is small except when v = v , where it has a sharp maximum. The case where the frequency v of the driving force equals the natural frequency v Q of the oscillator, when the response is large, is called the condition of resonance. The peak in the curve of amplitude against v is sharp if R m is small and is broad and low if R m is large, as is shown in Fig. 3. This figure also shows that the steady-state motion of the oscillator is not very sensi- tive to the value of the frictional constant except in the range of fre- quencies near resonance. The dotted curve for the amplitude of motion is for a value of R m eleven times that for the solid curve, yet the two are practically equal except in this frequency range. The motion is not usually in phase with the force, the angle of lag of the displacement behind the force being given by the angle &, which is zero when v = 0, is (x/2) when v = v , and approaches x as v approaches infinity (indicating that the displacement is opposite in direction to the force). The angle of lag of the velocity behind the force, <p = + (x/2) + #, is analogous to the phase angle in a-c theory. It is —k/2 when v = 0, zero when v = v , and +7r/2 when v is very large. In other words, when the frequency v of the driving force is much smaller than the natural frequency v of the oscillator, then the ampli- tude is small, and the displacement is in phase with the force. As v is increased, the amplitude increases and gets more and more out of phase with the force, until at resonance the amplitude is very large (if R m is small), and the velocity is in phase with the force. As v is still further increased, the amplitude drops down and eventually becomes very small. For these large values of v the displacement is opposed to the force. Figures 2 and 3 illustrate this behavior We use systems driven by periodic forces in two very different ways. One type of use requires the system to respond strongly only to par- 32 THE SIMPLE OSCILLATOR [II.6 ticular frequencies (examples of this are the resonators below the bars of a xylophone, the strings of a violin, and the human mouth when shaped for a sung vowel). In this case we must make the friction as small as possible, for then the only driving frequency that produces a large response is that equal to the natural frequency of the driven system. The other type of use requires the system to respond more or less equally well to all frequencies .(examples are the diaphragms of microphones and loud-speakers and the sounding board of a violin). Fig. 3. — Amplitude and phase of forced motion, as functions of the frequency v of the driving force. The frictional constant R m for the dotted curve is eleven times that for the solid curve, all other constants being equal. In some cases we wish the steady-state amplitude to be independent of the frequency; in others we wish the amplitude of the velocity to be constant; and in still other cases we would like the acceleration to have a constant amplitude. One or another of these requirements can be met within a certain range of frequencies by making the stiffness, the friction," or the mass large enough so that its effects outweigh those of the other two in the desired range of frequency. These three limiting types of driven oscillators are called stiffness- controlled, resistance-controlled, and mass-controlled oscillators, respec- (5.4) II.6J FORCED OSCILLATIONS 33 tively. Their properties and useful ranges of frequency are Stiffness controlled: K large ; Z m ~ i f - — h x ~ I -=z \ e _iat v considerably less than both ( ?r- I -» /— and ( n „ I \27r/ \ m \2trR m / Resistance controlled: R m large ; Z m ~ R m ; ( — J ~ ( — J e~ iut v considerably less than I -p-^- \ larger than ( _ „ J \2irm/ \2TR m / Mass controlled: m large; Z m ~ — Qnrivm; ( -r^ I ~ ( — ) e~ iwt v considerably larger than both \w-j -*/— and ( ^-^- ] It is to be noticed that every driven oscillator is mass controlled in the frequency range well above its natural frequency v , is resistance controlled near v (though this range may be very small), and is stiffness controlled for frequencies much smaller than v . It simply requires the proper choice of mechanical constants to place one or another of these ranges in the desired place in the frequency scale. It is also to be noted that there is always an upper limit to the fre- quency range over which an oscillator is stiffness controlled, a lower limit to the range over which one is mass controlled, and both an upper and a lower limit to the range over which an oscillator can be resistance controlled. We can move these limits about by changing the mechanical constants, but we never remove the limits entirely. Energy Relations. — The average energy lost per second by the oscillator due to friction, when in the steady state, is P = %R m U 2 = iR m (F/\Z m \) 2 [see Eq. (4.7) of the last section]. The rate of supply of energy from the driver to the driven oscillator is F 2 vF cos (2irvt) = j=-, cos (2irvt) cos (2irvt — <p) \Z n F 2 \Z m \ cos {2-irvt) [cos <p cos (2-Kvt) + sin <p sin (2Tcvt)] The average value of this energy supplied per second by the driver is HF 2 /\Z m \)cos<p = %R m (F/\Z m \) 2 (since cob* = RJ\Z m \), which equals the loss of energy to friction P. One can say that the amplitude and phase of the driven oscillator so arrange themselves that the energy delivered by the driver just equals the energy lost by friction. 34 THE SIMPLE OSCILLATOR [II.5 Electromechanical Driving Force.— There is one particular form of coupling between the driver and driven systems that is of particular interest to us in the study of sound: the electromechanical coupling between a vibrating diaphragm and an electrical driving circuit. In practically all acoustical work at present, the sound vibrations dealt with are transformed into electrical oscillations in order to amplify them. After amplification they are then changed back into sound waves, if need be, and an electromechanical coupling between the amplifier and the sound generator is used. Such coupling can have a variety of forms. For instance, the amplifier current can be sent through a coil wound on a magnet, varying the magnetic field and thus varying the force on a steel diaphragm (this is the coupling used in the ordinary telephone receiver). Or the current can be sent through a coil, attached to the diaphragm, which is placed in a magnetic field whose direction is perpendicular to the coil winding (this coupling is used in the so-called dynamic loud-speakers). Or the electric voltage may be impressed across a piezoelectric crystal, causing it to change size and shape (this coupling is used in crystal microphones and loud-speakers). In the magnetic forms of coupling the mechanical force exerted on the driven oscillator is proportional to the instantaneous current I through the coil, F = DI dynes, where the factor of proportionality D has a value that depends on the particular sort of coupling system that is used. For instance, if the coupling coil is fastened to the diaphragm and consists of n turns, each of radius b cm, and if the coil is placed in a radial magnetic field whose intensity is B gauss at the position of the winding, then D = 2irnbB/10, if the current is measured in amperes. In certain cases D may be a complex quantity, as, for example, in an ordinary telephone receiver when hysteresis must be taken into account. If the current is an alternating one, represented by loe -2 "*', the force on the oscillator is DIoe~^ ivt , and if the driven system behaves like the simple oscillator discussed in this section, its steady-state dis- placement and velocity will be, according to Eq. (5.2), x = ^° e- 1 '**, v = §^-° <r*"«, a = 2™ (5.5) The motion of the driven system produces a back emf in the coil. Owing to the reciprocal relation between the force on a current in a magnetic field and the emf produced by motion in the same field, we can immediately say that the back emf produced in the coil by moving II.6] FORCED OSCILLATIONS 35 the driven system with a velocity v is vD/10 7 volts, where D is the same constant that appeared in the expression for F. The back emf produced in the coil due to its motion is therefore Em = y~ he-*"' = 15-1 where the constant r = D 2 /10 7 is called the electromagnetic coupling constant. The effective electrical impedance in the amplifier circuit due to the motion of the driven system is the ratio between Em and I. This complex quantity Z M r Zm = Rm — iXM = 77- — TY m = T(G m — iB m ) <6m where G m — ( R m \ B _ _ (X m \ Y _ _1_ iv \ (5.6) toj' Bm ~ \\z m \*)' Ym ~ \z m \ e ( \Z m \*=-Kl + Xl; Z m = com-^ Z M is called the motional impedance of the coil. It is a true electrical impedance, measured in ohms, in contrast to the mechanical impedance Z m . The quantity Y m = {\/Z m ) = G m — iB m is called the mechanical admittance, with units seconds per gram or centimeters per dyne- second, having a real part G m , called the mechanical conductance, and an imaginary part B m , called the mechanical susceptance. For the case of electromagnetic coupling the electrical impedance due to the motion of the coil is proportional to the mechanical admittance of the coil. The constant of proportionality r has the units of ohm-grams per second. We label all electrical impedances with a capital-letter sub- script to distinguish them from the mechanical impedances Z m , Z r , etc. Equations (5.6) indicate that the larger the mechanical impedance of the coil (i.e., the harder it is to move), the smaller the motional impedance of the coil (the less the back emf due to its motion). A little thought will show that this is a reasonable result. Motional Impedance. — It is interesting to notice that an electrical circuit with an inductance whose magnitude equals (T/K), a resistance of magnitude (T/R m ) and a capacitance (m/T), all in parallel, will give an electrical impedance of just the value given in (5.6). As far as the electrical circuit is concerned, the loud-speaker mechanism is com- pletely equivalent to the circuit shown in Fig. 4. If the complex impedance of the coil when the loud-speaker is rigidly clamped is Re — 2irivLe, then its impedance when the loud-speaker is allowed to 36 THE SIMPLE OSCILLATOR [H.5 move is Z E — Z c + Z M , where Z c = Rc — iwLc and Z M = T/Z m . Z c is called the clamped or blocked impedance of the coil and Z E its total electrical impedance. When a current Z = Joe - ""* is sent through the coil and no external force F e is applied, there will be a total voltage drop E = Z E I across its terminals. This voltage can be considered to be the sum of the drop across the blocked impedance Z C I and the motional emf E M — Z m Im, which is the voltage drop across the part of the circuit enclosed in a dotted line in Fig. 4. Once the equivalent electrical circuit is determined, and the mechanical behavior of the coil, its velocity and r M echanical System _ Current /^Source F e /P 10 Im <r/K) ^O'OOWO'OOWH (r/Rm) -wwwvww- (m/r) E m . — ^OOOOO'OOW* — vwwwwv ' L c R c Fig. 4. — Electrical circuit equivalent to the driving coil of a loud-speaker or "dynamic" type of microphone. Effective electrical impedance of mechanical system is equal to that of circuit enclosed in dashed line. Effect of external force F e applied to coil is equivalent to constant-current source of strength (F e /D) applied as shown. displacement, the power Po lost owing to heating the coil and the power used in moving the coil P M can all be determined in terms of the voltage Em across and the current Im through the mechanical part of the circuit : E M = ImZm volts r V ~f X = (= Z M IMF e = 0) Em = VlO^T \~\ cm/sec cm; 2iriv Pm = %Rm(\Im\ 2 ) — -^rj?, Rc watts (5.7) \Z n \v\ iR m tkv watts In the last of these equations, we see the relation between the electrical and mechanical units. On page 26 we showed that the expression II.6] FORCED OSCILLATIONS 37 ii2 m |v| 2 is equal to the power dissipated by th'e mechanical resistance R m . Mechanical power is measured in ergs per second, however, and it takes 10 7 ergs per sec to equal 1 watt. The relationship between T and D 2 provides just this factor, so that Pm, computed from the equiva- lent electrical circuit, comes out in watts. There are several interesting corollaries from Eq. (5.7) which are worth mentioning. The mechanical impedance discussed in the example can be represented by a sum of impedances -torn, R m , and (K/—iw), as in a series circuit. The equations are more general than this, however: if the mechanical load on the coil can be represented by a mechanical admittance Y m = (1/Z TO ) which is any complex func- tion of co, then the motional impedance of the coil due to this load is TY m . For instance, if the mechanical load consists of a spring and a resistance in parallel, the mechanical admittance is (l/R m ) — (ioo/K) and the motional impedance is Z M = (T/R m ) — (iwT/K) ohms, as though an electrical resistance (T/R m ) and inductance (T/K) were in series. Later in the book we shall consider impedances due to radia- tion loads. Here again, to find the motional impedance Z M) we multi- ply the total mechanical admittance of diaphragm plus air load by I\ If the coil is short-circuited and is then moved by a mechanical force, the mechanical load is greater than the mechanical impedance, because the motion of the coil induces a current in the coil which pro- duces an additional reaction force. The additional mechanical imped- ance, due to current in the short-circuited coil, is Z s = T/Z c g per sec ana is the counterpart, for the mechanical system, of the motional impedance Z M for the electrical circuit. The total mechanical imped- ance of the diaphragm and shorted coil is Z m + Z 8 . The quantity Z a can be called the magnetomotive impedance. This magnetomotive impedance can be utilized to obtain an alter- native formula for the velocity of the diaphragm-coil system when driven by an emf E = Etfr™ % impressed across the coil DIm DE DE v = Z m z m (Zc + Zm) Z m z c + r = DE / Z c = Die Zm ~T" Z s Z m -\- Z s where I c is the current that would flow through the coil if it were clamped. This formula is of use in computing the change in velocity of a diaphragm when the mechanical load is changed but the electrical circuit is kept the same. In such a case D, I c , and Z„ would be unchanged and only Z m would change. 38- THE SIMPLE OSCILLATOR [II.5 When an external force F e is applied to the coil, as occurs when the system is used for a microphone, an additional term must be included in the equations. It turns out that the corresponding addition to the equivalent circuit is a constant-current generator, of strength (F e /D) amp, applied across the "mechanical" part of the circuit, as shown in Fig. 4. For mechanical resonance the mechanical reactance arm — {K/w) is zero, the magnitude \Z m \ of the mechanical impedance is a minimum, and the magnitude Y/\Z m \ of the motional impedance is a maximum. Other quantities being equal, at resonance \E M \ has its maximum value, and consequently the velocity magnitude |v| is maximum. At still higher frequencies there may be a series resonance between the induc- tance L c and the capacitance (jn/T). So many kinds of impedance have been discussed in this section that it might be well to list them together to contrast their meaning and the units in which each are expressed: Z E = total electrical impedance, in ohms Z c = clamped electrical impedance, in ohms Z M — motional impedance, electrical, in ohms Z m = mechanical impedance, in g per sec [ w-8) Z r = radiation impedance, in g per sec Z 8 = magnetomotive impedance, in g per sec Throughout the book, whenever there is danger of confusion, we shall use capital-letter subscripts for electrical impedances, ratios between volts and amperes; and we shall use lower-case subscripts for mechanical impedances, ratios between forces and velocities. Piezoelectric Crystals. — Rochelle salt crystals are also electro- mechanical transducers, transforming electrical into mechanical energy, and vice versa. For the "a>cut" type, for instance, the crystal is cut in a rectangular form. A force is applied between two faces (between face S and the rigid table T in the figure) by means of some mechanical system (such as a diaphragm) with effective mechanical impedance Z m . Voltage then develops between two other parallel faces, which can be measured across the terminals A and B. Alternately, a voltage applied across A and B will cause a force to be exerted on the mechanical system attached to face S. The dimensions of the crystal a, b, and d are shown in the figure. In this case the displacement from equilibrium of face S, which we. can label x, is due to the crystal expanding lengthwise. The expansion per unit length ij = (x/q) is called the strain. The mechan- n.6] FORCED OSCILLATIONS 39 ical force causing the expansion or contraction is the external stress X in dynes per sq cm. This stress, in a piezoelectric crystal, causes an electric polarization n, in_ this case perpendicular to the condenser plates. The polarization in turn causes a charge to be formed On the plates, in addition to that caused by the external applied field. For ease in initial calculation we shall use electrostatic units for the first steps. The relations between these internal quantities £, X, and n External Force F» echanical Load I Y't'/'/W i Source J Mechanical System Fig. 5. — Typical arrangement of piezoelectric crystal, showing electrical connections and direction of application of force F e - An equivalent electrical circuit is shown. For other crystal cuts the metal plates would be on the top and bottom, rather than on a pair of sides. and the external quantities F e , x, the potential difference E between the plates and the total charge Q on the plates are given by the set of equations F e = bdX + Z m v, v = — io)X = —iu£a (5.9) 40 THE SIMPLE OSCILLATOR [II.6 where ki is the dielectric susceptibility of the unloaded crystal, su is one of its elastic moduli, and du is the appropriate piezoelectric con- stant. The constants /c, s, and d vary markedly with temperature. The first of the fwe equations (5.9) gives the relation between the driving force F e , the internal stress, and the motion of the mechanical load. The second is the usual relation between velocity and displace- ment for simple harmonic motion. The third is the usual electrical relationship between surface charge density, applied field E/d, and polarization II. The last two equations are peculiar to piezoelectric materials. The equation for the polarization in the particular direc- tion we have chosen is related both to the applied field (which is usual) and also to the mechanical stress. The strain in the vertical direction depends in the usual manner on the stress; but there is an additional dependence on the applied field. Our equations are given for one particular cut of crystal, with force and field applied in a par- ticular way. Other arrangements with respect to the crystal axes will change the last two of Eqs. (5.9), the modified equations involving other constants k, d, and s labeled by different subscripts. We have also assumed that the frequency of oscillation is smaller than the lowest resonance frequency of the crystal; otherwise the stress, strain, polarization, etc., will not be uniform throughout the crystal. The more complicated case of high frequencies will be taken up later. After a large amount of algebra, we can obtain an expression for the current I = —iuQ in terms of the applied voltage t . n /o6\ /1 , ,, n ■ (bdg/su ZmjFe + ioi(abd\J sud)E 1 = " UaE \&Td) (1 + 4TKl) + ' 1 - (l/ia,)(db/asuZ m ) Still further manipulation can be applied to show that this relation between I and E is equal to that due to the equivalent circuit shown in Fig. 5, where ^rff 4 io" radg { (510) v d s 4 4 9 ' t = 300 (r^y 1 ) volts/dyne, a = 10 7 r £ ohm-sec/g and where the transition from electrostatic to practical units has been made. The constant a can be called the piezoelectric coupling constant, II.6] FORCED OSCILLATIONS 41 analogous to r in the electromagnetic case. It changes the mechanical impedance of the load into electrical impedance in the equivalent cir- cuit. Values of these constants for Rochelle salt, for various tem- peratures in degrees centigrade, are given in Table 1. Table 1.— Piezoelectric Constants for Rochelle Salt (See Fig. 5) Tempera- Capacitance, nni ture °C t, juv/dyne a, ohms/g tor T (Cd/ab) (Ccd/ab) (C p d/ab) br X 10 6 -10 35.6 15.9 19.7 315 0.992 - 5 25.6 13.3 12.3 370 1.369 19.4 10.5 8.9 425 1.806 5 16.7 9.2 7.5 462 2.134 10 16.7 8.8 7.9 454 2.061 15 22.2 10.9 11.3 396 1.568 20 42.2 16.7 25.5 291 0.847 23.7 Upper Curie poi nt 24.7 132.3 24.4 107.9 209 0.437 25.7 71.5 19.7 51.8 240 0.576 28.2 38.1 15.4 22.7 303 0.918 31.0 23.4 12.2 11.2 382 1.459 From H. Mueller, Properties of Rochelle Salt, Phys. Rev., 57, 829, and 68, 565 (1940). These constants depend strongly on temperature, particularly near the Curie points. The corresponding constants for quartz are much less dependent on temperature. We notice several important differences between this equivalent circuit and the one in Fig. 4. In the first place, the part of the circuit corresponding to the mechanical load is a series circuit, not a parallel one; so that if m, R m , or K is very large, the equivalent electrical impedance is very large, instead of very small, as is the case in Fig. 4. A very large value of Z m corresponds to loading or clamping the crystal so that its upper face cannot move. In this case the equivalent elec- tric circuit reduces to the condenser C c (the subscript c indicating "clamped"). If the crystal is unloaded and the external force F e is zero, the equivalent circuit consists of the two condensers in parallel, having a combined capacitance of C farads. In the present case the external applied force F e appears as a voltage source, in series with the impedance equivalent to the mechanical load, instead of as a current source parallel to this impedance, as in Fig. 4. The velocity of the top face of the crystal is proportional to the current in the "mechanical" arm of the equivalent circuit. The equa- 03 42 THE SIMPLE OSCILLATOR [H.6 tions giving this velocity and the power used in moving the connected mechanical system are v = 10 7 tI p , x = - 10 7 ( ^ j I P •*"• ' = 300 fe) i (511) f\v\ 2 \ Pm = i(rR m \I P \ 2 = iR m I t^7 ) watts We note that the equivalent capacitance C p corresponds to an addi- tional stiffness, which is the stiffness of the crystal. The effective series capacitance of the mechanical arm of the circuit is Cm = [(1/C P ) + Kt 2 ]~ x , and the corresponding total stiffness of the mechanical system is Mechanical resonance will occur when the equivalent reactance t 2 [ojw — (K t /a))] is zero. At this frequency, the maximum current will flow in the mechanical arm for a given applied voltage E. At a higher frequency, parallel resonance will occur with the shunt capaci- tance C c . 6. RESPONSE TO TRANSIENT FORCES In the previous section we dealt with the response of various simple mechanical and electromechanical systems to simple harmonic driving forces. In particular, we studied the steady-state response to such forces after the transient effects, due to the force starting, have died out. In this section 1 we wish to follow a little further the study of the transient effects, particularly when the driving force itself is transient and the system never does settle down into steady driven oscillation. To carry this out we shall use the techniques of complex integration discussed in Chap. I. Representation by Contour Integrals. — The general method of attack is as follows : we express our transient driving force as an integral of the form of Eq. (2.19) f(t) = f^° M F(w)er^ t da such as the ones given in Eqs. (2.15) to (2.18), where F can be deter- mined by the use of Eq. (2.20). We next assume that the solution x 1 This section need not be studied unless the other sections in the book on transient phenomena are to be studied. II.6] RESPONSE TO TRANSIENT FORCES 43 is also of this-form : ^ v : - x ® = f-„ "X(fi>)e~ iut 'da> (6.1) so that if X is known x can be determined; or, vice versa, if x is known, X can be obtained by the related equation 2xJ_ X(<a) = y I x(t)e iu>t dt By this mathematical trick we can change from a differential equation relating x and / to an algebraic equation relating X and F. Finally, Eq. (6.1) can be used to obtain x. This method may seem rather mysterious and long winded. Actually, it is a mathematical restatement of a rather simple principle. Equation (2.19) says that a transient force can usually be expressed as a limiting sum {i.e., an integral) of a whole series of component simple harmonic forces of differing frequency (w/2tt) and amplitude F((a) dca. The individual equations for each of the component forces are solved to find the steady-state amplitudes X(a)) du>, and these are then finally combined by the limiting sum of Eq. (6.1) to give the required displacement x(t). To obtain the transient motion, we analyze it into its component simple harmonic steady-state motions and then recombine at the end. In most problems of practical interest, we can assume that the system is at rest at equilibrium until t = 0, when the transient is suddenly applied. Therefore, we can assume (at least for the time being) that the functions fit) and x(t) differ from zero only f or t > 0. In this case we can use Eq. (2.21) to suggest a simpler form for X X(ip) = 2^1 x(t)e-*" dt (6.2) where p = —i<a and the real part of p is positive. There is a similar equation giving F(ip) in terms of f(t). One general relation of great usefulness is the expression for the Fourier transform of (dx/dt). This can be obtained from Eq. (6.2) by integration by parts, or from Eq. (6.1) by differentiation inside the integral sign, if this is permissible. 2xJ \dt) e~ pt dt = pX '= — icoX (6.3) Additional terms must be added if x is not zero for t < 0, 44 THE SIMPLE OSCILLATOR [II-6 Transients in a Simple System.— By the use of these formulas we can transform the general differential equation m w + Rm % + Kx = m * £» F(o})e ' iwt do} into the algebraic equation {K - ioiR m - co 2 m)X(co) = F(u) by multiplying both sides of the differential equation by (l/2ir)e~ pt and integrating over t. The differential equation, of course, corre- sponds to a simple mechanical system of mass m, resistance R m , and stiffness K, acted on by force f(t). The equation relating X and F is a familiar one, dealt with earlier in our discussion of steady-state response to a simple harmonic force. The equation X(co) = F(w) = F{0,) (6.4) K — iwRm — co 2 ra —ioiZ{u>) where Z(co) = R m — i (com J is simply the expression for the amplitude of the component X(co), having frequency co, in terms of the component i^(co) entering into the integral for the total transient force. The final expression for the displacement and velocity of the simple system acted on by force /(0 is x O = J -TTn^ y w = ty x du (6.5) J- =<, — 2coZ(co) J- «» Z(co) where F(p) = ^j Q fV) e - pt d *> V = -t« These equations are valid if the imaginary part of co is always positive in the integration and if x, v; and / are all zero for negative values of t [this incidentally means that F has no poles above the real axis; see Eq. (2.15)]. In order that these integrals converge there must be certain obvious restrictions on the way Z behaves as co goes to infinity- To solve the contour integrals for x and v we must investigate the poles of the function F/(—iaZ). The poles of F(w) cannot be investi- gated until we choose a form for f(t), but we can discuss the factor l/(— icaZ). For this factor to have poles, the function — z'coZ(co) must have zeros for some values of u>, real or complex. 11.6} RESPONSE TO TRANSIENT FORCES 45 Complex Frequencies. — We have thus arrived at a broadening of our concept of impedance. Originally Z was defined as the complex ratio between a simple harmonic force of frequency (co/2r) = v and the corresponding steady-state velocity. We now are keeping to the same definition, but are broadening the concept by allowing o> to be a complex quantity, with real and imaginary parts. In particular, we are searching for those complex values of co for which (— zcoZ) is zero. For a. simple mechanical system ( — ioZ) = — co 2 ra — iu>R m -f- K. This can be factored to make the zeros apparent (—iooZ) = — m(co - (0/ + ik)(ca + co/ + ik) (6.6) where k = (R m /2m), u> f = co 2 , — k 2 , co§ = (K/m), in accordance with Eqs. (4.1) and (4.2). The values of co for which this function goes to zero therefore have for a real part plus or minus (2x) times the frequency of free vibration of the mechanical system and have for an imaginary part minus the damping constant of the system. At these values of co the function F/(—io)Z) will have simple poles, unless by extraordinary coincidence F has zeros or poles at these same values of oj (which eventuality we shall not consider). Calculating the Transients. — The poles of the function F/(—ioiZ) that are due to Z are therefore closely related to the transient oscilla- tions of the system discussed in Sec. 5. The points on the complex plane where these poles occur are symmetrically placed with respect to the origin, are both a distance co = \/K/m from the origin, and are both a distance k below the real axis. The residues of Fe-' Mt / (—iwZ) at these poles are and 2mo)f g — Utft _ kt F(o> f -i k) 2mco/ at co = — oif — ik > at w = co/ — ik As long as jP(co) has only simple poles, and as long as none of these coincides with the poles of l/(— iuZ), we can utilize Eq. (2.15) to write down the result for the displacement of the system x(t) = ir 1 x(t) = — — tf-*'[e*»/«iP(— co/ - ik) — er^'WCa/ - (2iri) times the sum of the residues oiFer^t/i—iwZ) at all simple poles of F on and below the real axis of co (for t < 0) ik)] (for t > 0) (6.7) 46 THE SIMPLE OSCILLATOR [II.6 The first term is the transient motion of the system due to the sudden onset of F(t) at t = 0. The terms due to the poles of F(w) are a sort of generalized steady-state motion and depend on the specific form <*/(*). The corresponding expression for v{t) is v{t) = (for t < 0) v(t) = JL- e- kt [(a f + i^e^Fi-oif - ik) + (w/ - itye-^f'Ficof - ik)] ^ (6.8) {2iri) times the sum of the residues of F e -iut/z at al i po i es f p on or b e i ow (for t > 0) the real axis of o> Further discussion requires choosing specific forms for / and F. Examples of the Method. — We shall first take a case that we have touched on earlier, the case where a simple harmonic force of frequency v = (a/2r) is turned on at t = 0. The force function chosen is f /0 (*<0) J \F e- iat (t > 0) According to Eq. (2.18), the corresponding Fourier transform is (F /2ti) F(co) = - co — a To find the resulting displacement of the system we insert the expression for F into Eq. (6.7) and turn the algebraic crank. After a number of turns, the first part, due to the poles of l/(— icaZ), turns out to be F e- kt ( e™' 1 e~ iaft \ 2mo)f \co/ + a + ik o> f — a — ik) while the term due to the pole of F is F e- iat /[ — iaZ(a)]. Further sub- stitution of expressions for co , oi f , and k in terms of K, m, and R m results in the final equation for a; ... F e- iat F e~ kt , '... . , — iaZ(a) 2a) f [—iaZ(a)] ' + («/ + a + ik)e-™*\ (for t > 0) which is to be compared with the results of page 29. The final numerical results can be obtained by taking the real part of the expres- sions for f(t) and x (t). II.6] RESPONSE TO TRANSIENT FORCES 47 No doubt this result could be obtained with less trouble by the simpler methods outlined in Sec. 5. However, we shall find later that the present method is more useful in discussing transient effects in more complicated systems, where the simpler method breaks down. The present examples are discussed so that we can become familiar with the method. The Unit Function. — Another force function which will be useful to study is that of a force suddenly applied at t = and maintained at a steady value thereafter. From Eq. (2.16) we see that the Fourier transform of *«-««-{? J > g » ™ - " (as) (6.9) The function u{t) is called the unit step function, often used in transient problems. The resulting expression for the displacement of the same simple system due to the application of the unit step force is therefore (6.10) 1 1 x u (t) = I — — ~- e~ kt cos (<j) f t — (*<0) a) (*>0) tan a — — <*f \/w§ — k 2 The first term is the "steady-state" displacement from equilibrium caused by the unit force after t = 0, which comes from the residue at the pole of F(«) (at o> = 0). The second term is the transient oscil- lation due to the sudden application of the force at t = 0, which comes from the residues at the poles of l/( — icaZ). One sometimes wishes to know what happens to a system that is held aside from equilibrium for a long time and then is suddenly released at t = 0. A force producing this behavior would be f(t) = 1 — u(t), a unit force applied steadily until t = and then released. The response for such a force would be *w = (A) - «.<« = i w* ? , „ , ('<« \ K J I (wo/Ka> f )e- kt cos (<a f t -a) (t > 0) If the force holding the system away from equilibrium is F instead of unity, the corresponding motion is x = F x h (t). If the force is an impulsive one, applied instantaneously at t = 0, the integral of this force over time (the impulse of the force) must be proportional to the unit step function. We can define a unit impulse function 8(t) as one that is zero for every value of t except zero and 48 THE SIMPLE OSCILLATOR [H.6 that has there such a value that jl ^ 5(r) dr = u(t) (6.11) The function 8(t) is sometimes called the Dirac delta function. Look- ing back at Chap. I, we see that the function defined in Eq. (2.14) approaches the function 8(t) as a approaches infinity. It is therefore useful to compute the response of the simple mechan- ical system to a force of the type defined by Eq. (2.14). The corre- sponding Fourier transform is n/ \ a 1 ^(co) = - 2ri co + ia and the expression for the displacement for t > is ae~ at a 2 m + aR m + K aer kt r e %w f t e -u, f t i 2mco/ Leo/ + i(k — a) to/ — i(k — a) J The limiting case for a — > °° gives the response of the system to an impulsive force 8(t) (0 (*<0) (6.12) XsKl) \ {l/mo> f )e- kt sin (a f t) (t > 0) The responses of the system to the two unit functions u{t) and 3(0 might be considered to be the basic transient responses. They are closely related as Eq. (6.11) indicates. If it were not that u and 5 are discontinuous functions, we could say that 8(t) is equal to (du/dt). From a formal point of view this is correct, since the limiting value of the quantity © |J Mfl - <t ~ A)] as A goes to zero has the properties of the function 8{t). The relation- ship is also apparent between the two transient responses, for x$ = dx u /dt, as a little close reasoning can prove. General Transient. — The two basic transient responses can be utilized to compute the response of the system to any more compli- cated force function. The delta function has the general property that, for any function /, f~J<?) W -r)dr= f(t) H.6] RESPONSE TO TRANSIENT FORCES 49 and, by integrating by parts, the step function has the property /. _ M ^-r)^tfr=/(*) The meaning of these integrals (if they have any!) is that a smooth continuous function of time can be considered as being built up of an infinite sequence of impulse functions, the one at time r having ampli- tude /(t), and so on. Or the function can be built up stepwise, the increase at time t being proportional to df/dr. By utilizing the func- tions xs(t) and x u (t), the responses to the unit functions 8(t) and u(t), we can write down the response of the system to any force f(t) as either x(t) = j^ J{r)xi{t - r) dr or ■ > (6.13) x{f) = J ^ %j& Xu (t - r) dr The second form should not be used with discontinuous force functions (where df/dr is infinite) unless one adds terms proportional to x u (t — t„), where r n is the location of the discontinuity in time and the propor- tionality factor equals the magnitude of the discontinuous step in /. In a manner of speaking, Eqs. (6.13) form an alternative method to Eq. (6.7) for obtaining the displacement function x{t) — although Eq. (6.7) must be used to obtain the functions x s , x u used in Eqs. (6.13). Actually, as might be suspected, these equations are two equivalent ways of writing the same fundamental equation. This can be seen by writing out the integral for x u in terms of Eq. (6.5) ^ ) = 2^J_ -^^ do) C — Kilt and, by differentiation, o — lut if" e Substituting this into Eqs. (6.13) and changing the order of integra- tion yields and, similarly, ^ (6.15) 50 THE 'SIMPLE OSCILLATOR [II.6 These expressions might be -considered to be fundamental to both Eqs. (6.7) and (6.13), the final form depending on whether we inte- grate first with respect to r or first with respect to w. If we integrate initially with respect to w, we obtain the first of Eqs. (6.13), and the second equation is obtained by integration by parts. If we integrate initially with respect to t, Eq. (2.20) shows that the factor in brackets is F(u>), and the result is just Eq. (6.5), from which we have obtained Eq. (6.7). Which of the forms for calculation of x(t) is to be used in a given problem depends entirely on the relative ease of integration of the two forms. Some Generalizations. — We can now review the method of com- plex integration (sometimes called the method of operational calculus), so as to point the way to the application to the analysis of more com- plex systems. Suppose that we have a force f(t), which is zero for t < 0, applied at some point to a mechanical system. To obtain the response to the general force, we first obtain the steady-state response of the system to a simple harmonic force of frequency v = (co/2ir). All that is needed is the ratio between the force (applied at point 1) and the steady-state velocity of some point of the system (either the same point 1 or another point 2) which is called the impedance Zn(co) or Zu(eo), a function of co. The ratio between the force and the corre, sponding displacement is -to2n(w) or —io)Z 12 (a). Often it is more useful to obtain the reciprocal of the impedance, the mechanical admittance Fn(«) or Y 12(a), the ratio between the velocity and the force. The ratio between displacement and force would then be Y (&>)/(—•&'«). This is a minor point, however, the main objective being to obtain the relation between the response of the system and the force for steady-state motion of frequency (co/27r). A possible second step is to analyze the transient f(t) into its simple harmonic components by obtaining the Fourier transform F(cj), according to Eq. (2.21). The final response to force f(t) is then obtained in terms of contour integrals of the form given in Eqs. (6.5), where Z(«) may be either Zn or Z i2 , depending on whether the response of the point of application or that of another point is required. An alternative form of the second step is to find the response of the system to an impulsive force 5(t), applied at point 1 where, again, Z may be Z n or Z 12 . The response to any force f(t) II.6] RESPONSE TO TRANSIENT FORCES 51 applied to point 1 is then given by x ® = Jl „ /( T )**(* ~ T ) dr, v(t) = f_ n f(T) Vi (t - t) dr (6.17) These equations are of very general usefulness, as we shall see later in this volume. The important steps in the process, for any type system, are finding the impedance or admittance function, giving the ratio between the simple harmonic force applied at some point of the system and the corresponding steady-state response of the same or another point, and computing the resulting contour integral, in the form of Eqs. (6.5) or Eqs. (6.16). It should be pointed out that the integrals of Eq. (6.16) do not converge unless Z and coZ go to infinity faster than the first power of co as co goes to infinity; otherwise Eqs. (6.15) must be used to find the correct solution. For instance, if Z = R, a constant, Eqs. (6.16) diverge. However, insertion of f(t) = 8(t) in Eq. (6.15) and utiliza- tion of the definitions of the Fourier transform would indicate that v$(t) = (l/R)8(t) and, by integration, x&(t) = (\/R)u(t) in this case. Sometimes a few tricks must be used to obtain results. For instance, if Z were R + (iK/u>), we could utilize the obvious equations 1 = (co/fl) = 1_ _ (iK/R 2 ) Z co + i(K/R) R co + (iK/R) The integral for x« then becomes (using the "definition" of u) This integral is well-behaved, and we finally obtain, for t > 0, Xi(t) = 4 u(t) - 4 + 4 e-twt = ( ^ J e-c*/*)« R " w " R "*" R The corresponding velocity can be obtained by differentiation, or by using the same trick vs{t) = G) m ~ {§*) e ~ {K/m > {t > o) Later in the book we shall have further occasion to utilize this trick. Laplace Transforms. — Another way of describing the procedure utilizing the impulsive response is applicable to all systems we shall encounter in this book. We first compute the steady-state behavior of the system under consideration (simple oscillator, string, diaphragm, 52 THE SIMPLE OSCILLATOR [H.7 or room full of air) to a driving force (1/2tt) e~™ 1 applied at the point (or area or volume) we wish to subject to transient force. After com- puting the behavior (displacement, velocity, shape of string, etc.) we obtain the corresponding response to a unit impulsive force by integrating the expression for this behavior over a> from — oo to + » .. . The response for a general transient force can then be obtained by using Eq. (6.17). This technique is of general utility, and we shall refer to it from time to time throughout this book (keeping our eyes open to be sure that the integrals converge, of course). Equations (6.16) are of the general form f(t) =7^J_ n Qi-iah-^da (6.18) By Eq. (2.20) we see that $> can be expressed in terms of an integral of /. Whenever f(t) is zero for t < 0, we obtain the following integral for 3>(p), after substituting p for — iw. *(P) = f ~ e~ pt f(t) dt (6.19) which is related to Eq. (6.2). The function 3>(p) is said to be the Laplace transform of f{t) <*> = £(/) In many cases where it is difficult to evaluate Eq. (6.18) directly in terms of residues [when $ has other infinities besides simple poles, as for instance a factor Vco + a, or where the convergence of (6.18) is questionable] it turns out to be better to try to find an f(t) which, when substituted in Eq. (6.19), gives the required form for $. In order to aid in this procedure of working the problem backward, several tables of Laplace transforms [tables of f(t) with their corre- sponding $(£>)] have been published; these can be used just as tables of integrals are used. Such tables must be used if 3>(p) has terms with p = —iu involving radicals. This will be discussed again in Chaps. VI and VII./ /" - --- l ) 7. COUPLED OSCILLATIONS We must now treat the case that we avoided in the last section, the behavior of two oscillators coupled tightly enough together so that we cannot neglect the feedback of energy from the driven system to the driver. ■ In this case both oscillators are on an equal footing; we cannot call lone the driven and the other the driver, since each is affected by the other. We shall expect that the results obtained n.7i COUPLED OSCILLATIONS 53 from our analysis will not be much different from those of Sec. 5 except when the frequencies of the two oscillators are nearly equal, for in any other case the amplitude of motion of one oscillator will be much smaller than that of the other, and the feedback will not be large. The General Equation. — The general case will be treated first, however. We shall call one of the coupled oscillators No. 1, call its mass mi and its displacement from equilibrium #i,.and call the other oscillator No. 2, with mass m 2 and displacement x%. When x\ and x 2 are both zero, the system will be in equilibrium. We neglect friction , Equilibrium , I Positions \ \ rC"" 1 f fill ■ K 3\ ..; '"' Masses -■' \ "'Stiffness Constants m, \\ K 3 k; _|| — r^TR^p — I k; Fio. 6. — Simple example of two coupled oscillators; two masses connected by springs. The effective stiffness constant for mi is K\ = K\' + K%; that for m, is Kt = Kt' + Ks. The analogous electrical circuit is also shown. A force applied to mi is analogous to the voltage source F\, a force on mt to the source ^2. for the time being, since it will only confuse the problem. It can be considered later if need be. Suppose it turns out that if we keep x% equal to zero, the force on mass wii is equal to — K\X X (this will be true if Xi is small enough). Then if we clamp oscillator 2 at x 2 — 0, oscillator 1 will vibrate with a frequency v\ = (1/2jt) V-Ki/wi. Similarly, if we clamp mass mi at xi = 0, the restoring for ce on m 2 is — K^x%, and its frequency of vibration is v z = 0-/2w) \jK%/m^. A displacement of mass mi, however, produces a force on m 2 , for this is what we mean by coupling. Suppose that this force is Kzpc\. Then owing to symmetry of the system, the force on mi due to a displacement of m 2 is K^. The 54 THE SIMPLE OSCILLATOR [H.7 constant K s is called the coupling constant. It is usually very small compared with Ki or K 2 . We shall not lose generality by assuming that v\ is not less than v 2 . The equations of motion of the two masses are d 2 xi d 2 x 2 T . , _ m l ~^2 = —-^1^1 + K SX2, m 2 -J72 = -K 2 X 2 + K3X1 If we change the scale of the displacements to allow for the difference in mass of the two oscillators, letting x = xi V^i and y = x 2 Vmi, then we can write these equations in a simpler form. ~ + Wv\x = 4ttVV, ^f + ^v\y = 4xV 2 x 4tt 2 ^ = K, \ ( 7 -D If one of the oscillators is clamped down, then the other will oscillate with simple harmonic motion; but if both oscillators are allowed to move, the resulting motion will usually not even be periodic. Curves c of Fig. 8 show a typical case. The curves for x and y as functions of time are certainly not sinusoidal. This is annoying, for we do not like to have to develop and define some new sort of function to express the motion. Simple Harmonic Motion. — So we first ask whether it is not possi- ble to start the two masses in some special way so that the motion is simple harmonic, even though the motion in general is not. Of course if the motion is to be simple harmonic, both oscillators must be vibrating with the same frequency. We therefore try the solution x = Ae- 2 * ivt , y = Be- 2 ™* in Eqs. (7.1) and see that it satisfies these equations if the coefficients A and B are related in the following manner: (v\ - v 2 )A = ^B, (?f - v 2 )B = ^A. From these two equations we can find the ratio of A to B and the value of the frequency v. s Multiplying one equation by the other gets rid of both A and B and leaves an equation for v, v i — (v\ + v \)v 2 + v\v\ — n A = 0, having for a solution v = [*(»? + 4) +i VW - v\y + 4 M 4 ]* ± i V(K 1 m 2 - K 2 mxY +.4iqm 1 m s l (7.2) H.7] COUPLED OSCILLATIONS 55 Thus there are two possible frequencies of simple harmonic oscil- lations for the combined system, and we shall see that (unlike the simple oscillator) which of the frequencies it will vibrate with depends on how we start the system into motion (i.e., what the values of A and B are). Neither of the allowed frequencies is equal to either of the natural frequencies v x or v 2 of the individual oscillators taken separately. Suppose that we call the frequency involving the plus sign in front of the radical v + and that involving the negative v_. The value of v- is smaller than either v\ or v 2 , that of v+ is larger than both vx and v 2 . Coupling always spreads apart the natural frequencies. Normal Modes of Vibration. — We can say that although the gen- eral motion of the system is not periodic, nevertheless if the masses are started into motion in just the right way so that the amplitudes of motion of m x and m 2 are related by B+ = A + (v\ — v\)/n 2 = A + n 2 /{v\ — v\), then, and only then, will the system oscillate with simple harmonic motion of frequency v+, these ratios between the amplitudes of motion A+ and B + remaining the same throughout the motion. Similarly, if the masses are started so that their amplitudes are related by the equations #_ = A-(v\ — v 2 _)/y. 2 = A-/i 2 /(i>% — vt), then, and only then, will the system oscillate with simple harmonic motion of frequency V-. If the motion is started in any other way, there will be no permanent ratio between the displacements of the two masses, and the motion will not be periodic. These two especially simple ways of motion of the system are called its two normal modes of vibration. By a little juggling of terms, these motions can be represented as follows: If the frequency is v+, x = C+ cos a 6-2""+', xi = [a+ cos (%rv + t) Vmi + 6+ sin (2irv+t)] cos a y = -C+ sin a er 2 ***-**, x 2 = —= [a+ cos (2irv+t). V m 2 + b+ sin (2irv + t)] sin a If the frequency is i>_, \ (7.3) x = C_ sin a er 2ri '-*, xi = — j= [a_ cos (2tvJ,) VWi + b- sin (2rv-t)] sin a y = C_ cos a er™'-*, x 2 = —== [«_ cos (2ncvJL) \/m 2 + 6_ sin (2irv_t)] cos a 56 THE SIMPLE OSCILLATOR [H.7 where the angle a has been introduced to unify and simplify the nota- tion. It is related to the v's, etc., as follows: tan< = {v\ - vj) = _ ("i - vi) tan 2a = 2m 2 (v\ - v\) (v\ - v_) n - "2 C+cosa = A+, C+sina = — B + , C-Smet = A-, C-Cosa = J5_ One may ask why we are so interested in these normal modes of vibration when they are such a specialized way for the system to oscillate. The answer is that as soon as these normal modes are found the problem of determining the general motion of the system is suddenly seen to be quite simple. For it turns out that the general motion can always be represented as a combination of both the normal modes of vibration. The general solution of Eqs. (6.1) is (7.4) \ y \ \ \ \ \ a. \ \ x = C+ cos a e-*™* 1 + C- sin a er M '-* y = — C+ sin a e- 2wir+t + C_ cos a e' 2 ^"- 1 ■7 as can be verified by substitution in (011). The displacement of each individual mass is a combination of two oscillations of different frequen- ciesj which results in a nonperiodic mo- tion (except in the rare case where the ratio of v+ to v_ equals the ratio of two integers, when the motion will be peri- odic no matter how the system is started. Why?). We can represent the general mo- tion of the system in a still simpler manner. Let X = C+er 2 ™* 1 and Y = C-e- 2 ™-'. Then x = X cos a + Y sin a, y = — X sin a + Y cos a. These equations are just the ones used in analytic geometry to represent the transformation of the coordinates of a point in a plane with respect to the axes x and y to a new set of axes X and Y inclined at an angle a to the first set, as shown in Fig. 7. Suppose that we represent the position of the system at any time by a point on a plane whose abscissa is x, the dis- placement (in the proper scale) of the first mass, and whose ordinate is y, the displacement of the second. The motion of the system \ \ Fig. 7. — Transformation to nor- mal coordinates for two coupled oscillators. n.7] COUPLED OSCILLATIONS 57 corresponds to a motion of the point on the plane. What we have said above is that as the point moves, corresponding to a general sort of oscillation of the system, the projection of the point on the rc-axis (the displacement of the first mass) moves back and forth in a complicated nonperiodic way, and so does the projection of the point on the #-axis (the displacement of the second mass). However, the (c) f— " p IG g. — Motion of two coupled oscillators. Curves to the right show the dis- placements x and y as functions of time; those to the left show the path of the point representing the system in the x-y plane (configuration space). Cases (a) and (6) show the two normal modes of vibration, when the system point travels along a normal coordinate. Case (c) shows the general type of motion. projection of the point on the X-axis always moves back and forth with simple harmonic motion of frequency v+ (since X = C+er 2 ™* 1 ) with an amplitude of motion C+, and the projection on the F-axis moves with frequency ?_, with amplitude C_. Only when the system is so started that its point moves along the X-axis or along the F-axis is its motion periodic (see Fig. 8 for examples). The plane in which 58 THE SIMPLE OSCILLATOR [H.7 the point moves is called the configuration plane of the system, and the axes X and Y are called the normal coordinate axes for the system. The general motion of the system can be written in the form given in (7.4) or, less symbolically, as si = —7=^ [a+ cos (2tv + {) + b+ sin (2wv + t)] sin a , , n + —7= [«- cos (frcvJL) + &_ sm (2jri»_i)] VWi [A+ cos a cos (27t»' + ^ — f> + ) (7.5) + yl_ sin a cos (2x^_/ — $_)] a: 2 = _ -~Z. [o+ cos (Sto-v+O + 6 + sin (2irM)l COS £tf H 7= [0- cos (2x^-0 + 6_ sin (2jrv.J)l VW-2 = — 7=[~~ A + sina cos (2x^+2 — $+) + A_ cos a cos(2rv_^ — $_)] The general solution involves four arbitrary constants a+, b+, a_, and 6_, or A+, <£> + , ii_, and $_, whose values are fixed by specifying the initial displacements and velocities of the two masses. Energy Relations.— Another example of the special simplicity that is given to the equations of motion when normal coordinates are used is the expression for the energy of the system. The kinetic energy is, of course, frn^dx^dt)* + im 2 (dx 2 /dt)\ To find the potential energy, we find the amount of work necessary to push the system from equilibrium to the position where the displacements are X! and x 2 . Since friction is negligible, we can do this process in any order that we wish and get the same result. We choose to push m x out first, pushing against a force —K x xi and so requiring an amount of work JKiXt dx x = \K x x\. We next push m 2 out, pushing against a force — K 2 x 2 + K 3 X! (by our definition of K z ) and doing an amount of work \K 2 x\ - K3X1X2. The total energy is therefore mi \w) + m2 \w) + KlX * + K2X > ~ 2K ^^] A3f) + (off + w ^ + 47r2 ^ 2 " 8*V*v] an expression complicated by the term in xy. 2 = 1 2 H.7] COUPLED OSCILLATIONS 59 If we substitute the expressions for x and y in terms of X and Y, expressing the functions of a in terms of functions of 2a, and use the formula for tan (2a), we find a simple formula for W w - "2@ 2 + 3©' + ***+** + *•,._ r- (7.6) without any term in XY. If X = A+ cos (2irv + t — $ + ) and Y = A- cos {2-rcvJt — <£»_), as indicated in (6.5), then W = 2ir 2 (ulA% + vlAl) the sum of the energies of vibration along the two normal coordinates. The Case of Small Coupling. — Now that we have obtained a general solution for the motion of two coupled oscillators, it might be well to see how the results correspond to the discussion of forced oscillations given in Sec. 5. As we said at the beginning of this sec- tion, whenever the coupling is small and the frequencies of the two oscillators are not equal, the amplitude of motion of one will be much larger than that of the other. The oscillator showing the smaller motion can then be considered as the driven oscillator, and its ampli- tude of vibration will be given by Eq. (5.3) of Sec. 5. For instance, if m x is to be the "driver," its amplitude of motion should be larger than that of m 2 . If the coupling is small, a is small, and the case we should analyze is that for the frequency v + , where (vW cos(a) \ m 2 v\ — v\ \ m 2 Xl ~ I .. rzr ) cos(a)e- 2a " 4 '''+' and #2 K z xi £ 2 - m 2 (2irv + ) 2 Since a is small, x 2 will be smaller than xi unless mi is very much larger than m 2 . From the point of view of m 2 , the quantity K3X1 is just the force f(t) applied to m 2 through the coupling spring, due to the motion of xi. The driver is oscillating with a frequency v+, so that/ = F^e~^ iv ^ — K3X1 where F = (i^ 3 C + /\/wi) cos a. The divisor is proportional to the mechanical impedance of m 2 at the frequency v + , since — 2nriv+Z 2 — K 2 — m i (2rv + ) 2 60 THE SIMPLE OSCILLATOR [H.7 Therefore the expression for the displacement of m 2 when it is being "driven" by mi at a frequency v+ is — Zrlv + Zi2 which is to be compared with Eq. (5.3). The Case of Resonance. — However, our primary purpose in this section is not to check the formulas given in Sec. 5 but to find out what happens in the cases where the formulas of that section do not hold, where the feedback of energy from m 2 to mi is appreciable. As an example of this let us consider the case where the two oscillators have the same natural frequency (Vi = j> 2 ) and where the friction is negligible. If we try to use the formulas of Sec. 5 in this case, we find that they predict that the amplitude of the driven oscillator will be infinite. The infinity simply means that the amplitude of motion of the driven mass will become large enough to absorb a large fraction of the driver's energy, so that the formulas of Sec. 5 cannot be used. To solve this problem, we must use the formulas of the present section. According to Eq. (7.2), if v x — v i} V+ = Vv{ + M 2 = vi + ^ + ,2 v- = V4 - M 2 = vi - 2^ + * ' ' » a = 4 If ju is much smaller than v h only the first two terms in the series expansions of the square roots need be considered. Therefore, if two similar oscillators, each of natural frequency vx, are coupled together, they can no longer oscillate with frequency vi but can oscillate with a frequency either (^ 2 /2i'i) larger or this same amoun t smaller than v\. If they are started so that x\ = —x% v / m 2 /mi, then the system has only the higher frequency; but if they are started so that Xi = Xi -\/'mz/m\, then it has only the lower frequency. If the system is started in any other way, the motion will be a combina- tion of both frequencies. If the system is started by pulling mass 1 aside a distance x , while keeping aj 2 equal to zero, and letting both masses go at t = 0, then the values of the constants to be used in (7.5) are 6+ = 6_ = 0, a+cosa/VW = a-sma/s/mi = (x /2). The displacements of the two masses will be represented by the equations n.7] COUPLED OSCILLATIONS 61 = X COS I IT — t) COS (2TVit) (7.7) = VS Xo ^Cy sin(w) if the trigonometric equation for the sum or difference of two cosines is used. The second form of the equations shows that, if the oscil- Fig. 9. — Motion of two coupled oscillators having the same natural frequency. Solid curves show the displacements of the two oscillators as a function of time and illustrate the alternation in amplitude. lators are started in the manner described above and if the coupling is weak, the motion is like an oscillation of frequency v\ whose ampli- tude of oscillation itself oscillates with the small frequency 0u 2 /2i>i). This is illustrated in Fig. 9. Such motion is not simple harmonic motion, since the amplitude changes with time. As the first forms of Eqs. (7.7) show, it is a combination of two harmonic motions whose frequencies differ by a small amount, so started that they at first reinforce each other for Xi but after a while get out of phase and cancel each other out, and so on. We notice that when the amplitude of motion of mi is large, then that of w 2 is small, and vice versa. Transfer of Energy. — We can compute the approximate value of the free energy of oscillation of m x and m 2 by using formula (4.6), W = 2w 2 mv 2 A 2 (t). Of course, it is not strictly correct to speak of the energy of only a part of a system, and it is not right to use the 62 THE SIMPLE OSCILLATOR [II.7 formula for the energy of a simple oscillator to compute the energy of a coupled oscillator; but if the coupling is weak, the formula has some meaning. We find that the average "free" energy of mass mi is 27r 2 mi^x§ cos 2 (tt/xV i 'i) an d * na * °f m 2 is 2r 2 miv\xl sm 2 (ir(i 2 t/vi). It indicates that at t = all the energy of the system resides in mass mi but that as time goes on the energy of the first mass diminishes while that of the second increases until all the energy is 'transferred to m 2 . Then the flow of energy reverses, that of mi building up again, and so on. The total energy of the system is 2r 2 mivlxl, a constant, as it should be. This example illustrates how different from the usual forced oscil- lations can be the motion of two coupled oscillators if a large coupling or an equality of frequencies of the two vibrators makes possible a large transfer of energy from one to the other. It also illustrates an obstacle encountered in building a resonating system that will respond very strongly to only one frequency. In the last section it was pointed out that such critical response required a system with very small friction. We see now, however, that if we make a system with negligible resistance tuned to a natural frequency v\ and try to drive it by a system oscillating also with a natural frequency v\, the result of the coupling will be that neither driver nor driven system can oscil- late with frequency vi but must oscillate with frequency v\ ± (n 2 /2vi). The very fact that the response is so large at vi destroys the possibility of vibration at that frequency. Forced Vibrations. — We also wish to find the response of the sys- tem of two masses to an applied force. Here again the analysis goes from the simple harmonic case to the more complicated transient forces. We first find the response (i.e., the displacement and velocity of mi and m 2 ) to a force F = i'V - "" applied to m x or m 2 . If it is applied to m h Eqs. (7.1) become ^f + o*c - x 2 V = a*r"; -^ + o>\y - x 2 x = (7.8) co? = (ifi/rai); cof = (i£ 2 /m 2 ), x 2 = \K z /\/m~m^\ a x = Fo/y/rni Solving for the steady-state response consists in substituting x = xtf'™* and y = y e~ iut in Eq. (7.8) and solving for x and y . The results, when reduced back to expressions for the actual displacements, are _ Fo (cof - to 2 )*-*" _ Kitr 1 ** Fp •^l ... f O 9 \ / 9 \ 5 ^2 mi (co 2 - co 2 .)^ 2 - «i)' (co 2 - co2.)(co 2 - coi) mim 2 (7.9) H.7] COUPLED OSCILLATIONS 63 where co + — 2ttv + , w_ = 2irv-, with the v's given in Eq. (7.2). «i = i(" 2 i + «?) + i V(col - col) 2 + 4 X 4 «L = *(«? + col) - i V(«J - cof) 2 + 4x 4 If we had applied the force to ra 2 , the corresponding response would have been *! ™ /..2 .* \/..?> «\> # 2 — Wim 2 (to 2 - oj2.)(co 2 - »£.)' " m 2 (co 2 - ^ + )(<o 2 - «i) (7.10) The symmetry of the expressions is apparent. Another way of writing the response is specifically in terms of impedances corresponding to the circuit of Fig. 6. We define the blocked impedance of Wi as the impedance when m 2 is held fixed at equilibrium, Z± = —iwmx + i(K\/(a). The corresponding blocked impedance of m 2 is Z 2 = — *com 2 + i(K 2 /(o). The coupling can be expressed in terms of a mutual impedance M — i(K z /(S). By differ- entiating Eqs. (7.9) with respect to time, and by subsequent algebraic juggling, we obtain expressions for the ratios between the simple harmonic force applied to m\ and the corresponding velocities of m t and m 2 „ F ZrZ % - M\ F ZrZ 2 - M* ■ &\\ = — = ~ , ^12 = — = jj (.7.11) V\ Z 2 Vi M Z\\ is called the mechanical input impedance of the system at point 1; Z 12 is called the mechanical transfer impedance between points 1 and 2. If the force is applied to ra 2 , the corresponding impedances are z 22 F ZrZi - M\ V2 Z X 5/21 _ F _ Vi " Z X Z 2 - M 2 M Z12 (7.12) This brings out the interesting fact that transfer impedances are symmetrical. The response at point 2 due to a force at point 1 is the same as the response at point 1 due to the same force applied at point 2. Sometimes this result is referred to as the principle of reciprocity. Resonance and Normal Modes. — Each of the impedances given in Eqs. (7.11) and (7.12) becomes zero at the two resonance frequencies v+ and v-. At these frequencies the displacements become infinite, as is shown in Eqs. (7.9) and (7.10). The infinity is due to our neglect of frictional forces; if a small amount of friction had been included, the displacements would have been large but not infinite. The ratio between y and x, when the driving frequency is v + , is x 2 /( w i — o>+) 64 THE SIMPLE OSCILLATOR [H.7 = l?/(A - "+), according to Eqs. (7.9). A glance at Eqs. (7.3) shows that this is just the ratio maintained when the system is in free vibration at frequency v + . In other words, when the system is driven by a simple harmonic force of frequency equal to one of the frequencies of free vibration of the system, its response is large, and the relationship between the motions of the parts (i.e., the configuration of the system) is the same as if the system were in free vibration at that frequency. One of the methods of finding the possible frequencies of free vibration of a system and the configuration of the corresponding normal modes is to drive the system by a simple harmonic force. The resonance fre- quencies are the frequencies of free vibration, and the configurations at resonance are those of the normal mode. In the present case, there are two natural frequencies of free oscillation, so there are two resonance frequencies. For three masses connected by springs there would be three normal modes and three resonance frequencies, and so on. When the present system is driven by a force of frequency consider- ably smaller than v-, applied to m x , both masses move in phase with the force, the ratio between x 2 and x\ being (K3/K2) = Ks/(K 3 + K'^), so that X\, the displacement of the point of application of the force, is greater than x 2 by a factor depending on the relative sti^ness of the two springs connecting with m 2 . According to Eqs. (7.9), as the driving frequency increases, the amplitudes increase until the resonance at v_, which is a lower fre- quency than either vi or p 2 , the resonance frequencies of each mass taken separately. For frequencies just above v-, both displacements are opposing the force (180 deg out .of phase). At v 2f the displacement of mi, however, becomes zero (if friction is not zero, xi is small, but not zero). At this driving frequency the input impedance Zw of Eq. (7.11) becomes infinite, since the blocked impedance Z 2 of m 2 is zero. What has happened is that at this fre- quency m 2 resonates when mi is held rigid. If we attempt to move mi at this frequency, m 2 will immediately absorb all the available energy with only an infinitesimal motion of mi. This corresponds to a parallel resonance in the analogous electrical circuit shown in Fig. 6. Above v 2 , X\ is in phase with the force and x 2 is opposed. At v = v+, higher than vi or v 2 , the amplitudes are again infinite. For higher frequencies, X\ opposes the force and x 2 is in phase with it. Transient Response. — A few examples of the application of opera- tional-calculus methods to the coupled system will be useful to discuss H.7] COUPLED OSCILLATIONS 65 here, just to begin to show how the method works in more complicated cases. Suppose that we compute the response to the impulsive force 8(t) applied to mi. From Eqs. (7.9) we see that the quantities (1/— icaZ) needed to compute the integral of Eq. (6.16) for the coupled systems are / 1 \ = 1 i<4 ~ co 2 ) = \ — iwZ u J mi (co 2 — co^Xco 2 -r coi) / 1 \ = 1 X 2 = "2" \-ia)Zi 2 J y/mmz (« 2 - co 2 h )(« 2 - coi) -ico Since we are not considering friction, all the poles of these quantities are along the real axis, being at ± co_ and + co+. The residues of (l/2x) (e _iu '/ — icoZn) at w = ±« + are — Vji 47rmico + \co 2 h — coi/ + 6 The residues at the poles « = ± co_ are obtained by interchanging co + and co_ in these expressions. A great deal of algebra and utilization of the formulas „ (co 2 . - CO 2 ) . , (cof - w 2 ) cos 2 a = f-± 14-; sin 2 a = f-f =^ (C04. — col) (co^ — coi) Y 2 £ sin (2a;) = sine* cosa = —^ — ^ u + lead one finally to the expressions for the response of the coupled system to an impulsive force applied to mi t < 0, xu = x 2 & = — icaZj t > 0, xi S (t) = ^ J „ d» 1 [cos 2 ** . , . sin 2 a . , ..1 = ^T sm («+0 H sin (co_2) Wi L co + T co_ J Ar J-» — tcoZ 12 I mim 2 l 03 + r 1 a COS a — Leo- sin a cos a — sin (co+tf) sin (<a-i) J These formulas are to be compared with Eqs. (7.5). What the oper- ational calculus has done is to fix the constants a+, &+, etc., to corre- 66 THE SIMPLE OSCILLATOR [II.7 spond to the initial conditions of m 2 at rest and m x being given a unit impulse at t = 0. Just after the impulse at t - 0, x°& and v-a are both zero; Xi» is zero but v-a starts at a value (1/wi), as it should for a mass ra x which has just been given a unit impulse. The unit impulsive response when the force is applied to ra 2 is obtained by interchanging the subscripts 1 and 2 in Eqs. (7.13). The response of the system to any force f(t), applied to mi is, from Eqs. (6.17) Xl (t) = J*^ f(r)x lS (t - t) dr) x*(t) = f'^ f(r)x»(t - t) dr (7.14) This analysis of a simple coupled system has had no important practical applications, but it has served to indicate the direction our results will take for still more complicated systems. We should expect to find complex systems (at least those coupled together by springlike forces) to have a number of resonance frequencies. If the system is started in just the right way, it will oscillate with simple harmonic motion, being damped out if there is friction. There should be several ways of starting the system to get simple harmonic vibration, each different way corresponding to a different normal mode and to a different natural frequency. In general, the system will vibrate with nonperiodic motion which is a combination of several natural frequencies. If the system is driven at one point by a simple harmonic driving force, the whole system will respond at that frequency. The steady- state response of any part will depend on the frequency and point of application of the force, the ratio between force and response being given in terms of the input transfer impedances of the system. From these impedances, as functions of frequency, can be computed the response of the system when subjected to a sudden unit impulse at t = 0; and from the impulsive response can be obtained the response to any transient force. We shall work out these same results for a still more complicated system, the simple string under tension, in the next chapter. Problems 1. A vibrator consists of a 100-g weight on the end of a spring. The spring's restoring force is proportional to the weight's displacement from equilibrium; if the weight is displaced 1 cm, this force is 10,000 dynes. The frictional force opposing its motion is proportional to its velocity and is 100 dynes when its velocity is 1 cm per sec. What is the modulus of decay of the oscillator? What is its decrement? What "frequency" do the vibrations have? What frequency would H.7] COUPLED OSCILLATIONS 67 they have if there were no friction? If the weight were originally at rest and then were struck so that its initial velocity was 1 cm per sec, what would be its subse- quent motion? What would be its maximum displacement from equilibrium? 2. The diaphragm of a loud-speaker weighs 1 g, and the displacement of its driving rod 1 mm from equilibrium requires a force of 1,000,000 dynes. The frictional force opposing motion is proportional to the diaphragm's velocity and is 300 dynes when the velocity is 1 cm per sec. If it is assumed that the diaphragm moves like a simple oscillator, what will be its natural frequency, and what its modulus of decay? The driving rod is driven by a force of 100,000 cos (2irvt) dynes. Plot a curve of the real and imaginary parts of the mechanical impedance of the diaphragm as function of frequency, from v = to v — 1,000 cps. 3. The diaphragm of Prob. 2 is driven by a force of 100,000 cos (2irvt) dynes. Plot a curve of the amplitude of motion of the diaphragm as function of frequency, from v = to v — 1,000 cps. Over what frequency range is this loud-speaker mass controlled? 4. What is the mechanical impedance of a mass m without spring or friction? What is the impedance of a spring without mass? What will be the angle by which the displacement of the mass lags behind an oscillating force? What will be the angle of lag of the velocity of the mass behind the force? What are the corresponding angles for the spring? 6. The diaphragm of Prob. 2 is coupled electromagnetically to an electric circuit by means of a coil of negligible resistance and inductance, whose coupling constant r = 10,000. An alternating-current emf of 10 volts and variable fre- quency is applied across the coil. Plot the current through the coil and the ampli- tude of oscillation of the diaphragm as a function of frequency, from v = to v ~ 1,000 cps. Plot the real and imaginary parts of the motional impedance of the coil as function of frequency, from v — to 1,000 cps. 6. An a-c emf of 10 volts and variable frequency is applied to the coil of Prob. 5. Plot the current through the coil and the amplitude of oscillation of the diaphragm as a function of frequency, from v = to v = 1,000 cps. 7. A mass m is attached to the lower end of a spring of stiffness constant K. The upper end of the spring is moved up and down with an amplitude Be~ iu>t , and the frictional force on the mass is proportional to the relative velocity of the mass and the upper end of the spring (dr/dt), where r - x — Be~ iu>t . Show that the equation of motion of the mass is m (iO + Bm {§) + Kx = {K - *«««.)**-*" show that the steady-state motion of the mass is R m + i(K/co) _ . , x = — — ■ — - — — Be~ iwt R m — i[wm — {K/o))] and that the phase lag of x behind the displacement of the top of the spring is taU_1 l * R m — ~ J + tan_1 ( K / wR ™)- What is the amplitude of motion of xt What is the phase and amplitude of x at very low frequencies? At very high frequencies? 8. A loud-speaker diaphragm has mass and stiffness that can be neglected in the useful frequency range. Its motion is opposed by a force equal to 400 v, 68 THE SIMPLE OSCILLATOR HI.? where v is the diaphragm's velocity. The diaphragm is coupled to an electrical circuit by means of a coil of resistance 25 ohms, inductance 10 mh, and coupling constant r = 10,000. An a-c emf of 10 volts, with variable frequency, is applied to the coil. Assuming that all the energy lost by the diaphragm through friction is transformed into sound waves, plot the sound energy radiated per second (in watts) as a function of the frequency of the emf, from v = to v = 1,000. 9. A loud-speaker is coupled to the electrical circuit by means of a coil of resistance 25 ohms, of negligible inductance, and of coupling constant r = 10,000. It is found that the additional impedance due to the motion of the diaphragm is equivalent to a resistance of 25 ohms, an inductance of 10 mh, and a capacitance of 1 juf, all in parallel. What are the mechanical constants of the diaphragm? If all the energy dissipated by the diaphragm goes into sound energy, plot the over-all efficiency of the loud-speaker-coil system {i.e., the ratio of the power radiated as sound to the total power dissipated by loud-speaker and coil) as a function of the frequency, from v = to v = 1,000 cps. 10. The sharpness of resonance of a forced damped oscillator is given by the "half-breadth of the resonance peak," the difference between the two frequencies for which the amplitude of oscillation is half that at the resonance frequency v . Prove that if the natural period of oscillation is negligibly small compared with 2w times the modulus of decay {i.e., if (fc/47r) is small compared with v ), then this half-breadth equals (V3A) times the reciprocal of the modulus of decay of the oscillator. What is the half-breadth for the diaphragm of Prob. 2? What would the frictional force have to be in order that the half-breadth may be 20 cycles? 11. Two oscillators, each of mass m and natural frequency v {i.e., if one oscillator is held at equilibrium, the other will oscillate with a frequency v ), are coupled so that moving one mass 1 cm from equilibrium produces a force on the other of C dynes. Show that if C is small compared with £x 2 mvl, and if one oscillator is held 1 cm from equilibrium and the other at equilibrium and both are released at t = 0, then the subsequent displacements of the masses will be * = cos (i^bo) cos (w) ' Vt = sin G^To) sin (27r "°° 12. Suppose that each oscillator of Prob. 11 is acted on by a frictional force equal to R times its velocity. Show that the modulus of decay of the oscillations equals (2m /R). 13. Discuss the forced vibrations of the coupled oscillators described in Prob. 12. 14. Three masses, each of m g, are equally spaced along a string of length 4a. The string is under a tension T dynes. Show that the three allowed frequencies and the corresponding relations between the displacements for the normal modes are v = v y/l - V|, if yi = ?/3 = 2/2/V2 v — vo, if y\ = —2/2 and y 2 = v = vo \A + Vi, if 1/1 = 1/2= —I/2/V2 where (77a) = 2^v\m. 16. An "x-cut" Rochelle salt crystal, mounted as shown in Fig. 5, has the dimensions d = 1 cm, 6 = 3 cm, a = 2 cm. The external load is a diaphragm in n.7] COUPLED OSCILLATIONS 69 contact with water, which has an equivalent mass load of 10 g, negligible stiffness, and a resistance of 400,000 g per sec (radiation resistance). Plot the real and imaginary parts of the electrical impedance of the crystal from v = to v — 10,000 cps for a temperature of 15°C. 16. What will be the open-circuit voltage across terminals A and B of the crystal of Prob. 15 when an oscillating force of amplitude 10 dynes and frequency 1,000 cps is acting on its upper surface? What is the voltage if the frequency of the driving force is 10,000 cps? Consider the temperature of the crystal to be 15°C. 17. A voltage 100 cos (2irvt) volts is impressed across the plates of the crystal of Prob. 15. Assuming that all the power dissipated in the resistance R m = 400,000 is radiated into the water, plot the power radiated as a function of frequency v, from v = to v = 10,000 cps. The crystal temperature is held at 15°C. 18. A simple mechanical system of impedance Z m = R m — t'[«m — (A /«)] has applied to it a force (0 « < 0) F(t) = < sin (at) [0 < t < (mr/a)] [ [(nr/o) < t] Use the results of Prob. 15, Chap. I and of Eq. (6.7) to compute the displacement of the system. 19. A dynamic speaker of the type discussed in Fig. 4 has negligible coil resistance R c and mechanical stiffness K (the other quantities, R m , m and L are not negligible). An impulsive voltage E = S(t) is impressed across the terminals of the coil. Calculate the current through the coil, using the Fourier transform method [see Eq. (6.14)]. Show that the motional emf is M 2wLm J- <e (w + wo + ik) (w — « + ik) | j e ~ kt sin (u t) (t > 0) where k = (R m /2m) and o>l = (T/Lm) — (R m /2m) 2 . What is the velocity of the diaphragm? What are the expressions if (R m /2m) 2 > (T/Lm)? If « = 0? 20. The speaker of Prob. 19 has constants m = 50, R m = 10,000, T - 10,000, L = 0.02. Plot the displacement of the diaphragm as a function of time for an impressed emf E = 108(f) across the coil. 21. Use Eq. (6.13) to compute the velocity of the loud-speaker diaphragm of Prob. 19 for an applied voltage. !0 (t < 0) E (0 < t < T) (t- > T) Calculate the expressions for the three time ranges, t < 0, < t < T and t > T. Plot the result for the case of Prob. 20, for E = 10 volts, T = 0.01 sec. 22. A crystal of Rochelle salt, mounted as shown in Fig. 5, has a mechanical load which is pure resistive, Z m = R m . An impulsive force F e = 8(t) is applied to the upper surface of the crystal. Using the formulas related to Eq. (6.14), com- pute the current through the equivalent circuit if terminals A, B are open, and 70 THE SIMPLE OSCILLATOR [II.7 show that the voltage across A, B is ir /•" e- iut dco . ' 27rR m C c J-» (<*+ i[(C c + C P ) aR m C c C P ] V ° ltS (t < 0) (l/10 7 ri2 m Cc) exp [-(Cc + C P )t/SR m C c C P ] (t > 0) = 1 23. The crystal of Prob. 22 has the following constants: C c = C p = 10- 10 , t = 10-», a = 10, R m = 10« The applied force is !0 (t < 0) 10 dynes (0 < < < 0.001) it < 0.001) Plot the open-circuit voltage across A, B as function of time, from t — —0.001 to * = 0.005. 24. The driving rod of the loud-speaker of Prob. 2 is driven by a force of 10,000[15 sin (200irf) - 10 sin (600x<) + 3 sin (1000tt<)] dynes. Plot the displace- ment of the diaphragm during one cycle, and compare it with the curve for the force. 25. The driving coil of Prob. 5 is actuated by an emf of [15 sin (200n-J) — 10 sin (600ir£) + 3 sin (lOQOn-2)] volts. Plot the displacement of the diaphragm as a function of t for one cycle, and compare it with the curve for the emf. 26. The emf of Prob. 25 actuates the driving coil of Prob. 8. Plot the dis- placement of the diaphragm for one cycle, and compare it with the curve for the emf. CHAPTER III THE FLEXIBLE STRING 8. WAVES ON A STRING .... So far, we have been considering the vibration of a system whose mass is all concentrated near one or two points, so that the motion of the system is completely specified by giving the displacement from equilibrium of the one or more masses as a function of time. The vibrators we use are not usually of so simple a nature. Ordi- narily the mass is not concentrated at the end of a relatively weightless spring but is spread along the spring. The vibrating string of a violin cannot be considered as having all its mass concentrated at the center of the string or even concentrated at a finite number of points along the string; an essential property of the string is that its mass is spread uniformly along its length. Similarly, a loud-speaker diaphragm has a good portion of its mass spread out uniformly over its extent. In these cases, each portion of the system will vibrate with a somewhat different motion from that of any other portion. The position of just a few parts of the system will not suffice to describe its motion; the position of every point must be specified. Presumably, we could attack the problem of the string, for instance, by considering the motion of N equally spaced masses on a weightless string and then letting N go to infinity. We should then have an infinite number of equations of motion, whose solution would give the position of every one of the infinity of points on the string as a function of time. A solution of these equations can be obtained. The solu- tion shows, for one thing, that there are an infinite number of allowed frequencies of oscillation of the string, as we could induce from the discussions of the previous chapter. But this is a very awkward way of solving a problem that is essentially simple. What is needed is a new point of view, a new method of attack. The new point of view can be summarized as follows: We must not concern ourselves with the motion of each of the infinite number of points of the string, considered as separate points, but we must consider the shape of the string as a whole. At any instant the string will have a definite shape, which can be expressed mathematically by saying that y, the displacement from equilibrium of that part of 71 72 THE FLEXIBLE STRING [m.8 the string a distance x cm from one end, is a function of x. The motion of the string at any instant will depend on the shape of the string, and the subsequent shape will depend on the motion; what we must do is to find the relation between the shape and the motion. In other words, the string's displacement y is a function of both x and t, and we must discover the relation between y's dependence on x and its dependence on t. The same point of view will be necessary in study- ing the vibrations of diaphragms and of air, as we shall see later. The Wave Velocity. — Before becoming cluttered up with equa- tions, we shall utilize a simple, but rather clever, little trick to help us in gaining a picture of how the shape of a string changes. Suppose that we have a long flexible string of uniform mass e g per cm, unrolling from one reel, threaded through the glass tube shown in Fig. 10a, and being wound up on another reel. Suppose that the reels are rotating so that the string is traveling with a velocity v cm per sec through the tube and so that the string is under a tension of T dynes. Suppose that we pick out a small portion of the tube of length As and ask what force the string is exerting on this part of the inside of the tube (we neglect frictional forces). If a small enough length is picked, the shape of the part of the tube (and the string in this part) will be practically equivalent to the arc of a circle of radius R, where the value of R will, of course, be different for different parts of the tube. Now, if the tube is not bent too sharply anywhere {i.e., if R is every- where larger than the maximum deviation of the tube from the straight dotted line in Fig. 10a), the tension on the string will everywhere be T dynes. A study of Fig. 10b will show that the net force inward on the tube due to the tension of the string is AT = <pT = T As/72. In addition to this force is the centrifugal "force" due to the motion of each successive portion of the string around the curved arc As. This force is pointed outward, and, as we learned in elementary physics, its value is the mass of the portion of the string eAs times its velocity squared v 2 divided by the radius of curvature R. The net inward force on the portion of the tube under consideration is then (As/R)(T - ev 2 ). Fig. 10. — Forces on a moving string. m.8] WAVES ON A STRING 73 We notice that it is possible to run the string at a particular velocity c ?= \/T/e such that there will be no net force on this portion of the tube. And then we see that at this velocity c the string exerts no net force on any part of the tube! For the force is zero regardless of the size of R. If the string is run at the velocity c cm per sec, we can carefully break away the tube from around the string and leave the string moving with velocity c, still retaining the original form of the tube, a wave form standing still in space. Of course, running the string with a velocity c and having the wave form stand still is the same as having the string stand still and the wave form travel with a velocity c. What we have just proved is that as long as the string has a uniform density e g per cm, and as long as the displacement of the string from equilibrium is not too great, then a wave will travel along the string with a velocity c = y/T/e regardless of the form of the wave. The last phrase is the important part of the statement, for it means that, subject to the qualifications „ ., w ' * ^ Fig. 11. — Wave motion on a Stated, a wave travels along a String string. Curve (a) is the shape at without any change of form as it * = 0; curve (6) the shape at ' = 1 - goes. These two properties of a string, or, rather, two aspects of the same property, are peculiar to the uniform flexible string. They are not possessed by a string of variable mass density, or a string with stiffness, for example. This is just the property that makes the string so useful in musical instruments, as we shall see. The General Solution for Wave Motion. — The foregoing property can be expressed mathematically by stating that the dependence of the shape of the string on x and t when a wave is going in the posi- tive x-direction must be of the general form F(x — ct). Then at t = the shape of the string is given by the function F(x), as is shown in Fig. 11a. At t = 1 the wave form F is the same, but it has been shifted bodily to the right a distance c cm. The general form for any motion of the string must be y=F(x-d)+f(x + ct) (8.1) representing a wave of the form F going to the right and one of the form / going to the left. Every motion of the string can be considered as a superposition of two waves, each having a speed c, traveling in opposite directions. 74 THE FLEXIBLE STRING [HI.8 The symbol F(x — ct) simply means that, no matter what function F is, the quantities x and t enter into it in the combination (x — ct). Examples are A sm[k(x — ct)], B(x — ct) 3 er a < x - ct >% etc. With our convention of taking real values, another example is C exp [Hz (x — ct)] = Ce ikx e- ikct . A peculiarity of this sort of function of x and t, one that will be of use to us in further discussion, is the following: F changes in exactly the same way and by the same amount when t changes by an amount a as when x changes by an amount — ca. A plot of the shape of the string at a given instant of time is completely similar to a plot of the displace- ment of a given point on the string as a function of time. To find the shape of F we can either run our eye over the whole extent of the string at a given instant of time, or we can watch the motion of one piece of the string as the wave passes by. This is, of course, still another way of saying that the wave progresses along the string with- out change of shape. The same statement holds for f(x + ct), except that the direction of x is reversed. A mathematical way of stating the property of F and / discussed above is | F(x - ct) = -c A F{x - ct), j t f(x + ct) = c A/(* + ct) (8.2) where the symbol (d/dt) means the rate of change with respect to t when x is held constant, and (d/dx) means the rate of change with respect to x when t is held constant. A change in F due to a small change in t is equal to (— c) times the change in F due to a change in x of an equal size. Initial Conditions. — The particular forms of the functions F and / are determined by the initial conditions: the initial form and "velocity form" of the string. If the shape of the string at t = is yo(x), and its velocity at the point x is v (x), then F(x) + f(x) = y (x) and d/dt [F(x — ct) + fix + ct)] = v (x) when t = 0, or, what is the same thing, «[-?(*) +/(«)] -(i)..(x) using Eq. (8.2). Since v (x) must be integrated to solve the last equation, we shall define the function S(z) = r v (x) dx in.8j WAVES ON A STRING 75 where z = % ± ct. It is not difficult to see that the shape of the string that corresponds to the specified initial conditions is represented by the function - \ \yo(x - ct) + y (x + ct) - \ S(x - ct) + \ S(x + ct) j (8.3) At t = 0, y = y (x), and (dy/dt) = dS(x)/dx = v (x), as is required. The solution is built up of two "partial waves" going in opposite directions, which combine to give the required behavior at t = and spread apart thereafter. Two examples of this are given in Fig. 12, ^•-f I Fig. 12. — Motions of plucked and struck strings. The solid lines give the shapes of the strings at successive times, and the dotted lines give the shapes of the two "partial waves" traveling in opposite directions, whose sum is the actual shape of the string. one for a string pulled aside and started with zero velocity, and one for a string struck by a hammer so that it starts from equilibrium with a specified velocity. The spreading apart of the partial waves is apparent in the successive drawings. Boundary Conditions. — So far, we have been treating the string as though it had an infinite extent ; actually, it is fastened down some- where, and this fastening affects the motion of the string. The fact that the string is fastened to a support is an example of a boundary condition. It is a requirement on the string at a given point in space which must be true for alL time, as opposed to initial conditions, which fix the dependence of y and v on x at a given time. Boundary con- ditions are more important in determining the general behavior of the string, its allowed frequencies, etc., than initial conditions are. If the support is rigid, and the distance along the string is measured 76 THE FLEXIBLE STRING tDLS from it, the boundary condition is that y must be zero when x = 0, for all values of the time. If the support is springy, so that it is dis- placed sideward a distance CF cm for a sideward force F dynes, then the boundary condition is that y must always be equal to C times the component of the string's tension perpendicular to the equilibrium line: y = CT(dy/dx) at x = 0. Many other sorts of boundary conditions are possible. Reflection at a Boundary. — Let us take the simple case of the rigid support, requiring that y = when x = 0, and see what effect this has on the motion of the string. The solutions y = F(x — ct) or y = f(x + ct) cannot be used in this case, for they will not always be zero when x = 0. However, the solution y = -F(x - ct) + F{-x - ct) will satisfy the boundary condition. At x = 0, y = -F(-ct) + F(-ct) = 0, (a) ^ '-* (b) ^(c) ' Ky^ x\-_. Fig. 13. — Reflection of a wave from the end support of a string. The solid lines show the shape of the string at successive instants of time; the dotted lines, the imagi- nary extension of the wave form beyond the end of the string. for all values of t. To see what motion of the string this expression corresponds to, let us suppose that F(z) is a function that is large only when 2 is zero and drops off to zero on both sides of this maximum. Then when t = —10, the function F(—x — ct) will have a peak at x = 10c representing a single pulse traveling leftward along the string. The function F(x — ct) at this time would have its peak at x = — 10c if there were any string to the left of the support, but since no string III.8] WAVES ON A STRING 77 is there, the term —F(x — ct) is not apparent, in the shape of the string at t = — 10. It has been represented in Fig. 13a by a (Jotted line to the left of x = 0. As t increases, the wave in the actual string moves to the left, and the wave in the imagined extension of the string moves to the right until they begin to coalesce at x = 0. During the coalescing the displacement of the point at x = is always zero, for the effects of the two waves just cancel each other here. A little later, the waves have passed by each other, the wave that had been on the imaginary part of the string now being on the actual string, and vice versa. i^il_L --X^ Fig. 14. — Motions of plucked and struck strings fixed at one end. Dotted lines show the traveling partial waves; their sum is the solid line, the actual shape of the string. The succession of events is pictured in Fig. 13. What has happened is that the pulse which had originally been traveling leftward is reflected at the point of support x = and comes back headed toward the right, as a pulse of similar form but of opposite sign. The bound- ary condition at x = has required this reflection, and the particularly simple sort of condition that we have imposed has required this very symmetric sort of reflection. Most other boundary conditions would require a greater difference between the original and the reflected wave. When a wave strikes it, a rigid support must pull up or down on the string by just the right amount to keep y zero; and in doing so it "generates" a reflected wave. 78 THE FLEXIBLE STRING [III.8 The expression for the motion of a string satisfying the boundary condition y = Oat a; = and the initial conditions y = y (x),v = v (x) at t = is y = ^ I Y(x - ct) + Y(x + ct) -- c H(x- ct) + ~H(x + c<) (S.4) where Y(z) H{z) = f y («) (z > 0) \-</o(-z) (2 < 0) = (S(z) (z > 0) \S(-z) (z < 0) S(z) = f* v (z) dx These definitions of Y and H are necessary because y and y are defined only for positive values of x (where the string actually is), whereas the form of the partial waves used to build up the subsequent forms of the string must be given for all values of z = x + ct. The particular forms of Y and H are chosen so that they automatically satisfy the boundary conditions at x — for all values of t. Two examples of the way in which the motion of the string can be built up by the use of these partial waves are given in Fig. 14. Strings of Finite Length. — Actual strings are fastened at both ends, so that really two boundary conditions are imposed. For instance, the string can be fastened to rigid supports a distance I cm apart, so that y must always be zero both at x = and at x = I. The most important effect of a second boundary condition of this sort is to require that the motion of the string be periodic. A pulse started at x = travels to the other support at x = I in a time (l/c), is reflected, travels back to x = 0, and is" again reflected. If the supports are rigid, the shape of the pulse after its second reflection is just the same as that of the original pulse, and the motion is periodic with a period equal to 21/ 'c. The motion in this case is not, in general, har- monic, as we shall see, but it is always periodic. This periodicity of all motion of the string depends entirely on the fact that we have imposed a particular sort of boundary condition; if other conditions are imposed at a; = and x = I (i.e., if the supports are not perfectly rigid), then it may not be true that every motion is periodic; in fact it may never be periodic. The quantitative manner of dealing with the two boundary con- ditions is by means of the partial waves F. When the string is only IH.8] WAVES ON A STRINO 79 I cm long, we are free to give any shape to F(z) for z larger than I or smaller than zero. "Free" is not the correct word, however, for we must choose that form of F which satisfies both boundary conditions. If we start out at t = with a pulse of the form F traveling to the left, then, as before, we can satisfy the condition y = at x = by setting y = -F(x - ct) + F(-x - ct). To have y = at x = I, we must arrange the rest of the function F, beyond the limits of the Ky= V-/^*~T is^l 2s/ ^\^r ^" N Fig. 15.— Periodic motion of a string fixed at both ends. Solid lines give the shape of the actual string at successive instants; dotted lines show the imaginary extension of the wave form beyond the ends of the string. The motion is made up of two partial waves going m opposite directions, each being periodic in x with period 21. actual string, so that F (I - ct) = F(-l - ct), or, setting z = -I - ct, so that F(z) = F(z + 21) for all values of z. This means that the function F(z), which must be defined for all values of z, must be periodic in z, repeating itself at intervals of 21 all along its length. An illustration of how this sort of partial wave can be used to determine the motion of the string is given in Fig. 15. To satisfy the boundary conditions y = at x = and at x = I and the initial conditions y = y (x), v = v (x) at t = 0, we build up a combination similar to that given in Eq. (8.4): V = \\Y{x - ct) + Y(x + ct) - ±H(x - ct) + \H{x + ct)] (8.5) 80 THE FLEXIBLE STRING [m.9 where Y(z) = -yo(-z) (-1 <z < 0) 2/0(2) (0 < z < I) -Z/o(2Z - z) (I <z < 21) yo(z - - 21) (21 < z < 31) etc. S(-z) (-1 <z < 0) S(z) (0 < z < I) S(2l - z) (I <z < 21) S(z - 21) (21 < z < SI) etc. H(z) = S(z) = j" v (x) dx Two examples of the motion of such strings are given in Fig. 16. In Fig. 17 the displacement of a point on the string is plotted as a ^s S* ^v k^' < i > < ^ :>< ^ t" — v — > — % - e*^t Fig. 16. — Motions of plucked and struck strings fixed at both ends. The solid lines show the successive shapes of the string during one half cycle. Shapes for the other half cycle are obtained by reversing the sign of the curves. function of time, showing that the motion is periodic but not simple harmonic. 9. SIMPLE HARMONIC OSCILLATIONS It has been seen in the last section that the imposing of boundary- conditions limits the sorts of motion that a string can have and that if the boundary conditions correspond to the fixing of both ends of m.9] SIMPLE HARMONIC OSCILLATIONS 81 the string to rigid supports the motion is limited to periodic motion. This last result is an unusual one, for we found in the last chapter that even as simple a system as a pair of coupled oscillators does not, in general, move with periodic motion. It is not unusual for a system to oscillate with simple harmonic motion (which is a special type of periodic motion) when it is started off properly (we shall see that practically every vibrating system can do this); what is unusual in the string between rigid supports is that every motion is periodic, no matter how it is started. Our problem in this section is to find the possible simple harmonic oscillations of the string (the normal modes of vibration) and to see what the relation is between the frequencies of these vibrations that makes the resulting combined motion always periodic. The problem of determining the normal modes of vibration of a system is not just an academic exercise. For systems more complicated than that of the string between rigid supports we have no method of graphical analysis similar to that of the last section, and the only feasible method of discussing the motion is to "take it apart" into its constituent simple harmonic components. There is also a physiological reason for studying the problem, for the ear itself analyzes a sound into its simple harmonic parts (if there are any). We dis- tinguish between a note from a violin and a note from a bell, for instance, because of this analysis. If the frequencies present in a sound are all integral multiples of a fundamental frequency, as they are in a violin, the sound seems more musical than when the frequencies are not so simply related, as in the note from a bell. The Wave Equation. — To obtain the normal modes of vibration of a string, we must take up the problem that we laid aside in the last section, that of finding the equation relating the shape of a string and its motion. This equation was almost derived in the previous section. Looking back at Eqs. (8.2) given there, we find that, by repeating the differentiation, both F(x — ct) and /(a; -f ct) satisfy the equation Fig. 17. — Displacements of the points marked (c) on the strings shown in Fig. 16, plotted as functions of the time. d 2 y _ 1 d 2 y Bx 2 at 2 c = \/T/e (9.1) 82 THE FLEXIBLE STRING [III.9 Since every motion of the string is a combination of the two waves, every motion of the string must satisfy this equation. It is called the wave equation. We shall meet it often in this book. The wave equation corresponds to a number of statements con- cerning the motion of a string. We saw in the last section that it implies that the wave motion travels with its shape unchanged, at a velocity c, independent of this shape. Since the derivative (d 2 y/dx 2 ) is proportional to the curvature of the shape of the string at a given instant, Eq. (9.1) states that the acceleration of any portion of the string is directly ■ proportional to the curvature of that portion. If the curvature is downward, the acceleration is downward, and vice versa; and the greater the curvature, the faster the velocity changes. This second statement corresponds to the usual method of deriving the wave equation. The net force on a portion of string at any instant is proportional to the curvature of this portion at that instant. A study of Fig. 18 shows us that the net force perpendicular to the z-axis on a piece of string of length ds is T(sm<p 2 — sin<pi). If the displace- ment of the string from equilibrium is not large (and we have already had to assume this), the angles >i and <p 2 will be small, so that sin^>i will be practically equal to tan<pi, and similarly for <p 2 . How- ever, tan <pi is equal to the slope of the string at the instant in question, (dy/dx), at the point x. Similarly tan <p 2 is the slope at the point x + dx. Now, from the definition of the symbol (d/dx), the value of a function / of x at the point x + dx is equal to the value of / at the point x, added to dx times the rate of change of / with respect to x, * x+dx Fig. 18. — Forces on an elementary length of flexible string. f(x + dx) = f(x) + dx 1 dx (9.2) so that the net vertical force on the element of string is T \(dx\ _ (dy\ ] = Tdx ±(§y\ = Tdx ejy l\dy/x+dx \dx/x] dx \dxj dx 2 The mass of the element of string is e ds; and if the angles <p are small, this is practically equal to e dx. The equation of motion of the element is therefore HI.9] SIMPLE [HARMON IC OSCILLATIONS 83 edx^=Tdx^, or ^- € ^ edx dt2 1 dx dx2 , or ___ — which is identical with Eq. (9.1). Standing Waves. — But we are still looking for the possible ways in which the string can execute simple harmonic vibrations. To exe- cute such motion, every portion of the string must oscillate with the same frequency. The mathematical counterpart of this requirement is that the shape of the string y(x, t) must be equal to the exponential e -2*irt multiplied by a function of x alone: y = Y(x)e- 2rirt . If this expression is substituted in the wave Eq. (9.1), we find that Y must satisfy the equation d 2 Y 4tt 2 i/ 2 ^+-^"^ = 0, c*=(T/e) (9.3) Solutions of this equation are the exponentials e 2vivx/c and e- 2vivx/c . All the simple harmonic motions of the string must therefore conform to the expression y = Q e (.2rir/c)(x-ct) _J_ Q _ e (2iciv/c){-x-ct) = A+ cos -^ (x - ct) - <S> + + A- cos ^ (x + ct) - $_ (9.4) representing two sinusoidal waves, of different amplitudes, traveling in opposite directions. This result is another illustration of the relation between the shape and the motion of the string; when we require that the dependence of y on t be sinusoidal, the dependence of y on x must also be sinusoidal. The wavelength of these waves X = (c/v) is the distance between the beginning and end of each cycle of the sinusoidal wave form. In the special case when the amplitudes of the two waves are equal, the trigonometric formula for the sum of two cosines can be used, giving A = 2A+ = 2A_ y = A cosf — x — Q J cos (2irvt — <f>), = £($_ + $ + ) (9.5) $ = i($_ - $ + ) Here the traveling waves combine to form standing waves. At points on the string where cos ( — a — £2 J = 0, the two traveling waves always just cancel each other, and the string never moves. These 84 THE FLEXIBLE STRING [HI.9 points are called the nodal points of the wave motion. In the case that we are considering, where the density and tension are uniform, the nodal points are equally spaced along the string a distance (c/2v) apart, two for each wavelength. Halfway between each pair of nodal points is the part of the string having the largest amplitude of motion, where the two traveling waves always add their effects. This portion of the wave is called a loop, or antinode. Normal Modes. — So far, we have neglected boundary conditions. If we require that y = when x = 0, the general form of (9.4) can no longer be used; the number of possible harmonic motions is limited. The expression for y that must be used is the standing wave form (9.5) with the angle ft so chosen that a nodal point coincides with the point of support x = 0: y = A sin( x J cos (2-irvt — $) (9.6) This agrees with the discussion in the previous section. For the simple boundary condition that we have used, the reflected wave has the same amplitude as the incident wave; and when the incident one is sinusoidal, the result is a set of standing waves. Any frequency is allowed, however. When the second boundary condition y = at x = I is added, the number of possible simple harmonic motions is still more severely limited. For now, of all the possible standing waves indicated in (9.6), only those which have a nodal point at x = I can be used. Since the distance between nodal points depends on the frequency, the string fixed at both ends cannot vibrate with simple harmonic motion of any frequency; only a discrete set of frequencies is allowed, the set that makes sin ( -^ I ) zero. The distance between nodal points must be I, or it must be (1/2), or (1/3) . . . etc. The allowed fre- quencies are therefore (c/2l), (2c/2l), (3c/2Z) . . . etc., and the differ- ent allowed simple harmonic motions are all given by the expression y = A„sinfe) cos(™ t - ^ (n = 1, 2, 3, 4 • • •) (9.7) The lowest allowed frequency v x = (c/2l) is called the fundamental frequency of vibration of the string. It is the frequency of the general IH.9] SIMPLE HARMONIC OSCILLATIONS 85 periodic motion of the string, as we showed in the last section. The higher frequencies are called overtones, the first overtone being vz, the second v%, and so on. The equation for the allowed frequencies given in Eq. (9.7) expresses an extremely important property of the uniform flexible string stretched between rigid supports. It states that the frequencies of all the overtones of such a string are integral multiples of the funda- mental frequency. Overtones bearing this simple relation to the fundamental are called harmonics, the fundamental frequency being called the first harmonic, the first overtone (twice the fundamental) being the second harmonic, and so on. Very few vibrating systems have harmonic overtones, but these, few are the bases of nearly all musical instruments. For when the overtones are harmonic, the sound seems particularly satisfying, or musical, to the ear. Fourier Series.— To recapitulate: The string has an infinite num- ber of possible frequencies of vibration; and if the supports are rigid, these frequencies have a particularly simple interrelation. If such a string is started in just the proper manner, it will vibrate with just one of these frequencies, but its general motion will be a combination of all of them: y = Ai sin( -y J cosf -j t — 3>i J + A 2 sin( — p J cosf — j- t — # 2 ) + . . . or, symbolically, ■ - 2<^x)[ B "°K ! T*) + c ""K s r)] (9 ' 8) where the symbol 2 indicates the summation over the number n, going from n = 1 to n = «> . The value of A n is called the amplitude of the nth harmonic. Equation (9.8) is just another way of writing Eq. (8.5). The present form, however, shows clearly why all motion of the string must be periodic in character. Since all the overtones are harmonic, by the time the fundamental has finished one cycle, the second har- monic has finished just two cycles, the third harmonic just three cycles, and so on, so that during the second cycle of the fundamental 86 THE FLEXIBLE STRING [HI.0 the motion is an exact repetition of the first cycle. This is, of course, what we mean by periodic motion. Equation (9.8) is in many ways more useful for writing the depend- ence of y on t and x than is Eq. (8.5). For it gives us a means of finding the relative intensities of the different harmonics of the sound given out by the string (corresponding to the analysis that the ear makes of the sound) and thus gives us a method of correlating the motion of the string with the tone quality of the resulting sound. We shall have to wait until farther along in the book to discuss the quantitative relations between the vibrations of bodies and the intensity of the resulting sounds, but it is obvious that the intensity of the nth har- .monic in the sound depends on the value of the amplitude A n . Once the values of all the A n 's are determined, the future motion of the string and the quality of the sound which it will emit will both be determined. Initial Conditions. — The A n 's and $ n 's, or the B n 's and C n 's, are an infinite number of arbitrary constants, whose values are fixed by the initial conditions, corresponding to the infinite number of points along the string whose positions and velocities must all be specified at t = 0. Our analysis will not be complete until we devise a method for determining their values when the initial shape and velocity shape of the string are given. The initial conditions must satisfy the equations, obtained from (9.8) by setting t = 0, y(x,0) = y (x) = ^ B n sm[~) Series like the right-hand sides of these equations are called Fourier series. Now, the initial shape and velocity shape of the string, the func- tions yo(x) and v (x), can be any sort of functions. that go to zero at x = and x = I. It therefore must be possible to express any func- tion satisfying these boundary conditions in terms of a Fourier series of the type S5„ sin (rnx/l). Subtle. mathematical reasoning must be used to prove rigorously that this is true (in fact, a completely satis- factory proof has not yet been devised), but since we know as physicists that the motion of the string is definitely specified by its initial shape and velocity shape, we shall assume that it must be true. m - 9 ] SIMPLE HARMONIC OSCILLATIONS 87 The Series Coefficients.— The trick to obtain the values of the B n 'a and C n 's consists in multiplying both sides of the preceding equations by sin (irmx/l), where m is some integer, and integrating over x from x = to x = I. The utility of the trick lies in the fact that J o sin {irnx/l) sin (irmx/l) dx is zero unless n equals m, in which case it equals (1/2), so that the infinite series of integrals on the right-hand side has only one term not zero, involving just one B m or C m . For instance, X sin (^) y ° (x) dx = % B « £ sin (^) sin (^) ** = \2j Bm so that „ 2 C . (irmx\ . . . \ m = T ) sin \T/ y °w ) and, similarly, ( (Q Q , Cwi = ^ Jo sin V-T^o(x) rfz (m = 1, 2, 3, 4 ■ • • ) ) Equations (9.9) provide a means of determining the values of the Bn& and C n 's in terms of the initial conditions. Plucked String, Struck String.— A few examples will indicate how the method works. For instance, if we pull the center of the string out h cm and then let it go at t = 0, all the C m 's will be zero, and 8h . firmS = ^ sin V-27 {0 if m is an even integer (_1)(— d/ 2 if w is an odd integer Therefore, y - ¥ r n Vi7 cos \tJ - 9 sm \ir) cos {-r) +'.■■■ • (9-iP) 88 THE FLEXIBLE STRING [m.9 Computing this series for y as a function of x and t gives the same values for the shape of the string at successive instants as are shown in the first sequence of Fig. 16. .Figure 19 shows how the correct form is approached closer and closer the more terms of the series are used. At first sight, the foregoing series appears to be simply a more awkward way of finding the shape of the string than the method used in the previous section. However, the series can tell us more about the string's motion than the results of the last section can. It tells /\ First Te First Four* Terms Fig. 19. — Fourier series representations of the initial form of the string given in Eq. (9.10) and the initial velocity form given in Eq. (9.11). Successive solid curves show the effect of adding successive terms of the series; dotted curves show the actual form, given by the entire series. us, for instance, that the second, fourth, sixth, etc., harmonics will be absent from the sound given out by the string, for they are not present in the motion. It tells us that if, for example, the intensity of the sound emitted is proportional to the square of the amplitude of motion of the string, then the fundamental frequency will be 81 times more intense than the third harmonic, 625 times more intense than the fifth harmonic, etc. The absent harmonics correspond to standing waves that have a nodal point at the center, the point pulled aside. This is an example of the general rule (which can be proved by computing the # required integrals for B m ) that in the motion of any plucked string all those harmonics are absent which have a node at the point pulled aside. IH.9] SIMPLE HARMONIC OSCILLATIONS 89 If the string is struck, so that y = and ((*) ( 0< *<d (o Q<»<») then all the B m 's are zero, and iron ~ 2 f P z/4 A/xx\ . /xmx\ , so that . 1 + V2 . /3tx\ . (Zicct\ + — 27- sm VT-J sin V~ry "m^—jsm^—J- • • • J (9.11) ^ 7T°C 125 We note that the fourth, eighth, etc., harmonics, those having nodes at x = (1/4:), are absent in this case. Energy of Vibration. — There is a general analogy between the amplitudes A n of the various harmonics and the amplitudes of motion along the normal coordinates X and F discussed in Sec. 6. This can be illustrated by computing the energy of vibration of the string. The kinetic energy of the string is the integral of the kinetic energy of each element of length: ie I i-^Jdx. Its potential energy is equal to the amount of work necessary to move the string into its instantaneous form from the equilibrium form y = 0. Suppose that its form at a given time tis y, a function of x. Then we can imagine changing the string from equilibrium form to final form by making its intermediate form be ky, where k changes from zero to unity. The force on any element of string of the form ky is T f — 2 ky) dx, and the 90 THE FLEXIBLE STRINO [III.9 force that we must use to oppose this is equal to this value and oppo- site in sign. As we displace the string from equilibrium by changing k, the element of displacement is y dk, and so the work required to bring this element of string into place is J>(s) ^)kdxydk = -Ty[^ 2 (5)*J>--«>(S) dx The potential energy of the whole string, the work required to bring the string from equilibrium into the form represented by the function V, is if we integrate by parts. Since y = at x = and x = I, the first term is zero, and the potential energy is just (T/2)j(dy/dx) 2 dx. The total energy of the string is therefore When series (9.8) is substituted into (9.12), the expression for the energy becomes 12 -*(")' -I- 4AM2 s i n (-/ ^ nA n cosf^y- J cos(^ 3>„ J \ dx A\ sin 2 (?f - <h) £ sin 2 (y) dx + • • • $i ) sin Cr - * 2 ) Jo sin Vr) sin (nr) dx + • • • + A\ cos 2 fe* - * x ) Pcos 2 fe)^ + • ■ • } All the integrals of the sort / sin (ttx/T) sin (2rx/l) dx are zero, whereas integrals of the type of / sin 2 (rx/l) dx and / cos 2 (irx/l) dx are equal to {1/2), This simplifies the expression enormously,- and after adding sin 2 to cos 2 , term by term, we can write w = w.) 2 (™J A i = 2 2t ° (I) vlAl ' " -■■ (l) (9 - 13) 111.10] FORCED VIBRATIONS 91 The energy is therefore a series of terms, each term depending on just one of the harmonics, an expression similar to that given in Eq. (6.6). The different harmonics are the different normal modes of vibration of the string, and the quantity A n is the amplitude of vibration along the nth normal coordinate of the system. We can say that the energy of a vibrating string is equal to the total energy of an infinite number of equivalent harmonic oscillators, each having a mass equal to half the total mass of the string {It/ 2), one having frequency vi and amplitude A i, another having frequency v 2 and amplitude A 2 , and so on. 10. FORCED VIBRATIONS So far we have studied the particularly simple case of the free vibrations of an idealized string. The analysis was useful, for it gave us an insight into the general properties of the motion without entangling us in the algebraic complications that crop up as soon as some of the idealism is relaxed. Before we study the effects of motion of the end supports, friction, and nonuniformity on the behavior of the string, let us study the response of the idealized string to a driving force. Here again we find that a neglect of complications, at first, enables us to bring out the salient behavior with the least amount of mathematical camouflage. Wave Impedance and Admittance. — The simplest case is that of a string of infinite length stretched at a tension T between supports at x = and x = °o. The mass of the string per unit length is e, so that t he v elocity of wave motion is c = -y/T/e. The support is rigid in the x-direction, so as to sustain the tension of the string; but it is hinged so that it can move in the ^-direction, transverse to the string. A transverse force applied to the support will move the end of the string as well as the support, so that the mechanical im- pedance offered to the force is the sum FlG - 20 -- Forces on a flexible su PP° r *- of the transverse mechanical impedance of the support and the mechanical impedance of the end of the string to transverse motion. The impedance of the support is of the general type discussed in Chap. II, and need not concern us here. What is now of interest is the wave impedance of the string, the ratio of the transverse force applied to the end of the string to the transverse velocity of the end 92 THE FLEXIBLE STRING [111.10 of the string, when the driving force is simple harmonic and the trans- verse impedance of the end support is neglected. The displacement of the string at a distance x from the end and at a time t is y(x,t). The angle that the end of the string makes with the equilibrium line is 9 = tan _1 [(dy/dx)_o]. The longitudinal force on the end of the string is — T cos 0. Since we are assuming, every- where in this chapter, that (dy/dx) is small compared with unity, we can say that, to the first approximation, the longitudinal force is T, the constant tension, and the transverse force exerted by the support on the end of the string is ■ - r (2) -rt».--r.^-/«> (io.i) If the driving force is simple harmonic, / = Foe - *"', the shape of the string must be sinusoidal, as is indicated in Eq. (9.3). In the present case the wave must be going in the positive ^-direction, since it is being generated at x = 0. Therefore the space factor will be Ae+ ikx , where — A oihx — iut Incidentally, because we have chosen the negative exponential e -u>t f or our convention, it turns out that the sign of the exponential e ikx indicates the direction of motion of the wave. A wave in the negative ^-direction would be e - i( ~ kx+a>t) , with a negative sign in the a>part. The value of A is determined from the expression for the driving force in Eq. (10.1) Foe-™' = so that V = \dx) x= o 1 ** ° e ik{x-ct) — l(j) eC V ~ dt ec 6 (10.2) The input impedance of the string, the ratio between the driving force and the transverse velocity of the string at x = 0, is 111.10] FORCED VIBRATIONS 93 ec = (±\ = y/Te (10.3) This quantity, the input impedance of an infinite string, is sometimes called the wave or characteristic impedance of the string. We see it is real, being a pure resistance for a simple string. This implies that energy is being continuously fed into the string, energy of wave motion that never returns, since the string is infinite in extent. When we come to consider strings of finite length, we shall see that the input impedance differs from the characteristic impedance ec and is not purely resistive. The power input to the string is the average value of Fv at x = p = i &!! = i €C | y p (io.4) In many cases we wish to know the velocity of some other part of the string when the force is applied at x = 0. This is obtained in terms of the transfer impedance, already discussed in Sec. 7. It is usually easier to deal with the reciprocal quantity, the transfer admit- tance, the ratio between the transverse velocity of the string at point x and the simple harmonic driving force (in this case at x = 0). For the simple case of the infinite string, ^=r„(0,*;«)= jgl^i;^ (10.5) General Driving Force. — When the transverse force on the end of the string is more complicated than a simple harmonic force, the response of the string can be obtained by the methods of Sec. 6. We find the Fourier transform of the force function (W) = ^J- Then, in accordance with the analysis leading to Eqs. (6.15), we obtain the equation for the velocity shape of the infinite string when the end at re = is acted on by a transverse force f{t), v(x,t) = f Y m {0,x;o>)e-^ da> I ^ J ^ f(j)e™ dr J .. /» 00 (* 00 . = %T~ \ e (Wc) - i < tf ' du I /(r)e iwT dr 94 THE FLEXIBLE STRING [HI.10 Using the general formulas for Fourier transform given by Eqs. (2.19) and (2.20), we find that the velocity shape of the string is ^-sK'-O-M'-?) (ia6) This is a very interesting and simple result, one that we. should have been able to obtain without recourse to the machinery of Fourier transforms. It restates the simple fact that was discussed in Sec. 8, that the shape of the string, as function of x for a constant time, is simply related to the motion of one part of the string, as a function of time. Since the input impedance of the infinite string is constant and resistive, the velocity of the string at the point of application of the force (x = 0) is proportional to the force no matter what form f(t) has. Therefore for x = 0, v(0,t) = (l/ec)f(t), where ec is the input resistance. And since the motion of the string causes waves in the positive x-direction, the expression for v(x,t) must be that given in Eq. (10.6), with the characteristic quantity (x — ct) appearing. The corresponding expression for the displacement is obtained by integrating /, y(x,t) = \- c Q (t - fj; Q(Z) = j* f(t) dt (10.7) We can now verify our calculations still further. From Eq. (10.1) we have that . *>--*&L- -s[=«H)L which checks, if we use Eq. (10.7) for Q and the relation c 2 = (T/e). The reason that there is such a simple relationship between / and y in this case is that the input impedance is a constant, independent of «. There are no resonant frequencies, where the impedance is zero and the admittance has a pole, for the infinite string; since there is no reflection of waves from the far end, there is no periodicity of the wave motion and, therefore, no natural frequencies of free vibration. String of Finite Length. — When we come to consider the simple string of finite length with a rigid support at x = I, we find that the input impedance at x = is considerably more complicated. The string now has resonance frequencies, so that the impedance goes to zero for certain values of <a. The waves are reflected from the rigid support at the far end (x ~ I), so that no energy is lost. Therefore m.10] FORCED VIBRATIONS 95 if the support at x = I is rigid, and if there is no resistance to the string's motion, then the input impedance is purely reactive, with no resistive term. We find the displacement of the string from Eq. (9.3). The two simple harmonic waves of frequency (w/2tt) = v are combined to form a standing wave of zero amplitude at the rigid support x = I, y = A sin[k(l — xjje-^, k = (w/c) The amplitude o. the wave, A, is adjusted to fit the amplitude of the force at x = 0, , (dy\ . To, /<al\ . , 1-^1 — A — cos I — )e-" at \dx/ x= o c \ c ) Therefore the expressions for the displacement and velocity of a string of length I driven from one end are Foe-* * = El sin[(co/c)(S ~ x)] e _^ t ecu cos (col/c) .F 8w[(a/c)(l — x)] ec cos (ool/c) (10.8) and the expression for the input admittance and the transfer admit- tance (reciprocals of the impedances) are r(0,0; w )=-(i)tan(^) --©MtM?)--"©] (10.9) Y(0,x;co) = Foe-™ 1 This case, therefore, gives definite resonances; the amplitude of motion becomes infinite (since we are neglecting friction) whenever tan (o)l/c) becomes infinite. This occurs at the frequencies for which M/c) = (tt/2), (&r/2), (&r/2) • • • ; or Vn = (^) (2n + 1} (n = °' 1 ' 2 ' 3 ' ' ' > io n = 2irv n = fe)(2n + 1) For these frequencies the input admittance is infinite (the input imped- ance is zero). For the frequencies v = (c/2l), (2c/2Z), (3c/ 21) • • • the input admittance is zero, the input impedance is infinite (for the zero friction case). This is analogous to the parallel resonance in electrical circuits. At these antiresonance frequencies the motion of the end \k*~ JJ^~ 06 THE FLEXIBLE STRING [HI.10 point x = is infinitesimally small, though the rest of the string is in motion. For very low frequencies, the input impedance 1/F(0,0;«) has the limiting value i(ec 2 /lu) = i(T/fa>), a " capacitative " reactance, with an effective stiffness constant (T/l) and "capacitance" (l/T). The next step in this analysis would be to compute the response of this system to impulsive forces. This case will be left for a prob- lem, and we shall go on to a somewhat more useful example. Driving Force Applied Anywhere. — Often the driving force is not applied at one end of the string, but elsewhere along the string. Some- times, indeed, the force is distributed along the string, instead of being concentrated at a point. A solution of this general case is best obtained in terms of solutions for concentrated forces, however; so the next problem we shall study is that of the string of length I, held between two rigid supports, and acted on by a force F(£)e -iwt applied at the point x = £. When we obtain the steady-state response for a simple harmonic force applied at a point, we can use the operational calculus to obtain the response to an impulsive force applied to a point. Once this is obtained, the response of the string to a force that is any function of time and that is distributed in any manner along the string is com- puted in terms of integrations over | (the point of application) and r (the instant of impulse), analogous to the integrals of Eqs. (7.14). If a simple harmonic force of frequency (w/27r) is applied to the point x = £, the part of the string f or x < £ should be a part of a standing wave that is zero at x — 0, such as a sin (cax/c). The part for x > £ should be a standing wave that goes to zero at x = I, such as b sin[(<o/c)(£ — x)]. There will be a sudden change of slope at x = £, and — T times this change in slope must equal the applied force. Calculations similar to those made in the previous example show that the correct solution is >-- s \* \ eco sin(«Z/c) ecu sin {ool/c) \ c / F(£)e-™ 1 sin (w|/c) . {-:>-*>] (10.10) (x> The transfer admittance (v/f) for this case is i_ sm[(o)/c)(l - £)] sin (a>x/c) . < ^ _ i_ sin ((o^/c) sin[(co/c)(f - x)\ . > . ec sin (wZ/c) m.10] FORCED VIBRATIONS 97 The input admittance is obtained by setting x = £ in the expression for Fax;co). Alternative Series Form. — Equation (10.10) is a closed form for the shape of the string driven by a simple harmonic foree applied to x = £. Sometimes it is useful to express this function in terms of a Fourier series of the type given in Eq. (9.8). As shown there, if we set y = XB n sin (rnx/l) the integrals for B n can be worked out ?« = j I sin {jyJ v(x,t) dx where y is given in Eq. (10.10). After a great deal of algebraic manipulation, we obtain the final series n = 1 which is equal to the expression given in Eq. (10.10). While it is an infinite series, it is sometimes easier to use it for computation than is the closed form of Eq. (10.10). As a matter of fact we could have obtained this series in a much more straightforward manner by going back to the equation giving the force on each portion of the string. According to the discussion following Eq. (9.2), the force on the element of string between x and x + dx is T dx (d 2 y/dx 2 ), due to the tension. This is the only force on most of the string in the present case; except for the point where the driving force is applied, x = £. Here the applied force per unit length of string is very large, so that the integral over x of the applied force near x = £ is equal to .Foer**'. The applied force per unit length is therefore given in terms of the delta function discussed on page 48. The equation of motion of the string then becomes If we are interested in the steady-state response, we consider that the transient oscillations have died out, and all that is left is a shape that oscillates with frequency («/2x). In brief, we set y{x,t) = Y^e-^K The resulting equation for Y is d?Y dx 2 -® r -[¥\«?-o < io - is > 98 THE FLEXIBLE STRING [111.10 where c 2 =. (T/e). Now if we can find a Fourier series expansion for 8(x — £), we shall be able to find a Fourier series expansion for F. A series expansion for the delta function can be found by the methods of Eq. (9.9). The important property of the delta function K x ~ £) i s that for any function f(x) £f(x)8(x- Z)dx=f(® (10.14) Therefore if 8 can be expressed as a Fourier series SZ)„ sin {irnx/l), the series coefficients are D.-f X*-e-»(T)*-?W ! f). Formally, therefore, the series representing the delta function is «(* - e - r 2 ""Or) sin (t 1 ) (10 - 15) The only awkward part is that the series does not converge. This should not be surprising, however, for one would expect a peculiar sort of series to represent as peculiar a function as 8(x — £). It need not disturb us particularly, however, for we do not intend to use series (10.15) to compute 8(x — Q; we shall use it only as a means to com- pute the series for y. As long as we use the series for 5 only as a short- cut means of obtaining other series which do converge, we can perhaps justify our use of a nonconvergent series. Naturally we must use the short cut with care and check our results by other methods whenever we can. Going back to Eq. (10.13), we assume that Y = SA n sin (irnx/l) and substitute this and the series for 8(x — |) into the equation. The result is Equating coefficients of sin (mx/l) on both sides of the equality sign gives equations for A n , and the resulting series for y = Ye*™ 1 is just that given in Eq. (10,12). This justifies the use of the series for 111.10] FORCED VIBRATIONS 99 b{x — £), since the series for y converges, and also indicates how much shorter is the short cut than the earlier method. The series of Eq. (10.12) shows very clearly the phenomenon of resonance and the resonant properties of systems, which we began to discuss in Sec. 7 on coupled systems. The steady-state shape of the string is usually a combination of all the shapes of all the normal modes of free vibration of the string, sin (irnx/l). When the driving frequency (oj/2t) approaches one of the natural frequencies (nc/2Z), the contribu- tion of the corresponding mode to the shape of the string becomes larger than all the others. In other words, as the driving frequency approaches one of the natural frequencies, the amplitude of motion increases without limit and the shape of the string approaches that of the corresponding normal mode. The series also indicates the effect of the point of application of the force. If the force is applied at a node of one of the normal modes, the corresponding factor sin (irn^/l) is zero and that normal mode and resonance is absent from the motion. Distributed Driving Force. — To find the response of the string to a simple harmonic driving force that is distributed along the string, we need only integrate the expressions given in Eqs. (10.10) and (10.12) over the point of applications | of the force. If the previous expressions give the response to a delta functio nTiorce at £, it is not difficult to see that the response of the string to a force ^(De - ^ dynes per cm length distributed along the string is given by the formulas sin (tux/1) (im/iy - (a>/c) 2 _ e-**' (10.16) One example of the use of this equation should probably be given. When a plane wave of sound of frequency a)/2ir passes over a string, it produces a force on it which is in phase over the whole length of the string if the direction of the wave is perpendicular to the string, but which varies in phase from point to point if the wave front is at an angle. In general the force per unit length at the point £ cm from one end can be given by the general expression F(f)er-*"« = /Toe*"* - *"', a = (2t/X s ) COS^ where X s is the wavelength of the sound wave in air and # is the angle the direction of the wave makes with the string. The relation between 100 THE FLEXIBLE STRING [HI.10 F and the intensity of the sound will be discussed later in this book [see Eq. (29.6)]. Taking the simple case of normal incidence first, where the force Foe - "*' is uniform and in phase over all the string, we use the closed form of Eq. (10.10) in Eq. (10.16) to obtain ■fc(-!)] J? -lot ) C0S I = b^_ < ,_„ N -/-■ _ ! i (10 17) 9 €0) 2 \ COS(cdZ/2c) ) This has resonances at every other harmonic (v = co/2tt = nc/2l; n = 1,3,5,7 • • • ), the symmetry of the normal modes for the even harmonics precluding their excitation. The shapes exhibited by the string for different driving frequencies are shown in Fig. 21. For the more general driving force Foe*"* - ™', it is easier to use Eq., (10.12) for y{£,x,t), and the final result can be expressed in the series V °\ T J ^™* (xw) 2 - (aiy (10.18) (ttw) 2 - (a>Z/c) 5 This series is equal to the closed form of Eq. (10.17) for a = 0. Transient Driving Force. — To calculate the response of the string for a transient force, we use the operational-calculus methods again. We begin by computing the response of the string to an impulsive force at t = applied at x = £; f(t) = 5(x — £)8(t). Referring to Eq. (6.16), we see that the proper expression is given by the integral Using Eq. (10.11), this becomes , x . t) = _L f" *"*" s(»,z) " ' 2irec J - «, o) sin (coZ/c) where (sin(^)sin[(f) (*-*)] (*>{) The poles of the integrand are atu= (rnrc/l) where n is any integer, positive or negative (these correspond to the natural frequencies). Near one of the poles for an even n, the quantity sin (coZ/c) approaches 111.10] FORCED VIBRATIONS 101 v = 4v, £ 43 » s io»i Fig. 21. — Shapes of steady-state motion of an undamped string between rigid sup- ports, driven by a uniform force of frequency v. The fundamental frequency of the string is v\. Those parts of the string above the dashed equilibrium line are in phase with the force; those below the line are 180 deg out of phase. the value (l/c)[w — (ntrc/l)]. In other words, for <a very near (nirc/l) (n even) the integrand becomes 1 Q—inrct/l sin I * - T ) sm V" ) 2rel (rnrc/l) w — (nwc/l) (*< Q The residue of this expression, its limiting value when multiplied by [w — (nirc/Vj\, as this factor approaches zero, turns out to be -1 2ir 2 ncc o — invct/l sin -r) sm KT) (0 < x < I) 102 THE FLEXIBLE STRING [HI.10 For n odd, the factor sin (cd/c) approaches the quantity fc\ ( rarc\ but the factor sin(nir — mr^/l) turns out to be — sm(nir£/l), so the two minus signs cancel, and the result is the same as before. The final value for the integral, — 2wi times the sum of the residues on or below the real axis of <a, gives the value f or t > / (t < 0) Vitefi = ]zl ^ I ^(^) ^J^A Pr ^,i {t > o) l iwec ±^ n n \ I J \ I / Utilizing the equation sinz = {l/2i)(e iz — e~ iz ), we have / (t < 0) y s a,x;t) = ] _2_ ^ 1 S in (™*) sin (™) sin (^l) (t > 0) 1 n = 1 (10.19) Incidentally, this equation could also have been obtained by substi- tuting series (10.12) in the contour integral. The series for y& does not converge well, any more than the series for the delta function does. However, as with the delta function, we are not interested in computing y s , the behavior of the string after being hit by an idealized impulsive force at a mathematical point on the string. We only intend to use the series as an easy means of com- puting the behavior of the string when acted on by more realistic forces, distributed along the length of the string and spread out in time. For the most general type of force /(£,£), a function of time and of position, the response of the string is dr I d£ f(£,T)ys(£,x,t - t) -=3i|jJ>[T (, -' ) ]*-. (1 °- 20) This series does converge for reasonable forms of /((•,<)• The Piano String. — A reasonable approximation to the force of a piano hammer on a string is m.10] FORCED VIBRATIONS 103 /0 (-icr > *> fr; . a* - if > $ > a* + fr) *U0 = A cos [(^ - * )] cos(^) (10.21) \ (-iff < * < i<r; z - if < £ < x + if) The quantity a is the time duration of the application of force, and f is the length of the portion of string acted on by the force. The force starts acting at t = —io-, rises to maximum at t = 0, and goes again to zero at t = io-. The distribution of the force along the string is also like the positive half of the cosine curve, with the center of force, where it is greatest, falling at £ = x . Substituting this into Eq. (10.20) and carrying out the usual accompaniment of trigonometry and algebra, we can work out the expressions for the shape of the string. The integration over £ is from x Q — if to x + if, but the integration over t is a bit more elusive. When t is less than - i<r, y s (£,x; t - r) is zero for all values of r for which F(£,r) differs from zero, so the integral is zero, as it should be (since the string has not been hit yet). For t > io- the range of integration over t is from - io- to + io-; but f or - io- < t < io- the only range over which Fy s is not zero is between r = - io-, where F goes to. zero, and t = t, where y»(£,x; t - t) goes to zero. Conse- quently, we have three expressions for the resulting shape y(x,t) Fort<- io-, y(x,t) = For - io- < i < i<r, For t > io-, . U ( X t) = *M. ^S i l" cog(™rr/2Q l ["cos (n7rco-/2Z)l x^ec ^ n Ll - (nf/0 2 J |_1 - (WO 2 J D ;^ (TrnxA . (t71x\ . • (irnci\ 8m \—) sm \-r) sm \-r) This formula appears quite formidable, but it can be computed if need be. Certainly it is not very difficult to obtain from it the relative magnitude of the various harmonics in the free vibration after the end of the blow (t > io-). Incidentally, this example is a good one to show the power of 104 THE FLEXIBLE STRING [111.10 the operational-calculus methods. The algebraic gymnastics neces- sary to obtain Eq. (10.22) from the combination of Eqs. (10.20) and (10.21) are not particularly easy for one mathematically muscle-bound. Nevertheless, the calculations are only laborious, not subtle. On the other hand, to obtain the final formula for y(x,t) by any other method would involve still more labor, and more mathematical subtleties than we care to include in this volume. Our applications of operational calculus in Chap. II may have seemed rather like using a sledge ham- mer to drive a tack. We see now, however, that the problem of the simple string already provides a spike worthy of the sledge. To be honest, the resulting series for y(x,t) is not too good an approximation for the actual motion of an actual piano string when it is struck, partly because the actual piano string is not a perfect string but has stiffness. We shall indicate how to correct for this in the next chapter. The Effect of Friction. — In the foregoing analysis we have neglected friction, although it is present in every vibrating string. To complete our discussion we should show, as with the simple oscillator, that the effect of friction is to damp out the free vibrations and to change slightly the allowed frequencies. To show that this is so is not diffi- cult by the use of operational calculus, although it would be difficult by any other method. The difficulty lies with the nature of the frictional term. The resistive force per unit length opposing the string's motion is due to the medium surrounding the string, the medium gaining the energy that the string loses. Part of the energy goes into heating the medium, the amount depending on the viscosity of the medium; and part goes into outgoing sound waves in the medium, the amount depending on the radiation resistance of the medium. The medium also adds an effective mass per unit length, which may not be negligible if the medium is a liquid. The important point, however, the one that is responsible for our difficulties, is that the effective resistance due to the medium (and also its added effective mass) depends on the fre- quency of the string's motion. The equation Of motion for the string when friction is included is d 2 y_ T d 2 y MsdV e dF- T dx-*- R{03) M where R is the effective frictional resistance per unit length of string. To find the "normal modes" of the string involves a sort of circular process, since we cannot solve for the natural frequencies until we IH.10] FORCED VIBRATIONS 105 know the value of R, and we cannot obtain the value of R unless we know the frequency of motion. In this case it is actually easier to start with steady-state driven motion, for then the frequency is known and the value of R is definite. The equation for the string acted on by a simple harmonic force of frequency (a)/2ir) exerted on the point x = £ is *4 + 2 *(co) |f - c* g « m 8( x - fr** (10.23) c 2 = (f/e), *(«) = R(<a)/2e We are assuming that the added mass due to the reaction of the medium is negligible compared with the weight of the string. The more general case, where we must assume that e is also a function of <a, will be discussed later. As has been done previously, we shall assume that the steady- state motion of the string can be expressed in terms of a Fourier series V = ^j °» sin ( nr) e_i "' Substituting this in Eq. (10.23), multiplying both sides by sin (ttwz/Z) and integrating over x from to I, gives an equation for a m , from which one eventually obtains a series for y le ~{ C" + ** ~~ «*»)(« + tk + tow) where -w = [(T) 2 -^ We note that both w n and k are functions of w, the driving frequency. Characteristic Impedances and Admittances. — An interesting alternative method of writing this equation is in terms of transfer admittances. The ratio between the string's velocity v = — iuy and the driving force F{£)er iMt is Y m (S,x;o>) = 2 \~^ , 7T X \-Zm (£,x;a},n) - s \-zr) J csc v— ) csc \— ) ,106 THE FLEXIBLE STRING [111.10 The input admittance is obtained By setting x = f. Considering v to be analogous to a current and F analogous to a voltage, the reaction of the string (at x = £) is analogous to that of an electric circuit of an infinite number of parallel branches, the nth branch consisting of an inductance (le/2) csc 2 (rn^/l), a resistance of (IR/2) csc 2 (tw^/0> and a capacitance of (2l/ir 2 n 2 T) sm 2 (irn^/l), all three in series. The response of the string to a unit impulsive force concentrated at x = £ is obtained by computing the contour integral mU:X . t) .if { 2 < ?T* , \f* ( H!l ^ e ~n d » y vs ' ' ' irk J- „ ( ^-J (w + ik — w n ){w + ik + w n ) ) One pole for the nth term occurs when a; + ik(o>) — w n (u>) is zero. This may be difficult to solve algebraically if k(w) is a complicated function of «. However, it can usually be solved graphically or by successive approximations. We can write the solution symbolically as <a = io n — ikn', k n = k(w n — ik n ) \ HO 26") an = w n (<a n - ik n ) = [{nrnc/iy - A:*]* j It turns out that the other pole of the nth term is at co = — u> n — ik n . Taking residues at all the poles we finally obtain / (* < 0) .J ^sin(^f)sin(^)sin M (t > 0) y»(Z,x$) = 2\ e-*»< ^ (Tn%\ ^ (irnx\ ^ / A (t ^ ft ^ (10.27) which is to be compared with Eq. (10.19). This expression can be used in Eq. (10.20) to obtain the response of the string, with friction, to any transient force. The expression for ys gives the free vibrations of a string when started with an impulsive blow. It shows that the effect of friction is to introduce a damping term e~ knt into each of the component vibra- tions. The frequencies of free vibration (co„/27r) do not greatly differ from the harmonics for the undamped motion, (nc/2/), if the frictional constant k n is small. In a good many cases k n increases as n increases, so that the higher harmonics damp out more rapidly than the lower. In such a case the sound emitted by the string will be harsh just after the start of the motion, owing to the initial intensity of the higher harmonics, becoming "smoother" as the motion damps out. The amount of energy radiated by a string directly into the air is quite small compared with that which can be radiated by a sounding board attached to the string supports, as may be determined by com- in.il] STRINGS OF VARIABLE DENSITY AND TENSION 107 paring the intensity of sound radiated by a violin with and without backboard. The effect of the transmission of energy from a string to a sounding board, via the motion of the end support, will be discussed in Sec. 13. 11. STRINGS OF VARIABLE DENSITY AND TENSION The Fourier-series method discussed above is but one special case of a method of dealing with vibrating bodies which we shall use in all our subsequent work. We first find the shapes of the possible modes of simple harmonic motion of the system, the modes that satisfy the boundary conditions. In the case discussed above, these shapes were given by the functions A n sin ( irnx \ I ) but for other boundary conditions or for other systems {e.g., for a string with nonuniform mass distribution or for a stiff bar) the func- tions of x will be different ones, which we can represent by the symbols $n(x), A set of such functions, all satisfying the same equation and the same boundary conditions, is called a set of characteristic functions, and the corresponding allowed values of the frequency are called char- acteristic values. General Equation of Motion.— The most general equation for the string will involve a density e(x) and a tension T(x), both of which vary with x. The net force on an element dx of string can be found as before K2)L-[K2)1 The equation of motion of the string is finally where f(x,t) is the applied force per unit length. To study the normal modes of oscillation and to compute the char- acteristic functions and values, we consider the cases where f(x,t) = 0. For simple harmonic motion we set y(x,t) = Y{x)e~^ ivt . The equation dx 108 THE FLEXIBLE STRING [111.11 for Y is T{x) ^1 + (2tv)Hx)Y = (11.2) The solutions of this equation which satisfy the boundary conditions at x = and x = I are the characteristic functions \f/ n , and the cor- responding allowed values of the frequency are the characteristic values v n = (w«/2tt). Examples of these solutions will be given later in this section. The general free vibration of the system is then given by the series y = % A ^(x)e-^ (H.3) 71 = 1 where A n = B n + iC n = \A n \e^\ The determination of the con- stants B n and C n from the initial conditions is effected by the same methods that obtained Eqs. (9.9). The characteristic functions turn out to be orthogonal; i.e., J^(»)*.(*M«) «*«-&. (*-") (1L4) where the constant M n is called the normalization constant, or the effective mass, for the nth normal mode. We note that for the uniform string of Sec. 9, M n = iZe. Therefore if we are to represent the initial shape of the string, y , by the series 00 yo(x) = X B ^{x) n = X we obtain the values of B n by multiplying both sides by ^ m (x)e{x) and integrating over x from to I. The result is B m = Q-) f yo(x)*~(x)e(x) dx (11.5) which is analogous to Eqs. (9.9). Orthogonality of Characteristic Functions.— The property expressed by Eq. (11.4) is a general property of the characteristic functions discussed here. This can be shown in the following maimer : We multiply the equation for ^ n by \l/ m *4( r t) --»*■* m.ll] STRINGS OF VARIABLE DENSITY AND TENSION 109 Next we multiply the equation for i// m by \(/ n and subtract the two equations. The left side of the equation is Both sides are now integrated over #, giving [rw (*. f - *.*£)][ = (Bt - «a JJ *** * In all the cases we are to consider, the quantity in brackets on the left side of the equation is zero at both ends of the string, either because the ^'s or their derivatives are zero, or because T is zero there. There- fore the right-hand side of the equation is zero, and therefore the integral must be zero unless o»2, = <a\, i.e., unless m = n. This is just the orthogonality property given in Eq. (11.4). Driven Motion. — If the string is driven by a simple harmonic force of frequency (w/2r) which is concentrated at the point x = £, so that f(x,t) = F(£)8(x — %)e- iai , we find the steady-state motion by using a series of characteristic functions, just as we did in the previous sec- tion. We expand the delta function for the force into the series F(&S(x - = F(Q 2 (Jfc) <*)Mk)Mx) as can be proved by multiplying both sides by f m (x) and integrating over x. This series, of course, does not converge, but we are using it as a short-cut means of getting y. We also set y(t,x;o),t) = ^a n ^ n (x)e- iat and insert both series into Eq. (11.1), which gives 2 M*)(«i - «»)*•(*) = F(0 2 (^) <x)M&Mx) Equating coefficients of $ n (x), we obtain an expression for a n , such that the series for y is ^-J M n (« — o>«)(ci> -f «») v ' This is to be compared with Eq. (10.12) for the simple case. The corresponding transfer admittance (l/F)(dy/dt) is given by the series ■ - , 110 THE FLEXIBLE STRING [111.11 t j£1 M n (CO — «„)(« + <ti n ) n = l It is purely imaginary, indicating that thelmpedance (1/F) is purely reactive. At low frequencies (w < «i), y is in phase with the force, and the impedance is stiffness controlled (see page 33). The next stage is to use contour integration to obtain an expression for the response of the string to an impulsive force at t = 0, localized at x = £, symbolized by the expression S(t)8(x — £). The integral for this is obtained from Eq. (6.16) and is Each term in the series for Y m has two simple poles, atw = + o>„. Adding the residues for each term, we finally obtain the impulsive response i0 (t < 0) IS (-V) M&Ux) sinW) (t > 0) (11.8) This series does not always converge, any more than the series of Eq. (10.19) did. The final result, however, will converge. The desired final result is the shape of the string, y(x,t), when acted on by a general force per unit length /(£,£), distributed along the string in an arbitrary manner, and varying in an arbitrary manner with time y(x,t) = J dr J J^ y, (Z,x;t - t)/($,t) d$J - % tM {£ . * ™ M - ^ ] [ J>*-«> d "]} (lL9) We notice that the motion may involve components with each of the natural frequencies (co„/2tt). The relative magnitude of these com- ponents depends on the shape of /(£,£)• This general series is the formal solution for the motion of the string of variable mass and tension when acted upon by a transient force. In the special case of the uniform string, it reduces to the series of Eq. (10.20). For other cases, it will not be possible to go further until we have worked out the specific form of the characteristic func- III.11] STRINGS OF VARIABLE DENSITY AND TENSION 111 tions \p n for the case we happen to be interested in. We shall work out two cases in detail to show how these functions" may be computed and then will indicate briefly how friction can be dealt with. Nonuniform Mass. — Any variation in the distribution of mass along the string also disturbs the harmonic relation between the over- > tones. As an example of this we shall consider a string of length I, stretched between rigid supports, which is heavier at the center than at the ends. Taking the point x = at the center of the string, we specify that its mass per centimeter be e(x) = e [l — (x 2 /a 2 )], where a must be larger than (1/2) . (Why ?) The wave equation then becomes dx 2 \tJ\ a 2 ) dt 2 Waves do not travel with a constant speed along this sort of string, nor is the shape of the wave unaltered during its travel. This means that the graphical methods discussed in Sec. 8 cannot be used, and we must analyze the motion into its simple harmonic modes of vibration. If we set y = if/(x)e~ 2 * irt : , the wave equation reduces to dfy . 471-VeoA x 2 \ , We must find the solution of this equation and then pick out the values of v that allow the solution to be zero at x = ±(l/2). Equa- tion (11.10) was considered in Prob. 4 of Chap. I. The solution was shown to be z = 2ttv Veo/Tx and where D.(k,z) = oos(2) + i s « - ggg*' + ■ • ' Bo(M) = sin(*)+±*.-J| 2 T+... The function D e is even with respect to z = 0[i.e.,D e (k, —z) = D e (k,z)], and Do is odd [i.e., D (k, —z) = —D (k, z)]. Since we have picked the origin at the Center of the string, we must use either D e or D a as a possible solution to satisfy the boundary condi- tions. Any other combination will not have its nodal points placed symmetrically about x = 0. The shape of the first mode of vibration, which will vibrate with the fundamental frequency, will be given by 112 THE FLEXIBLE STRING [m.ll D e for the value of v that makes the first nodal point of D e fall at x = (1/2). Since D e is symmetric about x = 0, it will also have a node at x — — (1/2) and will thus satisfy both boundary conditions. The shape of the next mode, for the first overtone, will be given by D for the value of v that makes the first nodal point of D (the first one, aside from the one at x = 0) fall at x = (1/2). This function will also have » a node at x = —(1/2). Going back to D e , if we increase v until the second nodal point falls at x = (1/2), we obtain the third mode, or the second overtone, and so on. The Sequence of Characteristic Functions. — This illustrates a general property of modes of vibration of strings, no matter what their mass distribution or boundary conditions; if we arrange the charac- teristic functions giving the shapes of the different modes of vibration in a sequence of increasing frequency of vibration, then each function has one more nodal point between the points of support than the preceding one, the curve for the fundamental having none, that for the first overtone having one, that for the second overtone two, and so on. These functions can be labeled in order: fa, fa, fa ■ • • , a sequence alternating between D e and D for successively larger values of v = (u/2ir). The corresponding characteristic values or frequencies are v\, v 2 . . . , or a>i, o>2 .- • The functions are orthogonal, as we have shown on page 108. From them we can compute the motion of the string when acted on by a driving force, by substituting in Eq. (11.9). The general form of the string of variable mass which corresponds to the initial conditions that y = y (x) and (dy/dt) = v (x) at t = is given by the series oo V = % fa(x)[B n cos(2irv n t) + C n sm(2Trv n t) n = l where B n = Ij^J J yofae(x) dx Cn = ( o 1 \ T ) I v fae(x) dx, and N n = #€ dx \27rv„iV w / Jo Jo This can be shown by the method given in the first part of Sec. 10, now that we have proved that the functions fa are mutually orthogonal. The Allowed Frequencies. — The calculation of the allowed fre- quencies for the string corresponding to Eq. (11.10) is a somewhat m.ll] STRINGS OF VARIABLE DENSITY AND TENSION 113 tedious task, but the results will be given here. In Table 2 are given the values of the ratio between the frequency v n and the fundamental frequency of a string of uniform mass e , for different values of l/2a and of n. It should be noticed that the frequencies of the overtones are not integral multiples of the fundamental, so that the general motion of the string is not periodic. n= I n=2 n = 3 n = 4 Fig. 22.— Normal modes of a string of nonuniform density. Solid lines show the shape of the characteristic functions for a string of variable density eo[l - (x/a) 2 ] where (l/2a) = 0.8. Dotted lines show the corresponding sinusoidal functions for a string of uniform density eo. The shapes of the first four modes of vibration are given in Fig. 22 and are compared with the sinusoidal shape of the uniform string. It is seen that the change in shape due to nonuniformity is not great. It consists in a pushing out of those parts of the wave where the string is heaviest (i.e., near the center) and a consequent pulling of the nodes toward the center. 114 THE FLEXIBLE STRING Table 2. — The String of Nonuniform Density [in.li 0.2 0.5 0.8 1 1.003 1.016 1.042 2 2.012 2.071 2.181 3 3.019 3.117 3.299 4 4.026 4.161 4.411 5 5.033 5.203 5.521 6 6.039 6.246 6.630 7 7.046 7.289 7.738 8 8.053 8.331 8.846 Frequencies of the string of variable density e [l — (x 2 /a 2 )], stretched between rigid supports a distance I apart. The table gives values of j3 n for different values of n and of l/2a. The allowed frequency v n is equal to (1/2Z) -y/T/eo multiplied by/3„. Vibrations of a Whirling String. — There are cases where gravi- tational, or "centrifugal/' forces act directly on the various portions of the string, so that the tension varies from point to point along the string. In these cases the wave equation for the motion of the string takes on a different form. Returning to the discussion following Eq. (9.2), we see that when the tension is a function of x the net vertical force on an element of string is \ dx/ x+ d x \ dx/ x ~ dx\ dx) The resulting wave equation is d_ dx ( r S) d 2 y dt 2 (11.12) which reduces to the usual wave equation when T is independent of x. As an example of such motion, let us consider the vibrations of a string of length I pivoted at one end to a rigid support and whirling about this support. We shall neglect gravity, so that the string's motion will be in a plane. If we neglect the drag of the air, the position of equilibrium will be a straight line rotating in a plane about the central support with a uniform angular velocity co„ radians per sec. The string can oscillate about this position of equilibrium if it is dis- turbed, as long as its total angular momentum remains constant. In studying this oscillation we can neglect the uniform motion of the equilibrium line and deal only with the string's displacement from this IIL11] STRINGS OF VARIABLE DENSITY AND TENSION 115 line, #. The displacement is a function of t and of x, the distance from the support. At first we take y perpendicular to the plane of rotation. To find the expressions for velocity and acceleration in terms of these whirling coordinates, we study the behavior of a vector of vari- able length z, rotating with constant angular velocity co„ about its end, as shown in Fig. 23. The velocity of the outer end of the vector has two components: one along the vector, of magnitude (dz/dt), and one perpendicular to it, of magnitude co^. There are two components of acceleration parallel to z of magnitudes (d 2 z/dt 2 ) and - w *z and a component perpendicular to z of magnitude <a a (dz/dt). co a (az/at) c) 2 z/dt* Accelerations Displacements for the Whirling String Fig 23.— Velocity and acceleration components for a vector g rotating about one SJ. Jit t^ f, , Veloci . t y "«•.,.!» the ca «e of the whirling string, vector x (pointing along the line of dynamic equilibrium) rotates but does not change in length; vector y the displacement from equilibrium, changes length and rotates. In the case of the whirling string, we are to find the acceleration of a point that is represented by the sum of two vectors, one of con- stant length x and the other, perpendicular to x, of variable length y Both are rotating with angular velocity co a . If y is parallel to the axis of rotation (perpendicular to the plane of rotation), then the accelera- tion of the point is -calx along x (x is constant, so d 2 x/dt 2 = 0) and d*y/dt* along y. But if y is in the plane of rotation, the acceleration is -<a a x - ca a (dy/dt) along x and (d 2 y/dt 2 ) - a>ly along y. Since dy/dt is assumed smaU compared with w a x, in both cases the acceleration along x is -««*, which acceleration must be due to the force of the tension in the string. The force on an element dx of the string at a point x cm from the center will therefore be e dx x<S where 116 THE FLEXIBLE STRING [HI.11 e is the density of the string in grams per centimeter (which is supposed to be uniform). The tension at the point x will be the sum of the forces on all the elements of string beyond the point x out to the outer tip of the string T(x) = £ eulx dx = (?f\ (P - The wave equation for vibrations perpendicular to the plane of rota- tion will then be @*[<r -*>$-% To study the normal modes of vibration of the string, we set y = ^/{x)e~ 2vivt ' } and to simplify the equation, we change the scale of length, making x = h, so that z = is the center, and z = 1 is the outer end of the string. The resulting equation for yp is We can see without any further analysis that the frequencies of vibration will depend oh the angular velocity co but will not depend on the string's length or its density (as long as the density is uniform;. For increasing the length or density increases the mass, which tends to diminish the frequency; it also increases the tension, which tends to increase the frequency; and the two tendencies just cancel. For vibrations in the plane of rotation (coplanar) the additional term — uly comes into the equation of motion, making the equation for $ become The analysis is the same in both cases, since we are determining the value of v in either case. If we determine the frequencies v p and the modes of oscillation for the vibrations perpendicular to the plane of rotation (transverse) the shapes of the modes will be the same for the coplanar vibrations; but the allowed frequencies v a for the coplanar case will be related to the transverse ones, v p , by the relation -4-fey m.ll] STRINGS OF VARIABLE DENSITY AND TENSION 117 We shall first find the frequencies for the transverse vibrations, per- pendicular to the plane of rotation. The Allowed Frequencies. — To find the allowed frequencies of oscillation, we must obtain the solutions of Eq. (11.13) which are zero at z = 0. The easiest method of solving the equation is to assume that \f/ is given by a power-series expansion \f/ = a\Z + a 2 z 2 + azz z + • • • where we omit the constant term a in order to satisfy the boundary condition at z = 0. If we set a = (Sir 2 v 2 / col), the equation becomes I (1 - z 2 ) ^j = 2a 2 + 2(3a 3 - a x )z + 3(4a 4 - 2a 2 )z 2 + dz _ = — oaf/ = ■— adiZ — aa 2 z 2 Equating powers of z, we obtain a 2 = a 4 = a 6 = • • • = 0; a 3 = 2 — a 12 — a — g — ai; a 5 = — on — az ' ' ' • ^ ne solution of Eq. (11.13) which is zero at z = is therefore the series ,-4 + ii«, + p--)ff-.o , (2 - q)(12 - q)(30 - «) _ , 1 " 1 " 5,040 z -t- • • -J Since this infinite series satisfies the one boundary condition no matter what value a has, it would seem that all values of the frequency would be allowed, since v is proportional to the square root of a. On the other hand, when we pick some value of a and compute the values of \f/ as a function of z, we find that the series becomes infinite at z = 1. Since we cannot allow an infinite displacement of the tip of the string, this would seem to indicate that no value of the frequency is allowed! The dilemma is resolved when we notice that if a equals 2, or 12, or 30, etc., the series is not an infinite one but breaks off after the first or the second or the third, etc., term; in these cases ^ is not infinite at 2=1. The applying of the two boundary conditions, that ^ = at z = and that rp be not infinite at z = 1, rules out all but a discrete set of values of a and therefore allows only a discrete set of frequencies, corresponding to the set of values a = 2, 12, 30 • • • 2n(2n — 1) • • • . If we always choose a\ so that if/ = 1 at z = 1, then the allowed frequencies of vibration and the corresponding character- istic functions are 118 THE FLEXIBLE STRING [III.ll v 2 = 2.4495 f^\ fc(z) = U5z 3 - 3z) v z = 3.8730 (g^, * 8 (s) = |(63z 5 - 702 3 + 15s) / (U ' 14) v* = 5.2915 ( ( ^\, M*) = inr(429z 7 - 693s 6 + 315s 3 - 35s *-fe) Vw(2n - 1) The functions ^ n are equal to the Legendre functions P m (z) [denned in (Eq. 27.6)] of odd order (m = 2n - 1). The fundamental vibration goes through one cycle every time the equilibrium line rotates through 360 deg. The overtone frequencies, however, are not integral multiples of the fundamental and therefore bear no simple relation to the frequency of uniform rotation of the equilibrium line. The forms of the string for the first four normal Fig. 24. — Forms of the normal modes of vibration for the whirled string. The displacements are from the dotted equilibrium line, which is supposed to be rotating in a plane about its left-hand end with a constant angular velocity a> a . modes are shown in Fig. 24. It is to be noticed that each function in the sequence has one more node than the one preceding, also that these nodes tend to concentrate near the outer end of the string, where the tension is least. The Shape of the String.— It can be verified that these functions are mutually orthogonal; that ( (n ^ m) i *•(«)*«(«) dz = { 1 (H.15) { (4n — 1) v ' ni.ll] STRINGS OF VARIABLE DENSITY AND TENSION 119 We can show, by the methods discussed earlier, that if the string is started into oscillation by making its initial displacement from equilib- rium equal to y (x) and initial velocity (over and above the uniform velocity of the equilibrium line) equal to v (x), the subsequent motion will be given by the series y(x where A n — 00 »0 = ^| #» (j) l A n cos(2jr* n + B n sin {2wvJ)] (11.16) n = l (4n - 1) Since the overtones are not harmonics, this motion will not be periodic unless it is started in one of the normal modes, so that all the A's and B's are zero except one pair. It should be noticed that the funda- mental mode of this transverse motion corresponds simply to a shifting of the string's plane of rotation. For the case of vibration in the plane of rotation, we mentioned earlier that if a were made (8irV/coi[) + 2, instead of (8irV/«2), the discussions on pages 117 and 118 would still be valid. This means that the shape of the normal modes of vibration in the plane have the same shape as the transverse modes, so that Fig. 24 is good for both types of vibration. However, the allowed frequencies for the two types differ; instead of those given in Eq. (11.14), the modes in the plane have the frequencies V! .= 0, V2 = 2.2361 fe) '■ = fe) V(« - D(2n + 1) The equations for the ^'s are the same as given in Eq. (11.14). We note that the lowest "frequency" is zero, indicating that the funda- mental mode is not a vibration but a steady motion, corresponding to a change in angular velocity of the string. Only the lowest modes for coplanar and transverse motions repre- sent a change in the string's total angular momentum, the coplanar mode corresponding to a change in magnitude, and the transverse mode corresponding to a change of direction. The higher modes represent vibrations that do not affect this total angular momentum; nor are these higher modes periodic with respect to the rotation of the string about its support. Driven Motion for the Whirling String.— The general formulas worked out at the beginning of this section «an be applied to the prob- 120 THE FLEXIBLE STRING pi.ll lem of the transient or steady-state motion of the string about its dynamic equilibrium. We shall see in Sec. 27 that the characteristic functions that we have derived for the whirling string are proportional to the Legendre functions $ n {z) = P^n-iix/l). Equations (27.7) give some of the mathematical properties of the functions. One other property is of use here P (0) - ( 1V . 1 • 3 • 5 • • • (2n - 1) J2n)^ Equation (11.6) indicates that, if a whirling string is subjected to a simple harmonic force F(x)e~ i<ot dynes per cm distributed along the length of the string, the steady-state displacement of the string from its dynamical equilibrium will be n = l L where *•(*) = P ^-i(j\ M n = ^^rY «i = «J(» - D(2n + 1) when the force applied is in the plane of rotation and F(x)e^ iat is the component perpendicular to the equilibrium line of the whirling string. A simple example of the use of this formula is the problem of the effect of gravity on a string whirling in a vertical plane. The com- ponent of force per unit length perpendicular to the string, when the string is at an angle 6 with the horizontal, is eg cos 0, uniform along the string (g is the acceleration due to gravity). ■ Since the angular velocity of the string is «„, 6 = co a t, so that the effective driving force is the real part of ege-^K The formula above can therefore be used, with F(£) = eg. The driving force in this case has a frequency co equal to the frequency co of rotation of the string; but since w does not equal any of the natural frequencies w n of the whirling string, no resonance occurs, so that the motion of the string away from equilibrium due to the effect of gravity is not very great unless co is small. To obtain numerical results we must compute the integral fegif/ n d£. Using Eqs. (27.7) and the results given above, f *.(*) # = I f P*n-i(z) dz = -^zn l p ^~2(0) - P 2 „(0)] _ , _°n*-i Z 1-1-3 • • • (2n-3) " ^ L) l 2 • 4 • 6 • • • 2n 2n(2n-2)l ~ y L ) l (2»n!) 2 STRINGS OF VARIABLE DENSITY AND TENSION 121 Putting this all together into Eq. (11.17), we finally obtain a series for the displacement of the string from dynamical equilibrium y, or better the dimensionless quantity y/l, analogous to z = (x/l), (j)-m^-^ 1 1-3 (2n - 3) 2-4-6 • 2n 4:71—1 2n 2 - n - 2 *-©} This gives the steady-state effect of gravity on a string whirling in a vertical plane. We see that the relative magnitude of the effect depends on (g/lafi), the ratio be- tween the acceleration of gravity and the acceleration of the tip of the string in its circular path (as should have been expected). Figure 25 shows the shape of the string at the eight parts of the cycle t = 0, (««/8), (« a /4), (3«„/8) • • • (7««/8), for the ratio (g/la>l) = (i). The dashed lines show the equilib- rium positions of the string if grav- ity were not acting. We note that on the average the displacement is out of phase with the force, being above the equilibrium lines on both sides, ahead of them when it is going up, and behind them when it is going down. Physically, this means that the string goes slower at the top of its swing than at the bottom, which corresponds to the well-known rela- tion between kinetic and potential energy. Mathematically, this comes about because the "driving frequency" (o) a /2ir) is above the natural frequency v x = of the first normal node, so that this mode (which is the largest term in the summation) is out of phase with the force. The natural frequencies of all the other modes are higher than (w /27t), so they are in phase with the force. The presence of the higher modes accounts for the curved shape of the string. Physically, it is due to the fact that there is a greater variation in the speeds of the string, between the top and bottom of its swing, for the outer portion of the string than for the portion near the axis. Fig. 25. — Shapes of the whirling string acted on by gravity, at eight equally spaced parts of the cycle. Ratio (ff/wa 2 Q has been chosen to be 0.20, large enough to exaggerate the effect. 122 THE FLEXIBLE STRING [111.12 12. PERTURBATION CALCULATIONS The motions of some strings, with simple distributions of mass and tension, can be computed by the methods discussed in the previous section. An exact solution of Eq. (11.2) can sometimes be obtained without excessive labor, and the integrations of Eq. (11.9) can then be worked out for the general motion. However, in many cases an exact solution of Eq. (11.2) is not easy to obtain, so it will be useful to work out an alternative, approximation method. This approxima- tion technique turns out to be most effective when the variations in density and tension do not change greatly along the total length of the string, so that the string does not differ much from a uniform string. The deviations from uniformity are called perturbations, and the technique of calculations is often called the perturbation method. The Equation of Motion. — As long as we are at it, let us make the formulas as general as possible. Suppose that the mass of the string per unit length at a distance x from one end is e [l + b(x)], where 6 is small compared with unity over the whole range of x from zero to I. Suppose also that the tension is ^[l + h(x)] and that the trans- verse frictional resistance per unit length is V^Vo r (%), where both h and r are everywhere small compared with unity. We assume, however (until the next section), that the string supports are rigid. In one of the cases considered in the last section, b was equal to -(l/a) 2 [x — (1/2)] 2 and h and g were zero. When a is large com- pared with (1/2), b is everywhere small compared with unity. The case of the whirling string, also considered in the previous section, cannot be dealt with by perturbation methods, since the tension varies from zero to a maximum value, and the corresponding h would not be small compared with unity. The case of a string hanging vertically between two rigid supports would be a workable example, however. The tension at the bottom (x = 0) would be T Q , that at the top (x = I) would be To + (e lg) (where g is the acceleration of gravity), and the tension would vary linearly in between, so that h would equal (e g/T )x. If the tension at the bottom, T , is larger than the weight of the string, (e lg), then the perturbation method can be used. Many other examples, some of practical utility, can be thought of. The quantities b, h, and r are the perturbations. If t hey ar e all zero, the string is a uniform one, with wave velocity c = y/To/eo, and with normal modes and frequencies IH.12] PERTURBATION CALCULATIONS 123 , » . (irnx\ yl(x) = sin I — \ (12.1) If they are not zero, but are small, we can expect that the shape of the nth. mode will be yl(x) plus a small correction and that the correspond- ing natural frequency will be vl plus a small correction. We wish to find how to compute these corrections. The equation of motion of the string, with no driving force applied, is ' *U + «*)] + VT^, r(x) | = T ± J [1 + «,)] I j (12.2) To find the correction terms for the free vibrations, we assume that the corrected normal mode can be expressed in terms of a series ^ m (x)e- ia - f = \y° m (x) + 2) a n yl{x)\ er*""* c& = (2x^(1 + Vm ) = (x) 2 (5) (1 + Vm) (12.3) where the coefficients a n and the correction to the frequency, t] m , are supposed to be small compared with unity. The summation indi- cated by the 2 sign is over all values of n except m (i.e., the symbol 5* means "not equal to"). We insert Eq. (12.3) into Eq. (12.2) and drop out "second-order" terms (i.e., bt], ah, etc.) and finally obtain First-order Corrections. — The values of the coefficient a n can be obtained by the device of multiplying all terms by sin (rsx/l) (where s is some integer) and integrating over x from to I. The orthogonal properties of this function make a large number of integrals vanish; in fact all terms of the summation vanish except that for which n = s. The resulting equation is 124 THE FLEXIBLE STRING [m.12 nm — (n/m)h nm + i(l/irm)r nm a n <^ 1 — (n/m)' where b nm = (j\ J o sin^j sin (^r) b W dx ^ = (?)^in(^)sin(^)Kx)^ where we have changed back from s to n after performing the integra- tions. The value of the correction to the frequency is obtained by multi- plying Eq. (12.4) by sin (irmx/l) and integrating r\ m ^ —b m m + h mm ~ 1 \~ZZ J Tmm J a m = o) m - ik m ; ^ — ( 2) Tmm ( ( 12 - 6 ) C0 m CH ( — ; — 1 Vl + Jl mm — bmm J Since b mm is proportional to the average value of the mass perturbation, averaged over the wth mode, and since h mm is proportional to the simi- lar average of the tension perturbation, we can say that if the mass perturbation tends to increase the mass the value of b mm will be posi- tive and the natural frequency («,»/2w) will be thereby decreased. Conversely, if the tension perturbation is positive, the natural fre- quency will be correspondingly increased. (This is not surprising.) The free vibrations are damped, due to the resistive term r, the expo- nential time factor for the wth mode being e-*-*-*— «. The coefficients a n representing the difference (to the first approxi- mation) between the shape of the perturbed mode of vibration and the mode for the unperturbed (uniform) string are proportional to the "transfer" integrals b nm , etc., which measure the amount by which the perturbations " couple" the various/modes together. If the only perturbation is a f rictional resistance r independent of x, then all these transfer integrals are zero, the a's are zero, the perturbed modes have the same shape as the unperturbed, and the sole effect of the m.12] PERTURBATION CALCULATIONS 125 friction is to introduce damping into the amplitude of free vibration, via the factor e~ kmt . Other examples are not as simple. Examples of the Method. — We take up again the case of non- uniform density discussed in the previous section, in order to show how easily (?) the calculations go. The perturbation is and the other terms h and r are zero. The necessary integrals are bmm = ~ \iw) l 1 ~ v^yj b — mnl2 n + (—i) m + n ] ° nm ir 2 (m 2 - n*)*a* L + { } J and the corresponding natural frequencies and shapes of the normal modes are, therefore, , , N . Ux\ %l 2 . (3irx\ Mx) c* sm^J - ^-^ sm^—j (5ttx\ _ 5Z 2 34567r 2 a 2Sm| , . , . (2tx\ 2l 2 . /Wv V /io^ Mx) ^ sm \^)-2^ sm \~r) ) (12 - 7) _ SI 2 . (Qttx\ _ 512x 2 a 2Sin \ I ) . , , . (3irx\ . 271 2 . (tx\ _ 135Z 2 . (5tx\ _ l,024x 2 a 2Sm \ I ) Since the perturbation is symmetric about x = (1/2), the series for the normal modes contains either all even modes or all odd modes. (Why is this?) We note again that the overtones are not strictly harmonic in this case. Incidentally, these results check fairly well with the values given on page 114 and with the curves of Fig. 22. The amount of work required for the present calculations is much less than for the exact solution. The approximate formulas are no longer satisfactorily 126 THE FLEXIBLE STRING [111.12 accurate, however, when a becomes smaller than about (3Z/4), and the exact solution is then the only satisfactory one. The effect of other types of mass loading can be computed by this method. A few examples will be given as problems. The other perturbation considered here in detail will be that of the string hanging vertically between rigid supports, where the tension is (To + € gx), with x measured from the bottom end. The perturba- tion here is h(x) = (gx/c 2 ), where c 2 = (T /e ), and the required integrals are , _ feY h _ _ 2^o m 2 + n 2 _ ( _ 1)m+n] n mm - \2ToJ' * 2 T Q (m 2 - n 2 ) 2 L V J J so that _ 2lge mn(m 2 + n 2 ) , _ ,_.s m+n] an ~ t 2 To (m 2 - n 2 Y L V J-; J Therefore the natural frequencies and the first two modes are n [To L , (lge \ (tx\ . (2lge \\l0 . faA , 68 . (4rx\ -»(t s )+"-] sin sin I ,290 + 91261 Sm| etc. Here the overtones are still harmonic (at least to this order of approxi- mation), and the even modes are modified by additional terms of odd modes, and vice versa. The result is to move all the nodes toward the bottom end, as will be seen if one plots out the i^'s. Characteristic Impedances. — It is now time to expand on the analogy, touched upon at the end of Sec. 10, which is suggested by writing the reaction of each mode of the string as an impedance or admittance. In Eq. (10.25) we denned a transfer impedance Z m {Z,x)w)ri) for the nth mode, relating the motion at point x due to force at point £. Now that we have seen that modes can be coupled together by perturbations, we see that there must be a transfer imped- ance corresponding to this coupling, which can be labeled Z m (£,x;a>;n,m). ni.12] PERTURBATION CALCULATIONS 127 This symbol may seem overmuch endowed with labels and subscripts, but we shall see that its use "saves considerable time and space and suggests several useful analogies. (We note here that the subscript m denotes "mechanical" and the m in the brackets denotes an integer. We shall omit the subscript in this section to reduce complexities.) Before we come to the impedances, let us define an average mass, resistance, and stiffness of the string, averaged over the nth normal mode in a manner analogous to that of Eq. (10.25): M{x,Z;n) = [yl(x)yl(k)]- 1 j Q [^)J 2 e [l + b(u)}du = (|J (1 + U CSclyJcSclyj R(x,!t;n) = faftaOriK*)]- 1 J Q [yl(u)Y VnTo r(u) du = (f Z ) r nn csc(^) csc(^) }> (12.9) K(x,i-;n) = -te(x)M&\- 1 £ V°M £{w + *(«)] iy°M du (12.10) = y^r) (1 + hnn) csc \t) csg vi) One can also write a corresponding "transfer" mass, etc., as follows M(x,£;n,m) = I , f° ^J I Hu)yl(u)y° m (u) du = (*-Ab esc (—} esc (^) D/ . /e cA (irnx\ /VraA R{x,^;n,m) — I -^ \r nm cscl -y- ) cscl — j- J trr y \ /V 2 ranT \ , (irnx\ /VraA K{x,^;n,m) = I — ^7 — J h ™ csc l ~T~ ) CSC \~T~ ) If this analogous symbolization is to have any meaning at all, it should serve to express natural frequencies, etc., by formulas analo- gous to those for a simple oscillator. For example, from Eq. (4.1), we see that the expression for the damping constant of the simple oscil- lator is k = (R/2m). The corresponding expression for the nth mode of the perturbed string is, from Eqs. (12.9) and (12.6), 128 THE FLEXIBLE STRING [111.12 I" R(x,$;n) 1 /e\r r nn 1 ,M h l2M{x,Z;n) J \2j |_(1 + b nn ) J ~ W *** ~ " if we neglect all but first-order terms. Similarly, the expression for the natural frequency of the simple oscillator is co = (2irv ) = y/K/m. In the present case, we have \ K (*,M "1* _ (*nc\ f (l + U 1* Vl + Ann - K again neglecting second-order terms. Therefore the analogy is a fruitful one. Each normal mode is analogous to a simple oscillator, or analogous to a circuit of inductance M(x,£;n), resistance R(x,£;ri), and capacitance l/K(x,l-;ri) in series. The characteristic impedance of the nth mode of vibration of the string is .. Z(x,Swn) = -ia>M(x,Z;n) + R(x&n) + (m K(x,£;n) (12.11) This impedance is a function of the frequency (co/2x), not only because of the factors (— iw) and (i/<a), but also because the quantities 6, r, and h which enter into M, R, and X may be functions of the frequency. For instance, if the string radiates sound, then both r(x) and b(x) will have radiation components that depend on frequency, as we shall see in Chap. VII. There is also a characteristic transfer impedance between the nth and mth modes Z(x,£;o>;n,m) = —i(j}M(x,£;n,m) + R(x,£yn,m) + (Mi£(z,S;n,m) (12.12) which suggests that it. is a sort of mutual impedance, coupling the nth and mth modes, as indeed it turns out to be. We notice that if the string is truly uniform (6 = h = 0, r independent of x) then all the transfer impedances are zero and there is no coupling between modes, which is as it should be. Finally, the expression for the perturbed normal mode has a par- ticularly simple form when written in terms of these impedances. For instance, the transfer impedance between m and n at the frequency <a m /2T of the mth mode [see (Eq. 12.6)] is, to the first order, m.12] PERTURBATION CALCULATIONS 129 Z(x,£;a) m ;m,ri) m\—i ( ~yj [%) & ™ + \1T/ Tnm Similarly, the impedance for the nth mode at the frequency of the rath mode is, to the zeroth order, A comparison of these formulas with Eq. (12.5) shows that an alternative fashion of writing the series for the perturbed normal mode is This is interesting in that it shows that the amount by which the nth mode modifies the mth mode is proportional to the transfer impedance Z(x,£;o) m ;m,ri), which is in turn dependent on the perturbation integrals Omnj l^mn) and. T mn . Forced Oscillation. — The full usefulness of the characteristic impedances defined in Eqs. (12.11) and (12.12) becomes apparent when we take up the forced motion of the string. Suppose that we apply a concentrated force at the point x — £. The equation of motion is then *(i + 6) g* + V^To r I - T 1 [(1 + h) I] = F{&B(x - 0e- (12.14) Setting into this equation, multiplying it by sin (irmx/l), integrating over x from zero to I, and finally dividing by sin (irm^/l), we obtain the funda- mental equations for the forced motion of the string 130 THE FLEXIBLE STRING [m.12 U m Z(x,Z;a>;m) + X U n Z(x,£wn,m) = F(Q - (12.15) where U m = —io)A m sin ( Tmx\ The quantity A n sin (irnx /I) is that component of the amplitude of motion of the point x on the string which corresponds to the nth mode, for a driving force F(£) applied at the point £. Similarly, the quantities U n are the corresponding modal components of velocities. The electrical analogue is therefore a network of infinitely many paths in parallel, the self-impedance of the rath path being Z(x,£;o);m), with each path coupled to every other by mutual impedances Z(x,£;o);m,n). The analogue to the applied force is a voltage F(g) applied across the network ; and the U n 's correspond to the currents in the parallel paths. The frequencies of free vibration correspond to the allowed frequencies of vibration of the network when the generator is short-circuited. In order to calculate the values of the A' a or U's by successive approximations, we first neglect the transfer impedances Z(x,£;a);m,ri) which are supposed to be considerably smaller than the impedances Z(x,%;o};m). (Why?) Therefore, to the "zeroth*' approximation, A G in 6^— F ^ This can then be inserted back in the series in Eq. (12.15) and the equation solved again to get the first approximation a • f Tmx \ ~ to |\ - "5! ?&jwmo1 AmSin \ l ) ~~ -iuZixfr^m) l l £d Zfofcwjn) J — io)Z(x,£',ai;m) sin(xraa;/7) if we utilize (and generalize) the definition of Eq. (12.13). Second and higher approximations could be obtained by repeating this process; but we shall let well enough alone. Therefore the expression y(x,^,t) c* 2i -i.Z(x^;m) [' ~ Jj ~Z^^rJ (12 ' 16) is an expression for the steady-state shape of the string when driven by the force F(£)5(£ — x)e~ ia \ which is correct to the first order in the small quantities [Z(x,%;a>;m,n)/Z(x,£;o>;ri)]. 111.12] PERTURBATION CALCULATIONS 131 When the string is uniform (6 = h = 0, r independent of x), then the quantities Z(x,%;<ti)m,ri) are all zero, and the expression becomes an exact one. Fig. 26. — Sequences showing the successive shapes of a damped string, driven by a force of frequency v, during a half-cycle of the steady-state motion. The force is concentrated at the point marked by the circle, and the arrows show its successive magnitudes and directions. The three sequences are for the driving frequency v, equal to f , two and three times the fundamental frequency of the corresponding undamped string. Uniform String, Steady State. y(x,£ya,t) = > }V m <** —uaZ(x,S;u;n) Z(x,$;a,;n) = [ -ico (^) + (^f) } (12.17) + ©(^)M~M?) 132 THE FLEXIBLE STRING [111.12 which is identical with Eq. (10.25). Figure 26 shows the shapes of such a string when driven by a force applied at £ = (1/3). The center sequence shows a case of resonance, with large amplitude (not infinite because the resistance is not zero). The left-hand sequence shows a case that should be another resonance, were it not for the fact that the force is applied at a node of the corresponding free vibration (in which case sin (rn^/l) is zero and the corresponding Z is infinite) so only nonresonant modes appear. As noted in the discussion of Eq. (10.25), the analogy of the input admittance to that of a parallel network of impedances Z(£,£;co;n) is obvious. In the present case when the perturbations are not uniform, the individual paths are coupled by the impedances Z(£,£;co;ra,?i.) = Z(£,£;co;w,ra). Transient Motion. — The calculation of the transient motion of the perturbed string follows the same procedure as has been outlined in Sec. 11. We set up the steady-state response of the string to a con- centrated force, which has been done in Eq. (12.16). We then express the Z's in terms of co, b nm , r nm , etc., and perform the contour integra- tions necessary to obtain the response to a unit impulse at £ y*M;t) = ^ J_ m [f^] vOM;»,0 ^ ( 12 - 18 ) The details of computing the residues about all the poles of this function are quite wearisome but involve nothing particularly different from the cases discussed before. Each of the first terms in the series for y(x,£;o),t) has two poles at co = ±co TO — ik m , with values that have been given in Eq. (12.6). The second terms, which are summed over both m and n, have four poles: two at co = ±co m — ik m , and two at co = +co„ — ik n . When the residues have been computed and the results expressed again in terms of the impedances, we have the formula t < 0, y»(x,$;0 = t > 0, yb(x,%;t) ~ real part of Op—kmt—iumt {'2 ^m(03m,^m(0} m ,X) (xmeoc) Vl + h nn + b r , { — - o — kmt — iifmt ^Mix^nOKix^m) (12 " 19) TO f ^ Z(x£',o) m ;m,n) _ ^ Z(x,%;o) m ;n,m) T\ L Si Z M'>">»ri Si Z(x^ m ;n) J/ in.13] EFFECT OF MOTION OF THE END SUPPORTS 133 which is correct to the first order of the small quantities b, h, and r (but is not correct to the second order). Here we refer to Eq. (12.13) for the formula for the perturbed characteristic functions, to Eqs. (12.9) through (12.12) for definitions of the impedances, and we again note that k m c^ [ R(x,Z;m) 1 _ [ K(x,!-;m) ~\ l2M(x,^m)]' " m -lM(x,Z;m)] Thus we have arrived again, by the route of the operational calculus, to the result that ^ ro (w m ,x), as defined in Eq. (12.13), gives the shape of the wth mode of free vibration of the perturbed string; and that k m and w m , as defined in Eq. (12.6), are the damping constant and fre- quency of this vibration; all within the range of validity of the first- order approximation considered here. The formula of Eq. (12.19) corresponds to that of Eq. (11.8). The integral for the general motion due to a general force similarly corresponds to that of Eq. (11.9). 13. EFFECT OF MOTION OF THE END SUPPORTS At the end of Sec. 10 we mentioned that strings are poor radiators of energy by themselves. In fact the most effective method of radi- ating the energy of vibration of a string is to attach its supports to a sounding board which will do the radiating. Before we can get very far with a study of the effect of the sounding board, we must discuss the effect on the motion of the string of the motion of the end supports. Such an investigation will be doubly useful, for we shall find that the methods developed here will also be necessary to the study of sound waves in tubes and in rooms. We shall consider here only strings of uniform density and tension: further complications of nonuniformity can be treated, if need be, by the methods of the previous section. Impedance of the Support. — Presumably the supports are capable of bearing the string tension in the direction of the equilibrium line of the string. Any force on the support at right angles to this line, however, will cause some sidewise displacement of the support, small or large depending on the transverse mechanical impedance of the sup- port. This impedance, the ratio between the sidewise force on the support and the transverse velocity ( — iu times the transverse displace- ment) will be called Z for the support at x = and Zi for that at x = I. The real part of Zq is a resistance R , and the imaginary part is a reactance X (Z = Ro — iXo = \Z Q \e- iv o) ) the phase angle being <po = tan -1 (X /JSo). 134 THE FLEXIBLE STRING [111.13 When the string is in equilibrium, the supports are in equilibrium, and their sideward displacements y(0) and y(l) are zero. When the string is displaced, however, there is a transverse component of the tension T, which will tend to pull the supports sideward, as shown in Fig. 20 (on page 91). This transverse force on the support at x = is T y = T sin 0, and when is small (as it usually is) it is almost equal to T tan 0, which' is equal to T times the slope of the string, (dy/dx), at x = 0. The new boundary condition is therefore that T^dy/dx), at x = 0, is equal to the transverse impedance of the sup- port, Z , times the string velocity {—io>y), at x = 0. At x = I the force is minus T times the slope at x = I. The two conditions may therefore be summarized as follows, for strings oscillating with simple harmonic motion of frequency (co/2tt) : .,«*-«> "-ty®! (I8J) <*-» »- + (£)(£)) We see that when the supports have infinite impedance the boundary conditions reduce to the usual ones requiring y to be zero at x = and I. Reflection of Waves. — A wave coming from the right and meeting the support at x = would not undergo perfect reflection with change of sign. Therefore we must first reexamine the reflection of simple harmonic waves under these boundary conditions, in order to see what the mathematical expression should be for the shape of the stand- ing wave. We consider first the support at x = 0, The incident wave, coming from the right and traveling in the negative x-direction, is Ate-W") (x+ct) , where c 2 ={T/e) and a = 2tv. The reflected wave must also be simple harmonic, but its amplitude A r will differ in magnitude and phase from Ai. To emphase this we can set A r = A ie -2Ta +2Tip 0} w here the value of <x determines the reduction in ampli- tude of the reflected wave and the value of jS determines the phase change on reflection. The total displacement of the string is therefore y — (Aie-^ ivx/c) + A r e +(2Tivx/c) )e- 2 * i ' ,t cm 2) where X = (c/v) = (2tc/o)) is the wavelength of the incident (and reflected) wave. The values of «o and /3 must be adjusted so that the boundary condition at x = is satisfied. The slope of the string is m.i3] EFFECT OF MOTION OF THE END SUPPORTS 135 (— ) = — [— ^^(e-CaTisA) — ^-2*(« -i/j -fa/X)) e -&»* The boundary condition of Eq. (13.1) for x = therefore requires that * - a - * - (t) -5 e -2T(a -i^ ) = + <z i -r i + r 5 j-0 j-iK>d |io9y = e I- Z- 2- tr- -4 (13.3) -3-2-1 1 1 a = Real Part of q. Fig. 27. — Conformal transformation of the q = o — ib plane onto the f = — i\ plane, and vice versa (when the drawing is turned upside down). These equations display a very interesting reciprocal relationship between q, the ratio between the amplitudes of the reflected and inci- dent wave, and f , the ratio between the transverse impedance of the support causing the reflection and the wave impedance of the string defined in Eq. (10.3). Relationships between complex quantities 136 THE FLEXIBLE STRING [111.13 expressable by equations relating them can be represented graphically by what are called conformal transformations, from which approximate values of the relationships can be read off. For instance, in this case the line a = — i is a straight line on the g-plane, parallel to the 6-axis, one-half unit to the left of the origin; but on the f -plane it is represented by a circle of radius two units, with center at f = 2, as shown in Fig. 27. For the particular transformation symbolized by Eq. (13.3), circles on the f -plane go into circles on the <?-plane, and vice versa. A straight line, being a limiting form of a circle, also goes into a circle. The coordinate lines of the g-plane are drawn on the f-plane in Fig. 27 and constitute a representation of the transformation. From it we can read off the values of and x corresponding to values of a and 6. For instance, corresponding to q = — 0.5 + 0.'5iis{" = 1 — 2i, and corresponding to q = — i is f = i, and so on. Since the reverse transformation (f to q) has the same form, Fig. 27 need only be turned upside down to have the reciprocal representation. Several interesting things are apparent on studying Fig. 27. In the first place, as long as is positive (and it must be in any actual case, for the real part of an impedance must be positive) the magnitude of q will be less than unity. This means that in any actual case the reflected wave will never be larger than the incident wave (which seems sensible). In the second place, when f = 1 (Z = ec) there is no reflected wave at all (q = 0); when f = °o, q = —1; and when f = o, q = 1. When |f | = 1, the real part of q is zero, so that the phase change on reflection is ± 90 deg. However, when the real part of f is zero, \q\ = 1, i.e. , the reflected wave is as large as the incident one. Hyperbolic Functions. — The relations displayed by Eq. (13.3) are interesting enough, but a modified form of the transformation will turn out to be much more useful in further calculations. The equa- tions will utilize hyperbolic functions, so that it will be well to review their properties. The fundamental definitions are parallel to those of Eq. (2.7), which defined the trigonometric functions in terms of the exponential function, cosh (x) = \{e x + e~ x ) ; e x = cosh (x) + sinh (x) sinh (x) = \{e x — e~ x ) ; e-* = cosh (x) — sinh (x) -j- cosh (x) = sinh (x) ; ' -r- sinh (x) = cosh (x) (13.4) The relationship between these functions and the trigonometric functions are given by the equations in.13] EFFECT OF MOTION OF THE END SUPPORTS 137 cosh (x + iy) = cosh (x) cos (y) + i sinh (x) sin (y) sinh (x + ty) = sinh (x) cos (?/) + i cosh (x) sin (y) cos ix + tV) = cos (x) cosh (y) — i sin (x) sinh (?/) sin ( x + *2/) = sin (x) cosh (?/) + i cos (z) sinh (?/) (13.5) The amplitude and phase angle of cosh (a; + iy) and of sinh(x + iy) is given in Plate I at the end of the book. We also define a hyperbolic tangent and cotangent in an analogous manner tanh (x) = sinh (x) _ 1 — e~ cosh (x) 1 + e~ coth (x) tanh (x) (13.6) Tables of the hyperbolic sine and cosine are given on page 438. More interesting to us at present are the values of the hyperbolic functions of complex quantities. The values can be obtained by using Eqs. (13.5), which result in the useful relations given in Eqs. (13.7). cosh[ir(o! .— iff)] = |cosh[7r(a — #)]|er*o |cosh[x(a — i0)]\ = \/cosh 2 (7rQ:) — sin 2 (xj8); tan £2 = tanh(ira) tan(x)3) [*« -if- 0] cosh[ir(a — ifi)] = i sinh | ir(a — ifi = — cosh[7r(a — i(3 — i)] tanh[ir(a - i&)\ = 6 - i X = f = |f \<T* sinh(27ra) _ _ sin(2x/3) e = x = cosh (2ira) + cos (2*0) ' A cosh (2™) + cos (2irj8) 2 + [x + cot(2x/3)] 2 = csc 2 (2tt|8) [6 - coth(27m)] 2 + x 2 = csch 2 (2W) tanh [ir(a — ip)] = tanh [ir(a — i@ + in)] 1 (13.7) tanh [ir(a — ifi + in + i*')] (n = 0, ±1, ±2 ) cosh (2x0:) — cos(27T|8) _ _ T [* sin (2tt/8) T cosh (2™) + cos (2x0) '^ ~ tan [sinn ( 27ra ) J (1 + , )2 + x2]= ^ tanh _ i 4x L(l - 0) 2 + X 2 . l_sinh (2xa) r 2 " i. >-£*""' [l^l?]/ From these equations we see that the line on the f = 6 — ix plane which corresponds to ft = constant is a circle of radius esc (2x/3), with 138 THE FLEXIBLE STRING [111.13 center at f = i cot (2tt/3) ; and that the line corresponding to a-con- stant is also a circle, of radius csch(2xa) and with center at f = coth (2ra). This transformation is plotted in Plates I and II on pages 453 and 454. Plate III gives the magnitude and phase angle of the hyperbolic sine and cosine. The transformation for the hyperbolic tangent, shown in Plate I, is sometimes called the bipolar transforma- tion. - Tables III and IV, on pages 440 and 443, also give values for the transformation, if greater accuracy is needed than can be obtained from the plates. ( Both tables and plates are given only for values of /3 between and 1, and only for positive values of a and 0. Only positive values of and a are included, because for systems encountered in "real life " the resistive term is never negative. The range of fi is restricted because the whole transformation is a periodic function of 0, and values for other ranges of /? can be obtained from the ones given. Both and x can be considered as functions of a and /5; 0(a,/3), x(a,(i). Values of and x for /3 increased (or decreased) by a whole integer are equal to the original values: 0(a,/3 + n) = 0(«,/3); X (<*,P + n) = xK/3); n = 0,±l,+2 • • • We can also see from Eqs. (13.7) that changing from fi to n — changes the sign of x but leaves unchanged : 6(a,n - 0) = 0(a,P); x(<*,n - 0) = -x(«,/3) Therefore we see that 6(a,0. 2) = 0(a,1.2) = 0(a,27.2) = 0(a,-6.8) = 0(a,6.8) = 0(a,O.8) = 0(a,-3.2), etc.; and x(«,0.2) = x(M-2) = x(«,27.2) .'= x(«,-6.8) = -x(«,6.8) = -x(«,0.8) = - x ( a ,-3.2), etc. String Driven from One End. — We must now go back to our physi- cal problem to try to show why it is necessary to bring in such a lot of new mathematical machinery. Suppose that the string is fastened, at x = t, to a support having transverse mechanical impedance Zi = ec£i and that a sideward force is impressed on_ the end x = 0. We wish to compute the transverse impedance of the string at x = and its shape and motion when acted on by the force. The impedance of the support at x = Z will modify the reflected wave returning from that end of the string and will, therefore, modify the whole reaction of the string by an amount that we must now compute. If the driving force is simple harmonic, with frequency (co/2x), the motion of the string must be some combination of the two exponen- tials e {iax/6) and e~ (iax/c) , multiplied by the time term e~ iwt . Examination of Eq; (13.4) shows that this combination can be expressed as 111.13] EFFECT OF MOTION OF THE END SUPPORTS 139 y(x,t) = A cosher a + j(y) - ifa fe-™ 1 where X = (2irc/co) is the wavelength of waves on the string. The constants a and j8 are to be determined by the value of the impedance of the support at x = I, by use of Eq. (13.1) A cosh[7r(aj — ifa)] = A ( - — =- ) sinh[7r(aj — ifa)] \ia)cZiJ or -(f) tanh[x(ai - ifa)] = ( — ) = ft = 61 - ix t (13.8) where the constants for the end x = I are related to those for x — by the equations m = a ; -(f> *-* + (?) Therefore the calculation of the wave impedance of the string involves a knowledge of the hyperbolic tangent of complex quantities and is the excuse for the previous several pages of formulas. The wave impedance of the string at x = is Z(0,0;o>) = (J-)(^) = ectanh[7r(«o - *j8 )] = ec tanh<7raj — iir fa + ( y ) ( To go from the impedance of the support at x = I to the impedance of string plus support, we find the values of a and P, on Plate I, corre- sponding to an equal value of a but a value of j3 increased by the length of the string in half wavelengths, (21/X). We follow the circle corresponding to the fixed value of a, from fa to fa -f- (2Z/X), and the result is the transverse impedance of the string in units of (cc). Shape of the String. — If the transverse driving force at x = is Foe"^ 1 , the shape of the driven string is ^o cosh {ir[a + i(2x/\) - ifa ]} e _ iat — t'coec sinh[7r(a — iPo)] which corresponds to Eqs. (10.8) for the case of the rigid support. By using some of Eqs. (13.7) we can express this in terms of amplitude and phase angle: 140 THE FLEXIBLE STRING [111.13 c & m ft .2 > i 3 g <3" -dTS .C H ft "« § a > IS 5& £ ft O a] ^2 o E 3 3 +3 © Ui E +3 03 ® - c - o c E gS a. v .• 2 < o* o ^_ Ets o ^ -O 3 3 u>E.^ ft 03 C x Q. o |E -o d a.2<£ d ® v 3 111.13] EFFECT OF MOTION OF THE END SUPPORTS 141 = El / cosh 2 Qr«o) ~ sin 2 (fcc - TftO V y uec \ cosh 2 (xa ) — cos 2 (71-ft) ) exp <t tan _1 [tanh(xo:o) tan (Ave — irfio)] — t tan -1 [tanh (ira ) cot (fl-ft)] — iwf + ( - )> where k = (2tt/X) = (w/c) and ft = ft + (2Z/X). Figure 28 shows a time sequence of such wave motion for several different values of ai and ft. The one for ai = is for no energy loss to the support at x = I and is a true standing wave, with true nodal points spaced a half wavelength apart. When ai is not zero, energy is lost to the support, the reflected wave is not so large as the incident wave, and at no point on the string is the amplitude exactly zero. The amplitude of motion of the string, as a function of x, does show maxima and minima, however, spaced a half wavelength apart. The dependence of this amplitude on x is through the factor -\/cosh 2 (7rao) — sin 2 (fcc — xft) = VcoshVaj) - sin 2 [(27r/X)(a; - I) - rft]. The maxima come when (2ir/\)(x — I) = 7rft + irn, (n = 0, ±1,±2 • • • ), and the maximum nearest the support at x — I is a distance d = (I — x) = (X/2)(l — ft) from this support (we assume that the value of ft lies between and 1, as it can always be made to do). The minima come when (2ir/X)(x — I) = irfii + ir(n + i), and the mini- mum nearest the support comes at d = (I — x) = (X/2)(£ — ft) if < ft < £, or (X/2)(f - ft) if 1 > ft > i. Therefore the value of ft for the support at x = I can be determined by measuring what part of a half wavelength lies between the support and the nearest point of maximum amplitude and subtracting this fraction from 1. At the maxima, the factor in the amplitude depending on x has the value cosh(xaz) ; at the minima this factor has the value sinh(iraj). Consequently, the ratio between the minimum amplitude of motion of the string and the maximum is tanh(7raj), from which we can deter- mine ai. Therefore we can measure the transverse impedance of the support at x = I by observing the driven motion of the string attached to it. The ratio of minimum to maximum amplitude gives ai, the distance of the maximum from the support gives ft, and the impedance Zi can be obtained from Eq. (13.8) and the plates or tables at the back of the book. 142 THE FLEXIBLE STRING [111.13 This same sort of analysis can be made for sound waves in a tube closed at x = I by a diaphragm having mechanical impedance Z h as will be shown in Chap. VI. Standing Wave Ratio and Position of Minima. — An alternative technique is to use Eqs. (13.3) and Fig. 27 to correlate measurements of standing waves with the impedance of the driven support (the load) or the impedance that the string exhibits to a transverse driving force (the generator). The magnitude of q, \q\, the ratio between the amplitudes of reflected and incident waves, is called the standing wave ratio. At the points where the two waves are in phase, the amplitude of motion Fmax is largest and is proportional to (1 + \q\); at the points where the waves are out of phase, the amplitude 7^ is least and is proportional to (1 — |g|). Consequently, a measure of the ratios between minimum and maximum amplitudes serves to determine the standing wave ratio, or vice versa P" = TT^ ! I?! - y- I y* = «-" (13.9) -t max -L I j C^ | -l max r -1 min The phase angle 2ir/3 of q = \q\e 2Ti ? gives the phase shift on reflec- tion. If it is zero, the amplitude at that point is maximum, and the point of minimum amplitude is i wavelength away; if /? is ■£- (phase angle 180 deg) the amplitude is there a minimum. Consequently, the number of half -wavelengths measured from the load to the nearest minimum is equal to i - p lo&d ; or the number of half wavelengths measured from the generator to the nearest minimum is equal to i + iSgen (which is a restatement of the discussion on page 141). This suggests using the chart of Plate IV, which is the portion of Fig. 27 inside the circle = 0. The contours for the resistance and reactance and % are plotted. The standing wave ratio \q\ is then simply the radial distance from the center (0 = 1, % = 0) to the point (0,x). The value of /3 is the angle of this radius vector. If the value of the load impedance is known, the corresponding point can be found on the chart and the standing wave ratio deter- mined by measuring the distance from the center to this point. The impedance of the string for the generator can be found by traveling along a circle with center (0 = 1, x = 0), starting from the point (0ioad, Xioad), and traversing the circle through an arc in degrees equal to 360 deg times the number of half wavelengths from load to gener- ator. We note that we must traverse the circle in a clockwise direc- tion in this case (increasing /3) . 111.13] EFFECT OF MOTION OF THE END SUPPORTS 143 If the value of the string impedance for the generator is known, we find the load impedance by going from the point (0 gen , Xsen) along a circle in the counterclockwise direction (decreasing /3) through an arc equal to 360 deg times the number of half wavelengths from load to generator. If the ratio (Fmm/Fmax) is known and the position of one mini- mum (or maximum) is known, either the impedance of the load or the string driving impedance can be computed by drawing a circle of center (6 = 1, x = 0) and radius equal to \q\ = [1 — (Y^/Ym^)]/ [1 + Yrwn/Yma*)]. Going around this circle, from the point /? = i, in the counterclockwise direction through an arc equal to 360 deg times the number of half wavelengths from the minimum to the load, ends at a point corresponding to the load impedance. Going around the circle clockwise through an arc equal to 360 deg X (number of half wavelengths between the minimum and the generator) reaches the point corresponding to the driving-point impedance of the string. Both types of charts (for a, j3 and for \q\, /3) are given at the back of the book. One is more useful for some calculations, the other more useful for others. It is well to get used to both. Characteristic Functions. — To discuss the driven motion of the string further we shall have to obtain the characteristic functions that satisfy the boundary conditions at a given frequency. To make the problem as simple as possible initially, without leaving out the essential parts, we shall assume that the transverse impedance of the supports at x = and x = I are large compared witTi ec, the wave impedance of the string. In this case approximate methods can be applied, and the technique is not too different from that employed in Sec. 12. We consider that the string itself is uniform, with uniform tension and zero distributed resistance. (These additional complica- tions can be brought in later; at present the only perturbations con- sidered are those due to the motion of the supports.) We assume that the nth characteristic function is ^n(x,ca) = sin f — ) (x — a n ) (n = 1,2,3 • • • ) where w n and a n are to be determined by the boundary conditions. Setting this in Eqs. (13.1) and assuming that (T/uZ) is small enough to neglect quantities higher than the first order, we finally arrive at the approximate formulas 144 THE FLEXIBLE STRING [111.13 a "^{ ! f)[ 1 + (dk) + (sass)] where T is the tension (T = ec 2 ). Therefore the characteristic function is \l/n(x,w) ~ sin< (ira) 1 - ( -^J (<r + tJto) - ( ^J (<ri + «i) [(f) + (a) <•* + *•>]} s^ ( — 7- J (o- + Wo) (at a; = 0) ~ ( ~~ 1)n ~ l fe) (<Tl + ^ (at X = l) where (ec/Z) = (1/r) = k - *V; k = ed?/(fl 2 + x 2 ) = 6/(6* + x 2 ); <r = -ecX/(R* + x 2 ) = -x/(6 2 + x 2 ) The quantity (Z/ec) is the specific impedance of the end support, the impedance in units of the wave impedance of the string. The recip- rocal of this is the specific admittance of the support, here taken to be a small quantity. The real part of this, k, can be called the specific conductance and the imaginary part a, the specific susceptance of the support. The* limiting values of the function yf/ n correspond to the motions of the end supports produced by the motion of the string. The characteristic function ip n is a complex quantity and is a function of the driving frequency co, both explicitly and also through the depend- ence of the admittances k and a on co. Following the procedure leading to Eq. (11.6), we see that the shape of the string, when acted on by a simple harmonic driving force concentrated at x = £, is ^-J M n (co — w n F(j) M MMx t u) ^ )(« + 0) n ) where Mn ~ \2 ) l 1 \io>lZ ) [iuizj] — \2coJ Transient Response. — The calculation of the response of the string due to an impulsive force at t = involves the calculation of the resi- m.13] EFFECT OF MOTION OF THE END SUPPORTS 145 dues of y at its poles in the w-plane. This is a little tedious, since co„ depends on w; but it can be carried out to the first order in the small quantities k and <r. The most difficult point comes when we set the approximate values w = + (rnc/l) into u n to calculate the first order correction to the position of the poles. It is a general property of impedances to change the sign of their imaginary parts (but not of the real parts) when the real part of co changes sign. Consequently, if . «„(«) = y hrn + \^~1 ) ( KOn + Kln ~ t(r0n — *°"* n ) the two roots for « are, to the first order, c , . V J {irn — 0-Qn — (Tin — 1*0n — Win) t) (—Tnc\ c , . . .v — 7 — J ^ j {—irn + a 0n + <n n — *«0n — lain) where we understand that the admittances in the nth term, <r» and R n , are computed for the frequency (irnc/l). This behavior must also be taken into account in evaluating $ n at the poles. The final result is !0 (t < 0) Jj\imc) (13.10) n = 1 N ' [ypn{%, - )^n(x, — )e i( - e/l) c™-«>»-«»)* -^(^,+)^„(a:,+)e- i ^^-^-«»^] (t > 0) where &»(z,+) = sin<[7rn — (<r 0n + zVc 0n ) — («ri* + im*)] [(i) + (£) ('-+ *->]} and ^ n (a?,-) C~ Sinf ~) + COsf ^y- J ( 1 — J J ((Ton — tKOn) — J ((Tin — IK-In) I 146 THE FLEXIBLE STRING [111.13 The series in (13.10) is correct only to the first order and, there- fore, does not give exactly the correct answer near t = 0; but the gen- eral behavior is a close enough approximation to see what is happening. A string supported between nonrigid supports, started into motion by any arbitrary force, vibrates with a complex of damped, harmonic motions. The damping constant for the nth mode is (c/l) times the sum of the specific mechanical conductances (/c 0w + ni n ) for transverse motion of the two supports at the nth natural frequency. If the supports are rigid or if their transverse impedances are purely reactive, then these conductances will be zero and the string will not be damped owing to support motion; though it may be damped owing to the reaction of the air (which we have treated in Sec. 12, but are neglecting here). But if the supports are attached to sounding boards that radiate the energy of vibration, then the conductances will not be zero, and the motion will be damped. We note that, because of the factor (c/l), the longer the string is, the less will it be damped. It is easy to see why this is so : for the same support conductance, a longer string will feed a smaller fraction of its total energy to the supports in any second. We note also that the overtones of the string are not harmonic, for the nth natural frequency is (cn/2l) — (c/2wl) (cr 0n + <xi n ), which is only approximately equal to n times the fundamental frequency (c/2l) - (c/2irl)(aoi + an). If the supports are stiffness controlled their reactances are nega- tive, their susceptances <r = —ecX/(R 2 + X 2 ) are positive, and all the natural frequencies are lowered. This is because the supports will move in phase with the part of the string nearest them, and the outer nodes, which are at the supports, if they are rigid, will be some- what outside the supports (virtual nodes, not on the actual string). This means that the wavelength has been increased a bit, so the frequency is lowered. If the transverse impedance of the supports is mass controlled, the susceptances are negative and all the natural frequencies are raised. There is a node in the string a short distance in from each support so that the support can move out of phase with the transverse force from the vibrating string. If the conductances are not zero, none of the nodes will be perfect; but the amplitude of motion of the string will be a minimum at these points, and the phase lag of the supports' displacement behind the string will be less than 180 deg. This analysis will be taken up again when we study the behavior of sound waves in tubes, and more accurate solutions will be derived. 111.13] EFFECT OF MOTION OF THE END SUPPORTS 147 Recapitulation. — We have gone into considerable detail in studying the motions of the string, perhaps more detail than seems necessary. This has been done because the string is the simplest case of a system with an infinite number of allowed frequencies, and it is best to discuss some of the properties common to all such systems for as simple a system as we can find, lest the mathematical complications completely obscure the physical ideas. The effect of friction, both on the system and through the supports, and the phenomenon of multiple resonance are both properties that are true of systems more complicated than the string. The damping effect of the air's reaction will be more important than the effect of the supports in systems more extended than the string, but the general effect will be the same as that dis- cussed above. We also have been developing methods of handling vibration prob- lems which will be exceedingly useful in our later work, developing them on problems where the general method is not too much obscured by details. In particular, we have been giving example after example of the utility of the study of the normal modes of vibration of a sys- tem. Once the normal frequencies and corresponding characteristic functions have been worked out for a system with a given set of boundary conditions, we can determine its motion for any set of initial conditions and for any sort of applied force. We can also discuss, by methods similar to those developed in Sec. 12, the effect on the motion of the system of slight changes in the system's properties (such as its distribution of mass or its distribution of tension). And, by expressing the applied force in terms of the characteristic functions, we can work out the forced motion. We can show, for instance, that when the driving frequency is equal to one of the allowed frequencies of the system, then the system takes on the shape of the corresponding characteristic function with an amplitude that is infinite if there is no frictional damping force. (Compare this with the discussion in the last paragraphs of Chap. II.) All the methods discussed above will be used again later in more complicated problems, either in the text or in the problems. Problems 1. A string, clamped at one end, is struck at a point a distance D from the clamp, by a hammer of width (D/4). The head of the hammer is shaped so that the initial velocity given the string is maximum at the center of the head and is zero at the edge, the initial "velocity shape" of this portion being like an inverted V. Plot the shapes of the string at the times t = 0, (D/2c), (D/c)', (3D /2c), 148 THE FLEXIBLE STRING [111.13 (2D/c). Draw a curve showing the vertical component of the force on the clamp as a function of time. 2. A harp string is plucked so that its initial velocity is zero and its initial shape is (i*r) x i° < x < 20J KT)\2- X ) \& <x< 2o) S20h\ , n an . . \ KW) {x ' l) V20 < ' < l ) Plot the successive shapes of the string during one cycle of the motion. Draw a curve showing the vertical component of the force on one of the supports as a function of time. 3. What are the total energies of the two strings shown in Fig. 16? 4. Show that if a string, having a wave velocity c, is plucked at a point whose distance from one end is 1/3 the equation for its subsequent shape is y = w L sm \t) cos vt) + 4 sin \rr) cos \tt) - tV sm vt) cos \rr) , . (5ttX\ /5wct\ , . . /7rx\ /7irct\ . 1 - A sm {-j- j cos {— j + A ^ {—) cos ^ j + • • J The string has a mass of 0.01 g per cm, is 25 cm long, and is under a tension of 1,000,000 dynes. Find the energy of vibration of the first four normal modes when the string is pulled aside 1 cm (h = 1). 5. A uniform string with no friction is stretched between rigid supports a distance I apart. It is driven by a force F e~ iut concentrated at its mid-point. Show that the amplitude of motion of the mid-point is (F /2ecu) tan (ul/2c) . What is the amplitude of motion of the point x = (1/4) ? 6. A uniform string of small electrical resistance is stretched between rigid supports a distance I apart, in a uniform magnetic field of B gauss perpendicular to the string. A current Iae~ i03t amp is sent through the string; what is the force on the string per unit length? Show that the velocity shape of the string (assume zero friction) is BJoe-*" {cos[(»/c)(a; - $Z)] ,) , v = ^-. < / 7/0 ^ ~ 1 C cm /sec lOzwe I cos (wZ/2c) ) Use the formula E = 10~ 8 j Bv dx volts to compute the motional emf induced in the string by the motion and, therefore, the motional impedance. 7. A condenser is discharged through the string of Prob. 6, producing a current S(t) amp. Compute the subsequent shape of the string. 8. The string of Prob. 5 is acted on by a force F(£) u(t) concentrated at the point x — I [see Eq. (2.6) for a definition of u, and Eq. (6.10) for its use]. Show that the shape of the string after t = is HI.13] EFFECT OF MOTION OF THE END SUPPORTS 149 From this formula compute the shape of the string when it has been subjected to the constant (independent of time) force F(£) up to t = and then released to vibrate freely. 9. A string of length I and mass e g per cm is hung from one end, so that gravitational forces are the only ones acting. Show that if the free end of the string is taken as origin, the normal modes of vibration have the form y - Jo^.^) where v n = (|8„/4) y/gjl) the function J (z) is given by the series (see Prob. 3 of Chap. I) z 2 z 4 ■W-^-iT + ei and the constants /3„ are the solutions of the equation Jo(irp n ) = [see Eq. (19.6)]. Is the motion periodic in general? What are the ratios of the lowest three allowed frequencies to the frequency of oscillation that the string would have if all its mass were concentrated at its lower end? 10. Utilizing the general formulas of Eqs. (11.6) and (11.8), obtain specific formulas for the shape of the string of Prob. 9 when driven by a periodic force and when struck by an impulsive force. 11. Choosing suitable values for the properties of the string of Prob. 9, plot the shape of the string when it is vibrating at its lowest three allowed frequencies. Plot its shape when driven at its free end at a frequency 1.5 times its fundamental. 12. The tip of the whirling string is struck an impulsive blow 5(0 in a direction perpendicular to the equilibrium plane of motion. Calculate the series repre- senting the subsequent displacement. What is the amplitude of motion of the lowest mode? 13. A string of infinite length is acted on by a force F(t) concentrated at the point x = 0, where (t < 0) F(t) = { F (0 < t < to) (t > U) Plot the shape of the string at the times t = 0, (to/2), U, (32 /2), 2t . What is the total energy given to the string? 14. A string of steel (density 7.7 g per cc) is stretched between rigid supports. Its fundamental frequency of vibration is 500 cps. Sound of unit intensity and frequency v falls on the string, normal to its axis. Compute the amplitude of motion of the mid-point of the string for the frequencies v = 50, 100, 300, 450, 499, 550, 600, 1,000. Note. — The force per unit length on a string of cross-sectional radius a, due to an incident sound wave of intensity T and frequency v 2 , is 0.0074Ka 2 \/T e~ 2rin dynes per cm length. 15. Plot the shapes of the first three modes of the stretched hanging string, described in Eqs. (12.8), for (lge /To) = 0.25. Compute the series for shape of the string when driven by a periodic force concentrated at x = £. 150 THE FLEXIBLE STRING [111.13 16. What are the values of tanh[7r(a — i(5)] when a ■= 0, /S = 1.75; when a = 0.2, j8 = -0.6; a = 0.1, /3 = 0.45; a = 0, '/3 = 0.5? What are the values of cosh [ir (a — ip)] for these same values of a and /3? 17. A uniform string of length I and tension T is fastened at x = £ to a support having transverse mechanical resistance R = ec tanh (ira — ^r) and zero reac- tance. The string is originally at equilibrium, and the end at x = is suddenly acted on by a transverse impulsive force 8(t). Utilize Eqs. (13.10) and (6.14) to obtain the formula for the shape of the string: 00 1 ^C^ p-(ira/l)(.x+ct) I r^ "1 »<*» " k 2 (. + «■ + * I <" + §> sin [r <» + »<* + C,) J + a'cos ^ (n + %)(x + ct)\l - s; 2 (»Tw-T- 1 (w + j) 8in B ( " + t)(x - c<) ] n = + « cos y (n + |) (x — cf) [ -\ tanh (ira) 18. A string of density 0.1 g per cm is stretched with a tension of 10 s dynes from a support at one end to a device for producing transverse periodic oscillations at the other end. When the driving frequency has a given value, it is noted that the points of minimum amplitude are 10 cm apart, that the amplitude of motion of the minimum is 0.557 times the amplitude at the maximum, and that the nearest maximum is 6 cm from the support. What is the driving frequency, what are the values of on and ft, and what is the value of the transverse impedance of the support? 19. A string is stretched between two supports having transverse mechanical resistance R large compared with ec, and zero reactance. The string is driven by a periodic force concentrated at a; = £ . What are the amplitudes of motion of the end supports, and by how much do they lag behind the driving force? 20. The string of Prob. 19 is struck an impulsive blow at the point x =' £. Compute the subsequent motion of the two end supports. CHAPTER IV THE VIBRATION OF BARS 14. THE EQUATION OF MOTION It must have been rather obvious in the previous chapter that we were analyzing the motions of a somewhat idealized string. In the first place, we assumed that the string was perfectly flexible, that the only restoring force was due to the tension. Secondly, we made no mention of the possibility of longitudinal motion of alternate compression and tension, which can be set up in any actual string as well as in any other piece of solid material. This longitudinal wave motion will be disregarded for a while longer; we shall spend the whole of the last three chapters discussing it. However, we can no longer put off studying the effect of stiffness on the string's motion. And we shall begin the study by discussing the transverse vibrations of bars. There is no sharp distinction between what we mean by a bar and what we mean by a string. In general, tension is more important, as a restoring force, than stiffness for a string, and stiffness is more important for a bar; but there is a complete sequence of intermediate cases, from stiff strings to bars under tension. The perfectly flexible string is one limiting case, where the restoring force due to stiffness is negligible compared with that due to the tension. The rod or bar under no tension is the other limiting case, the restoring force being entirely due to stiffness. The first limiting case was studied in the previous chapter. The second case, the bar under no tension, will be studied in the first part of this chapter, and the intermediate cases will be dealt with in a later part. Stresses in a Bar.— To start with, we shall study the bending of a straight bar, with uniform cross section, symmetrical about a central plane. The motion of the bar is supposed to be perpendicular to this plane, and we shall call the displacement from equilibrium of the plane y. When the bar is bent, its lower -half is compressed and its upper half stretched (or vice versa). This bending requires a moment M, whose relation to the amount of bending we must find. To compress 151 152 THE VIBRATION OF BARS [IV.14 a rod of cross-sectional area £ and length I by an amount dl requires a force QS(dl/l), where Q is a constant, called Young's modulus. The values of this constant are given in Table 3, for some of the more common materials. Now, imagine the bar to be a bundle of fibers of cross-sectional area dS, all running parallel to the center plane of the bar. If the bar is bent by an angle $ in a length dx, then the fibers which are a distance z down from the center sur- face (it is no longer a plane now that the bar is bent) will be compressed by a length 2$, the force required to compress each fiber will be Q dS(z$/dx), and the moment of this force about the center line of the bar's cross section will be (Q$/dx)z 2 dS. The total moment of these forces required to compress and to stretch all the fibers in the bar will be Fig. 29. — Moment acting on a bent element of a bar. M = \dx) J dS (14.1) where the integration is over the whole area of the cross section. Table 3. — Elastic Constants of Materials Material Brass, cold rolled Bronze, phosphor Copper, hard drawn. . German silver Glass Iron, cast wrought Iron-cobalt (70 % Fe) . Nickel Nickel-iron (5 % Ni) . . Silver, hard drawn. . . Steel, annealed invar Tungsten, drawn Q 9 X 10 11 12 X 10" 10 X 10 11 11 X 10 u 6 X 10 11 9 X 10 11 19 X 10 11 21 X 10 11 21 X 10 11 21 X 10 11 8 X 10 11 19 X 10 u 14 X 10 11 35 X 10 11 8.6 8.8 8.9 8.4 2.6 7.1 7.6 8.0 8.7 .7.8 10.6 7.7 8.0 19.0 Values of Young's modulus Q in dynes per square centimeter, and of density p, in grams per cubic centimeter, for various materials. We define a constant k, such that k 2 = (l/S)jz 2 dS, where S is the area of the cross section. This constant is called the radius of gyra- IV.14] THE EQUATION OF MOTION 153 lion of the cross section, by analogy with the radius of gyration of solids. Its values for some of the simpler cross-sectional shapes are as follows: Rectangle, length parallel to center line 6, width perpendicular to center line a: Circle, of radius a: VV12) -GO Circular ring, outer radius a, inner radius b: k = £ Va 2 + 6 2 Bending Moments and Shearing Forces. — Equation (14.1), giving the moment required to bend a length dx of rod by an angle $, is then M = Q$Sk* dx (14.2) If the rod does not bend much, we can say that $ is practically equal to the difference between the slopes of the axial line of the rod at the two ends of the element dx: $ \dx/ x+dx \dx/ + ( ^ = -dx Therefore the bending moment is (14.3) M+dM F+dF Fig. 30. — Bending moments and shearing forces to balance. This bending moment is not the same for every part of the rod; it is a function of x, the distance from one end of the rod. In order to keep the element of bar in equilibrium, we must have the difference in the moments acting on the two ends of the element balanced by a shearing force represented by F in Fig. 30 (moment and shear are, of course, two different aspects of the single stress which is acting on the bar). The moment of the shearing force is F dx and this must equal dM for equihbrium, which means that dx dx 3 (14.4) 154 THE VIBRATION OF BARS [IV.14 This equation is not exactly true when the bar is vibrating (since a certain part of the moment must be used in getting the element of bar to turn as it bends), but it is very nearly correct when the amplitude of vibration is not large compared with the length of bar. Properties of the Motion of the Bar. — The shearing force F is also a function of x and may be different for different ends of the element of bar. This leaves a net force dF = (dF/dx) dx acting on the element, perpendicular to the bar's axis; and this force must equal the element's acceleration times its mass pS dx where p is the density of the mate- rial of the rod. Therefore the equation of motion of the bar is dx (dF/dx) = P S dx (d 2 y/dt 2 ), or a^ _p_av (145) dx* Qk 2 dt 2 K } This equation differs from the wave equation in that it has a fourth derivative with respect to x instead of a second derivative. The general function F(x + ct) is not a solution, so that a bar, satisfying Eq. (14.5), cannot have waves traveling along it with constant velocity and unchanged shape. In fact, the term "wave velocity" has no general meaning in this case, although it can be given certain special meanings. For instance, a simple harmonic solution of Eq. (14.5) is y = (7 e 2m(Ma:-,« = ^ COS [2tt(^ ~ Vt) — $] (14.6) where "-(ass?)' '- 2 " A >/? This represents a sinusoidal wave traveling in the positive direction, of just the sort one finds on strings. There is an important difference between the two waves, however; for in the case of the bar the velocity of the wave u = (v/n) = (4 7 r 2 Q/c 2 /p) i V~ v depends on the frequency of oscillation of the wave, whereas in the case of the string it does not. The velocity of propagation of a simple harmonic wave is called the phase velocity and is one of the special kinds of "velocity" that have meaning for a bar. In the case of the string the phase velocity is independent of v and is equal to the velocity of all waves c. For the bar the phase velocity depends on v, and there is no general velocity for all waves. This is analogous to the case of the transmission of light through glass, where light of different frequencies (i.e., of differ- ent colors) travels with different velocities and dispersion results. A bar is sometimes said to be a dispersing medium for waves of bending. IV.14] THE EQUATION OF MOTION 155 Wave Motion of an Infinite Bar.— Although the phase velocity is not a constant, nevertheless we should be able to build up any sort of wave out of a suitably chosen combination of sinusoidal waves of different frequencies, in a manner analogous to the formation of a Fourier series. At present, we are considering the bar to be infinite in length, so that all frequencies are allowed, and the sum is an integral y = .Co ^"W/O cos (7 VO + C(n) sin (7 VOJ dfi (14.7) where y 2 = 4tt 2 k VQ/p, and where the functions B(n) and C(jx), corre- sponding to the coefficients B n and C n of a Fourier series, are deter- mined by the initial conditions. This integral is analogous to the Fourier integral of Eq. (2.19). If the initial shape and velocity shape of the bar are y (x) and t> (aO, a derivation similar to that given for Eq. (9.9), for the Fourier series, shows that -B(m) = ( 4—3 J I y (x)e- 2 ™» x dx \ ) 1 \"f ( 14 -8) ° (m) = V&W/ J_ m voWe- 2 *^ dx J as was shown in Eq. (2.20). This may seem a rather roundabout way of obtaining solutions of Eq. (14.5) which satisfy given initial conditions; but since the useful functions F(x - ct) and f(x + ct) are not solutions of (14.5), there is no other feasible method. The case y = e -* 2 / ia \ v Q = is one that can be integrated and can be used to show the change in shape of the wave as it traverses the bar. Solving Eqs. (14.8) and (14.7), we obtain We can now see the utility of Eq. (14.7), for while we can show that (14.9) is a solution of the equation of motion (14.5) by direct substi- tution, we can also see that it would be exceedingly difficult to obtain (14.9) from (14.5) directly. The shape of this function, at successive instants of time, is shown by the solid fines in Fig. 31. For comparison, the sequence of dotted lines shows the variation in shape of a flexible string with equal initial conditions. In the case of the string, the two partial waves move out- ward with unchanged shape. The shape of each "partial wave" for V 156 THE VIBRATION OF BARS [IV.15 the bar, however, changes continuously as the wave travels outward. In particular, notice the formation of subsidiary "ripples" ahead of the principal "crest" of each wave. This is due to the fact, which we have already noted, that the high-frequency, short wavelength parts of the wave spread outward faster than the rest. ■p 1G . 31. — Comparison between the motion of waves on a bar (solid line) and on a string (dotted line), both of infinite length. Each sequence shows the shapes at suc- cessive instants, after starting from rest in the form given at the top of the sequence. The constant y was chosen to make the average "velocity" of the waves on the bar approximately equal to c, the velocity of the waves on the string. 15. SIMPLE HARMONIC MOTION We cannot pursue our study of the motions of the bar much further without examining its normal modes of vibration. As with the string, we must ask if there are any ways in which the bar can vibrate with simple harmonic motion. We try setting y = Y{x)e- 2vivt and find that Y must satisfy the equation d 4 Y dx* = 16.Vr, ,-_£-,- 4^ (£) (15.1) IV.16] SIMPLE HARMONIC MOTION 157 The general solution of this is y = de 2 *"* + Cze- 2 *** + Cge 2 ""* + Cie- 2 ™"* = a cosh (2irfix) + b sinh (2x,uz) + c cos (2x^r) + d sin (2tthx) (15.2) For a discussion of the functions cosh and sinh, see Eqs. (13.4), and see Tables I and II at the back of the book. This general solution satisfies Eq. (15.1) for any value of the fre- quency v. It is, of course, the boundary conditions that pick out the set of allowed frequencies. Bar Clamped at One End. — For example, if we have a bar of length I clamped at one end x = 0, the boundary conditions at this end are that both y and its slope (dy/dx) must be zero at x — 0. The par- ticular combination of the general solution (15.2) that satisfies these two conditions is the one with c = —a and d = — b Y = a[cosh (2ttmz) - cos (2w/ia;)] + 6[sinh (2t{jlx) - sin (2jr/is)] (15.3) If the other end is free, y and its slope will not be zero, but the bending moment M = QSn 2 (d 2 Y/dx 2 ) and the shearing force F = —QSK 2 (d s Y/dx 3 ) must both be zero, since there is no bar beyond x = I to cause a moment or a shearing stress. We see that two con- ditions must be specified for each end instead of just one, as in the string. This is due to the fact that the- equation for Y is a fourth- order differential equation, and its solution involves four arbitrary constants whose relations must be fixed, instead of two for the string. It corresponds to the physical fact that while the only internal stress in the string is tension the bar has two: bending moment and shearing force, each depending in a different way on the deformation of the bar. The two boundary conditions at x - I can be rewritten as (l/4ir 2 n 2 )(d 2 Y/dx 2 ) = and (l/8ir*n 3 )(d*Y/dx*) = at x = I. Sub- stituting expression (15.3) in these, we obtain two equations that fix the relationship between a and 6 and between n and /: a[cosh (%rnl) + cos (2r/*Z)] + 6[sinh (2^0 + sin (2thI)] = afsinh (2jr/d) - sin (2ir/d)] + 6[cosh (2rtf) + cos (2r/*J)] = or b = a sin (2Tfxl) - sinh (2thI) = _ cos (2^1) + cosh (2t(iI) cos (2ir(d) + cosh (2rfxl) a sin (2^0 + sinh (2jr/rf) ( ^ By dividing out a and multiplying across, we obtain an equation for /x: [cosh(27TMZ) + cos(2ttmZ)] 2 = sinh 2 (2rAiO - sm 2 (2rnl) 158 THE VIBRATION OF BARS [IV.16 Utilizing some trigonometric relationships, this last equation can be reduced to two simpler forms: cosh (2ir/j.l) cos (2v/d) = — 1, or coth 2 (7ryuZ) = tan 2 (x/iZ) (15.5) where coth(z) = cosh(20/sinh(z). The Allowed Frequencies. — We shall label the solutions of this equation in order of increasing value. They are (2irfj.il) = 1.8751, (2*7*20 = 4.6941, (2*7*80 = 7.8548, etc. To simplify the notation, we let (1/tt) times the numbers given above have the labels jS„, so that (I) (15.6) where 0i = 0.597, /3 2 = 1.494, /3 3 = 2.500, etc. It turns out that j8„ is practically equal to (n — i) when n is larger than two. By fixing /*, we fix the allowed values of the frequency. Using Eq. (15.1), we have (15.7) or 0.55966 fl = l¥~ j> 2 = 6.267 vi v z = 17.548vi Vi = 34.387^1 etc. Notice that the allowed frequencies depend on the inverse square of the length of the bar, whereas the allowed frequencies of the string depend on the inverse first power. Equation (15.7) shows how far from harmonics are the overtones for a vibrating bar. The first overtone has a higher frequency than the sixth harmonic of a string of equal fundamental. If the bar were struck so that its motion contained a number of overtones with appreciable amplitude, it would give out a shrill and nonmusical sound. But since these high-frequency overtones are damped out rapidly, the harsh initial sound would quickly change to a pure tone, almost entirely due to the fundamental. A tuning fork can be considered to be two vibrating bars, both clamped at their lower ends. The fork exhibits the preceding behavior, the initial metallic "ping" rapidly dying out and leaving an almost pure tone. IV.16] SIMPLE HARMONIC MOTION 159 The Characteristic Functions. — The characteristic function cor- responding to the allowed frequency v n is given by the equation + n = a n [cosh(^) - cos(^)] 6„[sinh(^)-sin(^) + where (15.8) _k _ cosh (xpn) + cos (xj8„) _ sJnhQrftn) — sin(ir|8 w ) n sinh (t0 u ) + sin (*r/3„) ~~ Un cosh (wj8„) + cos (w0«) We shall choose the value of a n so that j ty\ dx = (Z/2), by analogy with the sine functions for the string. The resulting values for a n and b n are ai = 0.707, &i = -0.518, a 2 = 0.707, 6 2 = -0.721, a 3 = 0.707, & 3 = —0.707, etc. For n larger than 2, both a n and b n are practically equal to (I/a/2)- Some of the properties of these functions that will be of use are J,, C (m ?± n) o M*)*.(x) dx = j/j\ (m = n) * n 0) = (-l)-i V2 (^«(-l)-V5(^) and 0. ~ „-*(«> 2) ^ (15 - 9) it n ~: — -= [ff-rPnX/l _|_ (_Dn-l e i/J B (ir-0/fl V2 V ^ 4/ + sin(^- -^ (n> 2) The shapes of the first five characteristic functions are shown in Fig. 32. Note that for the higher overtones most of the length of the bar has the sinusoidal shape of the corresponding normal mode of the string, with the nodes displaced toward the free end. In terms of the approximate form given above for \f/ n , the sine function is sym- metrical about the center of the bar; the first exponential alters the sinusoidal shape near x = enough to make 4/ n have zero value and slope at this point; and the second exponential adds enough near x = I to make the second and third derivatives vanish. Note, also, that the number of nodal points in yp n is equal to n — 1, as it is for the string. 160 TH~E VIBRATION OF BARS [IV.15 In accordance with the earlier discussion of series of characteristic functions, we can show that a bar started with the initial conditions, at t — 0, of y = y (x) and (dy/dt) = v (x) will have a subsequent shape n=( given by the series n=2 n»3 y = 2) ^n{x)[B n co^{2irv n t) + C n sin (2rr n t)] (15.10) n = l where B„ = yo(x)$n(x) dx Cm — \irv n lj Jo o(x)^„(x) dx Fig. 32. — Shapes of the first five characteristic functions for a vibrating bar clamped at one end and free at the other. Plucked and Struck Bar. — Two examples of such calculations will be given. The first example is that of a rod suddenly released from an initial shape y$ = (hx/l), an undesir- able case in actual practice (for it bends the bar quite severely at x = 0) but one that can be easily solved. The values of the coefficients B n turn out to be £* = ^- JXTp n (x) dx o 2h ( -g-p \a n [2 + ^(sinnx/^ — sinTrjSn) — cosh^n — cosx/3 w ] + &„[7rj3 n (cosh'7r/3 n + cos7Tj8„) — sinh-7r/3„ — sinir/Sn]? 4h On when the ratios between a n and 6„, given in Eqs. (15.8), are used to simplify the expression. The successive shapes taken on by the bar when started in this manner are shown in the first sequence of Fig. 33. Since the overtones are not harmonics, the motion is not periodic. The other case to be considered is that of a rod struck at its free end in such a manner that its initial velocity is zero everywhere except at x = I, where it is very large, large enough so that jv dx = U. This case corresponds to that of a tuning fork struck at one end. If IV.16] SIMPLE HARMONIC MOTION 161 the total impulse given to the end of the fork is P, then U = P/pS. The coefficients B n are zero and tM Jo 7ry„Z 7r 2 ^ \ Qk 2 B- H- Fig. 33.^ — Motions of a bar clamped at one end. Left-hand sequence shows the successive shapes of a bar started from rest in the shape given in the topmost curve. Eight-hand sequence shows the motion of a bar struck so that its outer tip starts upward at t =0. Since the overtones are not harmonics, the motion is not periodic. when the value given for ip n (l) in Eqs. (15.9) is used. The shapes of such a bar at successive instants of time are shown in the second sequence of Fig. 33. This motion is also nonperiodic, as are all motions of the bar that correspond to more than one normal mode of vibration. Clamped-clamped and Free-free Bars. — Other boundary condi- tions will give rise to other characteristic functions and frequencies. 162 THE VIBRATION OF BARS [IV.16 The bar may be clamped at both ends, in which case expression (15.3) must again be used for Y, but instead of Eq. (15.4) we must have h — sin (2tt/xQ + sinh (2ir ixl) _ _ cos (2^) — cosh (2x^0 ~ cos {2irid) — cosh (2irnl) si n (2rnl) — sinh (2rnl) and instead of Eq. (15.5) Ave have cosh(27TM0 cos {2tt id) = 1, or tanh 2 (717*0 = tan 2 (717*0 The allowed frequencies can be obtained from the formula j8i = 1.5056 Vn==J W = w4^7^ ft = 2.4997 (15.11) /3 n ~ n + i (n > 2) Still another set of boundary conditions are the ones for a com- pletely free bar, where the second and third derivatives must be zero at both ends. The characteristic functions (which we can call if/") for this case can be obtained from the ones for the bar clamped at both ends (which we can call \[/ n , to distinguish them from #,') by simply differentiating twice with respect to x, $" = (d 2 if/ n /dx 2 ). By using Eq. (15.1) we can show that (d 2 $"/dx 2 ) = (47r 2 ^p/Q/c 2 )^ n , so that if ip n and (dif/ n /dx) are both zero at x — and x = I, then (dhl/'Jdx 2 ) and (dhp'^/dx 3 ) are both zero at the end points. The functions ^" therefore satisfy the boundary conditions for a com- pletely free bar. It. is not difficult to see that $„ is a solution of the equation of motion, corresponding to the same allowed frequency as does the function ^„. The allowed values are given in Eq. (15.11). We thus obtain the rather surprising result that the allowed fre- quencies for a free bar are the same as those for a similar bar clamped at both ends and that the corresponding characteristic functions are related by simple differentiation, although their shapes are quite different. Energy of Vibration. — It is sometimes useful to know the energy of vibration of a bar. This energy can be computed by using the expressions obtained in the previous section. We saw there [Eqs. (14.2) and (14.3)1 that the moment required to bend an element dx of the bar by an angle 3? is M = QSn 2 ($/dx). The amount of work required to bend it from equilibrium to this angle is J>— * ^-^(3)'* IV.16] SIMPLE HARMONIC MOTION 163 Therefore, the potential energy of the bar, the work required to bend the bar into its final instantaneous shape y(x,t), is The kinetic energy is, of course, so that the total energy turns out to be The energy of a bar, subject to one or another of the various boundary conditions discussed above, is obtained by substituting y = 2A W *„ cos(2W - fl B ) in Eq. (15.12). This results in a super- fluity of terms, each containing a product of two trigonometric functions of time and an integral of a product of two characteristic functions, or two second derivatives of these functions. The kinetic- energy terms, containing integrals of the products of two functions, can be integrated by means of the first of Eqs. (15.9) and reduced to the single sum 2T 2 r 2 nP lSAlsm 2 (2TvJ - O n ) The integrals in the potential-energy terms can be integrated by parts twice: by using the equation of motion. The terms in the square brackets are zero no matter which of the boundary conditions discussed above are used. The resulting potential-energy series is therefore WvlplSAl cos 2 (2w - 0.). The sin 2 of the kinetic-energy terms combines with the cos 2 in the potential-energy terms and gives for the total energy the simple series W = %™(jf)<Al (15.13) 164 THE VIBRATION OF BARS [IV.15 Note the similarity with the corresponding expression for the energy of the string, given in Eq. (9.13). Nonuniform Bar. — Now that we know the properties of the normal modes of vibration, we can solve a number of different sorts of prob- lems. For instance, we can find the change in the allowed frequencies and characteristic functions when the bar is made somewhat non- uniform along its length. If the bar's density or its cross-sectional area or radius of gyration changes with x, a review of the derivation of the equation of motion shows that the correct equation should be iS(s)te»L w te'J Q dt2 which reduces to Eq. (14.5) when S, p, and k are all constant. To study the normal modes of vibration, we set y — Y(x)e~ 2wivt and have for the equation for Y, 1 d 2 S(x) dx 2 [«»*,) g]-p=^a]r (1 5.H) If the variation in p, S, and k is small, we can treat the problem by a modification of the approximation method discussed in Sec. 12. If we write S = Soil + a(x)], k 2 = 4[l + °{x)], P = Poll + g(x)] where So, k , and p are constants and a, <r, and g are small quantities, then, by neglecting the products of small quantities, (15.14) can be rewritten as d * Y to ^ ■ n d(a + <r) d*Y d 2 {a + <y) d 2 Y ~dx~*~ {2Tfi) Y + l dx ~dx^ + dx 2 ~dx^ - (2x/x)%-<r)F = where ju 2 = (v/2v) VpoA&l- If a > °"> and g were all equal to zero, the solution of this equation would be one of the characteristic func- tions ^ n given in Eq. (15.8), and the allowed values of n would be the values (j9 n /2Z) given in Eq. (15.6) or (15.11). Since a, <r, and g are small, we assume that Y n = ZA^*,, where all the Akn's are small except A nn , and that n n = (j8»/20 4 (l + Vn), where r\ n is small. Sub- stituting this in the equation, and again neglecting products of small quantities, leaves IV.15] SIMPLE HARMONIC MOTION 165 d 2 {a + a) d 2 1 2 J &» = + 2 d(a + cr) d*_ dx dx 3 dx 2 dx 2 But (dtyk/dx*) = (Tfa/l)y k . Multiplying through by ^ n or ^ and integrating gives the equations for the correction factors: -ij:4-'*+k0[»^& + dx 3 d 2 (a + <r) c?V« •<*&» — 2A nnJ 8 S)Jo**{ da; 2 da; 2 } da; 2081 - ft) Jo ^ ~ g) ^ + (_ly r 2 <*<« + «o ^v» W«/ L dx dx 3 (15.15) + d 2 (a + <r) dV da; 5 da^J/^ As an example, consider the case of a tapered bar of rectangular cross section having a constant thickness a in the direction of the vibration and a varying width b (l - y |Y j n this case, « and p are constant, and S(x) = a b(l - y |\ so that g = a = and a = —(yx/l). The correction to m» is Owing to the boundary conditions, the quantity in the brackets reduces to - (i) (d^/dx)^ when the bar is clamped at one end and free at the other. The corrected value of the frequency is therefore "* V Po 4P V3po L + ir'flt V"da7A=J 7ra(n + i) 2 4l 2 } r 27 l ^ L 1 + T . (n + i )2 J (n > 1) This equation states that tapering the bar so that its free end is narrowest raises the natural frequencies over those for a bar whose width is everywhere the same as at the clamped end. (This is hardly 166 THE VIBRATION OF BARS [IV.16 a surprising result.) If it is tapered so that the free end is three- quarters the width of the clamped end (7 = t), the fundamental frequency is 1.077 times the fundamental of the nontapering bar of the same width as that of the clamped end; the first overtone is 1.023 times the corresponding first overtone; the second, 1.008 times the corresponding second; and so on. Forced Motion. — Besides the case of the nonuniform bar, we can discuss the forced motion of a uniform bar, now that we know the characteristic functions. If the bar is driven by a force F(x)e- 2 ' i " t dynes per cm length, the equation for the shape of the bar during the steady state is &! ** - 4,rV7 = f{x), f = %&, y = Y(x)e-**« (15.16) 3 expand Y and / in series of characteristic functions, Y = Zhntnfr), f = ^Qntnix), Q n = \^jj J q F{x)^ n {x) dx we obtain for the steady-state motion y = & ^-^tv^n(x)]e-^ (15.17) .n= 1 J Resonance occurs whenever the driving frequency v equals one of the natural frequencies v n , unless the corresponding g n is zero. These results are analogous to those discussed at the beginning of Sec. 11. The equation determining ^„ here differs from that used in Eq. (11.1) ; but Eq. (15.17) has the same form as Eq. (11.6). In fact, if we use the new characteristic functions we can utilize Eq. (11.9) to determine the general transient behavior of the bar (provided we can perform the integrations). The effects of friction can be handled as they were handled for the string on page 104 and in Sec. 12. The results are so similar that we do not need to go into detail about them here. Some cases will be taken up in the problems. 16. VIBRATIONS OF A STIFF STRING When a string is under a tension of T dynes, and also has stiffness, its equation of motion is 1 dx 2 V dx* P dt 2 K This equation can be obtained by combining the derivations in Sees. IV ' 16 J VIBRATIONS OF A STIFF STRING 167 9 and 14. The constant 8 is the area of cross section, k its radius of gyration, and p and Q are the density and modulus of elasticity of the material. Wave Motion on a Wire.— Sinusoidal waves can travel along such a wire, for if we set y = Ce™^-"> we obtain an equation relating p and v: or TV + 47r 2 Q£/cV = P Su 2 n^a I r ?a enCy iS VeiT Sma11 (ix -' if v ' is ver y much small er than I /16tt P Q£V), we expand the radical by the binomial theorem and keep only the first two terms, and p 2 turns out to be approximately equal to Q>Sv*/T). The phase velocity {v/p) is practically equal to the constant value VT/p~S, which would be the velocity of every wave if the wire had no stiffness (if QSk 2 were zero). The phase velocity for the wire is not constant, however. It increases with increasing v until for very high frequencies it is \/2lTv (Q* 2 / P )* the phase velocity given in Sec. 14 for a stiff bar without tension.' In other words, the wire acts like a flexible string for long wavelengths and hke a stiff bar for short ones. This is not surprising. The Boundary Conditions.— The usual boundary conditions cor- respond to clamping the wire at both ends, making y = and also (dy/dx) = at x = and x = I. Setting y = Y(x)e-^\ we have for Y > (16.2) 7 2 = (v/2t) Vp/Qk> ) Setting Y = Ae 2 *»*, we obtain an equation for the allowed values of p.: M 4 - 2/3 V - Y 4 = 0. This equation has two roots for p 2 and therefore four roots for p: P = ±p h p\ = V/3 4 + t 4 + 2 \ or / P = ±ip2, p\ = V/3 4 + t 4 - |8 2 ( ^ 16 ' 3 ^ p\ = 2/3 2 + „l MlM2 = 7 2 / The general solution of Eq. (1G.2) can then be written tee 2 *" 1 * + be~ 2r > iix -f- cc 2ri " 2X -f- de~ 2Ti c* x } Y = < A cosh (2ttp iX ) + Bsmh(2Tp 1 x) > (16.4) ' + C cos (2tp 2 x) + D sin (27r/i 2 a;) 1 . 168 THE VIBRATION OF BARS PV.16 If the boundary conditions are symmetrical, it will be useful to place the point x = midway between the supports. The normal functions will then be even functions, yp(-x) = t(x)', « r the y w 111 be °^ ones > $(-x) = — iKz). In either case, if we fit the boundary conditions at one end, x = (1/2), they will also fit at the other end, x = —(2/2). The even functions are built up out of the combination Y = A cosh (2irmx) -f C cos (2wn$x) and the odd functions from the combination Y = B sinh (2ir/*i^) + D sin (2x/x 2 x) The boundary conditions F = and (dY/dx) = at x = (2/2) corre- spond to the following equations for the even functions: Acosh(xMiO = — Ccos(7r/x 2 0, mi^ sinh (tt/xiO = M2C sin (71W) The Allowed Frequencies.— By the use of Eq. (16.3), these reduce to tan fcrW = - >/l + (^r) tanh (ri VmI + 2/8*). (16.5) which can be solved for the allowed values of M2. The allowed values of the frequency are obtained from the equation v = 2X7 2 * /55 = 2^ 2 J(/*3 + 2/3 2 ) %- 2 \ P \ \ P These allowed values of v can be labeled v h v s , v 6 ... , in order of increasing size. The corresponding equation for /* 2 for the odd functions is ^l + (?^ tan (xZ M2 ) = tanh fcrf VmI + 2/3 2 ) (16.6) The allowed values of v obtained from this equation can be labeled vs, vi, v 6 . . . ; and then the whole sequence of allowed values will be in order of increasing size, v x being the smallest (the fundamental), v 2 the next (the fi rst overton e), and so on. When 0(= VT/8t 2 QSk 2 ) goes to zero (i.e., when the tension is zero) the equations reduce to those of a bar clamped at both ends, with the following allowed values of v. IV -16] VIBRATIONS OF A STIFF STRING 169 v% = 2.7565V1 = 3.5608 [Q? v 3 = 5.4039*! (16.7) Vx P V p ' v 4 = 8.9330vi etc. obtained from Eq. (15.11) of the previous section. When is infinity (i.e., when the stiffness is zero) the equations reduce to those of a flexible string, and the allowed values of v are v% = 2v\ 0-5000 PF v z = 3vi (16.8) I \ pS Vi = 4j>i etc. If is large but not infinite (i.e., if the wire is stiff, but the tension is the more important restoring force), it is possible to obtain an approxima te expres sion for the allowed frequencies. When is large tanh (tI \/20 2 + n\) is very nearly unity, and Vl + (2/37/4) is a large quantity for the lower overtones (i.e., as long as /*f is small com- pared with 0). Equation (16.5) for the even functions then states that tan (trim) is a very large negative number, which means that (irl^) is a small amount larger than (t/2) or a small amount larger than (3x/2). In general, (tIh 2 ) = (m + iV + 5, where 5 is a small quan- tity and m is some integer. Expanding both sides of Eq. (16.5) and retaining only the first terms, we obtain (1/5) = V20 2 - — l or 8 = tt(2w + 1) VQSk 2 /T1 2 , where (2m + 1) is any odd integer. Equation (16.6) for the odd functions states that tan (W) must be very small, which means that for these cases (irlfi 2 ) is a small amount larger than t or a small amount larger than 2w, or, in general, (irl/i 2 ) = kir + 8, where 8 is small and k is some i nteger. The corre- sponding equation for 8 is 8 = ir(2k) VQSk 2 /T1 2 , where (2k) is any even integer. We have shown that for either even or odd functions we must have (irlm) = (nir/2) + 8, where n is any integer and where 5 = im VQSk 2 /T1 2 . The even solutions correspond to the odd values of n (n = 2m + 1), and the odd functions correspond to the even values of n (n = 2k). The a llowed values of /z 2 are given approximately by \ ~^~ T \~T~j an( * ^ e anowec * values of the frequency are n 21 " n ~ 21 \J7S K 1 ~*~ 1 \l :±: f L )' A more accurat e formula can be " S( i+ ?^> 170 THE VIBRATION OF BARS [IV.16 obtained by retaining the next terms in the series expansions that we have made: ? + (* + =£)$=] • • ) (16.9) This formula is valid as long as n 2 is smaller than (l 2 T /tt 2 QSk 2 ) . When the "stiffness constant" {QSk 2 ) becomes negligible com- pared with l 2 T this formula reduces to (16.8), the equation for the allowed frequencies of a flexible string. As the stiffness is increased the allowed frequencies increase, the frequencies of the overtones increasing relatively somewhat more rapidly than the fundamental, so that they are no longer harmonics. To obtain values for the higher overtones or for the cases where (QSk 2 ) is the same size as or larger than (Z 2 T), we must solve Eqs. (16.5) and (16.6) numerically. Problems 1 Which of the materials given in Table 3 will give the highest fundamental frequency for a bar of given size? Which will give the lowest frequency? 2. An annealed steel bar 20 cm long is clamped at one end. If its cross section is a square 1 cm on a side, what will be the lowest four frequencies of vibration? If the cross section is a circle 0.5 cm in radius, what will they be? If the cross section is a rectangle of sides a and 2a, what must be the value of a to have the bar's lowest frequency be 250 cps? 3 A bar of nickel iron of length 10 cm, whose cross section is a rectangle ol sides 1 cm and 0.5 cm, is clamped at one end. It is struck at the mid-point of one of its wider sides, so that v (x) = except at x = (1/2), and Jv dx = 100. What is the shape of the bar a time (Ti/4) after the blow? (Z\ is the period of the bar s fundamental.) What is its shape at a time (7V2)? What is the motion of the end point of the bar? What is the amplitude of that part of the motion of the end point corresponding to the fundamental? To the first overtone? To the second overtone? . , 4. Plot the shapes of the first three normal modes of vibration of a bar clamped at both ends. Of a completely free bar. ■,,-,• 5 What is the energy of vibration of the bar, struck at one end, which is shown in Fig. 32? What is the ratio between the energy of the fundamental and that of the first overtone? Of the second overtone? 6 A cylindrical bar of radius a, clamped at one end, is damped by the air by a force equal to 1.25 X 10"* a*v* (dy/dt) dynes per cm length of bar where , is the frequency of vibration and (dy/dt) the bar's velocity. Show that the modulus of decay of the nth allowed frequency is 16 X 10* ( »p/o«,«), where ,„ is given by Eq. (15.7). IV.16] VIBRATION OF A STIFF STRING 171 7. A cylindrical bar of glass, 0.5 cm in radius and 10 cm long, is clamped at one end. If it is struck at its free end, how long will it take for the amplitude of the first overtone to diminish to one-thousandth of the amplitude of the funda- mental? (Use the results of Prob. 6 to compute this.) 8. What are the first four allowed frequencies of the bar of Prob. 3? What are the first four frequencies if the bar is loaded at its free end with a mass of 1 g? The vibration is supposed to be perpendicular to the wider faces. 9. What are the shapes of the first two normal modes of vibration of the bar of Prob. 3 when its free end is loaded with a mass of 1 g? 10. A bar of phosphor bronze, of length 10 cm and thickness 0.5 cm, is tapered from a width' of 2 cm at its clamped end to a point at its free end. What are approximate values for its first three allowed frequencies? Plot the shapes of the normal modes corresponding to these frequencies. 11. An oscillating driving force of frequency v is applied to the free end of a bar, clamped at the other end. Show that the mechanical impedance of the bar is . npSl \n = l 2^ 12. Plot the mechanical impedance of the bar of Prob. 3 when driven at its free end, as a function of v from v = to v = 2,000. The motion is perpendicular to the wider faces. 13. The bar of Prob. 3 is driven electromagnetically at its free end by a coil of coupling constant G = 1,000, resistance 10 ohms, and negligible inductance. An alternating emf of 1 volt is impressed across the coil. Plot the amplitude of motion of the end of the bar as a function of v, from v = to v = 2,000. 14. A bar is whirled about one end with an angular velocity «, in a plane per- pendicular to the axis of rotation. Show that the simple harmonic vibrations of the bar from its position of steady rotation are given by y = F(z)e- 2 *<>", where Y is a solution of -£ + £[<'--> £] + *r-o where «* = (Q K y P £«»Z«), z = (x/l), and M 2 = (&rV/«»). Discuss the solutions when co is small; when w is large. 15. Calculate the transient motion of a clamped-free bar acted on by an impul- sive force S(t) concentrated at x = $. Obtain the equation corresponding to (.lU.^&i )* CHAPTER V MEMBRANES AND PLATES 17. THE EQUATION OF MOTION We must next consider the vibrations of systems extended in two dimensions, whose equilibrium shape is a plane sheet. Surfaces whose stiffness is negligible compared with the restoring force due to tension are called membranes (examples are drumheads and the diaphragms of condenser microphones). When the stiffness is the important factor, the surfaces are called plates (examples are the diaphragms of ordinary telephone transmitters and receivers). The analysis of the motions of a membrane is more complicated than that of the motions of the corresponding one-dimensional sys- tem, the flexible string; for the membrane has much more freedom in the way it vibrates than the string has. The form of the curve giving the displacement of a given point on a string as a function of time has a close connection with the form of the same string at some instant, as we have seen in Sec. 8. The corresponding curve for the motion of a point on a membrane can have no similar relationship to the shape of the membrane at some instant, for the curve showing the displacement of a point as a function of time is a one-dimensional line, whereas the shape of the membrane is a two-dimensional surface. It is possible for a membrane to behave like an assemblage of parallel strings, having waves whose crests are in parallel lines, per- pendicular to their direction of propagation (like waves on the ocean). The behavior of such waves is exactly like the behavior of waves on a flexible string; the waves travel with unchanged shape (when friction is neglected), and every such wave travels with equal speed. But this is only the simplest form of wave motion that the membrane can have. It can have circular waves, radiating out from a point or in toward it; it can have elliptical ones, going out from or in to a line segment; and so on. This more complicated sort of wave motion does not conserve its form as it travels, nor does it have a speed independent of its form. Much of it is so complicated in behavior that we are unable to analyze it at all. In this book we shall content ourselves with a treatment of parallel and circular waves. 172 V.17] THE EQUATION OF MOTION 173 Tdy- Tdy If the sheet of material is not perfectly flexible (i.e., is a plate), then the motion is still more complicated than that for a membrane.' We shall deal only with the simplest possible case of such motion, in the last section of this chapter. Forces on a Membrane.— Our first task is to set up the equation of motion for the membrane. The procedure is similar to that for the string. We shall find that any part of the membrane having a bulge facing away from the equilib- rium plane will be accelerated to- ward the plane, and vice versa. In general, the acceleration of any portion is proportional to its "bulg- iness" and opposite in direction. We must find a quantitative mea- sure of this bulginess. Suppose that the membrane has a density of <r g per sq cm and that it is pulled evenly around its edge with a tension T dynes per cm length of edge. If it is perfectly flexible, this tension will be dis- tributed evenly throughout its area, the material on opposite sides of any line segment of length dx being pulled apart with a force of T dx dynes. The displacement of the membrane from its equilibrium position will be called 77. It is a function of time and of the position on the membrane of the point in question. If we use rectangular coordinates to locate the point, v will be a function of x, y, and t. Referring to the analogous argument on page 82 for the string and to the first drawing of Fig. 34, we see that the net force on an element dx dy of the membrane due to the pair of tensions T dy will be T(r+dr)dy Fig. 34. — Forces on an element of a membrane in rectangular and polar coordinates. T V [\dx) x+dx \dx)z. -T^dxdy and that owing to the pair T dx will be T(d* v /dy 2 ) dx dy. The sum of these is the net force on the element and must equal the element's mass adxdy times its acceleration. The wave equation for the membrane is therefore 174 MEMBRANES AND PLATES [V.17 The Laplacian Operator. — The left-hand side of this, equation is the expression giving the measure of the bulginess (or, rather, the negative bulginess) of the portion of the membrane under consideration. We now encounter one of the extra difficulties of the two-dimen- sional analysis; for we find that if we had picked polar instead of rectangular coordinates, so that 77 was a function of r and <p, then the resulting wave equation would have a different aspect. The net force due to the tensions perpendicular to the radius (shown in the second drawing of Fig. 34) is |_V d<p/ v+d<P \r d<P/<pJ r 2 dip 2 and that due to the tensions parallel to the radius is ^[('SL-('S)J-?K'S)'** The resulting equation of motion in polar coordinates is i±( r *i\ + L*i = L*! (172 ) r dr \ drj ^ r 2 dip 2 c 2 dt 2 K - The left-hand side of this equation has a different form from the left-hand side of (17.1). This does not mean that it represents a different property of the membrane; for the value of the left side of (17.2) for some point on a given membrane will have the same value as the left side of (17.1) for the same point on the same membrane. It simply means that when we wish to measure the bulginess of a portion of a membrane by using polar coordinates, we must go at it differently from the way we should have gone at it if we had used rectangular coordinates. We often emphasize the fact that the left side of both equations represents the same property of the surface 17, by writing the wave equation as V*„4|? (17.8) where the symbol V 2 is called the Laplacian operator, or simply the Laplacian. It stands for the operation of finding the bulginess of the surface at some point. In different coordinates the operator takes on different forms: V.17] THE EQUATION OF MOTION 175 d 2 d 2 r~2 + ^~2 (rectangular coordinates) ^ 2 — \l d( d\ Id 2 r dr\ r d~r)~ i ~?dv* (p ° lar coordinates ) etc. The fact that we have different forms for the Laplacian operator corresponds to the fact that the membrane can have different sorts of waves. The form for rectangular coordinates is the natural one for parallel waves; that for polar coordinates is best for circular waves; and so on. Although the Laplacian is a measure of the same property of the membrane, no matter what coordinate system we use, never- theless there is a great variation in the facility with which we can solve the wave equation in different coordinate systems. In fact, the known methods for its solution are successful only in the case of a few of the simpler coordinate systems. Boundary Conditions and Coordinate Systems. — It should be realized, however, that the difficulties with coordinate systems are not, in one sense, a complete barrier to the discussion of waves of com- plicated form. For just as with the string, we can make up compli- cated wave motions by adding together simple ones. Circular waves can be made up by adding together a large number of parallel ones, each going in a different direction; parallel waves can be made up of a suitable sum of elliptic waves; and so on. We shall see that it is possible to study any sort of wave motion on a membrane of infinite extent, by expressing the waves in terms of a suitable sum of parallel waves and studying the properties of the sums (or rather, integrals, since the sums usually turn out to be integrals). It often turns out that the integrals are very difficult to evaluate, so that an immense amount of numerical integration is required to find the wave form. When the membrane is bounded, the difficulties encountered often become insuperable. For the greater generality of the problem of the membrane over that of the string is again apparent when we deal with the effect of boundary conditions. In the case of the string we needed only to specify boundary conditions at two points; but in the case of the membrane we must specify conditions all along a boundary line — and, in addition, must specify the shape of the line. A change in shape of the boundary line will have as much effect on the motion of the membrane as a change in the boundary conditions along the line. It turns out that the only way we can deal with the effect of boundary conditions is to solve the problem in coordinates suitable to 176 MEMBRANES AND PLATES [V.17 the shape of the boundary: rectangular coordinates for a rectangular boundary, polar coordinates for a circular boundary, and so on. And if we cannot solve the wave equation in coordinates suitable to the boundary chosen, we shall not be able to obtain the numerical results that are necessary for our study of the vibrations. Reaction to a Concentrated Applied Force. — Another point of difference between the string and the membrane is in the reaction to an applied force. A string of length I pushed aside by a force concentrated at the point x has a form composed of the two line segments shown in Fig. 35. The shape is so arranged that the vertical components of the ten- sion at the point of application Th/x and Th/(l - x) add up to equal the force F. The displace- ment h = Fx(l - x)/Tl of the point of application is a finite amount and is proportional to F. The membrane, on the other hand, cannot support a force concentrated at a point, and the displacement of the point of appli- cation is infinite, no matter how small the force is. For instance, if the force is concentrated at the center of a circular membrane of radius a, the displacement 17 of a point a distance r from the center will be r\ — (2F/T) In (a/r), where the symbol In means "natural logarithm of." This expression is a solution of Eq. (17.2) for equilibrium conditions [i.e., when the right- hand side of (17.2) is zero]. It becomes zero at r = a and infinite at r = 0. This result means that the simplification of considering the force to be applied at a point, which is allowable in the case of the string, is too much of a simplification for the membrane. Actual forces are not applied at a point but along a length (for the string) or over an area (for the membrane), although the length or area can be small. The point of the foregoing discussion is that, while small changes in the length over which the force is applied make, very small changes in the shape of the distorted string, changes in the size of the area of application of a force to a membrane can make large changes in the Fig. 35. — Equilibrium shapes of a string and a circular membrane each acted on by a constant force concentrated on a small portion of the string and the membrane. v - 18 l THE RECTANGULAR MEMBRANE 177 value of the maximum displacement of the membrane, so that we must take into account the size of this area. We shall find it best, however, to solve problems of forced motion where the force is concentrated in a small area by first solving the problem with the force concentrated at a point. This will result in a form for the membrane having a sharp peak of infinite height at the point of application. Then we shall cut off the top of the peak to a height such that the area of the top of the truncated peak is equal to the actual area of application of the force. This is shown in Fig. 35. This discussion shows that it is not efficient to drive a membrane by a force that is concentrated in a small area, for the amplitude of oscillation of the rest of the membrane will be very much smaller than that of the area of application. If the sheet of material has a certain amount of stiffness, then a force concentrated at a point will not produce an infinite displacement. But unless the stiffness is considerable, the displacement of the point of application will be considerably larger than the displacement of the rest of the sheet. 18. THE RECTANGULAR MEMBRANE The wave equation in rectangular coordinates is w + w=7>dF' C = v^> (18.1) A solution of this equation is v = F(x - ct), just as for the string. It represents a wave moving with velocity c in the direction of the positive a>axis, with its crests parallel to the y-axis. Its shape is independent of the value of y, and the membrane behaves as if it were made up of an infinite number of strings, all parallel to the z-axis. Another solution is v = F(y - ct), having similar properties, except that the direction of travel is parallel to the y-asaa and the crests are parallel to the z-axis. A still more general form is V = F(x cos a + y sin a — ct) (18.2) which represents a parallel wave traveling with velocity c in a direction at an angle a to the z-axis. Combinations of Parallel Waves.— In all these cases the mem- brane behaves like an assemblage of flexible strings, and the analysis of the motion is the same as that given in Sec. 8. For instance, the 178 MEMBRANES AND PLATES [V.18 reflection of such waves from a straight boundary along the z-axis is given by the solution 7} = F(x cosa + y sina — ct) — F(x cosa — y sina — ct) which is always zero at y = 0. This shows that when the angle of incidence is a the angle of reflection is — a. As soon as two or more waves going in different directions are superposed, the membrane has to bend in more than one direction, and the corresponding motion indicates that the membrane is more than an assemblage of parallel strings. As we mentioned in the previous section, every possible sort of wave can be built up out of a suitable sum of simple waves. The mathematical form of this state- ment is that every possible motion of the membrane can be expressed in terms of the integral v = f 2 * F a (x cosa + y sina — ct) da (18.3) where F a can have a different form for each different direction a. The problem is to find the forms F a to correspond to a given motion of the membrane. This will be done for a few cases, later in this chapter. The simple harmonic solutions of Eq. (18.1) can be built up out of a sum of waves of the type A a cos 'Ittv (x cosa + y sina — ct) over different values of a. This sum can be chosen so that it reduces to a set of standing waves suitable for fitting to rectangular boundaries: v = ^ i cos — (x cosa + y sina - ct) - $i \2ttv + cos + COS + COS (x cosa — y sina — ct) — *2 — (x cosa + y sina + ct) — $3 c 2irv c (x cosa - y sina + ct) - $2 - *s + *i|| = A cos fejx cos a - J cos (^ 2/ sin a - Q y J cos (frrvt - where Q t = K* a + *»), fi * = ^ _ * z) ' * = ^ 3 ~ * l) ' (18.4) V.18] THE RECTANGULAR MEMBRANE 179 Separating the Wave Equation. — Another, and perhaps more straightforward, way of obtaining (18.4) is by separating the wave equation (18. 1). If we wish to fit rectangular boundaries, we must use a standing wave having a set of nodal lines (lines along which the dis- placement is zero) parallel to the x axis and another set parallel to the y axis. Now, the only way that we can have a nodal line parallel to the y axis {i.e., for i\ to be zero for a given value of x for all values of y) is to have a factor of rj which is a function of x only and which goes to zero at the value of x corresponding to the nodal line. The nodal lines parallel to the x axis would require a factor depending on y only. Finally, if the motion is simple harmonic, the time depend- ence must come in the factor e~ 2vivt . Therefore an expression for the form of the membrane satisfying our requirements must be r] = X(x)Y{y)e~ 2 * ivt where substitution in (18.1) shows that y^ = _±[V _ d*Y dx 2 c 2 dy 2 or 1 d 2 X = _ i_ X dx 2 Y / 4*-V d 2 Y\ \ c 2 dy 2 / The left-hand side of this equation is a function of x only, whereas the right-hand side is a function of y only. Now, a function of y cannot equal a function of x for all values of x and y if both functions really vary with x and y, respectively; so that the only possible way for the equation to be true is for both sides to be independent of both x and y, i.e., to be a constant. Suppose that we call the constant — (4x 2 f 2 /c 2 ). Then the equation reduces to two simpler equations: dx 2 c 2 ' dy 2 c 2 K U The solution of this pair of equations is V = A cos( -~ x - Q x ) cosf — y - Qy\ cos (frtt VFT7 2 - 3>) (18.5) where we have set t = s/v 2 — f 2 . This solution is another way of writing (18.4), with f instead of v cos a and r instead of v sin a. The Normal Modes. — If the boundary conditions are that t\ must be zero along the edges of a rectangle composed of the x and y axes, 180 MEMBRANES AND PLATES [V.18 the line x = o, and the line y = b, it is not difficult to see that fi* - 8„ = tt/2, that (2f a/c) must equal an integer, and that (2t6/c) must equal an integer (not necessarily the same integer). The character- istic functions, giving the possible shapes of the membrane as it vibrates with simple harmonic motion, are rj = A\f/ (x,y) cos (2Tv m J — 3>) :\ • (nmy\ T XU \T) . (irmx (x,y) = sm^— (18.6) where Vmn =w!^/fe^w (m = 1, 2, 3 (n = 1, 2, 3 m = l n = l m=l n=2 m = 2 -^0=2 Fig. 36.— Shapes of the first four normal modes of a rectangular membrane. Arrows point to the nodal lines. The shapes of the first four normal modes are shown in Fig. 36. We notice that the number of nodal lines parallel to the y axis is (m - 1) and the number parallel to the x axis (n - 1). The Allowed Frequencies.— The fundamental frequency is vu, depending on T and a in a manner quite analogous to the case of the string. Among the allowed frequencies are all the harmonics of the fundamental, v 22 = 2vn, v 33 = 3vu, etc. But there are many more allowed frequencies which are not harmonics. When a is nearly equal to b, 2 extra overtones, v u and v 21 , come in between the first and second harmonic; 6 extra, v 13 , v 3h v 23 , v S2 , v u , v A i, all come between the second and third; 10 extra come between the third and fourth; and so on. No matter what the ratio between a and b is, it is possible to show that the average number of overtones between the nth and the (n + l)st harmonics is hm(a 2 + b 2 )/ab, or that the average number V.18] THE RECTANGULAR MEMBRANE 181 of allowed frequencies in the frequency range between v and v + Av is (2rvab/c 2 )Av. In the case of the string the allowed frequencies are equally spaced along the frequency scale, but in the case of the membrane the allowed frequencies get closer and closer together the higher the pitch. The higher the pitch, the more overtones there are in a range of frequency of a given size. This property is true of all membranes, no matter what shape their boundary has. It can be shown, for any membrane, that the average number of allowed frequencies between v and v + A? is (2tv/c 2 )Av times the area of the membrane. We shall show in Sec. 32 how this can be proved. X ^3Z^ y^T-V Fig. 37. — Various modes of simple harmonic vibration of a square membrane, for the degenerate cases v\% = vi\ and v™ — vz\. Arrows point to the nodal lines. The Degenerate Case. — An interesting phenomenon occurs when the rectangular membrane is a square one {i.e., a = b), for then the allowed frequencies become equal in pairs, v mn being equal to v nm . There are fewer different allowed frequencies, but there are just as many characteristic functions as there are when a is not equal to 6. This is called a condition of degeneracy. There are two different functions \l/ mn and \l/ nm , each corresponding to the same frequency (except for the cases n = m, which are not degenerate). In such cases the membrane can vibrate with simple harmonic motion of fre- quency v mn with any one of an infinite number of different shapes, corresponding to the different values of y in the combination rpmn COST + ypnm SU1 J 182 MEMBRANES AND PLATES [V.18 Figure 37 shows the shapes of the normal modes of vibration of the square membrane corresponding to v 12 and v 13 , for different values of 7. The vibrations can be standing waves, corresponding to V = {i>mn cos 7 + \f/ nm sin 7) cos (2rvmJ) The nodal lines have a different shape for each different value of 7. It is also possible to have traveling waves, corresponding to the expression V — bPmn cos {2irv mn t) + ^ nm sin (2irv m J)] Fig. 38. — Successive shapes of a rectangular membrane struck at its center. Times are given in terms of fractions of the fundamental period of vibration. In this case the nodal lines go through the whole range of possible shapes during each cycle. It is only in degenerate cases that it is possible to have traveling waves of simple harmonic motion in a mem- brane of finite size. The Characteristic Functions. — The characteristic functions for the rectangular membrane have the following integral properties: XT*- n&m'n' dx dy = fO {(ab/4) (rn' = to or n 7 and n' n) n) (18.7) V.19] THE CIRCULAR MEMBRANE 183 The behavior of a membrane having an initial shape y]o(x,y) and an initial velocity v (x,y) is therefore co oo V = 2 X ^ n {x,y)[B m n aos{2icv mn t) + C mn sin (2irv mn t)] (18.8) m= 1 n— 1 #m» = -T I I l/O^nm dc <%, Cmn = ft& J I V<$ mn dx dy This expression is the correct one for both the degenerate and the non- degenerate cases. Figure 38 shows the motion of a membrane that has been struck so that a small area around its center is started downward at t = 0. In this case r?o = 0, and v is zero except near the point x = a/2, y = 6/2, where it has a large enough value so that jjv dx dy = U. The integrations for the coefficients C mn become simple, and the result- ing expression for 17 2U v = ^ 7~~ tmnl g' 2) ^ mn ( X,V ^ Sln ( 2TV «» < ) xa6 The series does not converge if m and n both run to infinity, a corollary of the fact that a concentrated force produces an infinite displacement of the membrane. However, if the series does not continue for m larger than some number M or for n larger than N, then there will be only a finite number of terms to be added together, and the result will never be infinite. Such a finite series corresponds approximately to a starting area of dimensions (a/M) by (b/N). The case shown in Fig. 38 is f or M = N = 10. The initial shape and the shapes for successive eighths of the fundamental cycle are shown. We see the initial pulse spread out and then reflect back to pile up in the center in a "splash." Since not all the overtones are harmonics, the motion is not periodic. We notice that the shape of the pulse changes as it spreads out to the edge. More will be said about this change of shape in the next section. 19. THE CIRCULAR MEMBRANE Rectangular coordinates are useful to describe parallel waves and normal modes for a rectangular boundary, but for the discussion of circular waves and for the study of the normal modes for a circular boundary we shall find it easiest to use polar coordinates. In this study we shall encounter more forcibly than before the essential 184 MEMBRANES AND PLATES [V.19 differences between wave motion on a string and that on a membrane. To bring out these differences we start with a discussion of the general vibrations of a membrane of infinite extent. Wave Motion on an Infinite Membrane. — Let the initial shape and velocity shape of the infinite membrane be vo(x,y) and v (x,y). By letting a and b go to infinity in Eq. (18.8), we find that we can express the subsequent shape of the membrane by the Fourier integral [see Eq. (2.20)]. v(x,y,t) = -j I I d$ dr e ( 2 ™/ c ><r*+^> C J — 00 J — 00 [/» 00 /» 00 cos (2xVr 2 + t 2 I I da' dy' rj (x' ,y')e-^ i/c ^^'+^ 2xVr 2 + r 2 J--J- + da;' dy 7 v (z',2/')e- (2iri/c)(f *' +TI/ ' ) By a series of transformations too involved to discuss here, this integral can be transformed into the following simpler and much more in- teresting form : v(x,y,t) = ^ RdRdd y/cH* - R 2 * c l dt Jo Jo J*ct flit Vo(x',y') o Jo RdRdd VcH* ld& I (19.1) r -ct ~r -ct- Fig. 39. — Dependence of the wave motion of a string and membrane on initial conditions. The displacement of the point (x,v) on the membrane at the time t depends on the initial con- ditions for that part of the membrane enclosed by a circle of radius ct with center (x,y). The displacement of point x on the string depends on the initial conditions for the portion within a distance ct on either side of x. , Impermanence of the Waves, corresponding expression for the where R is the distance between the point (x,y) and the point (x',y'), as shown in Fig. 39. Equation (19.1) shows that the displacement of the membrane at the point (x,y) at the time t depends on the shape and velocity at t = of all those parts of the membrane with- in a circle about (x,y) of radius ct. The area that affects the displace- ment r](x,y,t) spreads out as t in- creases. , — Let us compare this result with the infinite string given in Eq. (8.3): V.19I THE CIRCULAR MEMBRANE 185 y(x,t) = ^ \j t I Vo(x') dx' + I v (x') dx' where we have expressed the equation in a form like (19.1). The esse ntial differ ence between this expression and (19.1) is the factor l/\/c 2 t 2 — R 2 in the integral for the membrane, which is missing in Fig. 40. — Comparison between the behavior of a plucked string and a plucked membrane. The first sketches show the initial shapes; the lower ones the shapes at successive instants later. One quarter of the membrane has been cut away to show the shape of the cross section. the integral for the string. That the difference is important can be seen by the following argument: Suppose that the initial disturbance on the string is confined to a small range, from a; = X-Atoa; = X + A, y and v being zero elsewhere. Then, by the preceding equation, the string at the point x (we assume that x is smaller than X — A, to make the discussion less verbose) will be undisturbed as long as ct is less than (X — x — A). The wave will reach the point x at t = (l/c)(Z - x - A), and from this time to t = (l/c)(X - x + A) 186 MEMBRANES AND PLATES [V.19 the displacement at x will be changing. However, from the time t = (l/c)(X — x + A) on, the string at the point x will again be undis- turbed. This corresponds to the fact that the wave, as it travels, does not change its shape; the pulse, as it passes the point x, leaves no "wake" trailing behind it. For comparison, let us suppose that the initial disturbance of the membrane is confined to a small circular area of radius A, a distance D away from the point (x, y). As with the string, no disturbance will Fig. 41. — Comparison between the behavior of a struck string and struck membrane. The initial displacements are zero, and the center portions of both string and membrane are given an initial upward velocity. occur at (x, y) as long as ct is less than (D — A). The wave will reach (x, y) at t = (l/c)(D — A), and from this time until t = (l/c)(Z> + A) the displacement at (x,y) will change. However, unlike the string, the displacement at {x, y) will continue to change after t = (l/c)(Z> + A), owing to the factor l/\^c 2 t 2 — R 2 in the integral expression. The pulse, as it spreads out from its source, leaves a wake trailing behind it. The wave therefore differs from the waves on a string and also differs from waves on a bar, where ripples travel ahead of the "crest." Figures 40 and 41 illustrate this property of the membrane and V.19] THE CIRCULAR MEMBRANE 187 contrast it to that of the string. Figure 40 gives the shapes of the membrane at successive instants, after it has been pulled out over a small area and let go at t = 0. Figure 41 shows the resulting motion when the central area in the first sketch is given an upward velocity at t = 0. The corresponding motions of a string with an equal value of c are also shown. In each case, the outermost part of the wave on the membrane keeps its shape as it moves outward, being similar to the shape of the corresponding part of the wave on the string. The rest of the membrane, however, changes shape as it moves, more and more of the crest being left behind. The initial conditions for these two figures have been chosen to have very exaggerated forms, just in order to show the above properties as clearly as possible. Simple Harmonic Waves. — Now let us turn to the circular waves that vibrate with simple harmonic motion. As in the rectangular case, we separate the wave equation. In polar coordinates the equa- tion is r dr\ dr/ r 2 d<p 2 c 2 dt 2 and by setting rj = R(r)${<p)e~ 2vivt we obtain the equations for R and <£: d 2 $ —— = — /z 2 <£>, <£ = cosO<p) or sin(jit^) (19.2) a<p W + r dF + V^" "" T 2 ) R ~ ° (19 - 3) For these circular waves one "boundary condition" is required even before the shape of the boundary line is decided upon. The requirement is simply that the displacement rj be a single valued function of position; for the coordinate <p is a periodic one, repeating itself after an angle %r, and we must have rj(r, <p) equal to r\(r, <p + 2r). This restricts the allowed values of /x to integers: $ em = cos (m<p), $om = sin (m<p) (m = 0, 1, 2, 3 • • • ) This is not true for a membrane whose boundary is shaped like a sector of a circle, where <p cannot go from zero clear around to 27r, but such cases are of no great practical importance. The foregoing requirement is the third different type of boundary condition that we have encountered, the first type being the fixing of the displacement or its slope (or both) at some point or along some line (as with the string, the bar, and the rectangular membrane), the 188 MEMBRANES AND PLATES [V.19 second being simply that the displacement have no infinite values in the range of interest (as with the whirled string). The third, the condition of periodicity, will be used whenever any of the coordinates are angles that repeat themselves. Bessel Functions. — Equation (19.3) for the radial factor is Bessel's equation. We have solved it for n = 1 in Sec. 2 and for n = in Prob. 3 of Chap. I. The general solution J m (z), where z = (2wvr/c) and where m is an integer, has the following properties : Jm(z) = m\ \$) * (m+1)! W + I . (*) m+ * _ . . . \ (19.4) ^2!(m + 2)!\2/ / v t ,\ 1 2" ( 2m + 1 \ i C 2w Jm(z) = 7T~^ I e lzcoBW cos (mw) dw 2iri m Jo Jm-l(z) + J m +l(z) = — J m (z) z -T- Jm(z) = i [J m -\(Z) - J m +l(z)] ^[z m J m (z)] = z m J m ^(z), ^z- m J m (z) = -z- m J m+1 (z) I Ji(z) dz = —J (z), I zJ (z) dz = zJx(z) I s I Jl{z)zdz = Z ^[Jl(z)+J\(z)] Ji(z)z dz = 2 [Jl(z) - J m -i(z)J m+ i(z)] 2 J m (az)J m (pz)z dz = — — % [pJ m (az)J m _ 1 (fiz) 2 _ R2 - aJ m (pz)J m -i(az)] (19.5) All these properties are proved in books on Bessel functions. Values of Jo, J i, and J 2 are given in Table V at the back of this book. The function J m {2irvr/c) is not the only solution of Eq. (19.3), for it is a second-order equation, and there must be a second solution. This other solution, however, becomes infinite at r = and so is of V.19] THE CIRCULAR MEMBRANE 189 no interest to us at present. It will be discussed in the next section, in connection with forced vibrations. The Allowed Frequencies. — Coming back to our problem, we can now say that a simple harmonic solution of the wave equation which is finite over the range from r = to r = <», which is single valued over the range from <p = to <p = 2t, is - cc)J m (^J r] = cosm(<p — a)J m [ J cos(2-n-*Z — Q) If the membrane is fastened along a boundary circle of radius a, the allowed frequencies must be those that make J m (2Tva/c) = 0. For each value of to there will be a whole sequence of solutions. We shall label the allowed values of the frequency v mn , so that *oi, *02, *oi, etc., are the solutions of J (2Tva/c) = 0; *n, * X2 , *i3, etc., are the solutions of J i(2t va/c) = 0; and so on. The values of * OTn are given by the equations Vmn =£-0mn, 001 = 0.7655, 002 = 1.7571, 003 = 2.7546 • • • 2a 0n = 1.2197, 12 = 2.2330, 0i 3 = 3.2383 02i = 1.6347, 022 = 2.6793, 23 = 3.6987 0m« ^ n + -7T- — t, if n is large (19.6) The frequency p i is the fundamental. Another way of writing these results is voi = 0.38274 ij^, vn = 1.5933poi, *2i = 2.1355*01 I ( 19>7 ) * 02 = 2.2954*oi, *3i = 2.6531*oi, v 12 = 2.9173* i • • • ) It is to be noticed that none of these overtones is harmonic. The Characteristic Functions. — Corresponding to the frequency *o« is the characteristic function Wr, fp) = j(~f^j (19.8) and corresponding to the frequency v mn (to > 0) are the two charac- teristic functions *«- = cos (m<p)J Jfe^J, tfw = sin (m<p) J m (^f\ (19.9) 190 MEMBRANES AND PLATES [V.19 Except f or m = the normal modes are degenerate, there being two characteristic functions for each frequency. The shapes of a few of the normal modes are shown in Fig. 42. We notice that the (ra, n)th characteristic function has m diametrical nodal lines and (n - 1) circular nodes. where Fig. 42. — Shapes of some of the normal modes of vibration of the circular membrane. Arrows point to the nodal lines. The integral properties of these functions are obtained from Eqs. (19.5): |T ° f ^ temntem'n-r dr d V = Jq Jq' ^Omnlfc>»V T dr d<p ( (m 7^ m' or n ?± n') = \ra 2 A mn (m = m! and n = n') (19.10) A „ = [J"i(ir/3 OTC )] 2 , A TO „ = i[J TO _i(x/3 m „)] 2 (m > 0) The values of the constants A mn can be computed from the following values of J m : JfrM = +0.5191, Ji(tM = -0.3403, Ji(7t/3o 3 ) = +0.2715, JiOtfoO = -0.2325 JoOnSii) = -0.4028, JoOnM = +0.3001, JoWh) = -0.2497, JoOtfu) = +0.2184 /iCir/Sai) = -0.3397, J r i(x/3 22 ) = +0.2714, J^&s) = -0.2324, J" 1 (7r i 824) = +0.2066 With the values of the constants A known, it is possible to com- pute the behavior of a circular membrane when started with the V.19] THE CIRCULAR MEMBRANE 191 initial shape vo(r,(p) and the initial velocity v (r,<p). By methods that we have used many times before, we can show that this behavior is governed by the series 00 / 00 n= 1 v m = where [Bemn COS (2lTV mn t) + C emn Sm (2lTV mn t)\ + J) tom n [B 0mn cos(2nrv mn t) + C 0m „ sin (2rv mn t)] > (19.11) TO = 1 / 1 f ° f ^ ■B emn = —-T- — I I r)Q\p emn r dr d<p Td'Amn Jo Jo i /»a /»2tt C«mn = o~i — ^2l — I I V($ emn r dr d<p 4ir z v mn a 2 A mn Jo Jo A similar set of equations holds for B Qmn and C Qmn . Relation between Parallel and Circular Waves. — In Sec. 18 we stated that a circular wave could be built up out of a series (or rather an integral) of different parallel waves, and we set up a general expres- sion (18.3) for the form of the integral. We must now verify our statements, by finding the form of F a which is to be used in (18.3) to give circular waves. If the waves are to be simple harmonic, and if we decide to make the dependence of F on a also periodic, Eq. (18.3) becomes it = I cos (ma) e (25riy/c) ( * °°* a+v Bin a-ct) da Since x — rcos<p and y = rsin^>, a; cos a + ysina becomes r(cosa co8<p + sin a sin <p) = r cos (a — <p). Changing to the variable = a — <p, the integral becomes /»2* V = e- 2Tiyt I [cos (m<p) cos(m/3) — sin(m^) sin (m(3)]e 2irivr <*» v* dp The term with sin (ra/3) integrates to zero, but the term with cos (m$) can be transformed, by the aid of the last of Eqs. (19.4), into v = e -2ri« C0S ( m(p )(2Tl m ) J J— J Therefore the expression for a circular wave in terms of parallel waves is 192 MEMBRANES AND PLATES [V.19 cos (m<p) Jrn\^~) e~ 2 * M \2Tri m ) Jo 2ir cos (ma)e (2ri ' /c) ( * C03 a+y sia a - ct) da. (19. 12) There is an equivalent equation with sin (m<p) and sin (ma) in place of the respective cosines. Similarly, it is possible to express the simple harmonic parallel wave e (2«»/c)o- c «) _ e (2wiv/c) (r cos a-ct) m terms of a series of circular waves SA m cos (m<p)J m (2Trvr/c)e~ 2vi '' t . We use the series with the cosine terms only, because the parallel-wave function is symmetrical in <p; a change in the sign of <p will not change the value of the expo- nential. Now, from the point of view of the dependence on <p, the series of circular functions is a Fourier series, and by the Usual methods of Sec. 9 we obtain 00 e (8»ir/e>(roo.r-eD — Q-^ivi V C m {r) COS (nip) m = Q where C °-2* i c 2 ' i r 2 " — I e iMwr/e)m *da, C m = - I e (2 " v/c)ooa ^cos(m^) dtp wr Jo t Jo (m > 0) From the last of Eqs. (19.4) we see that the first integral is just Jo(2irvr/c) and the second 2i m J m {2irvr/c). Therefore, the series of circular waves that build up into a parallel wave is e (2 r i,/c)(x-ct) — | J | ~ n " r , j + ^2i m cos (m<p) J m r^ J e- 2 ™' (19.13) m = 1 \ / J The real part of this series is 2p'(*-e*)] = W^p) -2coa(2<p)jjfe^-\ + • • • cos(2rvt) + [ 2 cos (<p)Jr(^) ~ 2 cos (3*)J 3 (^) + sin (2tt^) V.19] THE CIRCULAR MEMBRANE 193 The Kettledrum. — It sometimes happens that a circular membrane is stretched over one end of a vessel that is airtight. This is the case for the kettledrum and for some types of condenser microphones. Here the tension is not the only restoring force, for the motion of the membrane alternately compresses and expands the air in the vessel, and this reacts back on the membrane, changing its natural frequencies and its general behavior. If the diaphragm is very light and the tension is extremely large, the speed of wave motion in the membrane will be as large as the speed of sound in air, or larger; and the problem would require for its solution all the techniques that will be developed in Chap. VII. If the velocity of transverse waves in the membrane is considerably less than the speed of sound, however, the problem is much simpler, for then the compression and expansion of the air in the vessel is more or less the same over the whole extent of the mem- brane and will depend on the average displacement of the membrane. When the membrane is displaced from equilibrium to the shape expressed by the function -q, the volume of the vessel is diminished by an amount I I rjr dr d<p. If the equilibrium volume inside the vessel is TV and the equilibrium density of the air is p , then when the alternations of pressure are rapid enough to be adiabatic changes, the excess pressure inside the vessel will be p = — I fj^r? J \ I rjr dr d<p dynes/sq cm where c a is the velocity of sound waves in air at the equilibrium pres- sure and temperature in the vessel. At normal pressures and tem- peratures (temperature 20°C, pressure 760 mm of mercury) in air the value of p cl is 1.44 X 10 6 ergs per cc. This expression for the excess pressure will be proved in the next chapter. The pressure is given with a negative sign because it is always in the direction opposite to that of the average displacement of the membrane. The equation of motion for the membrane is therefore and the equation for simple harmonic vibrations is V = Y(r,(p)e- 2vivt _2 7 ,2 / -»\ l' a l* 2r \Vot) Jo Jo V*Y + ^Y=\$%)\ I Yrdrd<p (19.15) 194 MEMBRANES AND PLATES [V.19 If Y = c ? 8 (m<p) J m (2rvr/c) (m > 0), the integral on the right-hand sin side of this equation will be zero (owing to the integration over <p), and the solution that satisfies the boundary conditions Y = at r = a will be the characteristic functions given in Eqs. (19.9), with the corresponding allowed frequencies. The presence of the airtight Vessel, therefore, has no effect on the normal modes of vibration which have one or more diametrical nodal lines {i.e., which have a factor cosine or sine of (m<p), where m is not zero). For the case where m = the integral on the right is not zero. Since the solution of the equation without the integral is J (2rvr/c), we try the function Y = J (2Tvr/c) - J (2Tvva/c), which satisfies the boundary condition Y = at r = a. The integral then reduces to where x = (irpocla 4 /VoT). Inserting this into Eq. (19.15) and utiliz- ing Eqs. (19.5) results in the equation Jo(w) = — 2 ■ vr J (w) Ji(w) w = (2irva/c) ) -4w w 2 (19.16) which determines the allowed values of the frequency for those normal modes that are independent of <p (i.e., have no diametrical nodal lines). Table 4. — Frequencies op the Kettledrum X Yoi 702 703 704 0.7655 1.7571 2.7546 3.7534 0.5 0.7880 1.7590 2.7550 3.7535 1 0.8097 1.7610 2.7555 3.7537 2 0.8510 1.7651 2.7566 3.7541 3 0.8899 1.7694 2.7576 3.7545 4 0.9265 1.7739 2.7587 3.7549 5 0.9604 1.7787 2.7598 3.7553 6 0.9914 1.7837 2.7609 3.7557 8 1.0445 1.7945 2.7632 3.7565 10 1.1101 1.8065 2.7657 3.7573 Frequencies of the symmetrical normal modes of vibration of a circular mem- brane closing an airtight vessel, as functions of x the effective restoring force of the enclosed air. The nonsymmetrical modes (to > 0) are all independent of x- The frequencies in cps are given by the formula v = (y on c/2a). V.20] FORCED MOTION. THE CONDENSER MICROPHONE 195 The constant x is a measure of the relative importance .-of the air confined in the vessel with respect to the tension, as a restoring force on the membrane. It is small if the tension is large or if the volume of the vessel is large. The limiting case x = is the one studied earlier, and the allowed frequencies are given in Eqs. (19.6) and (19.7). The allowed values v 0n of the frequency [or, rather, of y 0n = (2av 0n /c)] for other values of x are given in Table 4. This table shows that the presence of the vessel tends to raise the values of the allowed fre- quencies v 0n . The Allowed Frequencies. — The allowed frequencies and corre- sponding characteristic functions for the membrane plus vessel are therefore ^eOn = Joi T 2- J — Jo(iryOn) \ f emn = CO s(mv)J m (?^y ^ n = Bm{m^) jj^f^\ (19.17) *--(&)$ --fe)^ (W>0) ) where the values of y 0n are given in Table 4 and the values of /3 m „ are given in Eq. (19.6). An approximate formula for y „, valid for small values of x, is 7o« ^ jSo« + (2xAr 4 /3jjJ, and the corresponding approximate formula for the allowed frequencies is v 0n c* (vji.) [l + o_ 3T 71 a 2 ;..o ^4 I (19.18) 8T*Vo<rK4ny where v^ n is the value of v 0n when x is zero, given in Eq. (19.7). A series analogous to that of Eq. (19.11) can be set up to represent the motion of the membrane started in any manner, by the use of the integral properties of the characteristic functions given in Eq. (19.10) and in the following equation: j: tf'eontfwr dr = < 2 |_ (tt7o«) 2 {n f = n) (n' t* n) (19.19) 20. FORCED MOTION. THE CONDENSER MICROPHONE So far, we have been dealing with circular solutions of the wave equation which are finite everywhere. When we have to deal with circular waves sent out by a simple harmonic driving force concen- 196 MEMBRANES AND PLATES [V.20 trated at a point, we must expect to use solutions that become infinite at the point of application of the force (which we set at r = 0). These solutions are the other solutions of BessePs equation (19.3), for this equation is a second-order differential equation and must have two independent solutions. Neumann Functions. — The second solutions of Bessel's equation are called Neumann functions and are given the symbol N m (z). They have the following properties : I d ( dN m \ , ( \ m 2 \ N (z) ^ (-) In (0.890536z) = (?) Qnz - 0.11593) \ (20.1) Ar , . (m - 1)1 /2\ w , ^ m N m (z) -> - ^ - ( - 1 (m > 0) z-»0 7T \Z / o N m -i{z)J m (z) - N m (z)J m _!(z) = I — The properties given in Eqs. (19.5) for the functions J m (z) are also true for the corresponding functions N m (z). The function representing a circular outgoing wave caused by a force Fe~ 2vivi concentrated at r = is '-{$[<*?) + <*r). (20.2) From Eqs. (19.4) and (20.1) we can show that when r is very small t? becomes very large, having the value — (2F/T)\nr er 2rirt , which it must have in order to balance the force, as we showed on page 176. When r is very large, r\ will approach in value the function {{ J / — e (2«i-/c)(r-c«)-tv/4 j representing a circular simple harmonic wave spreading outward with velocity c. The real part of this function is -~- J ln(r) cos (2-irvt) r->0 ^(l)V^ cos *Z lr -cO+l V.20] FORCED MOTION. THE CONDENSER MICROPHONE 197 Although the motion of the whole wave with time is sinusoidal, only the part of the wave at large distances from the driving force approaches a sinusoidal dependence on r, and even at these distances the amplitude of the waves diminishes with increasing r, owing to the factor y/c/vr. Note, also, the phase lag (tt/4) of the outlying wave behind the driving force, which is to be compared with the lag of (ir/2) for the string discussed on page 92. These properties again emphasize the difference between one- and two-dimensional waves. Unloaded Membrane, Any Force. — The coupling between a mem- brane and the surrounding medium is generally much more effective than is the corresponding coupling for a string. Here the vibrating system is a surface, which must move a sizable portion of the medium every time it moves, whereas strings are usually thin enough to avoid disturbing the medium much by their motion. Consequently, a calculation of the free and forced motions of a membrane in vacuum is not so satisfactory an approximation to its behavior in air or water as are the analogous formulas for a string. The calculation of the effect of the medium on the motion of the membrane is correspondingly more difficult than is the case for the string, where the effects are always small and can be treated as per- turbations. Only when the membrane is heavy and the medium light are the effects of the medium small enough to treat successfully by the methods of Sees. 10 and 12. This case will be considered first, however, since it is more straightforward. As a matter of fact, we shall start with the simplest case, where the effects of the medium are negligible, even though this is not often applicable. By methods similar to those used in deriving Eq. (10.16), we obtain an expression for the steady-state motion of a membrane under the influence of a distributed force F(u ) v)e~ iat per unit area at (u,v) (or r,<p) : F(u,v)\f/ mn (u,v) du dv «L — O) 2 ba ^J A mn IJ J m,n rc . , , , /0 (m! 9^ m or n' 9^ n) II tmntmV du dv = < _ ' \*3*-*-<mTl \ffl/ lib ctllLL lb — ft (20.3) where the integration is carried out over the surface of the membrane. The functions \f/ mn are given by Eq. (18.6) for the rectangular mem- brane and by Eq. (19.9) for the circular membrane; A mn is given by Eq. (18.7) (i.e., it equals -J-) or by Eq. (19.10); and w mn = %rv mn is given by Eq. (18.6) or (19.6). S is the area of the membrane and S<r its total mass. 198 MEMBRANES AND PLATES [V.20 When the force is concentrated near the point (u,v) and its total value is Pe~ iat , this series becomes SO- ^J K m n 0>l n - CO 2 mn mn When the frequency is zero, the sum gives the shape of the membrane when pushed by a steady force P concentrated at the point (u,v). m = ^Y(u,v;x,y); Y = ^ ^ n M^ n (x,y) The function Y goes to zero at the boundary of the membrane and approaches the function - (Sa/^T) \n[(x — u) 2 + {y — v) 2 ] when (x,y) is close to (u,v). Using Y, we can reexpress the series for the steady-state driven motion due to a concentrated force " = s [ Y(u,v, x,y) + ^ jr *%-J? r~ ( 20 ' 4 > L mn -■ The dependence of the shape on frequency is given by the second sum, which converges much more rapidly than the sum written first. Figure 43 shows the shapes of a square membrane driven by a force concentrated near its center, for different values of the driving frequency. The resonance frequencies are the fundamental ^n ajid the odd-numbered overtones pi 3 = vu v5, vw = 3^n, vn = vu vl3, etc. The even-numbered overtones do not appear, since the corre- sponding characteristic functions have a node at the mid-point of the membrane and the terms in the sum vanish. The nodal lines are shown in the figure by dotted lines. These change their shape, spreading outward as v is increased, a new node being introduced, near the center, after each resonance frequency has been exceeded. Localized Loading, Any Force. — When we begin to consider the effects of the medium surrounding the membrane, we must first ask whether the wave motion in the medium is faster or slower than trans- verse waves in the membrane. If the wave velocity in the medium is much slower than that in the membrane, then the reaction of the medium on any given portion of the membrane depends entirely on the motion of that part of the membrane; for the different parts of the medium are relatively slow in letting each other know what motions they are undergoing. V.20] FORCED MOTION. THE CONDENSER MICROPHONE 199 This case is seldom met in practice, but the calculations of driven motion are not difficult. The load of the medium is expressible in terms of a resistive term R per unit area and an additional reactive load that can be added to the mass per unit area of the membrane to give an effective mass <r e . Both R and <x e may vary with the driving frequency; however, when the wave velocity of the medium is very much less than that for the membrane, R is approximately constant, equal to the product of the density of the medium with the wave velocity in the medium, (pc a ), and the additional mass loading becomes negligible. Fig. 43. — Shapes of a square membrane driven by a force of frequency v concen- trated on a small area near the mid-point of the membrane. The resonance fre- quencies are the fundamental vu, P13 = 2.2361yn, v\z = 3vu, etc. At these frequencies the amplitude will be infinite, since friction has been neglected. The nodal lines are indicated by the dotted lines. The equation of motion for a distributed driving force is dt dri t =TVh,-R^+ F{x,y)e- We substitute the usual series of characteristic functions, appropriate to the boundary, for 17 and eventually obtain the familiar equation for the steady-state motion: lOiZn (20.5) where fmn = ( or — ) I I F(x,y)\p mn dx dy over the membrane 200 MEMBRANES AND PLATES [V.20 and where (2nrv mn ) 2 <r e = R — t\ OXTe (for localized loading) is the effective impedance of the (m,n)th mode for transverse motion of the membrane in the medium (at the frequency <a/2ir). As before, the characteristic functions ^ m „ are given by Eq. (18.6) or (19.8). The methods discussed in Sec. 10 can be used to compute the transient motion of the membrane. No further" details need be dis- cussed here. Uniform Loading, Uniform Force. — The other extreme, where the wave velocity in the medium is much greater than that of the mem- brane, is more often encountered but is more difficult to solve. Here the effect of the motion of one part of the surface is transmitted rapidly through the medium to affect the other parts, so that in the limit the reaction of the medium is uniform over the membrane, proportional to its average displacement. This limiting case has already been dis- cussed in connection with the kettledrum behavior. Here we wish to discuss the driven motion. The case of greatest practical interest is that of the circular dia- phragm, which is related to the problem of the condenser microphone. The constructional details of the microphone will be discussed some- what later. All that is necessary to know here is that the driving force due to an incident sound wave is Fe~ i01t per unit area, approxi- mately uniform over the membrane, and that the reaction force per unit area of the medium, on both sides of the membrane, is propor- tional to the average displacement of the membrane, the proportionality factor being —iuz, where z is the effective specific acoustic impedance of the medium (counting both sides of the membrane). The real part of z, R, is composed in part of the radiation resistance of the medium next to the outer part of the membrane. Formulas for the dependence of this part of R on frequency and membrane size are worked out in Chap. VII. The side facing the inside of the microphone case may also have a resistive part in its reaction, par- ticularly if the case is pierced with small holes to equalize the pres- sure inside and out. Motion of the air through these holes produces viscous friction. The reactive part of the impedance z due to the outer air is masslike (i.e., X is positive), whereas the reaction of the air inside the casing is usually stiffness controlled, as was the case with the kettledrum. V.20] FORCED MOTION. THE CONDENSER MICROPHONE 201 The equation of motion of the membrane is therefore d 2 7} dt 2 = TV 2 v + iazv + Fer* 1 (20.6) where ' = («*) I dv i nTdr By analogy with Eq. (10.17) we set down the following expression '-4 / {t)-'{t)]'~ ^ for the steady-state motion of the membrane. This expression goes to zero at the edge of the diaphragm (r = a). The average displace- ment rj can be found by using the integral formulas and the recursion formulas given in Eq. (19.5) : = AJ (?) Setting these expressions in Eq. (20.6) serves to determine the value of the constant A A = (Fa 2 /T/JL 2 ) Join) + Q/itiJM (x = (ua/c) = ir/3oi(v/i'oi) ) r = (za/ac) = (z/W^oi) > (20.8) = 6 -ix= (a/<rc)(R - iX)) The amplitude A is therefore proportional to the driving force F and inversely proportional to the tension T. The dependence on frequency is through the parameter /*. When the driving frequency is equal to one of the resonance frequencies v 0n , of Eq. (19.6), the quantity Jo(m) is zero, and if the air impedance parameter £* is small compared to n, A will have its largest values at these frequencies. In calculating the response of a condenser microphone, we must compute the average amplitude of motion of the diaphragm. From the results we have obtained, this is (Fa 2 /Tti 2 )J»(n)e- i0,t _ Fa 2 Join) + {$/iv)JM T i/(ju)e- i(a "- Q) (20.9) where and H(n) = _ JM J M 2 I + " ©■*>r tan £2 = 6JM nJo(») — %JM 202 MEMBRANES AND PLATES [V.20 Figure 44 shows curves of the average amplitude function H and the average phase lag £2 for x = 0, for different values of 0, plotted as" functions of the frequency parameter n = (<aa/c) — irfio n (v/von)- These curves are interesting and important, for they are typical of average response curves for membranes of any shape. At low- frequencies (m < 1) the average response is fairly independent of fre- quency. Using Eqs. (19.4) and neglecting all but the first power of n, we have * a * F er*" [a < (c/a)] ~ i ° 3Zm / v , x > (20.10) This result is similar to that for a simple driven oscillator. The effective driving force is the area of the diaphragm times the force per unit area. The mechanical impedance is the area of the dia- phragm times the total specific acoustic impedance z of the medium on both sides of the diaphragm, plus the equivalent mechanical impedance of the diaphragm itself for low frequencies. This latter corresponds to an effective mass of four-thirds of the total mass of the membrane and an effective stiffness constant of 8t times T, the mem- brane tension in dynes per centimeter. The factors (i) and 8w come in because all the membrane does not vibrate with the same amplitude of motion, as does a mass on the end of a spring. Formula (20.10) breaks down when the frequency comes near the first resonance, and the exact formula (20.9) must be used. This first resonance comes at n = 2.405, or v = v i [see Eq. (19.7)], and the average amplitude is large unless R is large. As the frequency is increased still further, the average amplitude decreases rapidly, becoming zero at \x — 5.136, or c= vzi, no matter what value x and 6 have. Just above the first resonance v i the membrane vibrates nearly out of phase with the driving force (if 6 is small) ; and as the frequency is increased still further a circular nodal line appears at the outer edge and shrinks in toward the center, the part of the membrane inside the node remaining out of phase and the part outside the node being nearly in phase with the force. As the driving frequency increases and the nodal circle shrinks, the motion of the outer part cancels more and more of the motion of the inner part in the average displacement, until at n = 5.136 the two parts completely cancel each other out, and the average displacement is zero. At this point, V.20] FORCED MOTION. THE CONDENSER MICROPHONE 203 «/ 2 (m) is zero and therefore the effect of the reaction of the medium also goes to zero (as long as this reaction depends only on the average displacement fj). For this reason the frequency of zero average dis- placement is not affected by the reaction of the air, represented by the quantities x and 0. 180" /I 90° H 0.2 Fig. 44. — Response curves and curves of angle of lag of displacement for a damped condenser microphone as functions of driving frequency, for different values of the air resistance parameter 0. The quantity fj. is 2.405 times the ratio of the driving frequency to the fundamental frequency of the diaphragm. A little beyond this antiresonance frequency, -when n = 5.520, is the second resonance, and so on. Between each of the resonance frequen- cies v 0n and the next successive one is an antiresonance frequency v 2n . The Condenser Microphone. — The condenser microphone corre- sponds approximately to the case just discussed. The diaphragm 204 MEMBRANES AND PLATES [V.20 *-l- P J N -i, B Fig 45. — Simplified cross section of a con- denser microphone. of the microphone is metallic, and therefore it has stiffness; but the diaphragm is often so thin and is under such great tension that the effects of stiffness can be neglected (though we can take the effect into account if necessary, as we shall see in the next section). Figure 45 shows a simplified cross-sectional view of a condenser microphone. D is the diaphragm, which is usually thin enough and under large enough tension to be considered as a membrane. This is stretched over the end of a vessel B, which usually has vent holes represented by H, so that it is not airtight. Behind the diaphragm a short distance is a plate P, sometimes pierced with holes so that the air can penetrate easily, and insulated elec- trically from the diaphragm. This forms the other plate of the condenser. The driving force is the excess pressure, on the outside of D, caused by a sound wave in passing. Unless the wavelength of the sound wave is smaller than the size of the microphone (which happens only for frequencies higher than 5,000 cps, for most microphones), the excess pressure can be considered to be uniform over the surface of the diaphragm. The case for small wavelengths will be taken up in the chapter on the scattering of sound. We can also assume that the pressure varies sinusoidally with time. If the sound is a combination of waves of many different frequencies, the resulting motion of the diaphragm will be a com- bination of the motions due to each wave separately. If the intensity of a simple harmonic wave of frequency v is T ergs per sec per sq cm, then the excess pressure on the outside of the diaphragm will be F == F e- 2iriH , F = 9.2 y/T dynes per sq cm (for air at standard conditions). This expression is correct only if the presence of the microphone does not alter the motion of the sound wave appreciably. In Chap. VII this question will be discussed in detail. We have mentioned earlier that the intensity due to average conversation from a person 3 ft away is about 1 erg per sec per sq cm. Electrical Connections. — The output voltage of the condenser microphone is not proportional to the amplitude A but is proportional to the average displacement of the diaphragm. If the equilibrium distance between D and P is A, then the equilibrium capacity of the microphone is C = (wa 2 /4irA) in electrostatic units. If the displace- V.20] FORCED MOTION. THE CONDENSER MICROPHONE 205 ment from equilibrium of the diaphragm is ri(r,t), then the inverse capacity of the microphone is 1 = i (A - „)„ = i-(l - X J v*) Using the expression for rj given above, we have Figure 46 shows the usual circuit for connecting the microphone to the amplifier. Resistance R is made large enough so that the condenser can- not charge and discharge rapidly enough to follow the alternations of capacitance caused by the sound waves. The aver- age charge on the plates is E Q C , and the average potential difference between the plates is E . At any instant, however, the potential difference will be the charge E C (which will not change if R is large enough) divided by the value of the microphone capacitance at that instant: To amplifier lililmi Fig. 46. — Electrical circuit for the condenser microphone. tfi = (^-°) = E i-i<- (¥) The difference between E and this quantity is the emf impressed on the amplifier: E = B, - *, - (M\ j,(?22\ e — The magnitude of the output voltage is therefore E = 1.348 Eq o-A(i/ i) 2 Vt#0) (20.12) where E is the voltage impressed across the microphone, A the dis- tance between the diaphragm and backing plate, a the density of the membrane in grams per square centimeter, v i its fundamental fre- quency of oscillation, and T the intensity of the sound in ergs per second per square centimeter. The quantity H is the one given in Eqs. (20.9), and the angle of lag of the output voltage behind the 206 MEMBRANES AND PLATES [V.20 sound pressure is the angle fl defined in the same equations. There- fore the curves of Fig. 44 are also curves of sensitivity and phase lag of the microphone signal, provided that its construction corresponds reasonably well to the assumptions as to air reaction we have made in deriving Eqs. (20.9). Transient Response of Microphone. — According to the discussion following Eqs. (6.16) and (6.17), we can find the response of the uniformly loaded diaphragm to a unit impulsive force spread uni- formly over the diaphragm, by integrating (l/27rF) times the 17 given in Eqs. (20.9) over to from — ■ °o to + °° : - (A - _^L f °° J"2(m) e-^ /o) This integral is extremely difficult to compute exactly, but it can be evaluated approximately as long as |f | is considerably smaller than n(\z\ < < w, i.e., the load per unit area due to the medium on both sides of the membrane is small compared with the mass reactance of a unit area of the membrane itself). In this case the zeros of the denominator of the integrand are very close to /z = ±(T/3 0n ). To the first approximation, near \x = irl3on, «/o(m) — — (m — Trfion)Ji(jfio n ), which can be obtained from Eqs. (19.5) and from the Taylor's expan- sion, J (n) = Jo(irp 0n ) + (m — *Po»)[dJo(n)/dfi] r p. Also, using Eqs. (19.5), we can show that in this region J 2 (m) = (2/m)«A(m) — «/o(m) — [/i - ir|3on + (2/irj8on)]J"l(ir/3on) ^ (2/lT|8o») Jl^Pon) for JU ~ Tl^On- Therefore .the contour integral breaks up into a sum of contour integrals, each around the points m = ±7r/3o«. The one about \i = +7T/3o« is approximately ( — 2iri) times the residue of ca ( 1 Y M - T0on + (2/tt/3 *) e - (icilt/a) ( 1 V M - x, %rT\Tp Q J n - Tpon + [2tf/0r/3 «) s at its pole at n = (tj8o») - [2#/(t0o») 2 ]. The contours around the points fx = — 7t/3 „ can be similarly approximated, and the final result, for the average displacement of the diaphragm in response to a uni- form impulsive force of unit impulse per unit area, is rj & (t) ~ ^ ^ Q-) 3 e-^w^> sin6^) (20.13) where R n is the resistive part of the medium's impedance per unit area (z = R — iX) at the frequency v ». The response to a general impulsive force F(«) per unit area is, according to Eq. (6.17), V.20] FORCED MOTION. THE CONDENSER MICROPHONE 207 f, = §[ ^ F(T)rj s (t - t) dr The corresponding series for the actual displacement t\ of the dia- phragm is obtained by multiplying the nth term in the series for m by [7r/W2/i(ir/3on)]/o0r/3o«r/a). Figure 47 shows a time sequence of the shapes of the diaphragm after being shock excited. Unit Impulse at "t=0,spreaol uniformly over Membrane I I I t=Q • * * * Fig. 47. — Cross-sectional view of motion of a circular membrane with damping proportional to average velocity (uniform damping), when subjected to a uniform impulsive force. Compare with Fig. 38. The motion corresponding to this formula is quite different from the related motion of a string; for in this case the higher the overtone, the less the damping. The damping factor here for the nth mode is 2R n /<T(irPo n ) 2 , which rapidly decreases as /3 « gets larger. The result is due to the uniform nature of the coupling between the diaphragm and the medium and is true only if the wave velocity in the medium is large enough so that the reaction depends solely on the average displace- ment of the diaphragm, being uniform over the diaphragm. For then, since the higher modes have a small average displacement com- 208 MEMBRANES AND PLATES [V.21 pared with their maximum displacement, not much of their energy of vibration gets carried away by the medium. By the same token, the coupling between the higher modes and the uniform impulsive force is small, so that the amplitude of the higher overtones is quite small (it varies inversely as @\ n , as a matter of fact). Thus the series converges rapidly. The equation predicts that a kettledrum diaphragm, for instance, when set into motion by a sound pulse, will have motion as a whole (lowest mode) which is rapidly damped, but that small-amplitude high-frequency short wavelength ripples in the diaphragm will persist for a long time after the pulse. This is actually the way such a dia- phragm does behave, so our assumptions must not be far from the correct ones for the drum. In the case of the condenser microphone it is not quite such a good approximation, for the reaction of the air inside the casing is not completely uniform, but is partly localized; and the effect of localized damping is to damp out the higher modes more rapidly than is the case when no localized reaction occurs. The intermediate case, where the wave velocity in the membrane is neither much larger nor much smaller than the speed of sound in the medium, is too difficult to handle this early in the volume. We shall return to it in Chap. VII. 21. THE VIBRATION OF PLATES The study of the vibrations of plates bears the same relation to the study of the membrane as the study of the vibrations of bars does to the study of the flexible string. The effect of stiffness in both cases increases the frequencies of the higher overtones more than it does those of the lower overtones and so makes the fundamental fre- quency very much lower than all the overtones. However, the motions of a plate are very much more complicated than those of a bar, so much more complicated that we shall have to be satisfied with the study of one example, that of the circular plate, clamped at its edge and under no tension. The diaphragm of an ordinary telephone receiver is a plate of this type, so the study will have some practical applications. The Equation of Motion. — The increased complications encoun- tered in the study of plates come partly from the increased complexity of wave motions in two dimensions over those of one, but also come about owing to the complex sort of stresses that are set up when a plate is bent. The bending of a plate compresses the material on the inside of the bend and stretches it on the outside. But when a material is V.21] THE VIBRATION OF PLATES 209 compressed it tries to spread out in a direction perpendicular to the compressional force, so that when a plate is bent downward in one direction there will be a tendency for it to curl up in a direction at right angles to the bend. The ratio of the sidewise spreading to the compression is called Poisson's ratio and will be labeled by the letter s. It has a value about equal to 0.3 for most materials. This complica- tion was not considered when we studied the vibration of bars, for we tacitly assumed that the bar was thin enough compared with its length so that the effects of a sidewise curl would be negligible. The derivation of the wave equation for the plate involves more discussion than is worth while here (it is given in books on theory of elasticity). The equation is y4„ + 3p(l ~ s 2 ) gg = (211) v ^ Qh 2 dt 2 K } where p is the density of the material, s its Poisson's ratio, Q its modulus of elasticity, and h the half-thickness of the plate. Values of p and Q are given for different materials in Table 3 of Chap. IV. We shall not spend any time discussing the general behavior of waves on a plate of infinite extent but shall simply remark that, like the bar, the plate is a dispersive medium; waves of different wave- length travel with different velocities. Simple Harmonic Vibrations. — To study the simple harmonic motion of a plate, we insert the exponential dependence on time and separate the factors depending on the individual coordinates. The differential operator V 4 is difficult to separate in most coordinate systems, but for polar coordinates the results turn out to be sufficiently simple to justify our analyzing them in detail. Here, if we set r\ — Y{r,(p)e~ 2Tivt , where Y's dependence on <p is by the factor cos or sin (m<p), then the differential equation for Y can be written as ( V2_ 7 2 )(V 2 + t2 ) F = o, T 4 = 12TV ffi, " S2) '(21.2) Therefore, Y can be a solution either of (V 2 F + y 2 F) = or of (V 2 F - t 2 F) = 0. Since V 2 and Y are to be expressed in polar coordinates, the solution COS of the first equation which is finite at r = is Y = . (nap) J m (yr), ^ sin where m is an integer. This is the usual solution for the membrane, with 7 instead of (2tv/c). The solution of the second equation is obtained from the first by changing y into iy and necessitates a little 210 MEMBRANES AND PLATES [V.21 discussion of Bessel functions of imaginary values of the independent variable. Let us call these hyperbolic Bessel functions and define them by the equation I m (z) = r^J^iz). The properties of the function I m (z) can be obtained from Eqs. (19.4) and (19.5) for J m (z). The more useful formulas are I m -i(z) - I m+1 (z) = — I m (z), -^ I m (z) = i[Im-i(z) + I m +i(z)] fIo(z)z dz = zlriz), J7i(z) dz = I (z) (21.3) Values of the functions 7o, Ji, and Z 2 are given in Table IV at the back of the book. The Normal Modes. — Possible solutions for the simple harmonic oscillations of a plate are therefore given by the expressions F ( r >*> = 1 S W [AJ m (jr) + BI m (yr)] Sill The boundary conditions corresponding to a circular plate of radius a, clamped at its edges, are that Y(a,<p) = and (dY/dr) r=a = 0. The first condition is satisfied by making B = -A r jm(7«) i |_/ OT (7a)J and the second condition is satisfied by requiring that y have those values that make d d Im(ya) -T- J m (yr) — J m (ya) -=- I m (yr) =0 at r = a (21.4) This equation fixes the allowed values of the frequency, for y depends on v. We shall label the solutions of Eq. (21.4) by the symbols y mn , where y mn = {ir/a)8 mn , and where l8oi =■ 1.015, O2 = 2.007, O3 = 3.000) 0n = 1.468, 0i 2 = 2.483, 8 13 = 3.490 > (21.5) 21 = 1.879, 022 = 2.992, 23 = 4.000 ) m Pmn, *■ n ~J~ ~~jr The allowed values of the frequency are therefore ^ (8 Y 3p(1 - s 2 ) irk / Vmn 2a 2 '\ : ^ • 9342 &)^£? : S [ (2L6) rn = 2.091^01, "21 = 3.426^01, "02 = 3.909P01 ^12 = 5.983^01, etc. V.21] THE VIBRATION OF PLATES 211 The allowed frequencies are spread apart much farther than those for the membrane, given in Eq. (19.7). The overtones are not harmonic. The characteristic functions are L m \ a ) ImiirPmn) ™ \ « / J fmn = COS(m and a similar expression for ^ «n (for m > 0) where sin(m«?) is used instead of cos (m<p). Some of these functions are shown in Fig. 48. Fig. 48. — Shapes of a few of the normal modes of vibration of a circular plate clamped at its edge. The free vibrations of the plate corresponding to arbitrary initial conditions can be expressed in terms of a series of these characteristic functions. Forced Motion. — As an example of the method of dealing with forced motion, let us repeat the calculations for a fully damped con- denser microphone with a plate diaphragm instead of a membrane. The equation of motion to be used instead of (20.6) is dt 2 4x F B 7 ^^ + ^° e " 2X " R //GO r dr d<p (21.7) ■jra 2 hp where t 4 = [12t 2 v 2 p(1 — s 2 )/Qh 2 ]. We now choose the function r? to be A p — 2irivt V = 77 — r [Ii(ya)J (yr) + Ji(ya)I (yr) - Ii(ya)J (ya) - Ji(ya)I Q (ya)] which has zero value and slope at r = a and which has the following average value: 212 MEMBRANES AND PLATES [V.21 TO" J J vrdrdtp = 7 , . [Ii(ya)J 2 (ya) - J 1 {ya)l2{ya)]e- Mvt In order to compare the results of these computations with the ones given in the previous section for the membrane diaphragm, we shall use the same variables n = 2A05(v/v 01 ) &nd£ = 0.3828(Z/2hpi>oi) that were used in Eq. (20.8); the only difference being that for v i, instead of the value given in (19.7), we use the value given in Eq. (21.6) for the fundamental frequency of the plate. In terms of these variables, (aV) = 4.23 In or ay = 2.057 Vi*. Substituting the expression for n into Eq. (21.7), and changing to the new variables, we obtain an expression similar to Eq. (20.8) (F/a, 2 h P ) 1 A = 7 1 (2.06 Vm) l "(m) + (TAm)^0*) (2L8) L (m) = /i(2.06 VmVo(2.06 Vm) + /o(2.06 V^)J 1 (2.0Q Vm) L 2 ( M ) = /i(2.06 Vm)«/ 2 (2.06 Vm> - / 2 (2.06 Vm)«/i(2.06 Vm) The calculations for a condenser microphone with plate diaphragm go through in a manner similar to that for the membrane, and the expression for the output voltage is E = 1.348 , , E \ 2A VTHMe-w-™ (21.9) flp{voi) A r <0 .) = ^ {[lm -Ilm] 2 + ($l>m] tanfli = \jiLo(ji) - xL 2 (n)] Curves for Ht as function of n for different values of 6 are given in Fig. 49. These may be compared directly with the curves in Fig. 44, for the membrane diaphragm. The units have been so chosen that when the two diaphragms have the same mass per unit area and the same fundamental frequency then both horizontal and vertical scales are the same. We see that below the first resonance frequency the response curves are quite similar and that the chief difference above this frequency is that the point of zero response is higher on the fre- quency scale for the plate than for the membrane — so much higher that the second resonance peak for the plate is outside the range plotted. It is evident that the plate diaphragm has the advantage of having a longer range of frequencies below the point of zero response. This V.21] THE VIBRATION OF PLATES 213 advantage, however, is almost neutralized by the disadvantage that the average response of the plate is somewhat smaller than that of the membrane, owing to the fact that the displacement must be small near its edge. The chief disadvantage of the plate, however, is the practical one that it is difficult to obtain and properly mount a plate thin enough to give a small mass per unit area and yet stiff enough to have a high-frequency fundamental (although plate diaphragms have been successfully used on miniature condenser microphones). 0.3 r Fig. 49. — Response curves for the damped condenser microphone with plate dia- phragm, plotted as a function of the frequency parameter fi for several different values of the damping constant 0. We shall not go into the details of the solution of the motions of a plate under tension, the intermediate case between the membrane and the plate without tension. It is obvious that the response curve for a damped condenser microphone with such a diaphragm will be intermediate between the curves of Figs. 44 and 49. Problems 1. A membrane is made of material of density 0.1 g per sq cm and is under a tension of 100,000 dynes per cm. It is wished to have the membrane respond best to sound of frequency 250 cps. If the membrane is square, what will be the length of one side? What will be the frequencies of the two lowest overtones? 2. A square membrane, 20 cm on a side, of mass 1 g per sq cm, is under a tension of 10 8 dynes per cm. Its motion is opposed by a frictional force of 42(dq/dt) dynes per sq cm. Find the modulus of decay of the oscillations. What are the first four "frequencies" of the damped motion? 214 MEMBRANES AND PLATES [V.21 3. A square membrane, b cm on a side, of density <r and under a tension T, is loaded at its center with a mass of M g. Show that the allowed frequencies are approximately (, T[m 2 , nn [, AM (b b\l) as long as (AM/ab 2 ) is small. What will be the expression for the frequency if the membrane is rectangular but not square? 4. Show that the energy of vibration of a rectangular membrane of sides a and 6 is ^-«-jriT[(2) , +(g) , ^®>* and that when the motion is given in terms of the series « t] = 2j A mn sin I J sin I -=— J cos {2irv mn t + * mn ) m,n \ a / \ / the series for the energy is W = — (<rab) ^ 2 ni,n mn mn 5. A square membrane 20 cm on a side, with a = 1 and T = 10 6 , is started from rest at t = with an initial shape r, = lCr b a;(20 - x)y(20 - y). What are the energies of vibration corresponding to the fundamental and the lowest three overtones, and what is the total energy of vibration of the membrane? 6. A rectangular membrane is pushed aside at the point (x ,yo) by a. steady force F and is then suddenly released at t = 0. What is the expression for the subsequent motion of the membrane? Neglect the reaction of the air. 7. A circular membrane of radius 10 cm, with a = 1, T = 36, is struck so that a very small area A»S around its mid-point has an initial velocity (1/A/S) cm per sec, all the rest of the membrane being at rest at t = 0. Plot the shape of a cross section of the membrane (77 as a function of r) for t = 0, 1, 2, 3, 4, 5, 6 sec. 8. The tensile strength of aluminum is 2.5 X 10 9 dynes per sq cm, and its density is 2.7 g per cc. What is the highest value of fundamental frequency that can be attained with an aluminum membrane stretched over a circular frame 3 cm in radius? If the aluminum is 0.005 cm thick, what will be the maxi- mum tension attainable? 9. A square membrane 4 cm on a side with a = 0.01 and T = 10 6 , which is undamped, is driven by a uniform force of l,000e- 2iriI " dynes per sq cm. Plot the amplitude of vibration of the mid-point as a function of frequency from v = to v = 2,000. 10. Suppose that the membrane of Prob. 9 is acted on by a damping force of A2(dT)/dt) dynes per sq cm. Plot the amplitude of motion of the mid-point as a function of v from v = to v = 2,000. 11. Find the effect on the allowed frequencies of a circular membrane due to a small extra mass of M g attached to its mid-point. V.21] THE VIBRATION OF PLATES 215 12. An undamped circular membrane of radius 2 cm, a- = 0.1, and T — 631,700 is driven by a uniform force of 40,000e~ 2iriw dynes per sq cm. Plot the amplitude of motion of the mid-point of the membrane as a function of v from v = to v = 1,200. What are the lowest three frequencies of free vibration of the membrane? What is the shape of the membrane (rj against r) when v = 600? 13. The circular membrane of a kettledrum has a radius of 50 cm, <r = 0.1, and T = 10 8 . What is its fundamental frequency without the backing vessel? The backing vessel raises the fundamental frequency to 1.45 times this value. What is the volume of the vessel? What are the lowest five frequencies of the membrane- vessel system? 14. A condenser microphone diaphragm has a radius a = 5 cm, a = 0.005, and T = 3 X 10 6 dynes per cm; A = 0.005 and E = 100 volts. What is the lowest frequency of the undamped diaphragm? Suppose that the impedance of the air per square centimeter of diaphragm is resistive, z = R = 100, the reaction being uniform and proportional to y. Plot the response curve H for the micro- phone for the frequency range v = to v = 10,000. 16. A condenser microphone diaphragm is a membrane of radius a with local- ized air reaction, the resistive term being R(drj/dt) per sq cm, and the reactive term being Krj (with no additional effective mass). Show that the effective impedance to insert in Eqs. (20.5) is = R -(£)[(*?*)+*] This diaphragm is acted on by a distributed force 9.2 \/ye iat dynes per sq cm. Show that the steady-state displacement of the diaphragm is 5.86 Vxr"' 2)- Jo(Tr@o n r/a) _- fionJi (irl3on) ( — iuzon) 16. Compute the output voltage of the microphone of Prob. 15 as a function of E , A, a, <r, T, R, K and driving frequency. 17. Calculate the transient response to a general impulsive force for the mem- brane of Eq. (20.5). 18. A membrane with uniform air reaction is acted on by a uniform impulsive force 8(0 per sq cm, so that its motion is given by Eq. (20.13). Values of the constants are (c/a) = 1, (4ca/T) = 1, (R n /<r) = 1. Plot value of ij(t) as function of time t from t = to t = 10. 19. The diaphragm of Eq. (20.13) is acted on by a uniform force (0 (* < 0) F(t) = < A sin (ut) [3 < t < (»/«)] (0 [t > (wtt/co)] Compute the series for the average displacement of the membrane. 20. A plate diaphragm of steel Q = 19 X 10 11 , p = 7.6, s = 0.3 has dimen- sions a = 3 and h = 0.05. What is its fundamental frequency? It is driven by a pressure of lOOe -2 *"*''' dynes per sq cm uniformly distributed over the diaphragm. The mounting is so designed that the reaction due to the medium is negligible over the useful range of v. Plot the displacement of the mid-point of the diaphragm as a function of v from v = to v = 5,000. 216 MEMBRANES AND PLATES [V.21 21. Suppose the diaphragm of Prob. 20 is to be driven by a force of 100e _2iri,,t dynes concentrated at its mid-point. Plot the amplitude of motion of the mid- point as a function of frequency from v = to v = 5000. 22. Plot the shape of the diaphragm of Prob. 21 (tj as a function of r) for v = 1,000, 3,000, 5,000. 23. A circular plate diaphragm is loaded at its mid-point by a mass M . Com- pute the approximate expressions for the allowed frequencies of the loaded plate, valid when M is small. 24. Compute the transient response of a plate diaphragm to a uniform impul- sive force, and obtain a formula corresponding to Eq. (20.13). CHAPTER VI PLANE WAVES OF SOUND 22. THE EQUATION OF MOTION We now come to the study of wave motion in air, the most impor- tant type of wave motion studied in the science of acoustics. Sound waves differ from the waves that we have discussed heretofore in several important respects. They are waves in three dimensions and as such can be more complicated in behavior than waves in two dimensions or in one. Sound waves also differ from waves on a string or on a membrane by being longitudinal waves. So far, we have been studying transverse waves, where the material transmitting the wave moves in a direction perpendicular to the direction of propagation of the wave. Each part of the string, for instance, moves in a direction at right angles to the equilibrium shape of the string, whereas the wave travels along the string. The molecules of air, however, move in the direction of propagation of the wave, so there are no alternate crests and troughs, as with waves on the surface of water, but alternate compressions and rarefactions. The restoring force, responsible for keeping the wave going, is simply the opposition that the gas exhibits against being compressed. Since there are so many points of difference between the waves discussed earlier and the more complicated forms of sound waves, it is well not to introduce all the complications at once. Accordingly, we shall first study the motion of plane waves of sound, waves having the same direction of propagation everywhere in space, whose "crests" are in planes perpendicular to the direction of propagation. They correspond to the parallel waves on a membrane. Waves traveling along the inside of tubes of uniform cross section will usually be plane waves. Waves that have traveled unimpeded a long distance from their source will be, very nearly, plane waves. Waves along a Tube. — Suppose that we consider the air in a tube of uniform cross section of area S. When everything is at equilibrium let us color red all the molecules in the plane, perpendicular to the axis of the tube, at a distance xi along the tube from some origin; 217 218 PLANE WAVES OF SOUND [VI.22 color blue the molecules in the plane at z 2 ; and so on. When a sound wave passes through the tube, these planes will be displaced from their equilibrium positions back and forth along the tube. At some instant the red plane will be at x x + £Oi), the blue plane at x 2 + £(2:2), and so on. Each molecule of the gas originally a distance x from the reference plane will be displaced in the x-direction a distance £. This displacement depends on the time t and also on the particular mole- cule that we are watching {i.e., it depends on x). Of course, this is a crude picture of what actually happens; the real molecules are bouncing back and forth because of temperature agitation, even when there is no sound. The quantity £(x,t) actually measures the average displacement, due to the sound wave, of those molecules whose average position was originally x. The average velocity of this plane in the z-direction, (d£/dt), is called the particle velocity u. We must now find an equation giving the dependence of £ on x and t. This equation will be obtained by combining three equations, one of them Newton's equation of motion and the other two repre- senting two simple properties of a gas. One of these properties is just a restatement of the law of the conservation of matter: The amount of gas between the plane of red molecules and the plane of blue molecules will remain the same as the planes move. The other property is that relating the change in density of a perfect gas with its change in pressure in the case when the gas is so rapidly compressed that it cannot unload its gain in heat to the surrounding gas. To put these properties into mathematical form we must make a few definitions. We shall denote the equilibrium density of the gas by p and its equilibrium pressure by P - The actual density at the point x and time t will be denoted by p(x,t), and the relative change in density will be denoted by 8(x,t), where P (x,t) = P (l + 5), 8 = i p(x,t) - 1 (22.1) The difference between the actual pressure and the equilibrium pres- sure will be denoted by p{x,t). It is this excess pressure p that pro- duces the motion of a microphone diaphragm. The Equation of Continuity. — Returning to the colored planes of molecules, we see that unless the gas moves so much that turbulence is set up the colored planes will remain planes as the gas moves. Since the planes are made up of molecules of the gas, it is clear that, unless there is turbulence, the gas ahead of one plane will always be VI.22] THE EQUATION OF MOTION 219 ,>j(o) >|(x,)/g(x 2 ) /j(x) -dx-* ahead of that plane, and the gas between two planes will always be between those planes. Suppose that we consider the gas between two planes as shown in the second sketch of Fig. 50, which at equilibrium are at the distances x and x + dx. At equilibrium, the mass of the gas be- tween the planes will be its density p times the volume cut off by the planes and the surface of the tube, S dx. By the argument above, the mass of gas between the planes, when they are displaced by the sound wave, will still be pS dx. But when the planes are displaced, the volume be- tween the planes may be altered, for the displacement of one plane is |(x) and that of the other is £(x + dx) = £(x) + dx(d%/dx). The volume en- closed is therefore S[dx + £(x + dx) - £(*)] = Sdx + Sdx(d£/dx). The density of the gas between the planes must be altered so that the total mass pS dx can remain unal- tered: p(x,t)[S dx + S dx(d£/dx)] = P S dx. With reference to Eq. 4(x)-- (**> N £(x+dx) x x+dx Fig. 50. — Longitudinal displace- ment in a sound wave. Gas particles originally in the plane at x are dis- placed by an amount l-(x). (22.1), this gives pS dx = P (l + 8)Sdx + 2} When the change in density and displacement are small (and they are small in all but the loudest sounds), we can neglect the product of the two small quantities 8 and (d£/dx), so that Therefore, ( 1 + g (l + S)(l+^)^l + 5 + 1=1+5+ dx or 5 = — dx dx (22.2) This is a form of the so-called equation of continuity. It states that when the gas to the right of a given point is displaced more (to the right) than the gas to the left of the point is displaced (i.e., if the two colored planes are pushed apart by the displacement) then the density of the gas is diminished at the point. We note for future reference that if the cross-sectional area of the tube S depends, on x, the correct equation will be 220 PLANE WAVES OF SOUND [VI.22 * = -s!< s » ■ (22 - 3) Compressibility of the Gas.— The second property of the gas which is used in deriving the equation of wave motion depends on some thermodynamic properties of gases. All perfect gases obey the well- known relationship between the total pressure P, the volume occupied V, and the temperature T which is PV = RT. Differentiating both sides gives R dT = P dV + V dP, and on dividing by the undiffer- entiated equation we obtain dT dP ,dV , 99 ^ T = ~P + T C If the volume occupied by the gas is kept constant, and the pressure is increased by heating, dV is zero and dP = (P/T) dT. The increase of the heat energy of the gas during this process will be dQ = (dQ/dP) dP = (dQ/dP)(P/T) dT, so that the rate of increase of heat energy due to rise of temperature at constant volume is (dQ/dT)rconst = (dQ/dP)(P/T). This rate of increase is called the specific heat of the gas at constant volume and is denoted by CV. Therefore, dP"P Cv The temperature of the gas can also be raised while holding its pressure constant, by increasing its volume by just the right amount during the heating. In this case dP = 0, dV = (V/T) dT, and the rate of increase of heating energy due to rise of temperature at con- stant pressure is (dQ/dT) Pcoast = (dQ/dV)(V/T). This rate of increase is called the specific heat at constant pressure and is denoted by C P . Therefore, dV'V Cp If, now, we allow the gas to change both volume and pressure, the change in heat energy will be the rate of change of Q with respect to V times dV, plus the rate of change with P times dP: from the equations for CV and C P . If the gas is expanded slowly enough to enable the temperature to remain constant, then dT will be zero in (22.4), and the relation between VI.22] THE EQUATION OF MOTION 221 the change in pressure and the change in volume is (dP/P) = — (dV/ V) . However, the expansions and contractions in the sound wave are entirely too rapid for the temperature of the gas to remain constant. The alternations of pressure and density are so rapid that it is much more nearly correct to say that no heat energy has time to flow away from the compressed part of the gas before this part is no longer com- pressed. In this case, where the gas temperature changes but its heat energy does not change, the compression is said to be adiabatic. To find the relation between the change in pressure and the change in volume we must use the equation for dQ, with dQ set equal to zero: C P (dV/V) - =C v (dP/P). This is the equation that will be used to find the elasticity of the gas. To return to the study of the gas in the tube contained between the planes originally at af and at x + dx, the volume of the gas is S dx, and the change in volume is S dx(d£/dx) ; the pressure is P , and the change in pressure is p. The preceding equation relating change of pressure to change in volume in adiabatic compression takes on the form or, letting (C P /C V ) = y c , V = ~^ p of||j = ToPoS (22.5) from Eq. (22.2). The constant y c , the ratio of the specific heat at constant pressure to that at constant volume, has the value 1.40 for air at normal conditions. The Wave Equation. — From the requirement of conservation of matter we have obtained Eq. (22.2) relating the change in density to the rate of change of displacement; and from the thermodynamic gas laws we' have obtained Eq. (22.5), relating the change in pressure to the change in density. Each of the functions p, £, and 8 is a function of x and t. By using one more equation, that relating the acceleration of the gas between the planes to the net force on it, we shall have enough equations to solve for all three quantities. The force on the gas between the planes, due to the gas to the left, is [P + p(x)]S; and that due to the gas to the right of the planes is [Po + p(x + dx)]S = [P + p(x) + (dp/dx)dx]S. The difference be- tween these forces —Sdx(dp/dx) is the net force acting on the gas between the planes and must therefore be equal to the mass of the gas P S dx times its acceleration (d 2 £/dt 2 ). This gives us the needed third equation: PLANE WAVES OF SOUND a 2 i 9 at 2 du dp dx 222 PLANE WAVES OF SOUND [VI.22 (22.6) By combining Eqs. (22.2), (22.4), (22.5), and (22.6) we obtain formulas for £, p, and 5 and the change in temperature AT, in a plane sound wave: d 2 £ dx 2 1 d 2 £ d 2 p _ 1 d 2 p d 2 8 _ 1 d 2 8 ~c~ 2 W dx 2 ~c 2 dt 2 ' dx 2 ~ ~c 2 W c V -iW- -pc.fi AT -0-^--^- 1 )M' (22.7) The displacement, density, and pressure all obey the equation for propagation of waves with a velocity c. For the simple case of plane sound waves of not too large amplitude, these waves will be propa- gated without change of shape and with a velocity c independent of shape. The three waves are not independent, being related by some of the equations above; once the wave of particle displacement is known (for instance), the waves of density and pressure are deter- mined. Where the pressure is greatest in magnitude, there the rate of change of displacement with x is largest; and where the pressure is zero, there the displacement is greatest (or least). Where pressure is greatest, there density is greatest, and temperature is greatest, and so on. It is a very interesting and important result that plane waves of sound, longitudinal waves, obey the same wave equation as do the transverse waves on a string. All the results that we have worked out for waves on a string can be taken over for plane sound waves, except that the meaning of some of the terms — displacement, shape of the wave, etc. — must be changed somewhat. The velocity of sound in air at standard conditions (20°C, 760 mm of mercury) and the related constants are c = 34,400 cm/sec pc = 42 g/cm 2 sec p = 0.00121 g/cm 3 , Tc = 1-40 ) (22.8) pC 2 = Tc p = 1.42 X 10 6 ergs/cm 3 P = 1.013 X 10 6 dynes/cm 2 V1.22J THE EQUATION OF MOTION 223 The constant pc is called the characteristic acoustic resistance of the air, for reasons to be given in the next section. Energy in a Plane Wave. — We can now obtain an expression for the energy involved in a sound wave. The kinetic energy of any ele- ment of volume of the gas is ip(d£/d£) 2 dx dy dz. The potential energy of the element is the amount of work required to compress the element from its equilib- rium volume dx dy dz to its new volume [1 + (d£/dx)] dxdydz = (1 — 8) dxdydz. Figure 51 shows a graph of the volume as a function of the pressure. The work done in compressing the gas jPdV turns out to equal the area of the four- sided figure (V, V , b, a). When p is small the side (a, b) can be considered to be ■ a straight line, and the area is equal to (P + %p)(V Q — V). If the wave whose energy we are mea- suring is a simple harmonic one or one made up of a number of simple harmonic components, the average value of the term Po(V — V) will be zero. Since we are interested only in average values of the energy, this term can be omitted. By the use of Eqs. (22.7) the re- maining term ip(V Q — V) becomes $pc 2 (d%/dx) 2 dx dy dz. The aver- age energy of a volume of gas due to the passage of a plane wave is therefore Average energy = \p I I |(^)+ c2 (^r) dxdydz ergs (22.9) where u = (di-/di). For a simple harmonic wave of frequency v = (w/27r), this can be expressed in terms of p alone: v v v— Fig. 51. — Relation between pressure and volume in a gas. The work done compressing the gas from volume Vo to volume V is equal to the area enclosed by the figure (Vo, a, b, Vo). Average energy = 1 2 P c' ///[©'©■ + p 2 dx dy dz These equations correspond to Eq. (9.12) for the average energy of a string. The average energy per cubic centimeter will be denoted by W. Intensity. — The rate at which the energy is being transmitted along the wave, per square centimeter of wave front, is called the intensity of the sound wave and will be denoted by T. This will equal 224 PLANE WAVES OF SOUND [VI.22 the excess pressure p on the square centimeter, multiplied by the velocity of the gas particles : v (!) = " pc * JJ Tt er ^ seG ^ cm < 22 - 10 > The average value of this is the intensity T. If the wave is a simple harmonic one, with frequency v, the expres- sions for pressure, energy, and intensity are fairly simple. For a wave going to the right, having a maximum pressure P+, p+ = P +e *(*-«), £ + = A + e ik ^~ ct \ f^tj = EZ+e*'*^') P+ = pc(^) } (22.11) »-c m T + = ZirWpcA*. = (||) = ipcET* To obtain the values of W and T given here it is necessary first to take the real parts of the expressions for p and £ before substituting in formulas (22.9) and (22.10) and averaging over time. For a wave going to the left with maximum pressure P_, rp_ — p_ e -ik(x+ct) £__ _ A_ e -ik(.x+ct) (_lr- J = JJ_ e -ik(x+ct) A --(dfe) u ~ = -(^} p - = ~ pc (i) T_ = -2w 2 v 2 pcAl = - (pA = -ycUl The intensity in this case is negative, since the wave is going in the direction of the negative z-axis. The expression used in Sees. 20 and 21 for the pressure on a con- denser microphone diaphragm is obtained from Eq. (22.11), for if the wave is all going toward the diaphragm, the total intensity will equal the square of the maximum pressure, divided by 2pc. - Since pc = 42, (22.12) VI.22] THE EQUATION OF MOTION 225 the maximum pressure will be V84T = 9.2 V T, which was the expres- sion used. This is not the correct expression if the microphone is large enough to distort the wave as it passes, as we shall see in Chap. VIII. The quantity (c/v) is called the wavelength X of the waves, and the quantity (k/2w) = (v/c) = (1/X) is called the wave number. The wavelength corresponding to y = 500 is 69 cm, or about 2 ft; that corresponding top = 4,000 is 8.6 cm, or about 3 in. The Decibel Scale. — While physical instruments can be designed to measure intensity in ergs per sec per square centimeter or in micro- watts per square centimer (1 microwatt per sq cm = 10 ergs per sec per sq cm = 10 -6 watt per sq cm), the ear does not respond in a man- ner proportional to intensity or to amplitude. It is outside the scope of this book to go into detail concerning physiological acoustics, but it is necessary to include a brief outline of the subject in order to have an idea of the ranges of values of intensity and frequency encountered in acoustical work. 1 The human ear is a remarkably rugged yet sensitive organ. It responds to a frequency range of about ten octaves, whereas the eye responds to less than one. It responds to air vibrations whose ampli- tude is hardly more than molecular size; it. also responds without damage to sounds of intensity 10 million million times greater (sound much louder than this becomes painful) . The response of the ear is not proportional to the intensity, how- ever; it is much more nearly proportional to the logarithm of the intensity. If we increase the intensity of a sound in steps of what seem to be equal increments of loudness, we find that the intensities form a sequence of the sort 1, 2, 4, 8, 16 • • • or 1, 10, 100, 1000 • • • and not of the sort 1, 2, 3, 4 • • • or 1, 10, 19, 28 • • • • Consequently, a logarithmic scale is often chosen in which to express acoustic ener- gies and intensities. The unit is the decibel (abbreviated db). A sound is said to be a decibel higher in level (or one decibel more) than another sound if its intensity is 1.259 (= \/l0) times the inten- sity of the other. If the ratio of intensities is 10, the difference in level is said to be 10 decibels (or one bel) ; if the ratio is 1,000, the differ- ence is 30 db, and so on. The difference in level of two sounds in decibels is equal to ten times the logarithm to the base 10 of the ratio between the intensities: 1 Many of the data quoted in this section have been obtained from H. Fletcher, "Speech and Hearing," D. Van Nostrand Company, Inc., New York, 1929. The xeader is referred to this book or to Stevens and Davis, "Hearing, Its Psychology and Physiology," John Wiley & Sons, Inc., 1938, for further details. 226 PLANE WAVES OF SOUND [VL22 Level difference = 10 log 10 ( yM (22.13) Intensity and Pressure Level. — At times it is convenient to deter- mine the absolute value of the intensity of some sound in the decibel scale. This is done by giving its level above or below 10 -10 microwatt per sq cm (= 10 -9 erg per sq cm per sec). Thus the intensity level of the sound is Intensity level = 10 log 10 (10 9 T) =90+10 logic (T) (22.14) This reference level is chosen because it is approximately the minimum audible intensity at 1,000 cps. Unfortunately (or fortunately, perhaps) we seldom measure sound intensity; what is measured is usually root-mean-square pressure amplitude p ims . If the sound is a sine wave, traveling in one direc- tion, then Eq. (22.11) shows the relation between p r m S and T; T = (prms) 2 /42, for air at standard conditions, since (twO 2 = i|p| 2 . How- ever, if the wave is not a plane traveling wave, the relation between T and p is not at all as simple. Indeed there are many cases when we do not know enough about the sound field to be able to compute the intensity once we have- measured the pressure ; so that we cannot give the value of the intensity level if we have measured only pressure. To obviate this difficulty, we can define a level in terms of pressure squared, the reference level being 0.000200 dyne per sq cm which cor- responds (within 2 per cent) to the pressure amplitude in a plane wave of zero intensity level. The pressure level of a sound is Pressure level = 20 logio L ^7o-J = 20 r °g»W + 74 < 22 - 15 ) When the sound is a plane wave going in a single direction, then its pressure level is only 0.1 db higher than its intensity level; when the sound has a more complex distribution, the pressure level, may differ from the intensity level by many decibels ; but in any case if we change the amplitude of a sound but not its distribution in space the decibel difference between pressure and intensity level will remain constant, the value of the difference being characteristic of the space distribution chosen. The response of the ear is not exactly proportional to the decibel scale, though it corresponds much more closely than it would to the intensity scale. Corresponding to the physical quantities intensity and frequency are the physiological (or, rather, psychophysiological) quantities loudness and pitch. The loudness of the sound depends on VI.22] THE EQUATION OF MOTION 227 both its intensity level and its frequency; its pitch depends chiefly on frequency, but to seme extent on intensity. Contours of equal loudness for the average person are plotted in Fig. 52 in terms of intensity level and frequency. The bottom curve, for zero loudness, is the threshold of hearing, below which a sound of that frequency is inaudible to the average person. The upper contour is the threshold of pain, above which the sensation is more of pain than of sound (and the result is more or 150| ,--.:,.,.W, 20 50 100 5000 10,000 20,000 200 500 1000 2000 Frequency, cycles per second Fig. 52. — Contours of equal loudness, plotted against intensity level and frequency for the average ear. Contours are numbered by correspondence with intensity leve at 1000 cps. Unshaded portion, the auditory area, shows range over which sound can be heard. less damaging to the ear). About 5 db below this upper threshold the sound begins to "feel uncomfortable"; at the threshold plotted the sensation is perhaps best described as a "tickling sensation." About 5 db above this, the sensation is "pure pain." The area enclosed within the two thresholds represents the range of audible sound in frequency and intensity. We see that the ear can hear sounds of frequency as low as 20 cps and as high as 20,000, although at these extreme limits the intensity range perceptible as sound is very small. On the other hand, the range of perceptible intensity at 1,000 cycles is as much as 125 db. 228 PLANE WAVES OF SOUND [VI.22 This area covers the range of frequency and intensity that sound reproduction equipment must cover for perfect performance. The ear is most sensitive at about 2,000 to 4,000 cps. If our abscissa were pressure level at the ear diaphragm rather than intensity level just outside the ear, the wavy appearance of the loudness contours between 2,000 and 10,000 cps would not occur, for these fluctuations are due to "cavity resonance" in the outer ear canal, increasing sensi- tivity at about 4,000 cps and decreasing it at about 8,000 cps. In terms of pressure level at the ear diaphragm, the ear is most sensitive at about 1,000 cps. Cavity resonance will be discussed at the end of Sec. 23. An interesting result of the peculiar response of the ear, evidenced by the fact that loudness-level contours, as shown in Fig. 52, are parallel neither to the intensity-level contours nor to each other, is that a complex sound changes its quality when its over-all intensity is changed. Suppose that one hears a sound consisting of many different components, some at frequencies around 100 cps and some around 1,000 cps, first when most of the components have individual intensity levels of about 100 db and then when their intensity levels are about 20 db (owing either to the fact that the source is farther away or that the sound is recorded and played back at a lower level). In the first case, when the components are at about 100 db, all components are of approximately equal loudness (since the 100-db loudness level contour is roughly horizontal). In the second case, with the intensity level of the components about 20 db, the 100-cps components would not be heard at all (for this level is below the threshold for 100 cps), whereas the 1,000-cps components would still be heard. Therefore in order to change the loudness of a complex sound without changing its quality, the different frequency components should be attenuated by different amounts, according .to the loudness-level contours of Fig. 52. Sound Power. — The amount of power produced by sound gen- erators is very small compared with usual electrical powers. The average power produced by a person talking in an ordinary conver- sational tone is about 10 -5 watt, or 100 ergs per sec; although the range of power that can be produced by the voice varies from about 1 erg per sec for very weak speech to about 10 4 ergs per sec for loud speech. Greater powers can be produced by singing voices, the range from pp to ff corresponding approximately to the variation from 10 3 to 3 X 10 6 ergs per sec. VI.22] THE EQUATION OF MOTION 229 Some musical instruments can produce greater power than the human voice. The violin cannot, but the trombone and cornet can produce about 4 X 10 5 and the bass drum about 6 X 10 5 ergs per sec. The ratio of maximum to minimum power produced by a band or orchestra while playing may be as great as 10 5 . The important range of frequency is from 30 to 8,000 cps. If we assume that the acoustic power from any of these sources radiates out equally in all directions (which it certainly does not do in many cases), then the relation between intensity and power output II would be T = Jl/^irr 2 . To this approximation, the intensity level due to a source of power output II ergs per sec at a distance of 3 ft (100 cm) would be 10 logio(n) + 39. The intensity level of a person talking normally 3 ft away is therefore approximately +60, corre- sponding roughly to a loudness of 60, according to Fig. 52 (since most components of spoken sounds are between 500 and 5,000 cps). The relation between power output and sound-intensity level farther from the source, particularly in rooms, will be discussed in Chap. VIII. Frequency Distribution of Sounds. — Since the ear does distinguish between sounds of different frequencies, it is often important to analyze the sound into its frequency components. This is, of course, what we have been doing throughout the book, even when we have discussed transient sounds. Referring to Eq. (2.19); we see that we can express any dependence of pressure on time by the formula p(t) = f "^ P{v)e-^ ivt dv where we can determine the function P(v) by the reciprocal relation P(„) = J_ f p(*)e 2 ""d< (22.16) The value of P (v) = \2P(v)\ or \2P{-v)\ is the pressure ampli- tude of that part of the sound in a unit frequency band around v (i.e., between v — \ and v + \ cps). If the sound is a plane wave, the square of P , divided by 2pc, gives the intensity of this part of the sound, T(v) = Pl(v)/2pc. The frequency distribution of a sound is often given in a decibel scale, in terms of the so-called spectrum level, which is the intensity level of that part of the sound in a unit frequency band at frequency v, Spectrum level = 10 logi [T(^)] + 90 (22.17) ~20 1ogio[Po(»')] + 71db 230 PLANE WAVES OF SOUND [VI.22 where the second line is valid only when the sound is approximately a plane wave and when pc = 42. It should be pointed out that a measurement of spectrum level (which can be obtained from a sound analyzer) is not sufficient to give the actual shape of the pressure fluctuation, for it gives values of Po(v), which is proportional to the amplitude only of P{v) and of P( — v). It follows that the phase angle of the frequency component of pressure P(v) is not determined by the sound analyzer; there- fore Eq. (22.15) cannot be utilized to compute p(t), if we have only sound-analyzer data. This usually does not matter, for the ear seems to be less sensitive to the phase of P{v) than it is to its mag- nitude. This is, of course, not true of very short pulses, for here the phases must be related in just the right manner to produce the pulse. Consequently, specifying the spectrum-level curve of a pulse does not specify it uniquely. A "pure" noise corresponds to a constant value of P over a large range of frequency, whereas a note of one frequency has a P which is zero everywhere except at just one frequency, where it is very large. Musical sounds correspond to a series of sharp peaks in the curve for P , equally spaced in frequency, the peak of lowest frequency corre- sponding to the fundamental, the next peak, at twice the frequency, corresponding to the second harmonic, and so on. The area under each peak corresponds to the amplitude of that harmonic. The peaks for a musical sound are never in practice infinitesimally narrow, corresponding to exactly one frequency, for there are always slight variations in pitch and intensity during the production of the sound to give it a "spread" in frequency. As a matter of fact, only a sine wave that continues forever with undiminished amplitude and unchanged frequency can have a peak of zero width. Any starting or stopping of the vibration will spread out the peak. For instance, we can utilize Eq. (22.16) to show that a sinusoidal oscillation of finite length (0 (t < - £A<) p(t) = <P m cos (2rv t) (- $At < t < iM) { (t> iAt) has a distribution in frequency corresponding approximately to r> / \ f P A sinh-^o - v)M] VI.22] THE EQUATION OF MOTION 231 (where we have neglected a term involving (v + v)~\ since it is usually much smaller, at least when v is approximately equal to v ). The approximation is valid if v is larger than (1/At), i.e., if the pulse contains several cycles. The quantity P (v) has a peak at v = vo, of height (P m At/2T), it falls to about half this value at v = v ± (l/2At), and is very small for (v - v) large. Figure 53 Fig. 53.— Two pressure pulses, whose shape is plotted as a function of time, and their corresponding distribution-infrequency for intensity. shows curves of the distribution in frequency of the intensity [P§0)/2pc] as functions of frequency. In order to build a pulse of finite duration we must use waves of different frequency, so arranged that they all cancel out except for the duration of the pulse. The shorter the pulse the more different frequencies must be used to form it. A measure of the "spread" of frequency needed for a pulse is the half width Av of the peak in the distribution in frequency, the frequency difference between the two points on the curve that have half the maximum height. This half width is related to the length of the pulse b V the relation AvAt^l (22.18) This relationship is approximately true even if the pulse is not started and stopped abruptly, as long as A* is taken as the half duration of the 232 PLANE WAVES OF SOUND [VI.22 pulse, the length of time between two parts of the pulse where the amplitude is half the maximum amplitude. Thus, the peaks in the curve of distribution in frequency of music or speech have a width that is at least as wide as the reciprocal of the length of time during which the note is maintained unchanged in amplitude and frequency. In the case of music, the duration of the tone is usually longer than ^ sec, and the half width is quite small, the actual width of the peak being chiefly due to the vibrato effects of the musician, who usually makes the frequency fluctuate slightly during the production of the note. It may well be that this increased spread in frequency is the reason that the vibrato note sounds better than a pure tone; for we shall see in the last chapter that the wider the spread in frequency a note has, the less marked are its resonance effects in a room. In the case of speech, however, the duration of any particular sound is quite short, in the case of some consonants being as short as ^V. sec. In this case the half width of the peaks is large, and the feeling that the sound has a definite pitch is lost. The Vowel Sounds. — The vowel and semivowel sounds are more or less musical in nature, however; their distribution in frequency being a sequence of many separate peaks, corresponding to the first 10 or more harmonics of the natural frequency of the vocal cords, the width of each peak corresponding to the duration of the sound. A bass voice will have its fundamental at 100 cycles (or thereabouts) and will have 10 peaks below 1,000 cps. A soprano voice with its fundamental at 250 cycles will have only four peaks below 1,000 cycles. Ordinarily, most of the peaks are small in height, only two or three of them being large. The position of the large peaks in the frequency scale deter- mines the sort of vowel that is spoken. For instance, the vowel " ah " has the peaks near 900 cycles enhanced, whether they correspond to the eighth, ninth, and tenth harmonics for the bass voice with a funda- mental at 100 cycles or to the third and fourth harmonics for the soprano voice of fundamental frequency 250 cycles. What seems to be happening is that the oral cavity for all individuals shapes itself for the vowel "ah" so that it resonates to a frequency of about 900, enhancing whatever part of the sound, coming to it from the vocal cords, is near this frequency. The vocal cords differ from person to person, some sending to the mouth cavity pulsations that can be analyzed into a Fourier series with a fundamental frequency as low as 100 cycles, some with a funda- mental as high as 250 cycles, etc. The mouth, then, is shaped to pick VI.23] PROPAGATION OF SOUND IN TUBES 233 from these frequencies the ones that correspond to the characteristic frequencies of the vowel spoken; or, from another point of view, the mouth is shaped so that the air in it is periodically set into vibration at the characteristic frequencies of the mouth by the pulsations sent to it from the vocal cords. The two points of view are essentially the same, one corresponding to an analysis of the motion in time and the other to an analysis in frequency. The characteristic frequency of long "oo" is 400, that of long "o" is 500, that of " ah " is 900. Some of the other vowels have two charac- teristic frequencies: Those of the "a" in "tan" are 750 and 1,600, those of long "a" as in "tame" are 550 and 2,100, those of long "ee" are 375 and 2,400, and so on. 23. PROPAGATION OF SOUND IN TUBES A large number of sound-generating devices are tubular in shape, sound waves of large amplitude being set up inside the tube, and some of this stored-up energy being radiated out into the open. Organ pipes, wood-wind and brass musical instruments, and horn loud- speakers are sound generators of this type. Inside the tube the sound waves are approximately plane waves and can be treated by the methods of the preceding section. We shall discuss the behavior of sound in tubes of uniform cross section in the present section and deal with waves in tubes of changing cross section in the next section, leaving the more complicated radiation in the open to be treated in the next chapter. In many cases the various parts of the tube (valves, constrictions, etc.) are small in length compared with a wavelength of sound. When this is the case the behavior of sound in the tube is analogous to the behavior of electric current in a circuit with lumped circuit elements of inductance, capacitance, etc. We shall take up this case first and work out { he analogous electric circuits for a few examples. When the tube elements are longer than a wavelength of sound, the electric analogue is the transmission line. This more complicated case will be taken up later in the section. Analogous Circuit Elements. — A tube of variable cross section, where the length of the elements are shorter than a half wavelength of the transmitted sound, is analogous to an electric filter circuit. The analogue of the pressure in a tube element is the voltage across the corresponding part of the filter circuit, and the analogue of the current at some point in the circuit is the total flow of air, the particle 234 PLANE WAVES OF SOUND [VI.23 velocity times the area of cross section 8 of the tube at its correspond- ing part. The analogous impedance is thus not the specific acoustic imped- ance z (which is defined as pressure over velocity) but z divided by S. It must be emphasized that the electric circuit is merely an analogue, to help our analysis (since electric filters are more familiar to most of us than acoustical filters). The analogous current comes out in cubic centimeters per second, and the voltage is in dynes per square centi- meters. The capacitances and inductances are not in farads and henrys but in the analogous acoustical units, and the power trans- mitted comes out, not in watts, but in ergs per second. Tube elements are usually of two kinds: pieces of small cross- sectional area (opening at both ends into larger chambers), which we shall call constrictions; and chambers of larger cross section (feeding into constrictions at both ends), which we shall call tanks. The air in the constrictions is mass controlled, and therefore the analogous circuit element is an inductance; the air in a tank is stiffness controlled, so the analogous element is a capacitance. The physical reasons underlying these statements, and their proof, will have to be post- poned until Chap. VII, when we shall work out the necessary mathe- matics. In the meantime some very crude analysis will have to suffice to compute the values of the analogous elements. Constriction. — The air in a constriction has a total mass of pSl e g, where S is the cross-sectional area and l e the effective length of the tube. We use an effective length because it turns out that some air beyond the ends of the constriction moves along with the air in the constriction, and this must be included to obtain the effective mass. It will be shown in Sec. 28 that the effective length l e is related to the actual length I by the formula Z e ~ Z + 0.8 V£ (23.1) Therefore even a hole in a thin plate has a nonvanishing effective length, proportional to the square root of the area of the hole. The force accelerating the mass pSl e is the difference in pressure at the two ends of the constriction, multiplied by the area S. The acceleration -iwt* is therefore equal to S times the pressure drop, divided by ( P Sl e ); and the analogous impedance, the pressure drop divided by Su, is -ioWS). A constriction is thus analogous to an inductance of value (f)g/cm 4 (23.2) VI.23] PROPAGATION OF SOUND IN TUBES 235 where £ is the cross-sectional area of the constriction, I its length, and h is given by Eq. (23.1). Tank.— If total flow of air Su is analogous to current, the analogue of charge is S times the displacement £. If a volume of air S% flows in through one opening of a tank, the increase in pressure in the tank is obtained by using a variant of Eq. (22.5), p = -pc\dV/V), where V is the volume of the tank and -dV is the volume of air introduced, St The analogous capacitance is the ratio between analogous charge (££) and analogous voltage p Ca ~ (^) cm4 sec2 /g (23.3) It will be shown later [see Eqs. (23.14) et seq.] that, when the output end of a constriction opens into free space, the radiation load on the opening is represented in the equivalent circuit by an analogous terminal resistance R T ^ fe) (23.4) which is shunted across the termination of the circuit, in series with the analogous inductance L a of the output constriction. This resist- ance becomes very small for low frequencies, since a small opening radiates very poorly at low frequencies. The power dissipated in this resistance is the power radiated out of the acoustical net. Examples.— By use of these formulas in the frequency range where they are valid, we can build up various acoustic circuits and study their properties, by studying the behavior of the analogous electric circuits A few examples are shown in Fig. 54. Case a is the simple Helmholtz resonator, analogous to the simple L-C series circuit. The analogous impedance at A is and the resonance frequency is v = 2tt VL a C a cps (23.5) Case b is a simple low-pass filter. Any standard text on filter circuits will prove that the cutoff frequency, above which the sound will not be transmitted, is at 236 PLANE WAVES OF SOUND [VI.23 1 _1 lc*S Case c is a simple high-pass filter, where frequencies above this value vo will be transmitted. Other, more complex, circuits may be built up and analyzed in a similar manner. l]~ | ■*■!- Area S («) -a "-a |..J ^ AreoS Area 5- (c) Fig 54 —Acoustical circuits on left and analogous electrical circuits on right. Constrictions of length I and cross-sectional area 8 are analogous to inductances La ; tanks of volume V are analogous to capacitances C«; as long as all dimensions oi indi- vidual elements are less than a half-wavelength of the transmitted sound. Analogous impedance at A equals ratio of pressure to S times particle velocity at A. The lumped circuit approximation is not so good a one for sound as it is for electric currents, because of the considerably smaller speed of sound and the corresponding shortness of wavelength for a given frequency. JJsually the formulas given above cease to have any validity for frequencies above 1,000 cps or so (for when an acoustical system is longer than a half-wavelength it behaves like a transmission line, and the wave formulas must be used). We have now encountered three kinds of acoustic impedance, a confusing redundancy only partly excused by the fact that the different impedances are useful in different kinds of calculations: the analogous impedance when we deal with lumped-circuit elements at low fre- quencies, the specific acoustic impedance when we deal with transmis- VI.23] PROPAGATION OF SOUND IN TUBES 237 sion-line calculations, and the usual mechanical radiation impedance when we calculate the coupling between the waves in the tube and the driving piston or the load at the output end. We list the three impedances here for comparison; their ratio turns out to be the cross- sectional area S of the tube. Analogous impedance Z a — (pressure/volume flow) (Greek subscripts, see page 234). Used in analogous circuits. Specific acoustic impedance z = (pressure/velocity) = SZ a (Lower case z, see page 239.) Mechanical impedance per unit area. Radiation impedance Z a = (force/velocity) = Sz = S 2 Z a (Lower case subscripts, see page 238.) The part of the mechanical impedance due to the sound field. There is also the acoustic impedance ratio f = — ix = z/pc, the ratio of the specific acoustic impedance to that of the medium, as indicated on the next page. In each of these cases we can draw equivalent electric circuits to help us understand the behavior of the acoustical system. These circuits are only analogous, however, for the impedances do not have the dimensions of ohms; instead of wires and amperes and volts we are dealing with tubes and air flow and pressure. Since we are more familiar with electric circuits than with acoustic ones, the equivalent circuits are often useful. The circuits equivalent to the analogous impedances are usually most useful, for there is a direct relation between currents and voltages in each part of the equivalent circuit and the pressure and volume flow in the corresponding part of the acoustical circuit. Another way of stating the same property is that the analogous impedance does not change markedly with distances along the tube of less than a half wavelength, even though the tube cross section changes markedly. Equivalent circuits for z and Z a are more difficult to interpret, but are sometimes helpful. Characteristic Acoustic Resistance. — We are now in a position to study the behavior of plane waves of sound in tubes of uniform cross section, which are longer than a half wavelength. The treatment will be quite analogous to that of Sec. 10 for wave motion on a string. We first take the case of a tube of infinite length, with no reflected wave, to get the simplest sort of air reaction. We suppose that we have a tube of uniform cross-sectional area S, starting at the origin and extending to the right along the #-axis for such a great distance that no wave is reflected back to the origin from its far end. Suppose that we fit the end at x = with a flat-topped piston which can vibrate and generate waves, so that its displacement, as a function of time, 238 PLANE WAVES OF SOUND (VI.23 is E(£)- The wave produced in the tube must be such that the dis- placement of the air next to the piston (i.e., at x = 0) is equal to S(0- Such a wave would be -0-5) which travels to the right in the tube with velocity c. The reaction F of the air back on the piston will be equal to the area of cross section of the tube S multiplied by the excess pressure p due to the wave. Using Eqs. (22.7) for the pressure, we have F = Sp = -Spc (©--"*[*(' -31 <, (dZ\ . ( . _ dZ(z) (23.6) The reaction on the piston is therefore proportional to the velocity of the piston, a purely frictional reaction. The proportionality constant (Spc) is called the radiation resistance of the air on the piston. The resistance per unit area of the piston R = pc is called the charac- teristic acoustic resistance. Its value for air at normal conditions is 42, as we showed on page 222. This is the expression used on page 199 for the reaction on the diaphragm of a condenser microphone. We have just seen that when the tube is so long that there is no reflected wave the acoustic resistance on the piston is independent of the type of motion of the piston. When the tube is not infinitely long, however, there will be some wave motion reflected back from the far end which will also affect the piston. When this happens, the reaction on the piston will be different for different motions of the piston. It will then be necessary to analyze this motion into its simple harmonic components and to treat each component separately. Incident and Reflected Waves. — When the sound wave is not all radiated out of the far end of the tube, both incident and reflected waves will be present. By referring to Eqs. (22.11) and (22.12), we see that the pressure, particle velocity, energy density, and intensity of a plane wave in a uniform tube with both waves present are ~y _ p gik(x-ct) _J_ p_g—ik(x+ct) ( zl \ — Jl (P e iHx-ct) _ p _ e -ik{x+ct)-y J_ ,™ , ™s > (23.7) 2pc 2 J_ 2pc W-^Pl+Pl) t = ^r (P+ - p -) VL23] PROPAGATION OF SOUND IN TUBES 239 To simplify the equations we shall express the ratio of the ampli- tudes of the waves in the two directions by means of an exponential: -(P-/P+) = (A-/A+) = (U-/U+) = e-w, where $ is, in general, a complex number; in other words, $ = irao — iirfio. Twice the imagi- nary part of yf/, —2x^ , gives the phase angle between the two waves at x = 0; and e~ 2Re ^ ) [where Re{<f) = ira is the real part of ^] is the ratio between the amplitudes of the two waves, without regard to phase. The ratio between the numerical magnitudes of the intensities of the two component waves is e~* ReW = e~~ 4wa . The expressions for the pressure and particle velocity, and their ratio, then become = 2P + e-+- 2 * i > t sinhU + ^llA \Wai) = <x) = (?P±) e -*-^« ..... L . *•■«* > (23- 8 ) z ( x ) = \ a // aj ) = pc tanh Specific Acoustic Impedance. — The quantity z(x) in Eq. (23.8), the ratio between the pressure and particle velocity, is called the specific acoustic impedance at the point x for the frequency v. It is a most useful quantity, for, once it is known, the reaction of the air on a vibrating system can be determined. As an example of its utility, consider a uniform tube closed at one end by a cap that yields somewhat to pressure and having a flat-topped driving piston at the other end. If we know the mechanical impedance of the cap, the ratio of driving force to cap velocity, we can determine the specific acoustic impedance of the air in the tube next to the cap's inner surface. For the velocity of this air must equal the velocity of the cap, and the pressure times the area of cross section of the tube must equal the force on the cap. Knowing z at the cap, we can find ^; and knowing \j/, we can find the value of z at the piston end of the tube. If this value is known, we can find the ratio between the driving force on the piston and its velocity, which is the effective mechanical impedance Z m of the piston for the frequency v. If the piston is driven by a force of magnitude i^o and frequency v, the velocity amplitude of the piston will be F divided by the value of Z m for the frequency p. Similarly, if the driving force is a combination of several components of different frequencies, the piston velocity is a similar combination of velocities of different frequencies, each with amplitude equal to the ratio between the magnitude of the component force and the impedance for the corresponding frequency. Thus a consideration 240 PLANE WAVES OF SOUND [VI.23 of the behavior of the specific acoustic impedance of a plane wave will enable us to work out the details of its interaction with various mechan- ical systems. In a great many cases it is most useful to express the acoustic impedance in units of the characteristic impedance of the medium, pc. The dimensionless quantity f = (z/pc) = 6 — ix is called the acoustic impedance ratio, and its reciprocal 77 = (1/f) = k — ia the acoustic admittance ratio. Its value indicates the amount of impedance mismatch at any change of medium or change of cross section. Where sound strikes the surface of a medium other than air (as, for instance, the bounding surface of a tube or the wails of a room) then the value of the admittance ratio 77 for the surface is a measure of the effect the surface will have on the wave motion striking the surface, as will be shown later. Standing Waves. — Equations (23.7) and (23.8) are duplicates of those discussed in Sec. 13 for waves in strings. The properties of the hyperbolic functions were given in Eq. (13.7); and the relationship between the behavior of the waves, which was discussed there, can be applied to the present case, though the motion here is longitudinal instead of transverse. The air displacement corresponds to the string displacement and the pressure to the transverse force. We note that the measurable quantity here is usually the pressure, whereas the measurable quantity with the string is the displacement. We shall need to distinguish between the real and imaginary parts of the quantity in the brackets in Eqs. (22.14), as before, by setting * + fe^*\ = x(a _ ij3) a = (l/ir)ifc#); = -(lA)/m(^) - (2x/X) where Re(\p) is the real part and Im(4>) the imaginary part of rp, and X = (c/v) is the wavelength of the sound wave. In terms of these quantities the acoustic impedance ratio, the amplitude of the pressure fluctuations, the amplitude of the air velocity, the energy density and the sound intensity at the point x are, respectively, I — J = f = - ix = tanh[x(« - #)] |pj = 2P+e-* a Vcosh 2 ^) - cos 2 (7r/3) \ u \ = (— ) e~* a Vcosh 2 (ra) - sin 2 (x/3) ^ VI.23J PROPAGATION OF SOUND IN TUBES 241 The minima of p are at the same places the maxima of string dis- placement were in the equations of page 141. This is because p corre- sponds to (dy/dx) for the string, whereas £ corresponds to y, so that the quantities observed are opposite in behavior with respect to ft Plates I, II, and III at the back of the book can be used in the present case as well as for the string, however, as long as this difference is kept in mind. Suppose we consider the case where the "load" is at x = I and the driving piston is at x = 0. The average impedance ratio of the load is (Z m /pcS) = izi/pc) = 6i — ixi, which is related to the corre- sponding phase parameters for the load by the equation di — ixi = tanh[7r(a!j — ipi)]. We are neglecting attenuation along the tube so that a = ai, a constant. However, = /3o — (2x/\) = ft + (2/X)(Z — x), so that /3 changes as we go along the tube, increasing as we move from load to source, and decreasing as we go from source to load. Therefore as we go from load to source the point correspond- ing to the impedance ratio moves from (aj,ft) on Plate I or II along a curve of constant a, in the direction of increasing ft The point moves in a clockwise direction by an amount of complete circuits equal to the number of half wavelengths in the tube length. This motion eventually brings us to the point (a ,ft>), where a = m, 0o = ft + (2Z/X). From this we can compute the load on the driving piston due to the air column plus load Z r = pcS tanh {ir[ai — ft — (2il/\)]}, by means of Plates I or II. The amplitude of the pressure fluctuation is a maximum where j3 is a half -integer (•••—£,+ £,+ $•••). At these points the amplitude of the particle velocity is a minimum. Where /3 is an integer ( • • • — 1,0,1,2 ••• ) the pressure amplitude is a minimum and the amplitude of the particle velocity is a maximum. Values of pressure and velocity amplitudes and phase angles can be read from Plate I in the back of the book. The values of the real and imaginary parts of (z/pc), 0, and x, corresponding to given values of a and ft are given in Plate II and in Tables III and IV at the back of the book. Alternatively, the magnitude of f and its phase angle can be obtained from Plate III and Table III. Examples of the use of the figures for calculation were given on page 138. The usefulness of the foregoing analysis will become more apparent later in the chapter. At present, we can see from Eqs. (23.8) that in wave motion composed of two plane simple harmonic waves travel- ing in opposite directions, if the specific acoustic impedance at some point x can be represented by the values R and X or by the con- 242 PLANE WAVES OF SOUND [VI.23 stants a and /?o, then the impedance at some other point x is given by the values of X and R corresponding to a = « and 18 = /3 — (2/\)(x — x ). The different impedances for different points along the wave correspond to the points on a circle of constant a in the figures, with /3 equal to the distance along the wave in units of half wavelength. Both R and X repeat themselves each half wavelength increase of x. The expression for the intensity in Eqs. (23.9) is obtained from those for the maximum pressure and particle velocity by multiplying the product of yw a nd (dg/da pmax by the cosine of the phase angle between them, (R/\/R 2 + X 2 ). This gives the maximum intensity, and the average intensity T is one half of this. This intensity is independent of (3 and is therefore independent of x. Although the pressure, the particle velocity, and the phase angle between them all change with x, they change in such a manner that the flow of energy along the wave is everywhere the same. This must be, for otherwise the energy would pile up at certain points and would continually diminish at others. We have assumed heretofore that a is independent of x, as indeed it is in a true plane wave, when there is no energy absorption during the progress of the wave. In some cases in practice, however, energy is absorbed by the walls of the tube (or in the air itself) as the wave travels along, and a will also depend on x. This case will be discussed later in this section. Measurement of Acoustic Impedance. — We can use the results of Eqs. (23.9) to devise a method for measuring specific acoustic impedance. We fit on the end of the tube, at x = I, the material or device whose impedance is to be measured; and a source of plane waves is placed at the input end, x = 0. The mechanical impedance of the device, Z m ( = force divided by velocity), divided by the cross-sectional area of the tube, S, is equal to the average specific acoustic impedance z ( = pressure divided by velocity) at the end of the tube, x = I. This terminal impedance is related to the quantities a and of Eq. (23.9) by the usual formula (s)- = 0i — ixi = tanh[7r(orz — ift)] The dimensionless quantity f = (z/pc) is the acoustic impedance ratio of the load, and and x are called the acoustic resistance and react- ance ratios respectively. VI.23] PROPAGATION OF SOUND IN TUBES 243 The impedance, pressure, and particle velocity at any point of the tube are then given by the equations |p| = A -\/cosh 2 (xa) — cos 2 (7rj8) Id = (— I Vcosh 2 (7ro:) — sin 2 (x/?) \pcj m where a = oc i; j8 = ft + (2/X)(Z - a) As we have mentioned in our discussion of Eqs. (23.9), the pressure amplitude \p\ is minimum where is an integer, or (2/X)(Z — z) = n — jSi, and the value of \p\ at these points is A sinh (rat). Conversely, the pressure amplitude is a maximum where /3 is a half-integer, or (2/X)(£ — x) = n + i — ft, and its value there is A cosh (raj). Therefore we move a microphone (a small one to avoid distorting the wave) along the tube, measuring the distances from the output end of the tube (where the unknown impedance is placed) to the points where the pressure amplitude is a maximum or minimum, and meas- uring the values of \p\ at these points. The distance between minima (or between maxima) is a half wavelength (X/2). The distance between the output end and the nearest minimum, divided by (X/2), is 1 — ft. And the ratio between the value of |p| at its minimum to its value at its maximum is tanh(7ra!z)- Therefore both oti and 8 t can be determined, and the value of the mechanical impedance of the device, Spc(di — ixi), can be computed from Table III at the back of the book. The method is particularly accurate for values of imped- ance corresponding to values of a larger than 0.02 and smaller than 0.5. Damped Waves. — In actuality sound waves cannot transverse a tube without losing at least a small part of their energy, either by absorption at the walls of the tube or by the viscous friction impeding the air nearest the wall, or else by absorption in the air throughout the tube (or for all three reasons). Some of these effects will be discussed in more detail in the next chapter; here we are interested only in the result, which is that the amplitude of each wave is damped out as it progresses along the tube. The incident wave has a factor e -Kx/c anc j tn e reflected wave a factor e KX/c , where k is the damping constant for the wave. Therefore the correct expression for the pres- sure is p(x) = 2P + e-' a »+* T ^- iw ' sinh x I a — — — tjSo + i -^J 244 PLANE WAVES OF SOUND [VI.23 This means that in actual cases a, as well as /3, depends on x. The dependence is not usually so pronounced as with /3, for (k/c) is usually much smaller than 2w/\). However, when a depends on x we cannot simply take the ratio of the minimum value of \p\ along the tube to its maximum value to obtain the value of tanh (7ra) ; for each maximum and each minimum will have a different value. What is necessary is to plot the values of the maxima (against the corresponding values of x where they occur) on semilogarithmic paper and join the points by a smooth curve, also doing the same for the minima. The difference between the two curves, extrapolated to x = I, will then give the value of tanh^aj), which is used (via Table III) to obtain the acoustic impedance at the termination. The rest of the calculations of this section will be made tacitly assuming that a is independent of x. In general, and for most pur- poses, this assumption is a good enough approximation so that the additional calculations incident to the inclusion of damping are not worth while. If damping must be taken into account, however, the effects can usually be computed by considering that a varies linearly with x, inserting a = ao — (kx/tc) into the formulas that are to be developed. Closed Tube. — The simplest case for calculation is that of a uniform tube closed at x = I by a rigid plate. In this case the impedance at x = I is infinite, corresponding to ai = 0; ft = -g-. The specific acoustic impedance at x = is therefore ' , [ (l . 2l\~] . r/27rZ\ . tt] . /coZ\ pc tanh \ —tti I H + t"/ = —ipciem l-yj + o =*pccot(— J If a piston were placed at the input end of the tube (x = 0), it would experience, because of the air in the closed tube, a mechanical imped- ance Z r = iSpc cot (j) = iSpc cot (?~j (23. 10) This is a pure reactance (although there would be a small resistive term if we took damping into account). At low frequencies (wl/c) becomes small and we can use the first term in the series expansion for the cotangent ^6)W [Z«(c/co)] (23.11) VI.23] PROPAGATION OF SOUND IN TUBES 245 which is a stiffness reactance, as would be expected of air in a closed tube. As noted earlier, the analogous impedance is equal to the corre- sponding mechanical impedance divided by S 2 , so that the analogous impedance for a closed tube (for I smaller than \/%r) is -my This justifies Eq. (23.3). A closed organ pipe is a closed tube driven at the "open" end x = by a jet of air blown across the opening. The jet is so adjusted that it is very sensitive to flow of air into and out of the tube, so that a small change in its direction makes a change in the pressure at the open end. In addition, the jet of air is adjusted so that it strikes a sharp-edged "lip" at the opposite side of the opening, which tends to set up a periodic motion perpendicular to the boundaries of the jet (sometimes with periodic vortex formation), the period depending on the speed of the air in the jet and on the distance between jet and "lip." This motion (called an edge tone) produces a net pressure fluctuation on the air in the closed tube. If the fluctuations are at the frequencies for which the impedance near the mouth of the tube is small, the tube response is large and, by reacting back on the jet, the oscillations will settle down into a vibration that is nearly simple harmonic. The frequencies to which the closed tube most strongly responds are those for which the Z r of Eq. (23.10) is smallest, i.e., for (wZ/c) = (n 4- £V, or Xn =~; v« = xi n (n = 1,3 » 5 ' ' ' ) (23>12) If now the closed organ pipe is "blown" at just the right air speed, the frequency of transverse vibration of the jet will equal the funda- mental frequency of the tube vi = (c/4Z), and the sound will consist almost entirely of the fundamental; the odd overtones will be present to some extent because the jet oscillations are not purely sinusoidal even when aided by the tube resonance. When the pipe is blown more strongly, the jet frequency is first held near the fundamental free frequency of the tube by the strength of coupling, but when the edge tone by itself would exhibit a frequency close to the third harmonic of the pipe, the note suddenly changes to this overtone and "locks in" at the new frequency. The pipe is then said to be "overblown." As indicated in Eq. (23.12), only the odd 246 PLANE WAVES OF SOUND [V1.23 harmonics are present to any extent in the sound from a closed pipe; the dependence of the amplitudes of the higher harmonics on the dimensions of the driving jet of air and on the location and shape of the "lip" regulating the edge tone is too complex to analyze in detail in this book. Open Tube. — The other case of particular interest is that of a uniform tube (of circular cross section of radius a) which is open at the end x = I. In Ghap. VII [Eq. (28.6)] we shall show that if the open end is fitted with a flange that is wide compared with a wave- length (so that the open end is effectively a round hole in an infinite plane wall) then the acoustic impedance at the open end, looking out- ward, is Ji(w)] zi ~ pc 1 - 2 ^Jp I - ipcM{w) = pc(0 o - *xo) (23.13) where w = (2<oa/c) = (47ra/X), where a is the radius of the tube and of the open end and J\ is the Bessel function given in Eq. (2.5). Values of the acoustic resistance ratio O = 1 — [2Ji(w)/w] and of the acoustic reactance ratio M(w), for an open end, are given in Table VIII at the back of the book. Their behavior for limiting values of w is If Xo = M(w) 9o = 1 _ 2 ZlW^ («V8) <»<0.6) w [ 1 (w > 5) (4w/3t) (w < 1) (4/ttw) (w > 12) (23.14) The approximate formulas differ from the correct values by less than 10 per cent within the designated ranges of w. Consequently, the limiting values of specific acoustic impedance at the open end are - (a - • \ I (p« 2 ^ 2 /2c) - io)(8 P a/Sir) (X > &ro) zi - P c{d t X o) — \ pc _ (i/ w )(2pc 2 Aa) [X < fcra/3)] At low frequencies the resistive term is quite small, so that very little energy is lost from the open end. Open tubes having cross- sectional perimeter much smaller than the sound wavelength are therefore nearly as good hoarders of energy as are closed tubes, for only a small percentage of the stored energy can be radiated away in any cycle. The reactive term is a mass load, equal to a mass of air (8pa/37r)g per sq cm of opening. When there is no flange on the end of the tube, or when the flange is less than a wavelength in size, the VI.23] PROPAGATION OF SOUND IN TUBES 247 reactance is reduced somewhat in magnitude, changing from (8pa/3ir) to approximately 0.6 pa. Equation (23.1) for the effective length of a tube with two open ends, used an approximate correction inter- mediate between these limits. For short wavelengths, the impedance is almost entirely resistive, approaching in value the characteristic acoustic resistance pc. In this case there is practically no reflected wave; the incident wave radiates out of the open end with little subsequent spreading, as though the tube were still guiding it. The sound wave, as it emerges into the open, does not realize that it has left the tube and so sees no need for sending back a reflected wave. The values of a and corresponding to the impedance Zi for the open end will be labeled a p and /3 P . Their values as functions of w = (^irva/c) are given in Table VIII at the back of the book. For very large or very small values of w they become I 27r(a/X) 2 = (co 2 /8x) (X > &ra) ap — \ (l/2ir) ln(artyX) = (l/2r) ln(«ti>/2) [X < (ra/3)] . (16a/&rX) = (4w/3tt 2 ) (X > 8xa) (i) [X < (W3)] According to Eqs. (23.9), the specific acoustic impedance at the driving end of the tube, x = 0, is pc tanh LY a p - ip p - i — J (23.16) . tan(27rZ p /X) +z27r 2 (a/X) 2 ,. ^ , ~ lpC l-^ 2 (a/X) 2 tan(2x^/X) (X > Swa) pc-i^- e** il/ * [X < (W3)] Twa where l p = I + (8a/3x). Small-diameter Open Tube. — For wavelengths long compared with the tube perimeter 2xa, the input impedance is almost purely reactive, except at the resonances and antiresonances [where tan(27rZ p /X) is zero, or where it is infinity]. When the frequency is low enough (so that X > 2l p ) , then the input impedance becomes z — —ipcy^- 2 ) = —iuplp = —iupll + ^-)> X > 2l p ^> 4ra z = X In this case the impedance is a mass load equal to a column of air of length l p = I + (8a/37r) ; the correction for the effective length of the "248 PLANE WAVES OF SOUND [VI.23 column being proportional to the tube radius a (if the flange is not large, the correction is approximately 0.6a). If the tube were open at x = also, the correction would be twice this: l e = I + (16a/37r). The effective length of the inductive element given in Eq. (23.1) is intermediate between 1.7a and 1.2a (where S is set equal to xa 2 ). As* long as the wavelength is long compared with 4ira, resonance occurs (zq minimum) when tan (2tI p /\) is zero, i.e., when 27 r K = =g; v n = ^-n (n = 1,2,3 • • • ) (23.17) where l p = I + (8a/37r) for a large flange, ~ I + 0.6a for a small flange. The impedance at the input end for these frequencies is a pure resistance which is lowest for the fundamental frequency and increases as n 2 for the higher harmonics. An open organ pipe is driven by a jet of air similar to that for a closed organ pipe. When it is blown at the right jet speed, most of the energy is in the fundamental, but there are small amounts of all the upper harmonics present. With increasing jet speed the second or higher harmonics can be emphasized. A flute is also an open tube, set into oscillation by a jet of air, which the player blows across the end at x = 0. The various musical notes are produced by varying the effective length of the tube (by opening one or more holes in the side of the tube) and by varying the strength of the jet, so as to emphasize one or the other of the har- monics. The inner bore of the actual flute is not a uniform tube, but is conical in shape; the effect of the conical shape will be discussed later in this chapter. Reed Instruments. — Clarinets are tubes, fairly uniform in diameter, open at the output end, which are driven by a reed set into vibration by a blast of air into the tube. Coupling with the air column in the tube sets the reed into periodic oscillation, the air blast through the reed being modulated by the motion of the reed. The driving system here is more analogous to a constant-current generator than to a constant-voltage one; the greatest response is when the specific acoustic admittance (1/zo) is smallest, or z largest. The natural frequencies are when tan (2rrl P /\) is infinite, or when X m = ^; v m = 4- m (m = 1,3,5 • • • ) (23.19) in 4tj> VI.23] PROPAGATION OF SOUND IN TUBES 249 In other words the natural frequencies of a "closed pipe" (i.e., open at only one end) excited at the "closed" end by a reed are the same as those of a "closed pipe" excited at the "open" end by a jet of air, as given in Eq. (23.12). The impedance at the driving end of the tube at the antiresonance frequencies given in Eq. (23.19) is as long as X TO is considerably larger than 4ira. The air jet through the reed or reeds is modulated by the reed vibration, and the reed vibration is maintained by the pressure fluctu- ations caused by the modulation, so the complete analysis of the motion would be quite intricate. A crude approximation to the actual state of affairs for a clarinet can be obtained, however, by assuming that the force driving the reed or reeds is an impulsive one, with the fundamental frequency of the tube i ''-'•S-I- 15 ^} = gfo[ • • • - Kt + T) + 5(0 - 8(t - T) + 8(t - 2T) - • • • ] (where T = 2l p /c), corresponding to a positive impulse (tF /2) at the beginning of each cycle and a negative one — (wF Q /2) a half -period T later. This may be caused by the production of vortices at the tip of the reed; at any rate it is governed by the air vibration in the tube and possibly by the player's tongue. Motion of the Reed. — The reed has its own resonance frequency (a)r/2r), which is usually several times larger than vi. The reed impedance is therefore (m r / — io))(a)l — co 2 ) where m r is the equivalent mass of the reed, and the displacement of the reed from equilibrium is ^l (F /m r ) r , . , _ ire Ji = (23.21) The air flow through the reed is proportional to the difference between the pressure P maintained in the mouth of the player and the pressure p at the input end of the tube and to the width (y + y ) of the opening at the reed, where y is the equilibrium width. The particle velocity at the input end of the tube is therefore 250 PLANE WAVES OF SOUND [VL23 u^o = G(P - p)(y + y) where G is the "conductivity constant "-of the mouth piece. Jf u is expressed in terms of a Fourier series in odd harmonics, 00 w*-o = U + 2 #»e~~* ,,l(2n+1)< (23.22) w = then the pressure at x = is obtained by multiplying each term in the series for the velocity by the corresponding specific acoustic resistance, obtained from Eq. (23.20), °° |- -, P-o = fii 2 [ (2» + l).» J e_l ' wl(2n+1)< J «i = (8pcZ p 2 A 2 « 2 ) This is valid as long as [2l p /ira{2n + 1)] is larger than unity; otherwise our approximate formula for the impedance of the open end [first formula of Eq. (23.16)] is not valid. Ordinarily l p is approximately 50a, so that the series is valid for n smaller than 10 or 15 (2n + 1 smaller than 20 or 30) The resistance R ± is approximately equal to 10 5 , indicating that the pressure fluctuation at the mouthpiece is much larger than the velocity fluctuation (which corresponds to the fact that the mouthpiece is at a velocity node). We next insert the series for y [Eq. (23.21)] and p in the expression for u and solve to obtain the values of the constants U n . At first we shall neglect the small term Gyp and set u = Uo + 2 u » e ~™ l(2n+1)t = GP Vo + GPy ~ Gy p ^Li 1 — {a)i/ur) 2 {2n + 1) e — iui(2n+l)« — Gv a R\ ^? — e -wi(2«+i)i Equating coefficients of the different exponentials, we have Uo = GPy ) v = B{2n + l) 2 } (23.23) ^ n [(2w + l) 2 + M 2 ][iV 2 - (2» + l) 2 ] ) where 5 = (GPF /m»«!), M 2 = GRiy ; and where iV = (« r /«i) is the ratio between the reed frequency and the fundamental frequency of the tube (N ~ 10 for the clarinet). Pressure and Velocity at the Reed. — The pressure in the tube, a distance x from the mouthpiece is, therefore, to the first approximation, VI.23] n = PROPAGATION OF SOUND IN TUBES (_l)ng-io> l (2n+l)t [(2» + l) 2 + M*][N 2 - (2n + l) 2 ] ' 251 • sin \~ (2n + l)(l p -x)+i g (2n + l) 2 ] The expression for the fluctuations in the velocity can be obtained from Eq. (22.6), so that uc~ Uo + RiB pc j(— \\n e iat(2n+l)t » = [(2» + l) 2 + M 2 ][N 2 - (2ft + l) 2 ] Ul pc cos I ~ (2ft + l)(l p - *)> i j£ (2n + l) 2 and the pressure and velocity at the effective output end of the tube u One Period >| Time Fig. 55. — Motion of reed, wave pressure and air flow at mouthpiece for a simplified model of clarinet. Resonance frequency of reed is ten times the fundamental tube frequency. [x = l p = I -}- (8a/37r)] are Pwmh „ pcB *y __._(»» + l)HX-l)v*"<*+»« U + Ux—lv [(2n + l) 2 + M 2 ][N 2 - (2ft + l) 2 ] RlB ^ t '(_l)ng-i«i(2n+l)t pc [(2ft + l) 2 + M 2 ][N 2 - (2ft + l) 2 ] Curves for pressure and velocity at the mouthpiece are shown in Fig. 55. We confirm that at the beginning of each half-period there is a 252 PLANE WAVES OF SOUND [VI.23 peak of back pressure from the tube at the mouthpiece, and a corre- sponding reversal in the motion of the reed. During the rest of the cycle the oscillations correspond closely to those for the free vibration of the reed, which has a natural frequency ten times the tube funda- mental (N = 10) for the case shown. This justifies (approximately) the assumption made to obtain Eq. (23.21) for the motion of the reed. We notice several interesting facts concerning the air vibrations in this approximate solution for a clarinet. Here, in distinction from the flute and open organ pipe, the odd harmonics are excited and the driving end is at a minimum of the a-c component of particle velocity, rather than a minimum of pressure. At the open end the a-c com- ponent of velocity is large (because Ri is large) and the pressure is small, as it must be at an open end of dimensions smaller than a wavelength. The frequencies in the sound which are emphasized are the funda- mental of the tube and those overtones which are near the resonance frequency of the reed, which is near the ninth or eleventh harmonic for tones in the middle register of the clarinet. The total output power can be computed from the pressure and a-c component of velocity at either the mouthpiece or the output end, by multiplying the a-c velocity and pressure, term by term, averaging and adding and finally multiplying by the tube area, S = ira 2 : n ~ — — 2 RlB2 2{[(2n + l) 2 + MW- (2» + 1)»]} (23 ' 24) where the expressions for the constants R lf B, M, and N are given on the preceding pages. The individual terms in the series equal the power output in the different harmonics, and the square root of each term is proportional to the pressure component for that frequency in the sound which is heard. Even Harmonics. — In the actual sound from a clarinet there are present a few components of even harmonics, which are not included in the series given above. This is because the modulation of the air flow by the reed is not purely linear : in other words we have neglected the term Gyp in the expression for u. This is small, but it does bring in even harmonics. The largest even terms are due, of course, to the product of the largest terms in the series for y, which is for (2w + 1) ca N — (a>r/coi), times the largest term in p, which is for n = 0. Therefore the largest even-harmonic terms in the series for the velocity at x = are (using the formula for the product of two cosines) VI.23] PROPAGATION OF SOUND IN TUBES 253 {RyB 2 /2P) (1 + M 2 )(N 2 - 1)[N 2 - (2n r + l) 2 ] [cos (2n r o)it) — cos[(2n r + 2)w^]} where n r is the integer for which (2n r + 1) is closest to N (for the clari- net n r is 4 or 5). We have neg ected the small term (pc/Ri)(2n -f- l) 2 . We see that the most important even harmonics are those just above and just below the odd harmonic which is nearest the reed resonance. The pressure at x = for these frequencies is obtained by using Eq. (23.18) for the impedance for even harmonics. It is equal to (8 P 2 c 2 B 2 /P) (1 + M 2 )(N 2 - 1)[N 2 - (2n r + l) 2 ] ' • {n 2 cos[2n r coid — (n r + l) 2 cos[(2w r + 2)wifl} Other even harmonics can be computed by using the less important terms in the series for x and p, but their amplitude is too small to make it worth while including them. Other corrections would also come in, owing to the fact that the actual tube may be slightly conical rather than of uniform cross section. In the case of the oboe, the conical shape is so pronounced as to modify considerably the empha- sized frequencies, as will be indicated later. The behavior discussed in the example we have treated in such detail is typical of most wind instruments. The air in a tube, open or closed, is excited by the oscillations of a reed, a jet of air, the horn player's lips, the vocal cords, or some similar mechanism. The oscillations in the tube react back on the driving oscillator to modify its motion. Often the coupling is sufficiently close for the driving oscillator to move with the frequencies characteristic of the tube, as is the case of the reed instruments (although the reed instruments can also oscillate with the frequency of the reed when played by an unskilled player). In such instruments the natural frequency of the driving oscillator is usually higher than the tube fundamental. Other Wind Instruments. — In other instruments the frequency of the driving oscillator is adjusted to the fundamental or to some har- monic of the tube. This is the case of the flute, the organ pipe, and the horns; in such cases the resulting note depends on the tuning of the driver (which tube harmonic is picked out) as well as on the effective length of the tube. In the case of the singing voice, the fundamental frequency of the driver, the vocal cords, is lower than that of the cavity, and the coupling is weaker than in the other examples. Here the fundamental is determined primarily by the driver, and the resonating "pipe," the throat and mouth, are (more or less) adjusted to one of the harmonics of the driver. 254 PLANE WAVES OF SOUND [VI.23 In all these instruments the coupling between driver and tube is nonlinear, which makes it possible to extract energy from a steady blast of air, but which makes the analysis of the motion quite difficult. Once the oscillations start, they build up at the frequencies that allow the system to extract a maximum amount of energy from the blast. For instance, in the case of the driving jet across the tube end (flute, organ pipe) a vibration of the jet causes a pressure fluctuation, and the frequencies emphasized are those with the smallest input imped- ance (to make the velocity large), so that the driving end is at a pres- sure node and at a velocity maximum. When a reed is the driver, a vibration of the reed causes a velocity fluctuation; therefore the resonant frequencies are those with the largest input impedance (to make the pressure large) so that the driving end is at a velocity node and at a pressure maximum. In some cases, there are alternative choices of frequencies, either choice giving large energy, as for instance the low and middle registers in reed instruments. Once started in one or the other of these choices the energy builds up, and the air blast must be nearly stopped to change to the other type. • Part of the skill of the player comes in avoiding undesirable oscillations of the instrument. A great deal of experimental and theoretical work is needed before we can say we understand thoroughly the behavior of any of the wind instruments. Tube as an Analogous Transmission Line. — From our discussion at the beginning of this section we see that we can consider any tube of uniform cross-sectional area *S to be analogous to an electric trans- mission line, with distributed series inductance L T = (p/S) per unit length and distributed shunt capacitance C T = (S/pc 2 ) per uni t leng th. The analogous characteristic impedance of the line is Z a = \Z L T /C T = (pc/S), and the analogous propagation constant is —ia> \/L T C r = — ioi/c, as long as we neglect the energy loss of the wave as it travels along the tube. The analogous impedance of a round open end, of radius a, fitted with a large flange is Z„ — „ 1 & pc_ ira 2 L (2coa/c) J ira 2 \ c / ( (jk>*/2kc) - ^(8p/37r 2 a) [« « (c/2o)] m „_. \ (pc/Tra 2 ) - (;/a,)(2pc 2 Ar 2 a 3 ) (« » (c/2a)] V ' ' shunted across the end of the transmission line. VI.23] PROPAGATION OF SOUND IN TUBES 255 A change of cross-sectional area corresponds to a change of dis- tributed constants for the line; an increase of area corresponding to an increase in the distributed capacitance and to a decrease in the distributed inductance; and vice versa. Therefore a narrow portion of the tube corresponds more nearly to a pure inductance and a wide portion to a pure capacitance, as we assumed at the beginning of this section. A hole in the side of the tube will correspond to an inductance and resistance shunted across the line at the proper point. According to Eq. (23.25) the resistance should be approximately (p« 2 /27rc) and the inductance should be approximately (0.5p/b) where b is the radius of the open hole, and where we have taken into account the fact that the effective mass of air in the hole extends somewhat inside the hole, as well as outside, as was done in deriving Eq. (23.1). After the analogous impedance at the input end has been com- puted, the specific acoustic impedance can be obtained by multiplying by Si, and the mechanical impedance of the air column can be obtained by multiplying the analogous impedance by Sf, where Si is the area of the input end. Open Tube, Any Diameter. — We now return to the general problem of the piston and open tube and treat the intermediate case where the diameter of the tube is neither large nor small compared with the wave length. As we shall see in the next chapter, the acoustic imped- ance ratio at the open end of a cylindrical tube equipped with a baffle flange is ?-"»,- M - 2 *^M _ iM(w)\ (23.26) where w = (Anrva/c) = (4ra/X) and a is the radius of the tube. Values of the functions 1 — [2Ji(w)/w] and M(w) are given in Table VIII at the back of the book. Equation (23.26) is valid for all wavelengths; the two limiting cases, corresponding to w — » and w — > °° , have been discussed previously. According to Eqs. (23.9), the direct and reflected waves inside the tube must adjust themselves so that tanh $ + r-v»)l f%nP is equal to the value of f (l) given above. To facilitate this calculation, solutions of the equation tanh [ir(a p - #„)] = 1-2 ^^ - iM(w) 256 PLANE WAVES OF SOUND [VI.28 are also given in Table VIII. Once the values of a p and P are found for a given value of w = (47ra/X), calculation of the properties of the piston-tube system is a simple matter. The ratio of reflected to incident amplitude is e~ 2 * a . The specific acoustic impedance at the piston is pc tanh Ur ( a p — i$ p — i — ) \> and the radiation impedance of the piston itself is %r = pc (-q) tanh L- f a p — i$ p — i— J = R r — iX r R r = pee (fjf), x r = pcx (§) Values of e and x can be obtained from Plates I and II, or from Table III at the back of the book. If the piston has a mechanical impedance Z p = R p — iX p , then the total mechanical impedance of the piston is Z m = R m — iX m , where R m = R r + R P , X m — X r + X p . If the piston is driven by a force of amplitude F , then its velocity u and amplitude of motion So, the intensity T of the sound in the tube, and the power n radiated out of the end of the tube are given by the formulas Uq = (23.27) 2(RI + XIY \SJ 2S{Rl + Xl) As an example, the acoustic resistance ratio e and reactance ratio x at the piston end are shown in Fig. 56 for a tube whose length I is ir times its diameter 2a [so that w = (Ara/X) = 2Z/X]. The quantities are plotted as functions of (2Z/X) = (2vl/c). For the low frequencies the impedance at the open end is nearly all reactive, so that little of the energy escapes, and large resonance peaks can occur. The points of minimum impedance, where resonance occurs, are separated by points of high impedance. As the frequency increases, more and more of the energy reaching the open end is radiated out, less and less is reflected back to help in the resonance, and the peaks and valleys in the impedance curve get less and less pronounced. At very high frequencies the reactance is zero, the resistance is pc, and no resonance occurs. The integral values of (2Z/X), shown by the dashed vertical lines, correspond to the frequencies of resonance of a tube with an open end VI.23] PROPAGATION OF SOUND IN TUBES 257 which allows no energy to escape (i.e., a "perfect" organ pipe).. We notice that the actual frequencies of resonance are slightly lower than this for low frequencies and are considerably lower at high frequencies, approaching the points midway between two lines. The other two curves in Fig. 56 show the behavior of the acoustic conductance and susceptance ratios, the real and imaginary parts of the admittance (pc/z), at the end x = 0. If the driving piston at x = has a constant driving force to apply to the air in the tube [which would be the case in Eq. (23.27) if Z p <£ Z r ], then the power 3r to y y _J5 I I I Zr 2- \U 1 - "I n _• i 1 ,. ,1 1 1 4 6 8 Z 4 XllM (21 A) Fig. 56. — Acoustic resistance, reactance, conductance, and susceptance ratios for the driving end of a tube with other end open. The length of the tube is ir times its diameter. If the driving piston is a "constant velocity generator," the power radiated is proportional to the resistance; if the piston is a "constant pressure generator," the power radiated is proportional to the conductance. radiated would be proportional to the conductance ratio k — 0/(0 2 + x 2 ). The curve shows that if the piston were driven by a ".pure noise" force, having all frequencies present in equal magnitude, those frequencies corresponding (approximately) to the first three or four harmonics of the resonance fundamental (c/2l) will be strongly reinforced, but the higher harmonics will not be reinforced very much. If the piston impedance Z v is larger than Z r , the veloeity amplitude of the piston will be nearly independent of v (if Z v is nearly constant) and the power radiated will be proportional to 0, the acoustic resistance 258 PLANE WAVES OF SOUND [VI.2S ratio. In this case, as the curves show, the first three or four of the odd harmonics of the antiresonance frequency (c/4Z) will be reinforced, and we have the analogue of the clarinet, as was discussed earlier in this section. In either case, however, the higher harmonics will not be reinforced very much, so that a tube of this shape (I = 2ra), driven by a "pure noise" force, will give out sound having only the first three or four harmonics present in any appreciable intensity. A pipe that has a larger diameter for its length than this will have even fewer high resonance peaks and will give out a tone even poorer in harmonics. One having a smaller diameter will have more peaks, and the tone given out will be rich in harmonics. Cavity Resonance. — Another problem that can be solved by the hyperbolic tangent method is that of a plane wave falling on the open end of a tube fitted with a flange and closed at the other end. Suppose that the plane wave travels along the a>axis, which is the axis of the tube and is normal to the flange, that the wave has fre- quency v and that far from the tube it has pressure amplitude P f and velocity amplitude Uf = P//pc. When it strikes the open end of the tube, it agitates the air in the open end, sending a wave down the tube to the closed end where it is reflected. The reflected wave returns to the open end, and part of it radiates out into the open again. This method of analysis considers the wave reflected from the tube-plus-flange system to be made up of two parts : the wave reflected from the flange, and the wave radiating from the open end. For wave- lengths long compared with the radius of the opening a, the wave radiated from the mouth will spread out in all directions (as we shall see in the next chapter), while the wave reflected from the baffle will return along the #-axis, if the baffle flange is large compared with the wavelength. If the wavelength is small compared with a, both waves will return along the negative z-axis without appreciable side- ward spreading, and there will be interference between the two waves. For very short waves, the phase difference between the two reflected waves is just (4xZ/X). If the open end of the tube is provided with a large baffle flange, the incoming plane wave will be reflected from it; and close to the flange (which will be taken as the plane x = 0) the pressure due to the plane wave has the value 2P f e~ 2 ' iyt , and the corresponding particle velocity is zero. The baffle flange can sustain this pressure without moving, but the air in the end of the tube cannot. To the degree of approximation considered in this chapter, the air in the tube mouth VI.23] PROPAGATION OF SOUND IN TUBES 259 will act as though it were a plane piston without mass or thickness, driven by a pressure of amplitude Po = 2P f , sending waves down the tube and also out into the open. The specific acoustic impedance for each square centimeter of the piston is that due to a tube of length I closed at the other end plus that due to radiation from a piston set in a baffle. From Eqs. (23.10) and (23.13) the impedance ratio is To = [l - 2 1M] - i [M(w) - cot(^)]; w = 4WX The velocity of the air in the mouth of the tube is therefore (d%/dt) = (Po/pcf o^ -2 ™"', and the velocity and pressure inside the tube are u = _ Po ,{ P< L, sin \^-a-x)\ e- M * f o sin(27rZ/X) iPo fo sin(27rZ/A) sin y (Z - x)\ V = v ^/o_7/^ c os y (Z - x) by Eq. (23.8). The pressure just outside the open end is the driving pressure P e~ 2 * ivt given above, plus the pressure due to the outgoing wave caused by the vibration of the air in the mouth. The ratio of the pressure amplitude at the closed end to the pressure amplitude at the open end is obtained from the foregoing equations: ®-{[i-»^r-'(?) + [MMsin(^)-cos(^ -i (23.28) where w = (4ira/X). Tables of the functions involved are given at the back of the book. The phase difference between the pressure at x = I and what it would be at the same spot (x = I) if there had been no tube or flange present turns out to be q = cot" 1 cot(2TZ/X) - M(w) 1_2 Jl ^ w (t) When the wavelength is so long that w is very small, the pressure p at the far end of the tube becomes Pi -* " 7/ . . Resonance cos (2xi/X) occurs whenever the frequency equals one of the natural frequencies of vibration of the closed pipe v n = (nc/4Z), (n = 1, 3, 5 • • • )• What happens is that some of the wave enters the tube and is trapped 260 PLANE WAVES OF SOUND [VI.23 there, because of the low radiating efficiency of the open end for long wavelengths. If the wavelength happens to be the proper size, resonance occurs. When the wavelength is so small that w is large, the pressure at x — I becomes Pi — » P , independent of v. The efficiency of the open end for radiating sound is so large that no wave can stay long enough in the tube to produce resonance. The ratio Pi/Po is plotted in Fig. 57, for a tube whose length is t times its diameter (as in Fig. 56), as H> 2l A Fig. 57. — The ratio of pressure at the closed end to pressure at the open end, for the tube shown in Fig. 56 when I = 2ira. The peaks are examples of cavity resonance. a function of (2Z/X) == (2lv/c). It shows the transition from strong resonance at low frequencies to a nearly uniform response at very high frequencies. We have assumed above that the driving pressure at the open end of the tube P is twice the pressure amplitude in free space P f . This is true as long as the baffle flange is large compared with the wave- length of the sound; but when the wavelength is very large compared with the size of the baffle, P becomes equal to P f . The curve for the ratio of Pi to Pf will therefore be similar to the curve shown in Fig. 57, over most of the range of (2Z/X), except that the vertical scale will be doubled. If the flange is not very large, the curve for (Pi/Pf) will be more like the dotted curve at small values of (2Z/X), reducing to unit value (on the doubled scale) as a lower limit, owing to the reduc- tion of the value of (Po/Pf) from two to unity for long wavelengths. This will be discussed in detail in the next chapter. VI.23] PROPAGATION OF SOUND IN TUBES 261 Many condenser-microphone diaphragms are stretched by means of a ring clamp set in front of the metal membrane, forming a short tube with the diaphragm closing - one end. When the wavelength of the sound is equal to four times the length of the tube, resonance of the sort discussed above occurs and will be quite marked unless the radius of the tube is much larger than its length. Such resonance in condenser microphones is termed cavity resonance. It is interesting to note that the dips in the contours of loudness level, in Fig. 52, at 4,000 cps, are due to cavity resonance of the air in the outer ear. At this frequency the pressure at the ear diaphragm is several times the pressure just outside the ear; so that cavity resonance acts to make the ear as a whole more sensitive at 4,000 cps than at 1,000 cps or at 10,000 cps, although the inner ear is not more sensitive at 4,000 than it is at 1,000 cps. Transient Effects, Flutter Echo. — As an example of the application of the contour-integral methods of calculation to waves in tubes, we shall consider the case of a uniform tube of length I, fitted with a piston at the end x = 0, and fitted at x = I with a termination having a purely resistive mechanical impedance. The piston will be held at rest until time t = 0, when it will be suddenly moved inward a unit distance. This will cause a pulse to travel down the tube. The pulse reflected from the resistive termination will not be so intense as the incident pulse. By the time this reaches x = the piston is at rest, so that the pulse will be reflected back down the tube without loss of intensity. At every reflection from the end at x = I, how- ever, there will be a further reduction in amplitude. One can guess the general form the reechoing pulse wave must take. Presumably it can be expressed in terms of the following function: y-y [o<t<(i/ C )] _ e - 2 ™ 6 ^ _ 2 l^j [(l/c) < t < (2l/c)] Flu(x,t) = \ ^" «V --H IW C ) < « < <«/«>] (23.29) f-4±*) [(4Z/c) < t < (52/c)] etc. 262 PLANE WAVES OF SOUND [VI.23 This represents a pulse reflecting back and forth from one end of the tube to the other, reducing in amplitude by an amount e~ 2va every time it reflects from the end x = I. Such pulse echoes are called flutter echoes. They are encountered in rooms with plane parallel walls of high acoustic reflectivity. Any pulse wave generated between the walls will bounce back and forth many times before dying out. Proper auditorium design is aimed, in part, at the elimi- nation of flutter echoes, as will be explained in Chap. VIII. Of course we have not yet shown that Eq. (23.29) is the correct solution of the problem we outlined at the beginning of this section. Before we use the contour-integration method to show that this formula is the correct solution, it will be advisable to modify the form of the expression for the function Flu(x,t), so we can recognize our answer when it is obtained. A pleasant exercise for the reader would be to juggle the delta functions around to arrive at Flu(x,t) = <rW»<«<-> ^ « 6 " 2m ' c + * ) jgjA.a-LzJ^ (0<x<t) m = .g— Ora/0 (ct+z) The two sums represent periodic waves, the first going in the positive ^-direction, the second in the negative ^-direction. Each sum is modulated by an exponential damping term which in effect reduces the amplitude of each succeeding pulse by a factor e~ 2 ™. The sums, being periodic, can be expressed in terms of a Fourier series. The first sum, for instance, can be expressed in terms of the series The coefficient A n can be obtained, as on page 87, by multiplying both series by cos [(WO (^ - x)] and integrating over [t - (x/c)] from -(l/c) to +(l/c). This results in the equations 1 = (^)a ; l = f-Vn (n = 1,2,3 • • • ) = (?)-; -ev Therefore the flutter-echo function Flu(x,t) can be expressed in terms of the series VI.23] PROPAGATION OF SOUND IN TUBES 263 Ftu M = ^<« {£ + \ J cos [(=-<) (* - f)]} _ ^,-(ira/I)(c<+x) )2Z ' I _ provided, of course, that we waive questions of convergence (as we mentioned on page 98). To complete our discussion we must show that this series is, in fact, obtained by contour integration of the transient function. From Eqs. (23.8) and (23.9) we see that the air velocity in a simple harmonic wave in a tube with terminal acoustic resistance R at x = lis u = Ae'™ 1 cosh \ira — -~ (J — x) f iia I ira = Ae~ iat sinh \ira (I — x) (tic \ ira — -~ J = (pc/R) = k. If the piston velocity is e-™', the velocity in the tube is _ _ faf sinhfra — (ia>/c)(l — x)] _ _ e'™ 1 sinhfra — (ua/c)(l - x) ] ~ sinh [ia — (icol/c)] i sin [((ol/c) + *Va:] According to the discussion at the end of Sec. 6, if the piston velocity is 8(t), the air velocity in the tube is v ' Am J- a, sin[(wt/c) + lira] The poles of this integrand are at co = (c/l)(irn — iira), where n is any integer, positive or negative (or zero). Close to the nth pole the sine function is ( — l) n (l/c)[w — (irc/l){n — ia)] and the integral around this pole, for the first exponential, becomes [since ( — 1)" = e™ n ] _ _L_ ( - | (D e -i«>(t-x/c)+Ta-(iul/c)+iirn ' dC0 - ia)\ 4iri \lj J |_ w ~ (irc/l)(n — _£_ (,-(Ta/l)(ct-x)-(iTn/l)(.ct—x) There is another pole at the corresponding negative value of n, and the two exponentials will combine to form the term (c/Z)e~ (Ta/Z)(c ' -:r) cos\{irn/l)(ct — x)]. Similar calculations for the second exponential 264 PLANE WAVES OF SOUND [VI.28 give similar terms; finally, by putting together all the terms, including those for n = 0, we are able to show that u s (t) = Flu(x,t) (23.31) Returning to Eq. (23.29), we can use the relation p(du/dt) = — (dp/dx) to obtain the corresponding pressure wave p&(t) = pcFlp(x,t), where 5 (t - fj [0< * < (l/c)] e -2*a 8 L _ ^-~j [(I/C) < t <(21/C)] Flp(x,t) = < / 21 4- x\ (23.32) e -4™ 5 h _ ^^) [(3Z/c) < i < (4Z/c)] etc. The pressure wave does not change sign on reflection, but it reduces in amplitude on reflection from the end x = I by the same factor e -2wa = (r _ P c)/(R + pc) as does the velocity wave (as, of course, it must). As in our earlier discussions of transient effects, the wave generated by moving the piston in some arbitrary way, {0 (t<0) g(t) (0<t<A) (t > A) where g(t) is any arbitrary function of t, can be expressed in terms of the pulse waves: u{t,x) = f g(T)Flu(x,t — t) dr p(t,x) = pc J[ A g(r)Flp(x,t - t) dr If g(t) is a sensible sort of function of t, the resulting series for u and p will converge satisfactorily, so that the question of the convergence of the Fourier series for Flu is only academic, as was pointed out in Sec. 10. We note that if the duration A of the motion of the piston is less than (21 /c), the reaction of the air during the motion is as though the tube were infinite in length (in other words it exhibits its characteristic resistance pc). We note also that, if the specific acoustic resistance R of the termination were to depend on co, the reflected wave would not in general be a simple pulse, but would be spread out more than the VI.23] PROPAGATION OF SOUND IN TUBES 265 incident pulse. If the termination impedance is not pure resistive but has a reactive part that also depends on w, the reflected wave would also be "blurred out." 24. PROPAGATION OF SOUND IN HORNS As we have noted before, and as we shall treat in detail in the next chapter, a tube whose open end has a diameter smaller than the wave- length of the sound sent out is a very inefficient radiator of sound. This inefficiency is a desideratum in organ pipes and other wind-instru- ments, for in these cases resonance is important, and most of the wave must be reflected back into the tube from the open end to have sharp resonance. The small radiating efficiency of a small opening (or a small dia- phragm) is a very great detriment in a loud-speaker, however, for a loud-speaker should have no marked resonance frequencies. Two general methods are in use for building loud-speakers so as to radiate sound efficiently. One method is to make the vibrating diaphragm large enough to radiate well (dynamic speakers are of this type). This type of design has the advantage of compactness, but it requires comparatively heavy moving parts with their consequent mechanical inefficiencies. The other method is to use a small diaphragm and to magnify its effective size by using a flaring tube, a horn. One purpose of the loud-speaker horn is to spread the concentrated waves coming from the diaphragm out over a large enough area so that they can continue out from the mouth of the horn with very little reflection back to the diaphragm. In this way a light diaphragm can be used, and acoustic efficiency can be maintained at the same time. Another occasionally important purpose of the horn is to concentrate the sound into a directed beam, so that most of the radiated energy is sent out in one direction. The horn must not flare too rapidly, for the sound waves will then not "cling" to the inner surface of the horn and spread out but will act as though they were already out in free space, radiating from a small diaphragm and exhibiting a correspondingly small efficiency. Therefore in order that the mouth of the horn may be large in diam- eter, the horn must be long. One-parameter Waves. — The analysis of wave motion in a horn is a very complicated matter, so complicated that it has been done in a rigorous manner only for conical and hyperbolic horns. If the horn does not flare too much, however, so that we can consider the wave as spreading out uniformly over a cross section of the horn as it travels 266 PLANE WAVES OF SOUND [VI.24 outward, then we can use an approximate method of calculation and obtain fairly satisfactory results. For in this case the displacement of the gas molecules all over a surface perpendicular to the axis of the horn will be the same, and the displacement, pressure, etc., will be functions only of t and of the distance x along the horn. The statements made in the last paragraph require further dis- cussion to bring out all their implications; although a detailed discus- sion would lead us into the far reaches of differential geometry and would require a more complete knowledge of the properties of the wave equation in three dimensions than we shall have room to cover in the next chapter.' Nevertheless, a few paragraphs of general discussion are needed here, to indicate the approximations and limita- tions of the assumption that a wave in a horn is a one-dimensional wave. The shape of the inner surface of the horn of course affects the shape of the wave, and since the horn cross section is not independent of x, we should expect the wave to depend on the coordinates per- pendicular to x as well as on x, and thus not be a plane wave. What we must discuss are the conditions under which the wave will approxi- mate the behavior of a plane wave, and what we should derive (but will not because we have not in this book the requisite mathematical tools) are the criteria showing when these conditions occur. A plane wave, traveling in the positive ^-direction, has a phase which is the same everywhere over the surface of a plane perpendicular to x. In the usual exponential notation, p = Ae**-™ 1 (24.1) The quantity A is the amplitude of the wave and can be taken to be real. The quantity 4>, also a real function, is the phase of the wave. In the case of the plane wave <f> = {a/c)(x - x ), so that over a plane perpendicular to the x-axis (x = a, for instance) <f> is constant. In a plane wave the surfaces of constant phase are planes, perpendicular to the x-axis, moving along x with a speed c. Other waves can be set up which have the same general form as Eq. (24.1), though their surfaces of constant phase are not planes, and their amplitude is not constant. In the next chapter we shall see that a spherical wave, radiating out from a point source, has the form of Eq. (24.1). The amplitude A for the spherical wave is inversely proportional to r, the distance from the source, and the phase <t> = (w/c)(r - r ) is proportional to r. In this case, therefore, the surfaces of constant phase are spheres, expanding outward from VI.24] PROPAGATION OF SOUND IN HORNS 267 the source. Just as with the plane wave, however, both amplitude and phase are functions of only one coordinate. In terms of these concepts we can restate our requirements for a "good" horn. We first set up a system of mutually perpendicular .-H=I0 Fig. 58. — Coordinate surfaces corresponding to three types of horns. To the degree of approximation of Eq. (24.2), surfaces of constant phase coincide with the ju-surfaces and the particle velocity is parallel to the lines = constant. See Eq. (24.3). coordinates (/*,#) suitable for the horn, so that the inner surface of the horn corresponds to # = # s (a constant), the axis of the horn (the x-axis) corresponds to # = 0, and so that n = x when # = {i.e., the scale factor for fi is the same as for x along the axis of the horn). Samples of such coordinate systems are given in Fig. 58, for several types of horn shapes. 268 PLANE WAVES OF SOUND [VI.24 We then set up the wave equation in these coordinates (this is, of course, difficult and is part of the detailed discussion that must be omitted here) and see whether we can obtain a solution of the form given in Eq. (24.1), with amplitude and phase depending only on the coordinate /*. If this can be done accurately, the wave is not a plane wave, but it is a one-parameter wave, which can then be handled as we have been handling plane-wave expressions. In this case the problem is straightforward. It turns out that very few coordinate systems give rise to a wave equation simple enough to allow a solution that is a function of only one coordinate (only the first of the three systems shown in Fig. 58 has this property). When the coordinates do not have this property, the particle velocity will not be parallel to the /z coordinate lines, and the wave will tend to reflect from the horn surface as it travels along, rather than moving parallel to it. As we have seen in the previous section, and shall see again in the next chapter, any reflection of the wave during its' progress along a tube reduces the amount of energy traveling out of the tube and tends to trap some of the energy inside, causing resonance for some frequencies and poor transmission for others. This condition is satisfactory for musical instruments, where we desire strong resonances, but it is not desirable for loud-speaker horns, where we desire a uniform transmission, independent of fre- quency. Consequently, we can say that one criterion for a good horn is that it should be possible to set up one-parameter waves inside it; that a possible solution of the Wave equation in coordinates suitable for the horn should be a function of the coordinate n alone. For some coordinate systems the correct solution is nearly a one- parameter function, and the properties of the corresponding horn, deduced by using an approximate one-parameter solution, will be close to the true behavior. It is always possible (though often diffi- cult) to check the degree of approximation involved for a given horn by setting up the exact wave equation in the suitable coordinates, substituting p = AGu)^ 00- * 61 ' in the equation and determining by what amount this function does not satisfy the equation. An Approximate Wave Equation. — If we find that a one-parameter solution is approximately correct for some horn shape, we should be able to set up a one-dimensional wave equation that is approximately correct, from which we can obtain the one-parameter approximate solution. This approximate equation is not so difficult to obtain or to work with as is the exact wave equation, and we are now in a position to derive it. VI.24] PROPAGATION OF SOUND IN HORNS 269 We consider a thin shell of air between two neighboring surfaces of constant phase (between <t> and # + d<(>), rather than a plane sheet between x and x + dx, as we did for the plane waves. If a one- parameter wave is a good approximation to the correct solution, this is nearly equal to the air between the coordinate surfaces p and /x. + dft. The total area of that part of the coordinate surface p which is inside the horn is called S, which is a function of p only (if the horn is a "good" one, S turns out to be proportional to the area of the cross section of the horn which is tangential to the coordinate surface p) . Therefore the volume of the gas in the shell at equilibrium is S dp (or approximately S dx, since p = x along the horn axis). During the passage of the wave, the surface of air originally at p is displaced to p + £, and the new volume of the shell is giving a fractional change in volume of (1//S) ^- (S£), which corre- sponds to Eq. (22.3) and to the equation p = ~("o")r (^)> relating pressure and particle displacement. The net force on the shell of air is everywhere perpendicular to the ju-surf ace (if the approxi- mation is good), and therefore Eq. (22.6) still holds, since differenti- ation with respect to p is the same as differentiation with respect to x along the horn axis. We next differentiate twice with respect to time the equation for p given above, and substitute for (d 2 £/d£ 2 ) from Eq. (22.6), obtaining, finally l±(^dp\_L^p. §u _ldp ( Sdx\ dxj c 2 dt 2 ' dt P dx K } where, as before, c 2 = (Po7c/p) and where u is the magnitude of the particle' velocity (the direction is perpendicular to the ju-surfaces, parallel to the & coordinate). If the waves are simple harmonic, the relation between u and p is 1 dp tup ox Possible Horn Shapes. — If approximate Eq. (24.2) is valid for a given horn shape, then we should expect, to the same approximation, that a wave traveling out of the horn would be represented by the form given in Eq. (24.1). Moreover, we should expect that the 270 PLANE WAVES OF SOUND [VI.24 amplitude A will (to the same accuracy) be inversely proportional to the square root of S. For the intensity of sound in the wave should be proportional to A 2 , and the total energy flow out of the horn would therefore be A 2 S, which must be independent of x if energy is to be conserved. We set S(x) = ir[y{x)] 2 , where y is the effective radius of the cross section of the horn at a distance x from its small end. Inserting the expression p = {B/y)e i '^ iut into Eq. (24.2), we obtain (ejy iMA_M 2 \dxj ^ y\dx 2 ) \c) dx 2 The real and imaginary parts of this expression must be zero sepa- rately, for both y and <j> are real. The imaginary part indicates that (d<f>/dx) must be a constant, which we can call (tco/c) for reasons that will shortly be apparent. Finally, in order that the approximations we have made in the previous paragraph be all valid and self-consistent, the real part of the equation must also be zero, which means that the effective radius of cross section of the horn must (at least approximately) satisfy the equation S-(f)'o-^-« A solution of this equation, properly adorned with constants, is y = yo S = So T 2 = 1 ■ co 8 h(|) + rs inh(|) co S h(|) + rsi nh(|) \uhj \2rhJ (24.3) and the corresponding approximate solution for the plane wave travel- ing out of the horn is V Q> 1 pHa/c){rx—ct) (24.4) The shape of the horn represented by Eq. (24.3) is determined by the relative values of the constants T and h. The constants y and S = iryl fix the size of the throat of the horn at x = 0. The "scale factor" h determines the rapidity of the "flare"; the smaller h is, the greater is the curvature of the plot of y against x. The constant T VI.24] PROPAGATION OF SOUND IN HORNS 271 is the "shape factor," determining the general properties of the horn near the throat. When T = (h/x ) and h is allowed to go to infinity, the horn is a conical horn, with angle tan" 1 (y /x ) between the axis and one of the elements of the cone [the "angle of opening of the cone" is 2tan- 1 (Wzo)]. When T = 1, y = y e x/h , and the horn is an exponential horn. When T = the shape of the generator of the surface is a catenary, so the horn will be called a catenoidal horn. 1 This horn can join smoothly onto a uniform tube at x = 0, whereas the conical and exponential horns have a discontinuity in slope of y{x) at x = 0, making a worse impedance match with the uniform tube. For large values of x, the catenoidal horn is indistinguishable from the exponential horn. The shapes of all three types are shown in Fig. 58. When the exact wave equation for the coordinates appro- priate to the horns of Eq. (24.3) is worked out, it can be shown that Eq. (24.4) is a reasonably close approximation to an exact solution as long as the radius y of the small end of the horn (the throat) is much smaller than the scale factor h. We shall next discuss in detail the acoustical properties of each of the three main types of horns expressible in terms of Eq. (24.3), for cases where the big end of the horn (the mouth) is large enough so that no part of the wave coming out of the horn is reflected back toward the throat. Finally we shall take up the case where the open end is small compared with the wavelength (which is the case of some wind instruments), where reflection from the mouth, and consequent resonance, occurs. The Conical Horn. — In a conical horn of angle # s (angle of opening 2#„ area of throat irxl sin 2 & a ) the areas of the phase surfaces are S = So \l + yyl ; S = iryl y = 2*o sm(^J where x is the distance back from the throat to where the apex of the cone would be, if the cone were extended back to its apex. Equation (24.2) turns out to be 1 d \, , , 2 dp] 1 d 2 p (x + x ) 2 dxl K ' u/ dx] c 2 dt 2 which is the exact wave equation for a one-dimensional spherical 1 Although it would also be appropriate to call it a "Salmon" horn, after the person who first discussed the acoustical properties of the family of horns of Eq. (24.3). 272 PLANE WAVES OF SOUND [VI.24 wave. The outgoing-wave solution of this is p fp = gt(w/c) (x—ct) X + X The particle velocity at x is pz L x + ZoJ ^ In this case the phase velocity of the wave inside the horn is the same as it is in free space, c. Since the wave front spreads out as it travels down the horn, the impedance looking out of the horn from the small end is not the same as for a uniform tube, even if there is no reflected wave: (p\ _ pc _ pc uj ~~ 1 + i(c/oox ) ~ 1 + i(\/2rx ) (24 - 5) The analogous impedance of the throat of the conical horn, (z /S ), is equivalent to a resistance (pc/S ) in parallel with an inductance (pXo/So). Therefore any acoustical filter, or other circuit, that is terminated by a conical horn has an equivalent electric circuit ter- minated by this shunt combination. The power lost in the resistive arm represents the power radiated out of the open end. The shunt inductance has very little effect on the behavior of the horn at high frequencies (X < < 2inro), but at lower frequencies it shunts out the resistor (pc/S ). The specific acoustic impedance at the throat of the horn is there- fore z = R — iX = \z\e-^ R = pC Y = (pcX/27T3 ) 1 + (\/2ttXo) 2 ' 1 +. (\/2irXo) 2 PS: _ pcxpw ^ (24.6) \A + (X/27TX ) 2 VC 2 + (XOCO) 2 \z = <p = tan — i \2irx J \XouJ If the particle velocity at the throat of the horn, causing the out- going spherical wave, is Woe -2 ™"', then the power radiated out of the horn is IT - 1 ,,2PO _ i pCMJ}ff _ ! „ , (2t;eop) 9 VI.24] PROPAGATION OF SOUND IN HORNS 273 In this case the power radiated is small at low frequencies and rises steadily as the frequency is increased, approaching asymptotically the limiting value ?Sopcu$. The limiting value is the power that the same piston would send into a uniform tube of infinite length. Transmission Coefficient. — We are now in a position to define what we mean by the term "radiating efficiency of a horn" which has been so freely used above. A measure of this efficiency is the ratio of the power radiated out of a given horn to the power radiated by the same diaphragm, moving at the same velocity, into a cylindrical tube of infinite length, having the same cross-sectional area as the small end of the horn. This ratio, for any type of horn, is defined as 2n Sopcul © (24.7) and will be called the transmission coefficient of the horn for sound. This ratio is not, strictly speak- ing, an efficiency, for it is the ratio of the actual power radiated to the power radiated for a stand- ard case, not an ideal case. Con- sequently, t sometimes becomes greater than unity. In general, however, r varies between zero and unity. When it is small the diaphragm will have to vibrate with large amplitude to radiate much power. When the veloc- ity amplitude of the piston is independent of frequency, the power radiated is proportional to the transmission coefficient. In any case the power radiated is n = iuiSopcr The transmission coefficient for a long conical horn with wide mouth is 1 (2tvx ) 2 0.2 0.4 0.6 Fig. 59. — Transmission coefficient for a long conical horn with open end large enough to eliminate resonance. 1 + (\/2tx q ) 2 c 2 + (2irvx y (24.8) It is plotted in Fig. 59 as a function of (x v/c) — (x /\) = lb/\(a — b), where I is the length of the horn and a and 6 are the radii of its large and small ends, respectively. This function has* the value -| when (xq/\) = \/2ir or when the frequency equals c(a — b)/2rlb. For fre- 274 PLANE WAVES OF SOUND [VI.24 quencies above this the transmission coefficient approaches unity, and the horn is relatively efficient as a radiator. For frequencies below this the horn is quite inefficient. The smaller the flare of the horn [i.e., the smaller the value of (a — b)/l\, the lower the frequency at which the horn will be efficient. A Horn Loud-speaker. — We can now put some of these formulas to work by showing how they can be used in designing horn loud- speakers and in analyzing their behavior. The sketch at the top of Fig. 60 shows the longitudinal cross section of a simplified version of a conical horn loud-speaker. The piston P is considered to move as a single mass of mechanical impedance (without the air load) Z p = R P — i[wm p — (K p /a))] dyne-sec per cm. Its radiating surface has area S p . The area of cross section of the throat of the horn, S , is less than this, in order to improve the impedance match between the piston and the air. The portion of tube between the piston and the horn consists of a "tank" of volume V and a constriction of cross- sectional area So, of length I and of effective length l e ~ I -f- 0.5 \/$o [this is a modification of Eq. (23.1) suitable for the present case]. Since the tube cross section changes, we shall find it easiest to make our first analysis in terms of analogous impedances, Z a = (z/S), for pressure and volume flow do not change discontinuously at changes in tube cross section. The analogous impedance of the horn, looking out of its throat, is equivalent to an inductance (px /So) and a resistance (pc/So) in parallel. We use Eqs. (23.2) and (23.3) to complete the rest of the equivalent circuit, the impedance of which is equal to the ratio Z a between pressure and volume flow of air (uS p ) at the piston face. This circuit is shown in the middle part of Fig. 60. The power radiated out of the horn is represented in this circuit by the power dissipated in the resistance (pc/So). This power is diminished at the lower frequency end due to the shunting inductance (pxo/So), which should therefore be made as large as possible (by making xo large) to enable the horn to radiate well at low frequencies. The sketch shows that for x to be large the conical angle # 8 must be small; therefore the horn must be long in order that the mouth of the horn be large enough to radiate away the power. This property of conical horns has already been shown in Fig. 59. The power output for the horn of Fig. 60 is also limited on the high-frequency side by the shunting condenser (V/pc 2 ) and by the series inductance (pl e /S ). Therefore the volume V in front of the piston must be as small as possible and the constriction as short as possible in order that the horn be efficient at high frequencies. VI.24] PROPAGATION OF SOUND IN HORNS 275 To include the effect of the impedance of the piston itself in the analysis, we shall find it easier to turn from analogous impedance Z a = (pressure) /(air flow) to mechanical impedance Z m = (force) /(velocity) = SlZ a . We set up the equivalent circuit at the bottom of Fig. 60 to aid in calculating this quantity. The equivalent voltage E equals Longitudinal Cross Section of Conical Horn — Loud Speaker S =47TX 2 sin z (6 s /2) Sp=Areoi Piston J P °1 (ple/So) ,_V_ ■pc« Circuit for Analogous Impedance Z a atthe Piston P (pleSp/S fl m p R p Sjpc* " f/Kp — 1|— wvw- pXoSpi pcSp; Equivalent Circuit giving Mechanical Impedance of Piston plus Radiation Load "-Impedance due to Air back of Piston Fig. 60. — Idealized sketch of conical horn loud-speaker, with corresponding circuits for analogous impedance of air load Z a , and for total mechanical impedance Z p + S p 2 Z a . the force on the piston, and the current through m p equals the piston velocity. This circuit is not so convenient as the one above, for the currents in the various arms do not everywhere correspond to air flow nor do the voltages always correspond to pressures. The pressure at the horn throat, for instance, turns out to be equal to the equivalent volt- age across the resistance (pcS*/So) divided by the area S (for in this 276 PLANE WAVES OF SOUND [VI.24 circuit, voltage corresponds to force = pS). The air velocity in the throat equals the equivalent current in the inductance (pl e Sl/S ) multiplied by (S p /S ). The power radiated is, of course, still equal to the power dissipated in the resistance (pcSl/So)- We now can see why it is wcrth while to have the piston area S p larger than the throat area So, even though this introduces the unde- sirable capacitance (F/£|pc 2 ). The magnitude of the mechanical impedance of the piston is usually much greater than pcS , so that there r^JM^ v pe>S 2 p = px S 2 p Rn"l :pcS p Equivalent circuit giving Mechanical Impedance of Mass Controlled Piston plus Radiation Load Lc SMAAM Circuit giving Electrical Impedance z E =z c +(r/z m ) of Coil of Electro- dynamically driven Piston Motional Impedance Fig. 61. — Equivalent circuit at top gives mechanical impedance of piston of Fig. 60 when piston is mass-controlled. If piston is driven by an electrodynamic coil, as dis- cussed on page 34, the electrical impedance of the coil whe n moving is equal to that of lower cir cuit. Velocity of piston is equal to \Ao 7 /VE M . Velocity of air in throat equals ■\/10 7 /T(S p /So)Er. We have here neglected the air load Z b on the back of the piston. is a poor impedance match between the piston and its air load unless the effective air impedance is increased by the factor (S P /S Q ) 2 . If the horn is well designed the stiffness and mechanical resistance of the piston are small compared with the mass reactance over the useful range of frequencies; in other words, the piston is mass con- trolled. In this case the equivalent circuit, representing the mechanical impedance, is that shown at the top of Fig. 61. The inductance, representing the piston mass, is in series with a parallel combination representing the air load. We have omitted the inductance due to VI.24] PROPAGATION OF SOUND IN HORNS 277 the constriction, because this can be made quite small compared with m p . The large mass of the piston plus driving mechanism is the chief difficulty in providing for uniform radiating properties of the loud- speaker. By using a horn, we are able to make the piston smaller and therefore considerably lighter than the diaphragm-driver unit of the "dynamic" speaker. But it is extremely difficult, to reduce this mass below 10 or 20 g. Consequently, the reactance due to m p at 1,000 cps will still be approximately 100,000 dyne-sec per cm. Since pc is only 42 dyne-sec per cm 3 , we must make the factor (S%/S ) of the order of 1,000 to have a reasonable impedance match. If, for instance, So were 3 cm 2 , S p should be about 50 cm 2 (piston diameter ~ 3 in.). But this large a piston is difficult to make stiff if it weighs as little as 10 g. All these practical limitations mean that it is quite difficult to design a conical loud-speaker that is efficient over a wide range of frequencies. To illustrate these difficulties we shall calculate the horn behavior for values of the constants which are difficult to attain in practice, but which show clearly the general behavior m p = 8 g; So = 3 cm 2 ; S p = 27 cm 2 Xo = 11 cm; # s = 5 deg; V = 8 cm 3 K v = 12.6 X 10 7 dynes/cm; m r = 3.2 g; R r = 10,000 dyne-sec/cm The mechanical impedance of piston plus horn is plotted in the top curve of Fig. 62. The mechanical resistance is maximum at 1,000 cps, corresponding to the parallel resonance between K v and m r . The horn would respond better to high frequencies if this resonance could be moved to higher frequencies; but this requires either that #« or V be made smaller, both of which are difficult. The reactance is large, due to the large value of m p , which is almost impossible to reduce much further. Now suppose that the piston is driven by an electromagnetic coupling system, such as that described on page 34 having coupling constant T. The motional impedance of the coil, due to the motion of piston and air in the horn, is equal to T times the mechanical admittance of the piston-horn system. This motional impedance is in ohms and is to be added to the clamped impedance of the coil to give the total electrical impedance of the driving coil. The reciprocal of the series- parallel circuit at the top in Fig. 61 is the parallel-series circuit inside the dashed line in the lower sketch. Consequently, this lower circuit has the same electrical behavior 278 PLANE WAVES OF SOUND [VI.24 as the driving coil. We can show that the velocity of the piston in centimeters per second is equal to \/10 7 /r times the voltage across the conde nser ( m p /T) and that the air velocity at the throat of the horn is Vl0 7 /r (S P /S ) times the voltage across the condenser (m r /T) o ^ T5 — |^ I_-Q O a. - 10 Volts Impressed Across Driving Coil 10 200 400 600 1000 2000 4000 6000 32 V. -10 15 3.2 J 1.0 | cc 0.32 I o 0.! Q - 10,000 Fig. 62. — Mechanical impedance of piston-plus-horn of Fig. 60. Electrical imped- ance of driving coil if piston is driven by electromagnetic means (see page 34) . Power radiated when coil is driven by 10- volt generator. and the resistance (T/R r ). The power radiated out of the horn is, of course, equal to the power dissipated by the resistance (T/R r ). If we assume that r = 10 5 ohms-cm per dyne-sec R c = 0.5 ohm, and L c = 0.0008 henry (an improbably small value, chosen to show the coil resonance separated from the piston resonance), then we can use the equivalent circuit to calculate the total electrical impedance VI.24] PROPAGATION OF SOUND IN HORNS 279 of the driving coil. This is shown in the center graph of Fig. 62. The resistance is largest between 200 and 600 cycles; above 3,000 cycles the mass load of the piston (the shunt capacitance m p /T) prevents much radiation, and the resistance is that of the clamped coil. We have adjusted the coil inductance L c so that it resonates with the con- densers at 2,000 cps, so as to strengthen the response above 1,000 cps. Assuming that we drive the coil of the speaker with a constant voltage generator of 10 volts emf but variable frequency, we should obtain a plot of output power against frequency of the form given at the bottom of Fig. 62. This output is fairly uniform over the fre- quency range from 200 to 3,000 cycles, but it drops very rapidly above this range, due to the coil inductance L e and to the shunt capacitance (Wp/r). In fact, the equivalent circuit corresponds to a rather crudely devised low-pass filter, so it is not surprising to have the high frequencies cut off. If the coil inductance L e were larger, the cutoff would be at a still lower frequency. It should be apparent by now that conical horn loud-speakers can be made reasonably uniform in response over a range of frequency cor- responding to a factor of 10 (if they are carefully designed), but not over a very much larger range. If the horn is to be reasonably effec- tive at 100 cps, it will probably not be very effective above 1,000 or 2,000 cps; if it is to be effective clear out to 10,000 cps (such high- frequency horns are called "tweeters"), it will not radiate well below about 1,000 cycles. This general limitation on frequency range is true of horn loud-speakers in general (as, in fact, it seems to be for all loud-speakers). New developments in light stiff material (to reduce m p ), or in "acoustic transformers'' to improve the impedance match between piston and air load, are needed to produce a marked improvement in behavior. The Exponential Horn. — When the cross section of the horn has the form S = S e 2x/h , the horn is called an exponential horn. The length h is the distance in which the diameter of the horn cross-section increases by a factor e = 2.718. We have already seen in Eq. (24.4) that the pressure wave in such a horn can be expressed approximately by the formula ^ _ Pq~ (x/k)+i(oi/c)(TX— ct) I r = Vl - (XAo) 2 = Vl ~ {n/vY > (24.9) Xo = 2rh; v = (c/2ttK) ) This wave travels out of the horn with ever-diminishing amplitude, with a velocity (c/t) larger than the speed of sound in the open. 280 PLANE WAVES OF SOUND [VI.24 This velocity is different for different frequencies, which makes the exponential horn a dispersing medium for sound waves (see page 154). When the frequency is equal to v (called the cutoff frequency) , the velocity has become infinite; in other words, the air moves in phase along the whole extent of the horn [this is, of course, only approxi- mately true, since Eq. (24.9) is an approximate solution]. Below this cutoff frequency there is no true wave motion in the horn, and the equation for the pressure is V = P expj ^1 - \^-J - lj | - iaA ( w < Wo ) (24.10) where co = (c/h). Thus it turns out that the exponential horn is very ineffective below a certain minimum frequency (the cutoff fre- quency) which is smaller the larger h, the scale factor, turns out to be. Again a very long horn is needed to produce low frequencies effectively. The air velocity in the horn for frequencies above the cutoff is obtained by differentiating Eq. (24.9) : Consequently, the specific acoustic impedance at the throat of the exponential horn, for outgoing waves, is z Q = R — iX = \z\er**; (« > « ) (24.11) - / 0)ft\ I c\ I \z\ = pc; <p = sin" Below the cutoff frequency, no exponential horn can be considered long compared with the wave length (inside the horn), so that the impedance match at the mouth of the horn must be taken into account. This will be considered later. Above the cutoff frequency the impedance has a constant magnitude pc, its phase angle being 90 deg at the cutoff frequency and approaching zero as the frequency is increased. If the velocity amplitude of the driving piston at the throat of the horn is Uo, the power radiated is n = iulSopcyjl - (^\ [v >v = (c/2Th)] Very little power is radiated for frequencies less than v Q . VI.24] PROPAGATION OF SOUND IN HORNS 281 The transmission coefficient for the long exponential horn, the ratio of power actually radiated to the limiting value (ulSopc/2), is '• - V 1 - (t)' = V 1 - (as)' <'>*>■ < 24 - 12 ) The curve for r e is shown in Fig. 63. Comparison with Fig. 59 for the conical horn shows that for some purposes the exponential horn is better than the conical horn. Although it radiates practically nothing below the cutoff frequency, the transmission coefficient rises 0.6 0.8 0.4 (h/A)=(vh/c) Fig. 63. — Transmission coefficient for a long exponential horn with open end wide enough to eliminate resonance. much more rapidly to unity above the cutoff. If h is made large enough (i.e., if the horn does not flare too rapidly), r e will be constant over practically the entire range of frequencies. Neither r c nor r e ever become larger than unity, however. For this behavior we must turn to the catenoidal horn. The Catenoidal Horn. — When the cross section of the horn has the form S = So cosh 2 (x/h), it is a catenoidal horn. The scale factor h is again a measure of how slowly the horn flares out. At large dis- tances from the throat the catenoidal horn is indistinguishable from an exponential horn, but at the throat the rate of change of ~S with x is zero, in distinction to the exponential case.' The catenoidal horn can therefore fit smoothly onto a uniform tube of cross section So, whereas the conical and exponential types have a discontinuity in rate of change of S, which tends to cause reflection back into the uni- form tube. 282 PLANE WAVES OF SOUND [VI.24 Referring to Eqs. (24.3) and (24.4) we see that an outgoing wave is approximately represented by the one-dimensional form v = — p e iwo(A-ct) (24.13) r cosh (x/h) The wave velocity in this horn, (c/t), is larger than the velocity in open space, as in the exponential horn, and there is again a cutoff frequency v = (c/2irh), below which the horn is a poor radiator of sound. The corresponding particle velocity is 1 u = — pc + £*■*©' p [a > coo = (c/h)] so that above the cutoff frequency the specific acoustic impedance at the throat (x = 0) of the catenoidal horn is a pure resistance ' * " " VI -W-)' = "" VI - (X/frrfc)' (24 - W) In this case the specific acoustic resistance is greater than pc above the cutoff frequency vo, rising to very large values as v — v diminishes toward zero. For a constant velocity amplitude u at the throat, the power radiated is E = WoSopc l f ^ [v > vo = (c/2rfc)] The transmission coefficient of the catenoidal horn is therefore T - = ( l ) = /i \ /v {v > Vo) (24 ' 15) A comparison between the transmission coefficients of conical, expo- nential, and catenoidal horns of the same over-all dimensions is shown in Fig. 64. The catenoidal horn has a cutoff frequency a little higher than the exponential horn, but the transmission coefficient just above the cutoff is considerably larger. Evidently, starting the horn with a uniform cross section at the throat (dS/dx = at x = 0) can improve the output of the horn just above the cutoff frequency. For instance, for a catenoidal horn with h = 50 cm = 20 in. (cutoff frequency = 110 cps) the output at 150-cps is nearly twice that of an exponential horn of the same value of h and is very much larger than that of a conical horn of the same size. As we have seen earlier, it is difficult to VI.24] PROPAGATION OF SOUND IN HORNS 283 obtain sizable power outputs at the low frequencies, so this property of the catenoidal horn is often useful. It is also apparent from Fig. 64 that the conical horn is the least efficient of the series of shapes we have analyzed. Even though it has no cutoff frequency its effectiveness at very low frequencies is quite small and does not usually make up for the very slow rise of transmission coefficient with frequency. Frequency Fig. 64. — Comparison between transmission coefficients of long, conical, exponen- tial, and catenoidal horns of the same over-all dimensions. Length I is chosen to be 2.4 times the h for the exponential horn; I turns out to be 11 times the xo for the conical horn; and 3.1 times the h for the catenoidal horn. Reflection from the Open End, Resonance. — In many cases, horns are used with open ends that are not large enough to radiate all the sound coming to them from the throat. Many wind instruments, for instance, are approximately exponential or catenoidal horns, with open mouths designed to be small enough to , ensure resonance inside the horn. Even horns designed as loud-speakers, to avoid resonance at most frequencies, cannot avoid some resonance at the lowest fre- quencies. In this section we shall discuss the effect of such an imped- ance mismatch at the mouth of the horn. We might as well carry out the calculations for the general form given in Eq. (24.3), for any value of T. The formula for S can be given in another form: S = So cosh 2 (e) cosh 2 G~> T = tanh (e) (24.16) 284 PLANE WAVES OF SOUND [VI.24 When e = 0, the horn is catenoidal; when e = », the horn is expo- nential; and when e = (x /h) + i(ir/2), with h approaching infinity, then the horn is conical. Intermediate values of e correspond to horns of intermediate shape. This can be substituted in Eq. (24.2) to give the approximate wave equation and eventually to obtain a solution for the wave. In this case we cannot neglect the wave reflected from the horn mouth, so our solution has the form V Pe~ cosh sinh^ + UaTX .) V u = — pc z = pc .,«h(, + =) + £—(! + .) ■(■ t coth( \j/ + l —\ + ^ tanhl t •i (HI (24.17) where t 2 = 1 - (c/co/i) 2 . This is similar to Eqs. (23.8) for the uniform tube. The differences in the expressions for the specific acoustic impedance z lie in the factor t, corresponding to the fact that waves in these horns travel with a speed different from waves in the open, and in the term including tanh[(rr//i) + e], corresponding to the fact that the surfaces of constant phase are curved surfaces, not planes as in the uniform tube. Our first task is to fit the wave at the mouth of the horn, at x = I. Since the surfaces of constant phase are not planes, this fit is not so simple as that for the uniform tube, which resulted in Eq. (23.13). To a reasonable approximation, however, the joining of the wave from the horn with the wave in the open effectively cancels the term in tanh [(x/h) + e] and results in the equation for \J/ £ tanh(* + ^) = pc [l - 2 ^] - ipcMiw) (24.18) which is analogous to Eqs. (23.13). The quantity w is (2coa/c), where a is the radius of the mouth of the horn. This joining equation holds as well for frequencies below the cutoff frequency as for those above. When (wh/c) < 1, we set -'^-(t)' and carry on with the calculations. VI.24] PROPAGATION OF SOUND IN HORNS 285 We solve Eq. (24.18) for \f/ + (ioirl/c) = *-(«& — ifa) and from this compute the impedance at the throat of the horn: z = pc <t coth ir(a h — tj8*) — ( -^— J H r tanh (e) > For large values of w = (2«a/c), a h becomes large, and *« » -l / • / P ^ + u ^ (24.19) w-* oo t + {ic/a)h) tanh (e) which corresponds to Eqs. (24.6), (24.11), and (24.14) for the special cases discussed before. For these higher frequencies no reflected wave is present in the horn. For small values of w, the approximate expressions for ah and fih are similar to those of Eqs. (23.15) : ah teX?)"' *-©(-) with a the radius of the mouth of the horn. The effective length of the horn is l p = I + (8a /Sir), as was the case for the uniform tube. In terms of this we can write the approximate expression for the acoustic admittance ratio at the throat : * =* \-*™™' c) *^}f} + 4 tanh («) (24.20) z (T/2)(aco/c) 2 — t tan (corZp/c) <ah ' (OW/C) « 1, T 2 = 1 - (C/C0/1) 2 Wood-wind Instruments. — If the tube of the flute or open organ pipe has a shape corresponding to Eq. (24.16) for some value of c, then the frequencies picked out are those for which z is real and as small as possible. This corresponds to the requirement that (oyrl p /c) be approximately equal to (nx), (n = 1,2,3 • • •). By using the equation for t, we obtain the following equation for the resonance frequencies : In comparison with the uniform tube [see Eq. (23.17)] the overtones here are not strictly harmonic, though if the tube flare is small (h large), the discrepancy is small. The specific acoustic resistance at the throat for the nth. "harmonic" is l -~ i "l\'V) + WJ (24 ' 22) which is to be compared with Eq. (23.18) for the uniform tube. 286 PLANE WAVES OF SOUND [VI.24 For a reed instrument with tube shaped according to Eq. (24.16), the frequencies picked out will be those for which z is as large as possible. We must choose a value of (cotZ p /c) so that (pc/z ) in Eq. (24.20) is real and as small as possible. This is not obtained by setting the tangent equal to infinity, for the term (c/oh) tanh e is usually larger than (aco/c) 2 , and (pc/z ) would not be real or as small as pos- sible. Instead, we set (o)tI p /c) = ir(m + i) + 5, (m = 0,1,2 • • • ) where 5 is quite small. This corresponds to tan (o>tI p /c) ~ —(1/5), which is quite large, but not infinite. We then adjust 5 so that, to the first order in the small quant ties 5 and (aco/c) 2 , (pc/zo) is small and real. This corresponds to making 5 = (c/toJi) tanhe, which is small enough, since we are assuming that we are above the cutoff frequency (co > coo = c/h). Setting this value for r in the equations for co and for z , we have Zanti (iW m ; + © !+ P taiih ' C I p 1 I 1 ( p I p + Vi Xiram) |_ \irhmj ir 2 hm 2 (24.23) where ra = 1, 3, 5 • • • . This is to be compared with Eqs. (23.19) and (23.20) for the uniform tube. We note that for the catenoidal case (e = 0) the term (8l p /Tr 2 h) tanh e is zero ; for the exponential horn the term is (8l p /T 2 h) ; and for the conical horn this term is (SI p /t 2 x ) and the term (2l p /irh) 2 is zero. The introduction of a nonuniform cross section thus makes the overtones nonharmonic by a small percentage, makes the resistance at resonance (z minimum) somewhat larger than for the uniform tube, and makes the resistance at antiresonance (z maximum) somewhat smaller than for the uniform tube. A certain small amount of flare (h large but not infinite) does not alter v m and z very much, however, and the larger mouth radiates more efficiently. The case not included in this approximate solution is the one for a conical horn when x is smaller than the wavelength X. This is the case where the horn throat is extremely small compared with the open end and is approximately the case for the oboe and the English horn. For the conical horn, Eq. (24.20) is pc ^ 1 — (t'/2)(aco/c) 2 tan(coZ p /c) , ic_ Zq (l/2)(aco/c) 2 — i tan(co? p /c) o)X and when x is small enough, (c/W ) is large, instead of small, though it is not usually so large as (c/aco) 2 . In this case, tan(coZ P /c) turns VI.24] PROPAGATION OF SOUND IN HORNS 287 out to be approximately equal to — (wx /c), and the allowed frequencies and the throat impedance at these frequencies turn out to be ) Here the allowed frequencies include both the even and odd harmonics, as is the case with the flute, rather than the clarinet. The specific acoustic impedance at the throat is not particularly large, but it is larger than any value of the resistance for any other frequency; so the frequencies v n are the ones that are heard in the emitted note. This lack of a pronounced resonance peak may partly explain the difficulty of playing the oboe. As a final item in this section, we note that the impedance at the throat of a long horn, below the cutoff frequency, can be easily obtained from Eq. (24.19). It is a pure mass reactance z ~ -io)(ph) Jl — I — J + tanhe (w < c/h) It is approximately equal to the mass reactance of all the air in the horn within a distance h of the throat. Transient Effects. — Calculation of transients in horns of general shape is difficult because of the complicated behavior of the impedance function. We shall carry through one example here, however, to show the general nature of the results. We consider the conical horn, to keep the calculations as simple as possible. Suppose that we have a piston in the throat of the horn, moving with specified velocity C/(w)e _iw '. The pressure wave out of the horn, if it is long enough so that reflections from the mouth are negligible, is pCX e i(»/c)(x-ct) x + x 1 + (ic/coxo) We refer now to Eqs. (6.15) and (6.16) and the related discussion. The expression for the pressure wave due to any velocity motion u (t) of the piston is OCX* f°° e -^>[t-(x/e)] C °° 2ir(x + Xo) J- «, 1 + (tc/oiXo) J-« 288 PLANE WAVES OF SOUND [VI.24 To improve convergence, we shall utilize the trick mentioned in the last part of Sec. 6: 1 = 1 — (* c /zo) 1 + {ic/wxo) (a -f- (ic/xo) Therefore the integral becomes = _£opc_ ( 1 | e - iu[t - {x/c)] do} J u^ivr dT x + x Q {2t J_„ J-» r °° r °° e -Mt-T-(x/c)] - I w (t) <2t I i /• / \ dco oJ-» J-» o) + (*c/a;o) 27TiCn .]- We now use Eq. (2.20) to calculate the first integral and use contour integration to calculate part of the second: Xopc V = — r — X + Xo t--\- y -^ T Mo ( T ) e -(o/.o)(«-r)+(^o) dr c) (x + XojJ-* Therefore part of the wave has the same "shape" as the "velocity shape" of the motion of the piston at the throat. Behind this wave, however, is a "wake" represented by the second integral, which dies out exponentially. This behavior is best seen if we cause the piston to move suddenly forward at t = 0. Then u (t) = 8(t), and ( [t< (x/c)] p,(0 = J _p^o_ 8 ( t _A_ • /*!_ e-cc/^^/o] [t > {x/c)] \X + Xz \ c) (x + Xo) The original pulse moves out of the horn with velocity c and with amplitude diminishing as (x + x ) _1 . After the sharp pulse comes a negative pressure wave which dies out exponentially; the smaller the constant x is, the more rapidly does this "wake" damp out but the larger is its initial amplitude. Problems 1. What is the speed of sound inside a gasoline-engine cylinder just after com- bustion, when the pressure is two hundred times atmospheric and the temperature is 1000°C; if the gas mixture has a value of y = 1.35 and would have a density of 0.0014 g per cc at 0°C and atmospheric pressure? 2. A plane wave in air has an intensity of 100 ergs per sec per sq cm. What is the amplitude of the temperature fluctuation in the air due to the wave? 3. The specific acoustic impedance at a point in a tube is R = 42, X = —42 when v — 344. What is the specific acoustic impedance at points 25, 50, 75, and 100 cm farther along the tube? 4. A vibrating piston is placed at one end (x = 0) of a tube whose cross section is 10 sq cm, and a second piston whose mechanical impedance is to be VI.24] PROPAGATION OF SOUND IN HORNS 289 measured is placed at the other end (x = 30). When the pressure due to the sound wave is measured at different points along the tube, it is found that the pressure amplitude is a maximum at the points x = 3, x = 15, x = 27> having an amplitude of 10 at these points. The pressure amplitude is a minimum at x = 9, x = 21, with a value of 6.57. From these data find the mechanical impedance of the driven piston, the frequency of the sound used, and the amplitude of vibra- tion of the driving piston. 5. A vibrating piston is set in one end of a tube whose length is 86 cm and whose cross section is 10 sq cm. Closing the other end of the tube is a diaphragm whose over-all mechanical impedance is to be measured. The measured radiation impedance of the driving piston (not including the mechanical impedance of the piston itself) has the following values: V 100 200 300 400 500 600 700 800 R p 42 147 420 189 840 67 1,260 -X-p - 71 1,260 -365 126 -840 76 -1,680 What are the real and imaginary parts of the mechanical impedance of the diaphragm at these frequencies? Plot a curve of the magnitude of the impedance. 6. A tube of length 86 cm and cross section 10 sq cm is closed at one end and has a piston fitted in to the other end. The piston is driven electromagnetically by a coil whose coupling constant is r = 1,000. The impedance of the coil is measured at different driving frequencies, and the part of the impedance due to the motion of the piston is found to be V 100 150 200 250 300 350 400 450 500 Rm 1.92 1.17 1.68 1.47 0.54 1.75 0.67 Xm -0.37 0.98 -0.75 0.88 0.89 0.64 0.95 Compute and plot the real and imaginary parts of the mechanical impedance of the piston and of its radiation impedance. 7. Two similar pistons, each of mass 10 g, slide freely in opposite ends of a uniform tube of length 34.4 cm and cross section 10 sq cm. One piston is driven by a force 100e _2 ' r * > ' dynes. Plot the amplitude of motion of the other piston as a function of v from v = to v = 500. 8. A piston of negligible mechanical impedance is fitted into the end of a uniform tube of cross section 10 sq cm. The other end of the tube, 34.4 cm from the piston, is closed by a rigid plate. The piston is driven by a force : F v - 42 sin (Stt*) - 28 sin (24irt) + 8.4 sin (40ttO Show that the velocity of the piston is u P = ^ tan ( 2^5 J cos (8nrt) — ^ tan ( ^ J cos (24x0 + ^ tan (^q) cos ( 407rf ) Plot the values of F and of the piston displacement as a function of t from t => to I ■'* (I), What is the curve for piston displacement when the tube is 68.8 cm long? 290 PLANE WAVES OF SOUND [VI.24 9. How long must an open organ pipe be to have its fundamental frequency equal to 250 cps, if the pipe is a cylinder, with radius 2 cm? 10. When a gasoline exhaust valve first opens after the explosion cycle, the volume occupied by the gas in the cylinder is 200 cc. The effective area of the valve opening is 1 sq cm, and its effective length is 1 cm. Assuming the cylinder- valve system to act like a Helmholtz resonator, what will be its natural frequency of oscillation if the air in the cylinder is at normal pressure and temperature? What will be the frequency if the gas is at a temperature of 546°C and is at normal pressure? 11. The exhaust valve in Prob. 10 opens into an exhaust pipe of 5 cm inside diameter, 100 cm long. What will be the lowest natural frequency of the cylinder- valve-pipe system for air at normal pressure and temperature? 12. A Helmholtz resonator has a cylindrical open neck 1 cm long and 1 cm in diameter. If the resonating vessel is spherical, what must its radius be to have the resonance frequency be equal to 400 cps? 13. A piston of mass 10 g, frictional resistance 1,000 dynes cm per sec, is at one end of a tube 13.69 cm in diameter and 43 cm long, open at the other end. Plot the real and imaginary parts of the total mechanical impedance of the piston as functions of the frequency from v = to v = 2,000. 14. A simple "muffler" consists of a tank of volume V with input pipe at one end and a cylindrical output pipe, of length I and radius a, at the other end. Draw the equivalent circuit for the analogous impedance at the input end; include the analogous radiation resistance R. If the volume flow into the tank at the input end is simple harmonic, v e~ iat cm 3 per sec, what is the volume flow out of the output pipe? What is the power radiated into the open? How does it depend on frequency? What is the frequency above which the output radiation falls off rapidly? Above what frequency will the equivalent circuit be invalid? 15. The back of a loud-speaker diaphragm looks into a "tank" of volume V, which connects with free space by a constriction of negligible length, and area S of opening. Set up the equivalent circuit for the analogous impedance at the back of the diaphragm, and give the formula for the additional mechanical impedance load on the diaphragm. Over what range of frequencies will the motion of air in the constriction be in phase with the motion of the diaphragm with respect to the outside (i.e., move out as the diaphragm moves out; be sure to express the phase relations between input current. in the equivalent circuit and diaphragm motion correctly) and be larger than the diaphragm motion? If the area of the constric- tion is 100 cm 2 , what volume must the "tank" have in order to have this reinforce- ment of the diaphragm motion come at and below about 100 cps? Above what frequency will the equivalent circuit be invalid? 16. A tube 1 cm in radius and 68.8 cm long has a hole in its side, placed mid- way along its length, which is 1 cm in diameter and is provided with a short open tube of the same diameter, 1 cm long. Plot the specific acoustic impedance at one end of the long tube when its other end is closed. What are the lowest three resonance frequencies of the system? Plot the specific acoustic impedance at one end of the tube when its other end is open. What are the lowest three resonance frequencies for this case? 17. An air-conditioning system has a circulation fan that produces noise of frequency chiefly above 200 cps. Design a low-pass acoustic filter, consisting of two vessels and three narrow tubes in series, which will filter out the noise. The VI.24] PROPAGATION OF SOUND IN HORNS 291 narrow tubes cannot be less than 5 cm in diameter, and the vessels cannot have a volume larger than 30,000 cc. 18. A condenser-microphone diaphragm is stretched across one end of a tube 2.5 cm in radius and 1.2 cm long, open at the other end. Compute and plot the ratio between the pressure at the diaphragm and the pressure at the open end of the tube as a function of v from v — to v = 10,000. 19. When the damping of wave motion along a tube must be taken into account, show that the pressure amplitude at the point x is given by the formula [ V \ = A -\/cosh 2 Lvcci + (-} (I - z)l - COS 2 \irf3i + (y) (I - X) J where k is the damping constant. Plot the pressure amplitude as a function of x, for a tube having I = 100 cm, X = 50 cm, (£) = (j^A «* = 0.1, ft = 0.4 What is the acoustic impedance atz = Z? At a; = 0? 20. The pressure amplitude is measured in a tube of length 100 cm, as function of the distance re along the tube from the driving end. The values of the maxima and minima, and their positions along the tube are Distance from driver x 12.5 25 37.5 50 62.5 75 87.5 Pressure |p|, max 1.899 1.478 1.204 Pressure |p|, min 1.934 1.335 0.869 0.489 From these data compute X, a*, ft, the damping constant k, and the specific acoustic impedance at x = and x = 100. What is the pressure amplitude at x = 0? At x i- 100? 21. A tube of length I has a termination impedance such that ai and ft are both independent of frequency. The damping constant k for waves in the tube is likewise independent of frequency. The end at x = is fitted with a piston, which is given a sudden push at t = 0; uo = 8(t). Show that the "flutter echo" in the tube is represented by the expressions: e -KX/C g e -2x(aj-t/3i) ? — 4x(ai— i/3i) : i-ifld-(f)(2J-*) s f f _ 21 - x \ [0 < t < (l/c)] {{1/6) < t < (2l/c)) [{21/ c < t < {3l/c)} [(3Z/c) < t < {4l/c}} 22. A conical horn has an angle of opening fi = 20 deg. The area of the neck of the horn is 5 sq cm, and that of the mouth is large enough so that no sound is 292 PLANE WAVES OF SOUND [VI.24 reflected back into the horn. Plot the transmission coefficient of the horn as a function of v from v = to v = 1,000. When the air in the throat is set into vibration at a frequency of 500 cps, what will be the specific acoustic impedance in the throat? If the velocity amplitude of the air in the throat is 1 cm per sec, what is the intensity of sound in the horn 200 cm from the throat? 23. The conical horn of Prob. 22 is 200 cm long, and its open end has an area of 4,000 sq cm. What value of x must be used in the formula for an exponential horn that has the same area of throat and open end as the conical horn and is as long? Plot the transmission coefficient of such an exponential horn as a function of frequency from v = to v = 1,000. What is the specific acoustic impedance in the throat of the horn at x = 500? If the air in the throat vibrates with a velocity amplitude of 1 cm per sec, what is the intensity of the sound in the horn 200 cm from the throat? 24. The conical horn of Prob. 22 has its small end fitted to a driving piston of area 25 sq cm and mechanical constants to = 1, R = 1,000, K = 1,000. Assum- ing that the horn is large enough so that Eqs. (24.6) are valid, compute and plot the total mechanical impedance of the piston as a function of v from v = 100 to v = 5,000. Is the radiation impedance an important part of this? A similar piston is attached to the exponential horn of Prob. 23.. Plot its total mechanical impedance as a function of v from v = 100 to v = 5,000. 25. The pistons of Prob. 24 are each driven by a force of 10,000e- 2Xi,,< dynes. Plot the power radiated by each of the horns as a function of v from v = 100 to v = 5,000. In the light of these curves discuss the relative advantages of the two types of horns. 26. What are the constants for a catenoidal horn having the same over-all dimensions as the horn of Prob. 23? Plot the transmission coefficient of such a horn over the frequency range from to 1,000 cps. What is the specific acoustic impedance in the throat at v = 500? What is the intensity at x = 200, if the air in the throat vibrates at 500 cps with a velocity amplitude of 1 cm per sec? 27. The loud-speaker of Fig. 62 has its conical horn replaced by an exponential horn of cutoff frequency v = 100 cps. What is the value of h? Recompute the curves of Fig. '62 for frequencies from 100 to 4,000 cps for this horn. Is this a more satisfactory loud-speaker? Can you suggest changes in values of the constants to improve results? 28. Repeat the calculations of Prob. 27 for a catenoidal horn having a cutoff frequency of 100 cps. Suggest further improvements. 29. One unit of a "tweeter" loud-speaker consists of a catenoidal horn of cutoff frequency 1,000 cps. The driving piston, of effective mass 1 g, is set in the throat of the horn, which has area So = 3 cm 2 . The piston is driven by electromagnetic means, having a coil of inductance L c = 0.001 henry, resistance R c = 1 ohm, and coupling constant r = 10 5 . Set up the equivalent circuit for the total electrical impedance of the driving coil. Compute the power radiated for a 10-volt driving emf for the frequency range 1,000 to 10,000 cps. What changes in constants will improve this behavior? 30. The "tweeter" unit of Prob. 29 has a conical horn with x& = 5 cm. Calcu- late the power radiated over the frequency range 1,000 to 10,000 cps. 31. The piston of Prob. 29 is driven by an x-cut crystal of Rochelle salts, of dimensions a = 2 cm, b = 2 cm, d = 1 cm (temp. = 15°C). Set up the circuit for the total electrical impedance of the crystal. Suppose that the crystal is VI.24] PROPAGATION OF SOUND IN HORNS 293 driven by an emf of 100 volts, compute the power radiated from v = 1,000 to v = 10,000. 32. A catenoidal horn of shape formula S = So cosh 2 (x/h), of length I, is used as a "speaking trumpet" for a dictating machine. Assume that the impedance of the diaphragm at the throat is infinite and that a plane wave strikes the mouth parallel to the axis of the horn. Modify Eqs. (24.18) and (23.28) to show that the specific acoustic impedance presented to the plane wave at the mouth (x = I) is *o = pc j [l - 2 ^pH - i \m(w) - i cot (corZ/c)] | where « = (4xa/X) = (2cm /c), t 2 = 1 — (c/coh) 2 , and a is the radius of the mouth of the horn. Show that the pressure in the horn a distance x from the throat is ipc po cosh (l/h) cos (utx/c) t Zo cosh(x//i) sin (oxrl/c) where po is the driving pressure at the mouth of the horn due to the incident plane wave. Plot the ratio of the pressure at the throat (x = 0) to this driving pressure -gdh 1 - 2 ^]Mt) + [*<•>-(?) — mv for a horn of I = 34.4 cm, h = 17.2 cm, a = 3.44 cm as a function of « = 2wv from v = to 500. 33. Use the relation X' / : — pr 6 V P 1 + a 2 e-* Jo(« Vt 2 - b 2 ) dt = / V p 2 + a 2 to help calculate transient sound in exponential or catenoidal horns. For a long horn of shape given by Eq. (24.16), show that if a unit pressure impulse starts at the throat at t = it will travel out of the horn according to the equation 1 cosh V = 2tt cosh [(x/h) + e] coshe /- e <(?)^RIy-^ M< : V4-!)^^4V(iRly]} cosh [(x/h) + e] } \ cj \ c) y/ t % - ( x / c ) where S and u are defined in Eqs. (6.9) and (6.11). 34. Use the equation quoted in Prob. 33 to show that the velocity wave out of a long catenoidal horn due to an impulsive motion of a piston at its throat is "* - o \tt-7i\ f" T 1 +—■ } C/k) tanh(s/ft)~|. 2x cosh (x/h) J- « L -*« VI - (c/coh) 2 J . e< („*/c)Vi-(^) 2 -^ do , c sinh (x/h) O-CKV^F© 1 ) h cosh 2 (x/h) Obtain an expression for the pressure pulse accompanying this velocity pulse. CHAPTER VII THE RADIATION AND SCATTERING OF SOUND 25. THE WAVE EQUATION We must now consider the more complicated case of sound waves that are not plane waves. When we studied the motions of a mem- brane we found that waves other than parallel waves are very difficult to analyze; in the case of sound wave's we shall find an even greater increase in complexity. Nonplane waves do not all travel with constant speed or shape. Their motions must be expressed in terms of coordinate systems which often are too complicated to handle. We shall content ourselves, in this book, with a discussion of cylindrical and spherical waves. The Equation for the Pressure Wave. — The first task is to obtain a wave equation for three dimensions. This can be obtained by a simple generalization of the argument of Sec. 22. The air particle at the point (x,y,z) can move in any direction, so we must find the three com- ponents of its displacement from equilibrium. This displacement is a vector d with components £, 17, and f in the x-, y-, and 2-directions, respectively. The corresponding particle velocity u = (dd/dt) has components u, v, and w. All these quantities, together with the pressure p, are functions of x, y, z, and t. When displaced by the sound wave, the elementary volume dx dy dz becomes a parallelepiped of volume J dx dy dz, where J = dx dx dx dj dy 1+p- dy it dy dj dz dr] dz dz ~ 1 + div(d) where div(d) = — + — + ^- is called the divergence of the particle displacement d. Therefore the counterpart of Eq. (22.2) for the equation of con- tinuity is 294 VII.26] THE WAVE EQUATION 295 14-5 = 1; 5 ~ -^div(d) Corresponding to Eq. (22.5) for the adiabatic compression of the gas is the relation 1 + ■?- = (1 + 8)t = ^- e ; V ^ -VcPo div(d) Finally, Newton's equation of motion is = -grad(p) ■©- where grad (p) is a vector, called the gradient of the pressure, with com- ponents (dp/dx), (dp/ By), and (dp/dz). These equations can be combined as follows: ^ = - 7c Po div(^ = ^ d iv[grad(p)] = Wp (25.1) where c 2 = (7<JVp) and where divferad( P )] = §* + $ + g = V*p is called the Laplacian of the pressure. This is the wave equation in three dimensions for the pressure. The particle velocity for simple harmonic waves can be obtained from the pressure by using the relation \io>p/ grad(p) (25.2) The potential energy of a unit volume of the gas at any instant can be given in terms of the absolute temperature pCvT ~ R (1 + 8) " ( 7 . - 1) K + } (7c ~ IV- 1 The equilibrium value of this, Po/(y c — -1), is the internal energy of the gas at equilibrium and is not included in the expression for the energy of the sound wave. When we expand the quantity (1/J 1 " -1 ) in a series of the small quantities (d£/dx), etc., we can leave out the terms in the first powers of these quantities and terms involving cross products, such as (d£/dy)(dn/dx), etc., because these terms have zero average value. The only terms that are not zero on the average 296 THE RADIATION AND SCATTERING OF SOUND [VII.25 (unless we go to terms smaller than second-order ones) will be those involving the square of the quantities (drj/dy), etc., (7. - i) v**- 1 / ~ r~TT) | ~ 2 l)y ° div 2 (d) + neglectable terms! The total average energy of the sound wave is therefore the integral W ~ ip f f f [Y^ + c 2 div 2 (d)l dx dy <fc = JJJ Up" 2 + 2p7 2 P 2 J rf z dy dz (25.3) Curvilinear Coordinates. — The equations for pressure, particle velocity, and energy have been given in terms of the rectangular coordinates (x,y,z); they can also be given in terms of other coordi- nates. The differential operator V 2 in Eq. (25.1) can be written as d 2 d 2 d 2 -z—z +.-^-5 + t~i m rectangular coordinates dx 2 By 2 dz 2 d 2 d 2 . t— 5 + r-:in cylindrical coords V 2 = ( ■'■ °> \ or/ r- d<i> 2 dz' ^ (25 4) 1 d r dr\ drj r 2 1 d 2 +■ , • o « t - 5 in spherical coords. r 2 sm 2 #d99 2 The operator V 2 is called the Laplace operator, or the Laplacian: it measures the concentration of a quantity (or, rather, the negative of the concentration). The value of V 2 p at a point is proportional to the difference between the average pressure near a point and the pressure right at the point. When this is negative, there is a con- centration of pressure at the point, and when it is positive there is a lack of concentration there. The wave equation simply states that, if there is a concentration of pressure at some point, the pressure there will tend to diminish. Compare this with the discussion on page 175 for the membrane. In order to compute the particle velocity in curvilinear coordinates we must have available expressions for the components of the vector grad(p) along the coordinate axes. If, for instance, we label the VTI.26] RADIATION FROM CYLINDERS 297 component of the gradient parallel to the cylindrical radius, grad r ( ), that perpendicular to r and to z, grad^,( ), and so on, we have grad,( ) = A; gradj/ ( ) = |. ; g r ad 2 ( ) = A in rectangular coordinates grad r ( ) = A; grad,( ) = A; grad,( ) = i A in cylindrical coordinates / v^^.o) grad r ( ) =-; grad^ . ) = - -; 1 8 grad»( ) = — : — r — - in spherical coordinates rsm&d<p These expressions can be substituted in Eq. (25.2) to compute the components of particle velocity in these coordinates. Finally, to calculate the sound energy in terms of particle displace- ment, we must give expressions for the divergence of a vector A : dA x 6A y dA z . . . .. x -^ r -r h -t— m rectangular coordinates in cylindrical coordinates in spherical coordinates With these formulas giving the different mathematical forms of the physical quantities V 2 (p), grad(p), and div(d) when expressed in these three coordinates, we are ready to study plane, cylindrical, and spherical waves in free space. 26. RADIATION FROM CYLINDERS We have already discussed circular waves on a membrane in Sec. 19 and have developed there the set of functions needed to discuss cylindrical waves. We showed there that circular waves have a more complicated behavior than plane waves — for instance, that circular waves change shape as they spread out, leaving a "wake" behind them. Cylindrical waves, the three-dimensional generalization of circular waves, show the same behavior. We shall spend most of our time here in discussing simple harmonic cylindrical waves. 298 THE RADIATION AND SCATTERING OF SOUND [VII.26 The General Solution. — The general solution of the wave equation in cylindrical coordinates is a combination of functions of the type p = cos (m<f>)[AJ m (kr) + BN^krW*- 2 ™' (26.1) Sill where v = (c/2ir) s/k 2 + k 2 . The variable z measures distance along the cylinder axis, r the perpendicular distance from the axis, and is the angle that r makes with the reference plane. The functions J m and N m have already been discussed; their properties are given in Eqs. (19.4), (19.5), and (20.1), and the values of some of them are given in Table III at the back of the book. Uniform Radiation. — For waves spreading uniformly out from a cylinder, we use the function for k z = and m = 0, which represents outgoing waves [see Eq. (20.2)] : >aJ-%- ^r-cD-iWA)^ k = (2tv/c) = (2tt/\) > (26.2) — l (v) ln ^ e ~ 2 * irt r— >0 Suppose that we have a long cylinder of radius a which is expanding and contracting uniformly in such a manner that the velocity of the surface of the cylinder is u = U er 2irivt . The constant A, to corre- spond to the radiated wave, must be chosen so that the velocity of the air perpendicular to the cylinder surface u r = (l/2Trivp)(dp/dr) is equal to u at r = a. If a is small compared with the wavelength, this velocity is (A/ic*vpa)(r 2rivt } so that A must equal (T 2 v P aU ). The pressure and velocity at large distances from the cylinder are then \(*V V -> TrpaUo J— e*<~»-«*/*> Mr yjcr TaU * - e*<~»-*<* /4 » The product of the real part of each of these expressions gives the flow of energy outward per second per square centimeter, and the average value of this T~U*pa 2 U 2 0?\ (26.3) VH.2G] RADIATION FROM CYLINDERS 299 is the intensity of the sound at a distance r from the cylinder's axis. The total energy radiated in ergs per second per centimeter length of the cylinder is n = T* P va*Ul Radiation from a Vibrating Wire. — A somewhat more complicated wave is generated by a cylinder of radius a vibrating back and forth in a direction perpendicular to its axis, with a velocity U e~ 2Tipt . If the plane of vibration is taken as the reference plane for <f>, the velocity of the part of the cylinder's surface at an angle <£ from the plane of vibration has a component U cos <j> e- Mvt perpendicular to the sur- face. In this case we take the radiated wave to be p = A cos0[J"i(At) + iNi(kry\er M '*, k = (2rv/c) .Ac „ . , » — i—j— cosd>e~ 2rtrt r _»o i r'rr y (26.4) > A J~ ^(r~ct)-i^/i) cog( x r-*co \TT l vr ^ If a is small, the radial velocity at r = a is cos<f>e~ 2Tipt \2ir 3 v 2 pa 2 J which must equal the radial velocity of the surface, so that A = (2ir z v 2 P a 2 U /c). The radial component of the particle velocity at large distances from the cylinder is Ur > I — J J-^ e^(r-c*)-*(3x/4) cog ^ j— >«> \pc/ \7r J vr There is a component of particle velocity perpendicular to the radius r, but it diminishes as r-* at large distances and so is negligible there compared with u r . The intensity at large distances and the total power radiated per centimeter length of vibrating wire are t-?^3co..* n = ^W3 (26 .5) The amount of sound energy radiated by a vibrating string therefore diminishes rapidly as the frequency of vibration of the string decreases and diminishes very rapidly if the string's thickness is decreased. As we have mentioned in Chap. Ill, a vibrating string is a very inefficient radiator of sound. 300 THE RADIATION AND SCATTERING OF SOUND [VII.26 The reaction of the air back on the vibrating wire is obtained from the expression for the pressure at r = a: p ~ — iupaUo cos0e -iw< The net reaction force on the wire per unit length in the direction of its motion is •J". 2*- p cos <pd<t> = -Foe-™ 1 = —i(ap(ira 2 )UQer i » t The ratio of this to the velocity of the wire is the mechanical impedance per unit length of wire, due to sound radiation, (jff) ~ - to(*-a 2 P ) (a « c/«) which is equivalent to the reactance of a mass of air of volume equal to that of the wire. The resistive part R of this impedance is too small to be included in this approximation when a is small. We can find the resistive part from Eq. (26.5) for the power radiated, for of course II = %RUl, Z tad c* -i^a^p) + ( 7 ^) (a « oh) This can be used to compute the effect of the presence of air on the motion of a vibrating string, as discussed on page 105. Radiation from an Element of a Cylinder.— To solve more com- plicated problems, where the velocity of the surface of the cylinder is a less simple function of <p than the preceding examples, it is convenient first to solve the problem where only a single line element on the surface of the cylinder does the vibrating. Suppose that the radial velocity of the surface r = a is (da , da\ - — < 4> < + -g J ( , da ^ , . da\ (^ + -2 < <t> < 2tt - T J The Fourier-series expansion for this function of 4> is \ / L OT= 1 U a = 2rivt V1I.26] RADIATION FROM CYLINDERS 30 1 To fit this distribution of velocity at the surface we choose a general sort of outgoing wave 00 V = 2 Am cos (- m ^ J ^ kr ) + iN m {kr)-\e-^\ k = (2rv/c) m = The corresponding radial particle velocity at r = a is a = ( X 02) = \A^1 e'y + ^ AmCm e 1 ^ cos(m<f>)\ e~ 2 * irt \2irivpo drj |_ 2p c -£J pc J u a = where 2 ^ [Jo(n) + iATo(M)] = iCoe**; ^ [/*(/*) + ii\T„Gz)] = iC«e^- (m > 0) and where p. = ka. Therefore, using Eqs. (19.5), Ji(ka) = %Co sin (70) ; Ni(ka) = — iC cos (70) J m+ i(ka) — J m -i(ka) = 2C m sin (y m ) (m > 0) N m -i(ka) — N m +i(ka) = 2C m cos (y m ) (m > 0) Note the additional factor of two in C , to anticipate the factor ^ in the sum for ra = 0. Limiting values of the amplitudes C m and phase angles y m are given by the following approximate formulas: When (ka) = (2ira /\) » m + £ Co c^ \/8/irka; y ~ka — (t/4) C m ~ y/2/wka; y m ~ ka — hf(m + i) (m > 0) When (ka) = (2ira/\) « m + i ) (26 6) C ~(4Ma); 7o^7r(fca/2) 2 „ m!/2V +1 **» (ka\ m Values of some of the C's and 7's are given in Table X at the back of the book. To fit the expression for u r at r = a to the expression for the velocity of the cylinder u a , we must make . pcU da . A m = - — ~ — e tym Since, at very large distances from the cylinder J m (kr) + iN m (kr) ~ ^-| e f [^-i^+i)] 302 THE RADIATION AND SCATTERING OF SOUND [VII.26 we have the following expressions for the pressure, particle velocity, and intensity of sound at the point r, <f> (when r is a large number of wavelengths) and for the total power radiated by the element per unit length of cylinder: a U da */- e ik{r - ct) \{/(<p) ; p ~ pcu \w 3 ka ^u (J m r 00 _ / .N r Tm + £ (2m+1) T pc\Uda) 2 *S? cos (m<f>) cos (n0). } (26.7) 2r*pr ^J n C m C n •cos [y m — y n + ^7r(m — n)] n = pc 2 (£/^) 2 [_2_ , ^ J. 1 Wv LCI £{ ell To find the intensity we have, of course, multiplied the real parts of u r and p together and averaged over time. The total power II per Long- and Short-wave Limits. — When the wavelength is quite long compared with 2ira, we can use the second part of Eqs. (26.6) to compute the radiation. The largest values of (l/C m ) are for m = 0, so that to the first approximation ^(0)c- A /pe-** /4 (26.8) T ~ (*£-') (IT *»)•; TL^(^)(Uday At these low frequencies the sound radiates out with equal intensity in all directions, and the amount radiated is small. The expression for intensity is the same as that given in Eq. (26.3) for a uniformly expanding cylinder, if we substitute for U in the earlier expression the average velocity (U da/2-n) of the surface. Values of T and II are plotted in Fig. 65. Polar curves of the intensity are shown for different values of n = ka = (2-ira/X) and a curve is given for II as a function of /z. We notice that at long wave- lengths very little power is radiated, and the intensity has very little directionality. As the wavelength is decreased more power is radi- ated, and the intensity has more directionality; the cylinder begins VII.26] RADIATION FROM CYLINDERS 303 to cast a "shadow," and a smaller proportion of the energy is sent out on the side of the cylinder opposite the radiating element. For very short waves the intensity is large and uniform from <£ = — {ir/2) to d> = +(x/2) and is zero from <t> = (ir/2) to <t> = (3tt/2), in the shadow. In the intermediate range of n, where the wavelength is about the same size as a, interference effects are noticeable. The polar curve for ju — 3 shows that a fairly intense beam is sent out from the cylinder in a direction diametrically opposite to the position of the line source (<£ = 180 deg). The general properties illustrated by this set of curves are a charac- Cylinder 2k<* teristic of all wave motion when it strikes an obstacle. When the wavelength is large compared with the size of the obstacle, the wave pays hardly any attention to its presence. The first polar curve in Fig. 65 shows that for long waves the intensity is distributed in approximately the same manner as it would be if the line source were not in the side of a cylinder but were all by itself, radiating into free space. On the other hand, when the wavelength is very small compared with the size of the obstacle, the motion resem- bles the motion of particles, the waves traveling in straight lines, and the obstacles casting sharp- edged shadows. Light waves have this raylike property in most cases; geometrical optics is a valid approximation because the light waves are very much shorter than the size of most of the obstacles that they encounter. When the wavelength is about the same size as the obstacle, com- plicated interference effects can sometimes occur, and the analysis of the behavior of the waves becomes quite complicated. Radiation from a Cylindrical Source of General Type. — If the line source is not at <f> = on the surface of the cylinder but is at <f> = a, the pressure and velocity at large distances are, by Eqs. (26.7), Fig. 65. — Power radiated and distri- bution-in-angle of intensity from a vibrat- ing-line source set in a rigid cylinder, for different values of n = 2ira/X. 304 THE RADIATION AND SCATTERING OF SOUND [VH.26 p — > pc(U da) .J- \}/(<j> — a)e ik< - r - ct \ U r -+(Uo da) J- 4>(4> - a )e mr - c » the axis of the polar diagram being turned through an angle a. If several sources are distributed over the surface of the cylinder, each for a different value of a, the resulting radiated wave will be the sum of all the waves for the individual sources taken separately. This fact can be used to express the radiation from a cylinder whose surface vibrates with any arbitrary distribution of velocity amplitude. If the distribution is such that the surface at <f> = a has the radial velocity TJ{a)e~' iTivt , then the wave may be considered to be the result of an infinite sequence of line sources, the one at <f> = a having the velocity amplitude U(a), etc. The pressure and radial velocity at large distances are then obtained by integrating the expressions given above for a single line source: p~ p cj® e ik ( r - c » J iK0 - a) U(a) da (26.9) -fe) For instance, if a section of the cylinder between a = —a and a = -\-a is vibrating, so that U(a) = U for — a Q < a < a , and is zero for the rest of the values of a, then the pressure wave at large distances from the cylinder is V = pcUo J^ e ik (*-°» J ° iK0 - «) da = 2pcU /7 tt(fWrf) >h sin (map) cos (m<ft) g -i[\m+f(2m + i)] t 2 \j/r ^J mC m (26.10) where we use the convention that - — - — — = a when m = 0. m The intensity and total power radiated are _ 2pc 2 Ul >ri sin (map) sin (na ) cos {m<j>) cos (n<f>) T*vr ^Li mnC m Cn • cos 7m — 7n + (m — n) I Vn.26] RADIATION FROM CYLINDERS 305 _ 2pc 2 U 2 [2a| ^h sm 2 (m<x ) ] 11 ~ tt 3 ? C 2 "^ 2j m*Cl J When the frequency is very small, so that X is much larger than a, the intensity and power radiated are T -> P^H n->7rpm 2 t/^ to the first approximation. The expression for T is quite similar to Eq. (26.3) for a uniformly expanding and contracting cylinder. Even though the velocity of the surface of the cylinder, in the present case, is not symmetrical about the axis, nevertheless the radiation at very long wavelengths is symmetrical, behaving as though it came from a uniform cylindrical source whose velocity amplitude is (ZJo<xo/t) (the average velocity amplitude of the actual surface). This is another example of the fact, mentioned above, that wave motion is insensitive to details smaller in size than the wavelength. Transmission inside Cylinders. — Now that we have discussed the behavior of cylindrical waves of general type, it is appropriate to return to the subject discussed in Chap. VI, the transmission of sound inside tubes, and indicate what happens when the sound is not a one- dimensional wave. As an example, we take a tube of circular cross section of radius a, driven at the end x = 0. The three coordinates are r and 4>, giving position in the cross-sectional plane, and x, the distance along the axis (we use x instead of z to correspond to the usage of the last chapter). The solutions for the pressure waves inside the tube will be some combination of the characteristic waves ™(»*V«(Av)« l( *^" l> ; *? + *2= , (fY The functions N m cannot be used here, for they become infinite at r = 0. The values of k r are determined by the boundary conditions at the inner surface of the tube, r = a. Ordinarily the tube is fairly solid, but it does yield slightly to the pressure of the waves on the inside, and in many cases this yielding has important effects. At any rate we cannot always neglect it entirely. We shall assume that each portion of the tube moves out- ward with a velocity proportional to the pressure of the wave at that point: u r = —) at r = a Zt 306 THE RADIATION AND SCATTERING OF SOUND [VII.26 The factor of proportionality z t is called the specific impedance of the inner tube surface and the ratio of this to pc is the impedance ratio of the wall, ft. It may be due to the elasticity of the tube walls (in which case z t is mostly reactive), or it may be due to the fact that the inner surface of the tube is covered with porous material which allows a small amount of air to penetrate the pores (in which case z t is often mostly resistive). Of course z t will depend on frequency. The reciprocal of the impedance ratio, the admittance ratio, will be written (pc/z t ) = (1/ft) = (i<t — i<r t ), where a t is the conductance ratio of the tube and c t is its susceptance ratio. We can compute the radial velocity from the pressure wave by using Eqs. (25.2) and (25.5). Confining ourselves here to the radially symmetric waves, the boundary condition is p = AJoiKrfeW**-^; u r = i l — J AJi(k r r)e i(kxX - iut ^ Jo(k r a) = i(^f)ji(k r a); k = (£), ft = (ji\ (26.11) If the specific impedance z t is large compared with the characteristic impedance of air (as it usually is), then ft is large compared with unity, and the first approximation to the solution for k r is that —Ji(kra) = jfj^x [Jo(kra)] is equal to zero. Solutions for this can be obtained from tables of Bessel functions: fc r a~xa „; aoo = 0; a i = 1.220; a 02 = 2.233 • • • (26.12) (For further values of the a's, see page 399.) The case a o = is the plane wave case we discussed in the previous chapter. We note that a 0n equals the quantity /3i„ given in Eq. (19.6) for the circular membrane. The next approximation is obtained by expanding the functions J and J i about k r a = wa 0n and taking the first terms: + {k r a)Ji(k r a) = (xaon)«/i(7ra n) + (k r a — ira 0n ) -j- zJ x (z) Therefore Eq. (26.11) becomes, to the next approximation (assuming that ft is large), 7 ■ I ka . \ . / oioa ,\ kra ~ ira n — 1 1 U ] ~ ira 0n — * [ U I \ira 0n / yraion / (26.13) k = ( w /c); ft = (z t /pc) VII.26] RADIATION FROM CYLINDERS 307 except in the case n = 0. In this special case, Eq. (26.11) reduces to 1 = li(k*a$i/k), or *'~V!s ; M+ fe) (n=0) ■ The general expression for axially symmetric waves in the tube is (26.14) + 2^.[(^)-<^) . g— (iKi/o)+i[ (&)T 0n /c) + (<rj/a) Jx— iu « where (1/ft) = x t — *Vt = (pc/z t ). Wave Velocities and Characteristic Impedances. — These waves all travel with different velocity, even in the case of a rigid tube (f t = oo ). This is due to the factor ro n entering into the term (r „oja;/c) and is fundamentally due to the fact that all the waves except that for n = (a o = 0) have components of motion perpendicular to the tube axis. The lowest mode {n = 0) represents the plane wave solution we discussed in the previous chapter, for the case of ft = oo . When the tube wall impedance is riot infinite, even this wave has a small com- ponent of motion in the radial direction, for p depends to some extent on r. To the second approximation the pressure wave for this lowest mode is (from the first two terms in the expansion for J ) V ( r 2 \1 \ _1_ £ w I 1 g— (xKt/a)+i[(a/c)+(<rt/a)]x— iwt \2acf t /J Therefore, when the tube wall impedance is taken into account, the pressure wave is not quite uniform over a plane perpendicular to the z-axis; there is a part of the wave, out of phase with the usual plane part, which is zero at the axis r = and reaches a maximum at r = a. If the tube admittance (l/z t ) = (1/pcft) = (l/pc)(/c, — ia t ) has a real part {n t /pc = conductance) not equal to zero, the wave dies out exponentially as it travels down the tube. If the tube susceptance (<r t /pc) is not zero, the wave velocity differs from that in free air, even for this "plane-wave" mode. To the first approximation, the wave velocity is c[l - <r t (\/2ira)], as long as f< is very large. A stiffness reactance corresponds to a positive susceptance (f ~ ipc{K/u) ; (1/f) — —if = ~i((a/Kpc), so that a tube with "stiff" walls has a 308 THE RADIATION AND SCATTERING OF SOUND [VII.26 wave velocity somewhat less than c, whereas a tube with walls pre- senting masslike reactance has a wave velocity somewhat larger than c. The departure from velocity c is greatest at long wavelengths (unless the variation of <x t with co cancels this effect). The higher modes travel with a wave velocity which is larger than that in free air. Even when the tube impedance is infinite the veloc- ities are (c/ron) = c/s/l — (Xa n/2a) 2 . Below a certain frequency v 0n = (ca 0n /2a), called the nth cuioff frequency, the nth mode cannot be transmitted down the tube as wave motion. The quantity t » becomes imaginary, and the pressure dies out exponentially for any lower frequencies. In other words, for a given frequency v only the first n modes are propagated down the tube, where n is such that a 0n is just smaller than (2av/c). For a tube of 1 ft diameter only the lowest, "plane-wave," symmetrical mode is transmitted for fre- quencies below about 1,300 cps, and for a smaller tube the higher modes are damped out for even higher frequencies. At the cutoff frequency the wave motion for the "cutoff" mode is purely transverse. Only at higher frequencies does this mode have any motion "to spare" for travel along the tube. The nonsymmetrical modes, involving cosine and sine of (ra<£), also have cutoff frequencies v mn = (ca mn /2a), where aio = 0.586; an = 1.697; a 12 = 2.717 a 20 = 0.972 • a 3 o = 1.337 } (26.15) TYh 1 a mn — w + 9 + 7 ( n large) (see also page 399) Only the "plane-wave" mode (0,0) is transmitted at all frequencies. We note that these a's are not equal to any of the /3's in Eq. (19.6) for the circular membrane. The ratio of pressure to air velocity in the direction of wave motion for a wave traveling in one direction is defined as the char- acteristic impedance for the mode in question. For free air we have found that it is pc. In a tube with walls of impedance pc{ t the char- acteristic impedance for the "plane-wave" mode is approximately pc[l — (\/2Ta)((Tt + iKt)]. Therefore if the tube impedance is purely reactive, the characteristic impedance is purely resistive; if the tube impedance has a real component, the characteristic impedance has a reactive term. The difference between the characteristic impedance for this mode and that of free air is proportional to the tube specific admittance (pc/£ t ) as long as f t is large. VH.26] RADIATION FROM CYLINDERS 309 The nth symmetrical mode has a c haracteristic im pedance (pc/T 0n )- •[1 — i(X/2iraT 0n ^t)], where r 0n = \A — (Xaon/2a) 2 . Therefore the characteristic impedance for the higher modes differs from pc even when the tube impedance is infinite. At frequencies well below the cutoff for the mode, the impedance is reactive, being approximately- equal to that of a mass of (pa/ira n) g per sq cm of piston surface. If a reflected wave is also present, due to mismatch at the far end, the impedance for each mode has a hyperbolic tangent factor in it, analogous to the cases worked out in the previous chapter. Generation of Wave by Piston. — The sort of wave that travels along the tube is determined by the nature of the driving piston at x = 0. To illustrate this part of the calculation, we consider a rigid tube and confine ourselves to symmetric oscillations of the piston; hence only the symmetric modes will be present. The extension to nonsymmetric modes and to nonrigid tubes will then be evident. Suppose that the velocity of the driving piston is given by u x = Ua{r)e~ iat at x = 0, and suppose that the tube is long enough so we can neglect reflected waves. The functions Jo(ira 0n r/a) are orthogonal characteristic functions, analogous to those discussed in connection with Eq. (19.10). Their integral properties are jJ*J^)j (^) rdr = K 2 (26.16) o \ a / \ a / )-x JKiraon) (to = n) Therefore the function u (r) can be expanded into the series U n Jo(Tra 0n r/a) n = where Uo(r) = > =-) f ^4 Joiiraon) n = ^» ~ TTl "T I u (r)J ( 7 ^IL_ \ r dr la 2 Jo{ira Qn )_\ Jo \ a / The coefficient Uo = (2/a 2 ) f u (r)r dr is the average velocity of the piston. We next fit the series of Eq. (26.14) (for K t = a t = 1/ft = 0) to this series as a boundary condition at x = 0, by first computing u x and then equating term by term. We find that A n = {pc/tq^} U n /J (Traon), so that the pressure wave traveling away from the piston is 310 THE RADIATION AND SCATTERING OF SOUND [VII.26 (pc/r 0n ) U n Jo(Tao n r/a)e i( - a/c) fro.*-*) -2 n = Jo(Traon) •Jo (~~) e^Wo**-^ (26.17) TOn = Vl — (TTCQIOn/wa) 2 If the motion of the piston is uniform over its surface, then only the first term (n = 0) remains, and we are back to the plane-wave case discussed in the previous chapter. If the piston is not perfectly stiff, the motion will not be uniform and some of the higher modes will come in. As an example, suppose that the driver is a membrane, driven so that its motion is u (r) = UJ (Tt3 ir/a), where O1 is adjusted so that Jo(irpoi) = [see Eq. (19.6)]. Then, from formulas for the integration of two Bessel functions, it turns out that TT _ (2C7/3oiA) t ( r \ 0.432E/ Un ~ ~Q2 -2- «/ lW30lj = "H /i or . a Ton POI — «0n U — (1.306o!0n) 2 ] The average velocity of the piston is U Q = 0.432 U. The average pressure back on the diaphragm is the coefficient of the first term in the series, pc times the average velocity. The pres- sure at a point on the diaphragm a point r from its center is oo Voir) = pcUo > pj n Qnft — '' . r — (26.18) ^-Jll — (1.306o:oJ 2 l/n(7ra;n„.)Tn» ' n = which is not equal to the average pressure. In fact this pressure is no t exactly in pha se with the velocity because of the factors t „ = \/l — (caon/2av) 2 , which become imaginary for the larger values of n. At this point we can refer back to our discussion of the air reaction on a membrane, given on page 193. There we said that if the velocity of waves on the membrane was very much smaller than the velocity of sound waves in air, then the pressure was fairly uniform over the membrane and was proportional to the average velocity of the mem- brane. In the present case, if the membrane is in free vibration at its lowest frequency, the driving fr equency will be v = (c TO /3 i/2a), so that the factors t » will equal \A — (ca 0n /c m ^oi) 2 (where c is the sound velocity and c m the velocity of waves in the membrane). If c is VII.27] RADIATION FROM SPHERES 311 very much larger than c m , then all the t's, except t o = 1 (for apo = 0), will be large and imaginary. Consequently, the most important term in the series (26.18) for the pressure will be the one for n = 0, which is just pc times the average velocity of the membrane, the higher terms being out of phase and smaller the larger the ratio (c/c m ) is. On the other hand, if (c/c m ) is quite small, a large number of t's will be approximately equal to unity; therefore series (26.17) will become p c^pcwo(r)e i(o,/c)<:af ~ c<) , except for small terms for higher n. In this case, therefore, the reaction at a point on the membrane is just pc times the velocity at that point. This is the other limiting case mentioned on page 198. The intermediate case is, as we see, quite complicated. The same sort of calculation can be made for waves radiating into the open (we shall discuss this in the next section) and the same general conclusions concerning wave-velocity ratios and reaction on the membrane can be reached. The transmission of sound through ducts will be taken up again in this chapter, after we have discussed the nature of wall impedance more in detail. 27. RADIATION FROM SPHERES More important for radiation problems than cylindrical coordi- nates are the spherical coordinates r, &, and <p; r being the distance from some center, # being the angle between r and a polar axis, and <p the angle between the plane through r and the polar axis and a reference plane. Few radiators of sound are so much longer than they are wide that they behave like long cylinders, but many radiators behave like spherical sources, especially when their dimensions are small compared with the wavelength of the sound emitted. Uniform Radiation. — The simplest sort of outgoing spherical wave is one that is radiated by a sphere uniformly expanding and contract- ing, so that the wave does not depend on # or <p. The wave equation in this case is d 2 p dt 2 A general solution of this equation which is finite everywhere except at r = is r 2 dr\ dr ) c 2 p =±F(r-cQ+±f(r + cQ consisting of a wave of arbitrary form going outward from the center and another wave focusing in on the center, 312 THE RADIATION AND SCATTERING OF SOUND [VII.27 This solution, except for the factor (1/r), is similar to Eq. (8.1) for waves on a string and to the equation for plane waves of sound given on page 238. This means that spherical waves are more like plane waves than they are like cylindrical waves. Plane waves do not change shape or size as they travel ; spherical waves do not change shape as they spread out, but they do diminish in amplitude owing to the factor (1/r) ; whereas cylindrical waves change both shape and size as they go outward, leaving a wake behind. Figures 40 and 41 show that, if a cylinder sends out a pulse of sound, the wave as it spreads out has a sharp beginning but no ending; the pressure at a point r from the axis is zero until a time t = (r/c) after the start of the pulse, but the pressure does not settle back to its equilibrium value after the crest has gone by. With both plane and spherical waves the wave for a pulse has a sharp beginning and ending, the pressure settling back to equilibrium value after the pulse has gone past. This behavior is another example of the general law (proved in books on the mathematics of wave motion) that waves in an odd number of dimensions (one, three, five, etc.) leave no wake behind them, whereas waves in an even number of dimensions (two, four, etc.) do leave wakes. Spherical waves do resemble circular waves on a membrane, how- ever, in that they have infinite values at r = 0. As we have seen on page 176, this simply means that the size of the source must be taken into account; every actual source of sound has a finite size, so that the wave motion never extends in to r = where it would be infinite. Suppose that a sphere of average radius a is expanding and con- tracting so that the radial velocity of its surface is everywhere the same function of time U{t). The rate of flow of air away from the surface of the sphere, in every direction, is Anra % U{t) = Q(t). To obtain an expression for the pressure wave radiated from the sphere, we write Newton's equation as p(du r /dt) = — (dp/dr). If p is chosen to be an arbitrary outgoing wave p = P(r — ct)/r, the requirement at the surface of the sphere, - -^ 5 = — -r 1 — - 9 —> at r = a serves to 7 r or r z 4xcr at determine the shape of the wave P. The Simple Source. — If the vibrating sphere is very small (more specifically, if a is small compared with the wavelength of the sound radiated), the sphere is called a simple source of sound. In this case (P/r) is much larger than (dP/dr) at r = a, and P ~ (p/4nr)(dQ/dt) at r = a. The pressure wave at a distance r from the center of the simple source is therefore VH.27] RADIATION FROM SPHERES 313 (27.1) &«■('-§) where Q'(z) = -r- Q(z). The function Q gives the instantaneous value of the total flow of air away from the center of the source. The pres- sure at the distance r is proportional to the rate of change of this flow a time (r/c) earlier. In some cases the source of sound is small enough or is so placed with respect to boundaries that source plus boundaries (if they are present) can be replaced by a distribution of simple sources, such that the element of volume dx' dy' dz' at (x',y',z') has an equivalent outflow of air q(x' ,y' ,z' ;t) dx' dy' dz' cm 3 per sec. The function q is called the source function or density of sources. This method of calculation will be particularly appropriate for the calculation of the radiation from a piston and for the calculation of sound radiation in a room. It is not difficult to see that a logical extension of Eq. (27.1) gives for the pressure wave at the point (x,y,z) < a d C C C dx' dy' dz' ( , , , R\ /nfr „ N V^y,z;t) = p - j J J —JL- a [x',y',z';t - -) (27.2) where R is the distance between (x,y,z) and (x',y',z'). The differential equation for the pressure wave in the presence of a source distribution of density q is An important special case is when the simple source has a simple harmonic flow of air, so that Q(t) = Qoe-™*, where Q , the magnitude of the volume flow of air at the surface of the source, is called the strength of the simple source. In this case, at large distances from the source (i.e., many wavelengths away) the pressure wave is p ~ -iw U~\ Qoe^o-"), k = (u/c) = (27I-/A) (27.4) The corresponding particle velocity at great distances is (p/pc), and the intensity and total power radiated from a simple source are 8cr 2 2c When the wavelength of sound is much longer than the over-all dimensions of the radiator, the radiation will be much the same no 314 THE RADIATION AND SCATTERING OF SOUND [VII.27 matter what shape the radiator has, as long as the motion of all parts Of the radiator is in phase. In such limiting cases the formula (27.4) for the simple source can be used. For instance, the open end of an organ pipe, or of any wood-wind instrument, is usually small enough to be considered as a simple source. If the average velocity of the air in the mouth of the tube is U e- 2rivt and the cross section of the tube has area S, then the strength of the equivalent simple source is UoS, and the power radiated is (wp/S V £7g/2c) . Spherical Waves of General Form.— To discuss any more com- plicated waves radiating from a sphere, we must consider the solutions of the wave equation that depend on # as well as on r and t. In this book we shall limit the discussion to waves that depend on # but are symmetric about the polar axis and therefore are independent of <p. To consider the waves that depend also on <p would add more compli- cation than is needed for the problems to be treated. The wave equation to be solved is r 2 dr V dr J ^ r 2 sin tf d& \ * d&) c 2 dt 2 If p = R(r)P(&)e- 2wirt , the equation becomes The left side of the equation is a function of r only, and the right- hand side is a function of # only. Since they are equal for all values of r and #, they must both have the same constant value, which we can label C. Legendre Functions. — We shall first solve the equation for P 1 * ( shi »§) + CP =° < 27 - 5 ) sin # d& or, letting cos# = x, d [d " **) ^] + CP = dx\ dx We solve this equation by setting P equal to an arbitrary series p = a + aix + a 2 x 2 + • • • and substituting this in the second equation. We find that the coefficients a must be such that VII.27] RADIATION FROM SPHERES 315 P = a 1 _ £ X 2 _ C(6 ~ O x < _ C(6 - 0(20 - O r6 2! 4! 6! + ax he H g|— x s + i ^ i a; 5 + The equation and solution have been treated in Prob. 5 (Chap. I) and on page 118. We have found that unless C has certain specific values the function P will become infinite at x = ±1. The only cases where P stays finite are for the following sequence of values of C and a or a x : C = 0, a x = 0; C = 2, a = 0; C = 6, a x = 0; C = 12, a = 0; etc. The allowed values of C are therefore C = m(m +1) (to = 0,1,2,3 • • • ) The solution of Eq. (27.5) which is finite over the range of x from —1 to +1, corresponding to C = m(m + 1), is labeled P m (x). It can be obtained by substituting the proper value for C in the series given above, making one a equal to zero and giving the other a the value that makes P m (l) = 1. The resulting solutions are m = o, C = 0, P (x) = 1 Po(cos#) = 1 w = 1, C = 2, Px(x) = X Pi(cos#) = cos# to = 2, C = 6, P 2 (x) = i(3z 2 - 1) P 2 (cos#) = -K3 cos 2* + 1) to = 3, C = 12, P,(x) = -H5* 3 - Sx) P 3 (cos#) = |(5 cos 3^ + 3 cos)* (27.6) The function P TO is called a Legendre function of order to. It can be shown to have the following properties: d 2 P dP {x2 ~ l) ^ + 2x ^~ w(m + l)Pm = (* = c °s*) 1 d m Pm{x) = ¥^.dx^ {x2 ~ l)m (2m + l)a;P m (aO = (m + l)P m+1 (z) + mP m ^{x) (2to + l)P- m (s) = *- [P m+1 (x) - P m -i(x)) (n J* 5 m) (n = to) (27.7) 316 THE RADIATION AND SCATTERING OF SOUND [VII.27 Values of some of these functions are given in Table IX at the back of the book. The last equation of (27.7) shows that the functions P m (x) constitute a set of orthogonal characteristic functions. Any function of x in the range from a; = 1 to a; = — 1 can be expanded in terms of a series of these functions: CO F(x) = 2) B„P m (x), B m = (m +i) f 1 F(x)P m (x) dx (27.8) m = The expression for the coefficients B m can be obtained by the method discussed on page 108. Bessel Functions for Spherical Coordinates. — We must now solve the equation for the radial function R: Only the allowed values of C can be used; otherwise the related func- tion of # will not be finite everywhere. Changing scale, to get rid of numerical factors, we obtain g+si+O-^)*-*-^-* (2 , 9 ) This equation looks very much like the Bessel equation (19.3), and in fact solutions of the equation are R = (l/\/z)J m+ i(z). The solu- tions can be expressed in a simpler way, however, in terms of trigono- metric functions. Direct substitution will show that two solutions of Eq. (27.9) are the spherical Bessel and Neumann functions j m (z) and n m (z), where 3 »(*) = smz z n (z) = — cosz z 3 iGO = sin z COS2 z n\(z) = sinz z cosz 3< .(*) ft- - J smz 2/ 3 k cosz z 2 ni{z) = 3 . 5 sinz z 2 _(3. \z 3 (27.10) - I cosz (27.11) VH.27] RADIATION FROM SPHERES 317 These functions have the following properties: 3m(z) = yJ^Jm+itz), n m {z) = J^N m+i (z) jm( - z) T^l'3-5 ••• (2m + 1) „.(,) _ - 1-1-8-5 u;( 2m-l) ... 1 / m-f 1 \ 3m(z) ~^~ cosl 2 2"— T ) . . 1 . / m+ 1 \ n m (z) > -sinlz £ — t ) Z— » oo Z \ if fft(z)z* dz = ~ [jl(z) +' no(*)iiOO] 2 3 /n§(2)2 2 dg = - [ng(«) - io(2)n!(2)] n m _i(2) < 7* OT (2) - n m (2)j m _i(2) = f-^J and have the following properties, for either j m or n m : 3m-AZ) + Jm+l(2) = J m (z) J -J di* 7 '" 1 ^ = 2m + 1 W»-i(*) ~ ( m + l)i-+i(2)] g^V-OO] = ^ + %-i(2), ^[2-^(2)] - ~z-j m+1 (z)) ( 27 -12) Jj'lCa) dz = -jo(z), Jjo(z)2 2 cfe = 2 2 Ji(2) 2 3 Jim(2)z 2 dz = g \.Jl(z) - i»-i(2)y*+i(2)] (m > 0) In all the equations (27.12) the function n can be substituted for j. Values of some of the functions j and n are given in Table VII at the back of the book. Returning to the problem of wave motion, we can now see that the general solution of the wave equation which is symmetrical about the polar axis and is finite everywhere except at r = is a combination of the functions V = P.(cos^) [Aj m (?^) + Bn m (^] e -w (27<13) This corresponds to Eq. (26.1) for cylindrical waves. 318 THE RADIATION AND SCATTERING OF SOUND [VII.27 The Dipole Source. — The first example of the use of the functions discussed above will be that of the radiation from a sphere of radius a whose center vibrates along the polar axis with a velocity Uoe-~ 2 * ivt . The radial velocity of the surface of the sphere is Uo cost? er 2Tivt } where # is the angle from the polar axis; and this must equal the radial particle velocity of the air at r = a. Therefore the pressure wave must be p = AP!(cos#)[ji(/cr) + in x {kr)]e-^ ivt , k = (2*-/X) (27.14) > — A ( ^— ) cost? e ik( - T ~ ct) r—>«> \lirvr/ The radial particle velocity is - * dP r 2wivp dr = Acos * [j (kr) + in (kr) - 2j 2 (kr) - 2in 2 (kr)]e-** i * oipc cost? e~ 2 * ivl »o \ pc ) \kr) If the radius of the sphere is small compared with the wavelength of the sound radiated, the limiting expression for u r can be equated to Uo cos t? e~ 2wivt at r = a to determine A : A ~ /4fVW \ A small vibrating sphere of this sort is called a simple dipole source. The intensity and the total energy radiated are 2irV 4 a 6 ^oCOS 2 <? (27.15) J»2ir Pit d<p o Jo Tr 2 smt? dt? = — ~x — - 6c 3 We note that the dipole source is less efficient than the simple source [given in Eq. (27.4)] for radiating at low frequencies, for II diminishes as v 4 instead of as v 2 . To find the reaction back on the source due to its motion, we first obtain the pressure at the sphere's surface. To the first approxima- VII.27] RADIATION FROM SPHERES 319 tion, this is p r==a ^ —rivpaUo cos# e~ 2irivt , for a small. The net' force on the sphere is in the direction of vibration and is F = ° 2 Jo' d<p Jo ( pr=a cos ^ sin ^ d * — ~ (^'"Xl^pa 3 ) Uoe- 2 ™ 1 This force is entirely reactive (since it is 90 deg out of phase with the velocity) and is equivalent to the reaction of a mass of air having volume equal to half the volume of the sphere. The resistive reaction, the part in phase with the velocity, is not given by this approximate expression, which simply means that the resistive part of the reaction for the dipole is so much smaller than the reactive part that it is not included in the first approximation. To find it we must compute the second approximation for A and p r=0 . However, since we know that the total energy radiated must equal (E/o/2) times the real part of F, we can find this real part from Eqs. (27.15). The radiation impedance of the simple dipole is therefore 7 F-Ww*pv*a? , -.-w-i ax - '"■' r = U 3c 5 (2Tiv)(hrp a*) Radiation from a General Spherical Source. — We take up next the general case of a sphere, not necessarily small, whose surface vibrates with a velocity U{d)e- 2Tivt , where U is any sort of function of #. We first express the velocity amplitude £/(#) in terms of a series of Legendre functions m = U m = (m + i) £ U(&)P m (cos&) sin# d& (27.16) To correspond to this we also express the radiated pressure wave in a series 00 p = 2) AmPmicoB^lj^kr) + in m (kr)]e- 2 " rt m = where the values of the coefficients A m must be determined in terms of the known coefficients U m . The radial velocity of the air at the surface Of the sphere is = - ^ A TO Z> m P m (cos#)e* 8 »'- i pc n» = 320 THE RADIATION AND SCATTERING OF SOUND [VII.27 where - j- [jm(ij) + in m {n)] = iD m e^ m , and n = (ka) = I -^ J Therefore, using Eqs. (27.11) and (27.12), mn m -i(ka) — (m + l)n m+ i(ka) = (2m + l)D m cos 8 m (m + l)jn^i(ka) — mj m -i(ka) = (2m + \)D m sin 8 m When ka » m + -g- D m ~lj-); 8 m ~ ka — £n-(m + 1) When k«m + | ^ 1 - 3 - 5 -';f; + r 1)(m+1) («>o) 5 ™ — 12 . 32 . 52 . . . (2m - l) 2 (2m + l)(m + 1) Values of the amplitudes D m and of the phase angles 8 m are given in Table XI at the back of the book. The radial velocity of the air at r = a must equal that of the sur- face of the sphere, and equating coefficients of the two series, term by term, we obtain equations for the coefficients A m in terms of U m : (27.17) Am — ( pcU m \ \D m ) The pressure and radial velocity at large distances {i.e., many wave lengths) from the sphere can then be expressed -© w r ~ C/ [ - J e ik( - r - ct) ^(&) ; p ~ p cu r \ka)^i *(*) = ( h ) ^j^P^cost^-^-^+u where we have multiplied and divided by Uo, the average velocity of the surface of the sphere. The air velocity near the sphere is, of course, not in phase with the pressure, nor is it entirely radial; but far from the sphere the velocity is radial and in phase with the pressure. When kr is very large, the intensity at the point (r,#) and the total power radiated are Vn.27] RADIATION FROM SPHERES 321 F 'W = (jzrr) 1 2 ^#^(cos^)P n (cost?) • . / \ kaU °/ TO ^o ^"^ > (27.18) • cos [5 TO — 8 n + ir(m — n)] n= fd* r r 2 Tsint? ^ = pg!_/^ ffl Jo Jo 2™ 2 -£J (2m + l)D 2 m The function /?,(#) is called the angle-distribution function for radiation from a sphere. When ka = (2ira/\) is quite small, all but the first terms in these series can be neglected, and, using Eq. (27.17), we have for the long- wavelength limit p ca - ia ( -~ J (4ra 2 Uo)e ik( - r - c ^ (27. 19) which is identical with Eq. (27.4) for the simple source, if Q , the strength of the source, is set equal to (4wa 2 U )- Radiation from a Point Source on a Sphere. — To show graphically how the radiation changes from symmetrical to directional as the frequency of the radiated sound increases, we shall work out the details of the radiation problem for two cases. The first case is that of a point source at the point # = on the surface; i.e., the surface velocity of the sphere is zero except for a small circular area of radius A around & = 0. The definition of U(&) is 77<W = i Uo (0 ^ * < A/a > K) \0 (A/a<#<C7r) and the coefficients U m are U m = (m + i)u f^ a P m (coad) sm&dfr — >$(m + i)u (A/a) 2 J0 A-+0 since P m (l) = 1. The intensity and power radiated can be obtained fromEqs. (27.18): • cos[3 w — S„ -f- -j7r(m — n)] 322. THE RADIATION AND SCATTERING OF SOUND [VII. 27 , / A 4 \ 4tt ^O 2m + 1 /27ra\ At very low frequencies, only the first- terms in these series are important, and the pressure wave again has the familiar form for the simple source V '--^(^) (7rA 2 Wo)e ifc(r-c0 with a strength equal to the velocity w times (xA 2 ), the area of the radiating element. ' Figure 66 gives curves ;f or the distribution in angle of the intensity radiated from a point source on a sphere for different ratios of wave- "0 90° 180° Angle from Spherical Axis ,9 Fig. 66. — Distribution in angle of intensity radiated from. a point source set in the surface of a sphere of radius a, for different values of /x = (2tto/X). Curves also give mean-squared pressure at point (a, #) on surface of sphere due to incident plane wave traveling in negative x direction; VH.27] RADIATION FROM SPHERES 323 length to sphere circumference. We see again the gradual change from radiation in all directions to a sharply directional pattern as the frequency is increased. These curves are of particular interest because of their dual role, as required by the principle of reciprocity, which will be discussed in Sec. 29. As computed, the curves give intensity or pressure amplitude squared at a point (r,#) , a considerable distance r from the sphere (when kr » 1, \p\ = \/2pcT) at an angle # with respect to the line from the center of the sphere through the radiating element of area at point (a,0). But the principle of reci- procity says that the pressure at a point Q due to a unit simple source at point P is equal to the pressure at point P due to a unit source at point Q. Consequently, the curves of Fig. 66 also represent the square of the pressure amplitude at a point (a,0) on the surface of the sphere due to a point source of strength (7rA 2 w ) at the point (r,#). Therefore the curves are useful as an indication of the direc- tional properties of the ear plus head, or of a microphone in a roughly spherical housing. Radiation from a Pistion Set in a Sphere. — The other example to be worked out is that of a piston of radius a sin (# ) set in the side of a rigid sphere. As long as # is not too large, this corresponds fairly closely to the following distribution of velocity on the surface of the sphere : The general formulas in Eq. (27.18) can be used, with U m = (m -f- i)w f P m (x) dx = %u [P m -i (cos#o) — P m +i (costfo)] Jcos&o where, for the case m = 0, we consider P-i(x) = 1. These expressions for U m can be substituted in Eq. (27.18) to give series for intensity and power. Curves for T as function of # are given in Fig. 67. When the wavelength is long compared with (2xa), the pressure and intensity are those for a simple source of strength 47ra 2 Wo sin 2 (t? /2). The radiation impedance for the piston set in a sphere can be computed by integrating the expression for the pressure at r = a over the surface of the piston. Alter quite a little algebraic juggling, we. find 324 THE RADIATION AND SCATTERING OF SOUND [VII.27 Z p — E p — iXp = pdira 2 sin 2 f -~ J (0 P — ix P ) d = i^ [Pm-i(cost?o) -P TO+ i(costV)] 2 X P = *2 TO=0 M 2 (2m + l)Dl [P m -l(C0S ff ) - P OT+ i(cos flp)] 2 (2m + l)D m (27.20) • [jm(ir) sin (5 m ) — n m (n) cos (5 m )] where fi = (2tt va/c) = (%ca/\). The quantities 8 P and Xp are the average acoustic resistance and reactance ratios at the diaphragm. These are plotted, in Fig. 68, as 2 ju 4 6 Fig. 67. — Distribution in angle of intensity and total power radiated from a piston set in a sphere. functions of the ratio of equivalent diaphragm circumference to wavelength, (2ira p /\) = {wa p /c) [a p = 2a sin (#p/2)], for several differ- ent values of #o = 2 sin -1 (a p /2a). We notice that the resistive terms all are small at low frequencies, increasing as o 2 ; rising to a value of approximately unity at wavelengths a little smaller than one- third VH.27] RADIATION FROM SPHERES 325 times the equivalent diaphragm circumference; and then, for very high frequencies, approaching the usual limit of 1 (z = pc). The reactance is always positive, representing a mass load. It first increases linearly with frequency, as the reactance due to a constant 2 4 6 Ratio Piston Circumference to. Wavelength, (Zircip/A) Fia. 68. — Values of radiation, resistance and reactance ratios (R r , X r /Tra p 2 pc), where a p = 2a sin (#<,/2), as function of (27ra p /X) = (wa p /c) for different values of t? , for radiation of sound from a piston set in a sphere of radius, a. mass would, but then it reaches a maximum at X~ 2xa, and for higher frequencies it diminishes more or less rapidly. The value of the initial slope is responsible for the factor in the end correction given in Eq. (23.1) and on page 247. These curves indicate the sort of radiation load one would expect on the diaphragm of a dynamic loud-speaker, set in a spherical case. 326 THE RADIATION AND SCATTERING OF SOUND [VII.28 The diaphragm of tho dynamic speaker is not usually the surface of a sphere but often has the shape cf an inverted cone in order to increase its mechanical strength. The radiation from such a cone would naturally differ from that from a section of a sphere; but it turns out that the average radiation impedance on a piston is approxi- mately the same, no matter what its shape, as long as its circum- ference is not changed, and as long as the volume of the mounting case is not changed. Therefore the radiation reaction on the dia- phragm of a dynamic speaker of outer circumference 2ira p , set in a cabinet of volume V P = 4x« 3 /3, is approximately given by {tca\pc) times the curves of Fig. 68 for # = 2 sin _1 [apOr /6 V ',)*], with the fre- quency scale equal to (2Trva p /c). Not only is the radiation impedance for a diaphragm, set in a cabinet of roughly spherical shape, given by the curves of Fig. 68; the average specific acoustic impedance at the open end of a tube or horn, of mouth circumference (2nra f ), is approximately equal to (pc) times the curves of Fig. 68 for # ^ 2 sin -1 (Tra p /2a f ) . This is the impedance referred to in Sees. 23 and 24. A plane piston, or an open end of a tube, of radius a p , set in an infinite plane rigid wall, corresponds to the case of a — » °o*, &o — > 0. Fairly simple expressions for the distribution in angle of the radiation and for the average specific acoustic impedance at the opening, can be obtained for this case by the use of Eq. (27.2). This will be dis- cussed in the next section. 28. RADIATION FROM A PISTON IN A PLANE WALL In this section we shall consider a useful example of the way we can build up the radiation from an extended source by considering it to be an assemblage of simple sources. This method has been used for cylindrical waves to obtain Eqs. (26.10); in the present discussion we shall build up the radiation out of spherical waves. Suppose that a circular flat-topped piston of radius a is set flush in an infinite plane wall and vibrates with a velocity u e- 2iri,, \ radiating sound out into the space in front of the wall. It does not matter, for the purposes of our analysis, whether the piston is an actual one or is simply the vibrating layer of air at the open end of a tube or horn; the effect on the air outside is the same. Actually, the air in the open end does not all vibrate with the same velocity, as the top of a piston does, but the velocity is nearly uniform in many cases, and the results of the following analysis will be nearly correct when used for VH.28] RADIATION FROM A PISTON IN A PLANE WALL 327 the open ends of pipes. Later in this section we shall indicate how the approximation can be improved. Calculation of the Pressure Wave.— The method that we shall choose to solve our problem is to consider each element of area of the piston as a simple source of sound and to add all the waves from all the elements together to obtain the resulting wave. The strength of the simple source corresponding to the element dS is Q = 2u dS; the factor 2 is used because the amount u dS can radiate only into the space to the right of the wall. The effect of the wall can be replaced by an image elementary source u dS radiating to the left of the plane of the wall. The actual element and its image together form a simple source of strength 2w dS. The radiation pressure at the point P in Fig. 69, due to the element dS, is obtained from Eq. (27.3) : dp = . fpvuo dS ■ty-JT^)* ( h -ct )} k = ( 2 *Vc) The particle velocity due to this wavelet is entirely radial, so that the velocity along the surface of the wall is everywhere parallel to the surface, as it must be. The quantity h is the distance from the element dS to the point P. If the point P is far from the piston, so that a is much smaller than r, then (l/h) is practically equal to (1/V), and h is approximately equal to r — y sin & cos yj/, where y is the dis- tance of dS from the piston center, # is the angle between r and the per- pendicular from the center cf the piston, and \p is the angle between the plane defined by this perpendicular and r and the plane defined by the perpendicular and y. Therefore, when r is much larger than a, the expression for the pressure wave at P due to dS is, approximately, Fig. 69. — Radiation from a piston set in a plane wall. The wave at P is the resultant of all the wavelets radiated from all the elements of area of the piston. dp ~ -<(*=*) Qik{r— cV)q— iky&ir>-dcos$y £y ^r where dS = y dy d\p. The total pressure at P due to all the simple sources making up the piston is the integral 328 THE RADIATION AND SCATTERING OF SOUND [VII.28 p ~ _ ih]^l J e * ( r-c«) J ydy\ e- 11 *™*™* d$ = -%riv(^Je ik(r -^ f J (kysm&)ydy = -™ P u a 2 -^[ ^^ J < (28.1) using Eqs. (19.4) and (19.5). The radial velocity at this large dis- tance is (p/pc). Distribution of Intensity. — Using the expression developed above for p, the radiated intensity at large distances turns out to be • 2Jx(MBintf) -j' (2jrtl/x) (28 . 2) T = Ipculu 2 Values of [2J x {x)/x~\ can be obtained from Table V or VIII. It is unity when x is zero, remains nearly unity until x is about (x/2), goes to zero at x about 1.27T, falls to about —0.13 at a; about l.lir, goes to zero again at x about 2.27T, and so on, having a sequence of maxima and minima which diminish in size. When the wavelength X of the sound radiated is longer than the circumference 2wa of the piston, then the value of (n sin &) is less than (t/2) even for # = 90 deg (i.e., even along the wall), and the term in brackets is practically unity for all values of #. Therefore, long wavelength sound spreads out uniformly in all directions from the piston, with an intensity four times that due to a simple source of strength ira 2 u . If the wall were not present and the "piston" were the open end of a pipe, this end would act like a simple source of strength ira 2 u for long wavelengths; so that the wall, or baffle plate, produces a fourfold increase in intensity. The sound reflected from the baffle reinforces the sound radiated outward, thereby doubling the amplitude of the wave and thus quadrupling the intensity, which depends on the square of the amplitude. Of course, to have the baffle give this considerable increase in amplitude, it must be con- siderably larger than the wavelength of the sound radiated, so that it will act as though it were infinite in extent. If X is smaller than 2xa, then the reflected sound still reinforces the sound radiated straight ahead, and the intensity has its maximum value pv 2 (ira 2 u ) 2 /2cr 2 at # = 0. At points off the axis, however, the reflected sound interferes with that radiated directly, and T diminishes in value as & increases, falling to zero when sin # is about equal to 0.6(X/a), then rises to a secondary maximum (where there VH.28] RADIATION FROM A PISTON IN A PLANE WALL 329 is a little reinforcement) of about 0.02 times the intensity for # = 0, then goes again to zero, and so on. Therefore, high-frequency sound is chiefly sent out perpendicular to the wall, with little spread of the beam. Most of the intensity is inside a cone whose axis is along the axis of the piston and whose angle is about sin -1 (X/2a). This main beam is surrounded by diffraction rings, secondary intensity maxima, whose magnitude diminishes rapidly as we go from one maximum to the next away from the main beam. Values of the intensity as a function of # are given in polar diagrams in Fig. 70, for different values of ju = (2tt va/c). The increased direc- tionality with increase in frequency is apparent. It is this direc- \-5a ^ A. -2a Fig. 70. — Polar diagrams of distribution in angle of radiated intensity from a piston set in a plane wall, for different ratios between the wave length X and the radius of the piston a. In the curve for X = o, the small loops at the side are diffraction rings. Compare this with Fig. 67. tionality that makes it difficult to use a single loud-speaker in outdoor public-address systems: although the low-frequency sound is spread out in all directions, only the people standing directly in front of the loud-speaker will hear the high-frequency sound. In rooms of moder- ate dimensions the sound is scattered so much by the walls that the directionality does not matter particularly (unless it is very marked), and one loud-speaker is usually adequate. Effect of Piston Flexure on Directionality. — The equations derived on the last few pages have been for a plane rigid piston, which moves as a unit, every part of the surface having the same velocity. It is of interest to see what effect variations in motion from point to point over the piston surface will have on the sound radiation. This will be particularly useful when the "piston" in the plane wall turns out to be the air in the open end of a tube or horn; for it is then not a very 330 THE RADIATION AND SCATTERING OF SOUND [VTI.28 good approximation to assume that the velocity is uniform over the open end. In a previous section we discussed the transmission of various modes down a tube of circular cross section. There we used (for the case of rigid walls) the following set of characteristic functions: J a{irao n y / a) *n(y) Jo(iraon) ^cmn{y,4>) _ COS / ,\ \/ , 2ira mn J m (Tra mn y/a) Vsmniyrf) Sin J m (Ta mn ) VWmn) 2 ~ W 2 dJ m (ra mn ) = ^ Eqg (26 12) and (26 15)] f * dty J ^f 2 y dy = ira 2 to describe the distribution of the waves across the tube. These functions will be useful for expressing the motion of any sort of piston, with any sort of distribution of velocity u (y,i/)e~ iat over its face: 00 00 U0\y } Y) = ?\ U n¥n ~T~ ^, \U cmn™cmn \ U smnXsmn) n = m,n — \ U " = J ^\ W \ My,t)*n(y)y dy > ( 28 - 3 ) ira 2 Jo Jo [ Ucmn hi r^ rMy,+)Z cmn (y,<t>)ydy ™ Jo Jo U sm „ -wa 2 Jo Jo ' ^ smn For instance, we can set this series into Eq. (28.1) and utilize the properties of the Bessel functions given in Eqs. (19.4) and (19.5) to obtain a series for the radiated pressure wave at considerable distances (many wavelengths) from the piston, at the point (r,&,<p) in spherical coordinates centered at the center of the piston: p ik(r—ct) I <r-> p ~ -irvpa* —^- \ >► ^«*».W (28.4) 'n = 00 . + ^ i m [U cmn cos (nap) + U smn sin (imp)] 2 **<n — ^(^ I S 2 — (Ta m n) 2 J m,n= 1 , 2sJi(s) . , A . _ fJm-lis) - J m+1 where s = ka sin # = (2ira/\) sin #. The term for Uo gives the radiation due to the average motion of the piston and turns out, of course, to be equal to the final formula VII.28] RADIATION FROM A PISTON IN A PLANE WALL 331 of Eq. (28.1), since a o = 0. The other terms give the modification of the angular distribution due to nonuniform distribution of velocity over the surface of the piston. The more important terms are those for m = 0, for the velocity distribution is usually radially symmetric (independent of \f/). Curves for some of these are shown in Fig. 71, as functions of s. We notice that the functions <£„ for n > add very little to the central part of the main beam (# ~ 0) but tend to modify the shape of the diffraction bands at the edge of the main na \ 0.6 \ V$o 02 - A^ A i ^ ,*6 n - -02 -04 V' *h- $5-'" C t > L \ c > i s i 3 1 2 1' \ 1 3 1 3 20 Angle Parameter s = (2ira/Msin0 Fig. 71. — Angle distribution factor <£>„ for radiation from a piston with velocity distribution proportional to the characteristic function Jo(.irao n r/a). <I>o gives dis- tribution for uniform velocity (rigid piston). beam. The zeros of each function $ n (s) coincide with those of the others, except that each function has one zero missing, the nth one; $o is not zero at s = 0, $i is not zero at s = 7ra i, and so on. All the radially symmetric waves are in phase, unless the U n 's are not in phase. The intensity at {r,&,<p) is, of course, (l/2pc)|p| 2 , as long as r » X. By juggling with the various $„'s it is possible to adjust the shape of the radiated beam to any desired distribution (within certain limits) and then to work back from the corresponding U n 's to deter- mine the distribution of velocity of the piston which will give this 332 THE RADIATION AND SCATTERING OF SOUND [VII.28 beam. For instance, the first "side lobe" in the diffraction pattern may be suppressed, or the first three or so can be drastically reduced in value (which is important if one needs a sharply defined beam). The limits of possible variation of the beam distribution are strin- gent, however, and can soon be learned by attempting to "tailor make" a beam. The most important restriction is that the main lobe (0 < sin# < a iX/2a) can never be made narrower by juggling the velocity distribution of the piston. The beam can be made broader by such juggling, but it can be made narrower (for a given wave- length) only by increasing the size of the piston. The minimum angular size of the main lobe is therefore given by the inequality (valid for X < a/2) Angular width of main lobe = (& for first zero) is larger than (35X/a) = (1, 200,000/ va) where the angle is in degrees, X and a are in centimeters and v is in cycles per second. Radiation Impedance, Rigid Piston. — Often it is more important to compute the reaction of the air back on the piston than it is to com- pute the distribution in angle of the radiated sound. The radiation impedance of the piston is necessary in calculating the mechanical and electrical properties of a loud-speaker, for instance. We have already given curves in Fig. 68 for the impedance ratios for a piston of effective radius a p = la sin (#o/2) set in a sphere of radius a. The limiting case of #o -* corresponds to a plane rigid piston of radius o p set in a sphere of infinite radius {i.e., in a plane wall). This case can be computed in closed form directly from the analysis developed in this section, as will be shown. To find the reaction of the air on the piston we must find the pressure at the point (r,<t>) on the piston, due to the motion of the element of area of the piston at (y,i). From Eq. (28.1) we see that this is dp = -IPV fy J e mh-ct) y d y fy where h = -\A 2 + y 2 — 2ry cos ty — 4). The total pressure at (r,<£) is p = —ipvu<>e- iu>t J>"J>*(f) (28 - 5) VII.28] RADIATION FROM A PISTON IN A PLANE WALL 333 where we are now using a, instead of a p , to denote the radius of the piston. To obtain the total force on the piston this must be inte- grated over the whole surface of the piston: F = —ipvuoe-™' i % *r rf *r^f e Kf) (28.6) The calculation of this integral involves some special mathe- matical tricks which will not be elaborated on here. The results give the total radiation impedance of a rigid piston of radius a set in a plane wall: ■p Z* = — =uTt = Ta 2 pc(0o — ixo) = tci 2 (Ro — iXo) d = 1 - (-1/1(10); w = 2ka = (4ra/\) = (2<oa/c) Xx/2 sin (to cos a) sin 2 a da U(ka) 2 (ka-*0) f(8kd/ZTr) (ka -> 0) "* (1 (ka -* oo ) ; Xo ~~* ((2/ik) (ka ~> oo) The properties of. these functions have already been discussed in the previous chapter, and their limiting values have been given in Eq. (23.14). These are the functions we have used to compute approxi- mate values of the average impedance of the open end of a tube. Tables of values for different values of w are given at the back of the book. Curves of d and xo as 'functions of (w/2) = (27ra/X) are given in Fig. 68, labeled # = 0. Comparison of the curves in Fig. 68, and of the curves of Figs. 67 and 70, show that a change from plane to spherical baffle makes a large change in distribution in angle of the radiated intensity, but makes little change in the average radiation impedance load on the piston. Sound is sent out in different directions, but its reaction back on the piston depends more on the ratio of piston circumference %ca (2wa p in Fig. 68) to wavelength X than it does on baffle shape. Distribution of Pressure over the Piston. — In many cases (the cases discussed in Chap. VI, for instance) it is sufficient to know the average reaction of the air on the rigid piston, which is given by the functions do and xo. To go further into the details of the problem, or to study the reaction of the air on a nonrigid piston, we must know the depend- ence on r of the reaction pressure p given in Eq. (28.5). The integral of Eq. (28.5) is a complicated function of r, which can most easily be expanded in a series of characteristic functions appropriate to a cir- 334 THE RADIATION AND SCATTERING OF SOUND [VH.28 cular piston of radius a. These functions have already been given in Eq. (26.17). In terms of them we can set n = L Jo(ir<XOn) . where the coefficients in the expansion for the case of the rigid piston are = -i I d$ I # \ r , ' r dr -r-ydy (28.7) Jo Jo Jo xXoVoCiraon) Jo fr The coefficient (do — ixo) has already been discussed. It gives the average acoustic impedance ratio over the surface of the piston. The coefficients f„ = (0„ — *x?0 for n > represent corrections to the distribution of radiation load over the rigid piston. Their average effect is zero, since the average value of each of the higher characteristic functions , Jo (iraonr/a) , m Jo{Trao n ) over the surface of the piston is zero. Expressions for these imped- ances can be given in terms of simple integrals: 1 J\(u)u du On = % = 2 M P - Jo [u 2 — (iraon) 2 ] V/* 2 - u 2 -( M 4 /37r 2 a 2 J 0»->O) (7a n/2/i 2 ) (m->°°) J\(u)u du {, (28.8) Xn = 2M J„ '[u 2 - (Traon) 2 ] V« 2 " M 2 _^(-9nH 0*-*0) \(2A/i) (m-*«) M = (2ira/X) = fca, 0i = 0.092, 02 = 0.0356, g 3 = 0.0194, £ 4 = 0.0116 • • • Some of these functions are plotted in Fig. 72, and values are given in Table XII at the back of the book. We see that for very small and for very large values of n = ka = (<aa/c) both d n and x« (f° r n > 0) vanish, being important only for values of ka near ira 0n (i.e., for frequencies near ca 0n /2a). Therefore VH.28] RADIATION FROM A PISTON IN A PLANE WALL 335 at very high and very low frequencies the reaction pressure of the air on the front of a uniformly vibrating piston is uniform over the face of the piston; only for frequencies such that the wavelength is the same general size as the piston dimensions is the pressure markedly nonuniform over the piston. 10 4 6 8, 10 "0 2 4 6 Frequency Parameter p.=ka=.(2iraA) Fig. 72. — Coupling resistance and reactance ratios between radiation and circular piston of radius a in infinite plane wall. Ratio f „ = 6 n - i'x» gives coupling between zero (uniform) velocity mode and the nth pressure mode. Nonuniform Motion of the Piston. — If the piston is not rigid, the analysis of the radiation reaction must be carried still further. We can express the velocity of the piston surface in terms of a series of characteristic functions of the sort we have been using, ^„(r) = Jo(Trao n r/a)/Jo(jra n) ' u (r) = ^UnVnir); u n = (~\ I u(r)* n (r)r dr n = as in Eq. (26.17). By analogy with Eq. (28.7) we can expand the reaction pressure of the air at a point on the piston a distance r from its center: oo oo p(r) = pC ^ ^ Untnm^m(r) 71 = m = 336 THE RADIATION AND SCATTERING OF SOUND [VII.28 where the impedance ratio f„ ro = (d nm - i Xnm ) = f«« are obvious generalizations of the impedance ratios £« (now written £ n) given in Eq. (28.7): Un = ^ J d * Jo ** Jo * m(r> dT Jo * n{y) T V dy The average pressure over the piston is the coefficient of ^ (r) = 1 in the series Pav = pC 2) U n £ n0 , (f n0 = fn) » = At low frequencies the quantity fro == f o is larger than any of the other f's. Therefore at frequencies small compared with (c/2tto) the pressure on the piston is approximately uniform, nearly equal to pcfo = pc(6o — ixo) times the average velocity u of the piston. This approximation was used in Chap. VI for the calculation of radiation out of the open end of a tube. For frequencies high compared with (c/2wa) the coefficients f mm are approximately equal to unity and the terms £ nm (n ^ m) are quite small. Therefore at high frequencies the pressure at r is 00 P(r) ~ pc ^ MmSmmtmW ~ pcu{r) (since f mm ~ 1) l m = which is pc times the piston local velocity at r. This is the result mentioned on pages 198 and 311. Radiation out of a Circular Tube. — We can now indicate how the radiation of sound out of the open end of a circular tube (fitted with a wide flange) can be calculated with greater accuracy than was pos- sible in Chap. VI. As an example we take the case where the tube radius a is smaller than (ca i/2v), so that the velocities (c/T 0n ) [see Eq. (26.17)] of all the higher normal modes are imaginary and only the plane wave (n = 0) mode is transmitted without attenuation. Therefore by the time the wave has gone from the piston to the open end, these higher modes have become small, and to the first approxima- tion only the plane wave mode S^o survives. In this case, to the first approximation, the velocity amplitude of the air at the open end is uniform and in a direction parallel to the tube axis (perpendicular to the plane of the open end). Suppose that this velocity has amplitude u . According to Eq. (28.7) the VH.28] RADIATION FROM A PISTON IN A PLANE WALL 337 pressure in the plane of the open end, a distance r from the tube axis, due to the radiation of the plane wave mode into the open, is « n = Vn(r) = JoiTaonr/^/Joiiraon) The average pressure due to this radiation is of course pcu (6 — ixo), which we have used in the previous chapter for calculating the acoustic impedance at the other end of the tube. But we now see that there are other, smaller, nonuniform terms in the reaction pressure, which cause local irregularities in the sound field near the open end. These corrections are very small for low and for high frequencies, so that it is usually not worth taking them into account. They are discussed here to indicate how these and similar details may be calculated if need be. The additional terms in the pressure modify the velocity distri- bution over the open end, by sending back down the tube waves corresponding to the higher modes V n . Since, by our assumption, these waves attenuate rapidly, they will not extend back into the tube any appreciable distance, but they will cause a modification near the open end. We have seen on page 309 that the nth mode of trans- mission in the tube has a characteristic impedance pc/ron = -iaiap/iraonTn), where T n = y/l - (v/p 0n ) 2 , v^ = (ca 0n /2a) Therefore, corresponding to the higher terms in the series for the pressure, we have a correction to the velocity distribution at the open end. Insteafl of being u it is, to the first approximation in the small quantities 0„ and x«, 1*3=0 ~ Uoe-™ 1 ~ 2 fe) ^k- + **-)*.(r) where the sign in front of the summation symbol is minus because these higher modes are being sent back into the tube (negative z-axis). In turn, each correction term for the velocity gives rise to a second- order correction to the pressure. The average value of this correction n = l ^ ' 338 THE RADIATION AND SCATTERING OF SOUND [VII.28 and therefore the average acoustic impedance of the open end for the plane wave mode is, for the case we are considering, pc {e - ixo + 2 (S) V 1 " {wj [26nXn + i{el " X " )] J to the second order of approximation in the small quantities n , Xn- Usually the summation can be neglected in comparison with O and xo- We can also utilize Eq. (28.4), together with the equation for w z =o given above, to obtain a somewhat more accurate expression for the distribution in angle of the wave radiated out of an opening in a plane wall. Transmission Coefficient for a Dynamic Speaker. — We have shown in the preceding pages that a dynamic speaker of radius a set in a plane wall, moving with velocity Uoe~ io>t , radiates the total energy fr-a 2 pcU?,d = bra 2 pcUf } ( 2toa / c ) I into the open. The quantity do is therefore the transmission coeffi- cient for a piston in a baffle, and its curve should be compared with those in Fig. 64 for various horns. As with the earlier cases, the coefficient is small at low frequencies, which means that the velocity amplitude must be increased at the low frequencies to obtain uniform response. This can be done by making the piston mass-controlled (which is not difficult) for then the velocity amplitude is inversely proportional to v over the frequency range of mass control. Since do is also the transmission coefficient for the open end of a horn, Fig. 72 shows that the open end of a horn must have its circum- ference larger than the longest wavelength that it is wished to trans- mit; otherwise, the open end will reflect an appreciable part of the sound back into the horn, causing strong resonance. Most dynamic speakers are designed so that over the useful fre- quency range the inertial reactance of the piston itself is the largest part of the total mechanical impedance (i.e., the speaker is mass con- trolled). Therefore, above some minimum frequency the velocity amplitude of the piston driven by a force of amplitude F is u — (F /2t vrrip). Above this minimum frequency the energy radiated by the speaker is n ~ — cm \n*Q>)[i-¥**\ (28-9) VH.28] RADIATION FROM A PISTON IN A PLANE WALL 339 The values of f — 5 ) * <A(w) are plotted in Fig. 73 as a function of w = (47ra/X). The curve shows that the response of this type of loud-speaker is good for low-frequency sound (as long as the fre- quency is above the lower limit of the mass control) but that as soon as the wavelength becomes smaller than the circumference of the speaker the power radiated diminishes. If the speaker is resistance controlled and is driven by a force whose amplitude is independent of v, the power radiated will be pro- portional to (do), shown in Fig. 72. In this case the higher frequencies in the useful range will be radiated best. 1.0 8R0 pew 2 0.5 flower limit of mass-con trot 2 4 6 8 W Fig. 73. — Power radiated by a "dynamic" speaker which is mass controlled and which is driven by a force with amplitude independent of the frequency. Design Problems for Dynamic Speakers. — To review some of the material we have developed in this section, and to show how it can be applied in the solution of practical problems, we shall discuss the design of a dynamic speaker. We cannot go into very many of the tricks of design used to improve the behavior of present-day dynamic speakers; for it is not our aim to discuss engineering practice. 1 We shall discuss some of the major problems encountered, however, to bring out the methods of analyzing the problems and the way in which the analytic techniques we have developed can help us in the design of acoustical equipment. We of course wish the dynamic speaker to reproduce passably well sound of frequency between 80 and 4,000 cps, a range of about six 1 For a discussion of the engineering aspects of this problem, see Olsen and Massa, "Applied Acoustics," The Blakiston Company, Philadelphia, 1934, for instance; or M'Lachlan, "Loud Speaker Design," Oxford University Press, Oxford, England, 1934. 340 THE RADIATION AND SCATTERING OF SOUND [VH.28 octaves; and we wish it to produce 10 6 ergs of sound energy per second if need be. (A person speaking with an average conversational tone produces about 100 ergs per sec and when shouting produces about 10,000 ergs per sec.) We shall assume that we can obtain a magnetic field as large as 10,000 gauss at the driving coil. We must first decide on the size of the loud-speaker cone, or piston. Since the piston is to be mass controlled (see the comments on page 338), we shall expect the output of the speaker to be more or less constant as long as the radiation resistance is increasing with the square of the frequency and to fall off when the radiation resistance becomes constant. Figure 73 shows that this output is greater than half its maximum value as long as w is less than 4 or as long as (va) is less than 10,000. For this limit to be at v = 4,000 we should have a = 2.5, or about 1 in. However, a piston of 1 in. radius would have to vibrate with an amplitude of more than a centimeter at v = 200 to radiate 10 6 ergs per sec from its small surface, so that we must sacrifice some of the intensity at the higher frequencies to obtain a piston large enough to radiate efficiently. We shall choose a to be 10 cm; in this case the intensity at c = 4,000 is one-sixteenth of the intensity at v = 1,000, a considerable diminution but perhaps not too great. The next problem is the driving coil. Increasing the number of turns on the coil increases the electromagnetic coupling constant G, but it also increases the mass of* the moving parts and the resistance of the coil. We choose the coil to be 1 cm in radius and to have 160 turns of wire of such a size as to make the resistance of the coil 5 ohms. In such a case it may be possible to make the mass m p of the coil- piston system as small as 30 g. If the magnetic field on the coil is 10,000 gauss, the electromagnetic coupling constants are D — 10 6 , G = 10 5 . It may be possible to keep the self-inductance of the coil as low as 0.2 mh. The next question is what to do about the space in back of the piston. We can, of course, let the back radiate into another room, but this is not usually done. If we enclose the back of the speaker, the enclosure will form a resonator that will add stiffness to the piston. If the inside of the enclosure is made of hard material, standing waves can be set up, and the enclosure can have a large number of resonance frequencies; so we shall line the interior with absorbing material to destroy most of the resonance due to standing waves. This will not appreciably alter the resonance of the Helmholtz type (see page 235) however, so we must make the volume of the enclosure large enough so that this resonance frequency is below the range that we wish to VTI.28] RADIATION FROM A PISTON IN A PLANE WALL 341 reproduce. The stiffness constant of the piston due to an enclosure of volume V is given in Eq. (23.3) : K p = (tci 2 ) 2 (pc 2 /V). If we choose V to be 20,000 cc, th e natural frequency of the resonator-piston system (l/27r) y/K p /m p will be 76 cps, which is just below the lower limit that we have set. This resonance will increase somewhat the response at lower frequencies. It will be assumed that the frictional forces on the piston, other than radiation resistance, are negligible (although in actual pistons the frictional resistance is usually larger than the radiation resistance). The constants assumed for the loud-speaker are therefore m p = 30, K p = 6.9 X 10 6 , a = 10, V = 2 X 10 4 \ D = 10 6 , G = 10 6 , Re = 5, Lc = 2 X 10~ 4 / /a \ / \ ( (28.10) ™v = i.3x 10 <, «> = (¥) = (m) I These are not recommended values of the constants, nor are all of them typical values. They have been chosen to show, as clearly as possible, how each property of the loud-speaker enters into the radiation process. Behavior of the Loud-speaker. — To compute the behavior of the speaker we first obtain its mechanical impedance: \Z m \ = s/Rl + XI R m = Tra^o = 1.3 X lO^Tl - 2^1 -*^a_, + 1rtlfXo > (28.11) 11 y 106 ' A + 190 v + 1.3 X 10W (w) X m = -jr — ? + 2rvm p + ira 2 X Zttv where R , X , and M are given in Eq. (28.5). Curves of R m and Z m are given in Fig. 74. We notice that over most of the range of fre- quency (for v larger than 200) Z m is practically equal to the reactance due to the mass of the piston Z m ~ 190j>. This is, of course, what is meant by saying that the loud-speaker is mass controlled. The sudden drop in Z m on the low-frequency side is due to the resonance of the vessel enclosing the back of the piston. The electrical impedance of the coil is \Z,\ = VfiJ + XI Rb ~ 5 342 THE RADIATION AND SCATTERING OF SOUND [VII.28 Curves of these three quantities are shown in Fig. 75. Since the resistance of the coil is fairly large, Z E stays fairly constant over most of the specified range of frequency. The motional reactance due to the piston just cancels that due to the impedance of the coil at about 600 cps. If the coil resistance were lower, this would cause a pro- nounced resonance peak in the response. The impedance rises on 10,000 Fig. 74. — Total mechanical resistance and impedance of a "dynamic" loud-speaker as a function of the driving frequency. The constants of the loud-speaker are given inEqs. (28.10). the high-frequency side owing to the inductance of the coil and on the low-frequency side because of the mechanical resonance of the loud-speaker plus enclosure. This rise in electrical impedance due to mechanical resonance is more than enough to cancel out the corre- sponding peak in the piston response (see the discussion on page 38). We shall assume that an emf of \0e- Mvt volts is applied across the ; coil. In actuality, the voltage amplitude will also vary with fre- quency, but we must cease adding complications somewhere and so VH.28] RADIATION FROM A PISTON IN A PLANE WALL 343 shall assume that the voltage amplitude is constant. The current through the coil is then (10/|Z*|), the force on the piston is(10D/\Z*$, and the velocity amplitude of the piston is u p = (10D/\Z E \\Z m \): The power radiated is \u\R m . Curves of u p and of n, the power radiated, are given in Fig. 76. They show that II is reasonably constant over the range 100 < v < 1,000 and that u p is inversely proportional to v over most of the frequency range. Below the mechanical resonance at low frequencies the curves for u p and II drop off. The falling off of u p \ below the straight line of the (1/v) curve, for high frequencies, is due to the inductance of the coil. The drop in II for high frequencies' is 5000 Fig. 75. — Total electrical resistance, reactance, and impedance of the driving coil of the loud-speaker whose constants are given in Eqs. (28.10). due partly to the inductance but is also due to the fact that the radi- ation resistance is not rising any more, so that it cannot make up for the decrease of u p . In actual loud-speakers the drop is not so rapid, because at high frequencies only the central portion of the piston vibrates, and the effective mass is reduced. Over the useful range of frequencies the over-all efficiency of the loud-speaker is about 1 per cent. At v = 100 the amplitude of motion of the piston (u p /2rv) is about 0.2 cm, so that the mechanism should be designed for an ampli- tude of about 0.4 Cm if it is to radiate 0.1 watt without rattling. The efficiency of the loud-speaker could be increased by decreasing the coil resistance, by increasing the magnetic field, by increasing the 344 THE RADIATION AND SCATTERING OF SOUND JVIL28 number of turns on the coil, or by decreasing the mass of the moving system. If we decrease the coil resistance much, the electrical imped- ance will not be constant over the useful range, and the resonance between piston mass and coil inductance will become prominent. If we increase the number of turns on the coil, we increase the resistance and also the mass, which is not desirable. It would be quite difficult to increase the magnetic field to any extent, though if this could be done the efficiency could be increased without any concomitant ill effects. If we decrease the mass of the system much, it will no longer be purely mass controlled, and the response will not be so uniform. -io 5 n - 100 1000 10,000 Fig. 76. -Velocity amplitude u p and power radiated II when the loud-speaker of Eqs. (28.10) is driven by an emf of 10 volts amplitude. The curves of Fig. 76 are for the loud-speaker out in the open with no obstructions near by. The effect of the resonance of a room on the response curves will be discussed in the next chapter. Transient Radiation from a Piston. — The general equation (27.1) for radiation from an elementary source can be used to compute the transient pressure wave from a piston in a plane wall. Suppose that the velocity of the piston is U(f) and its acceleration is A(t) = dU/dt. Study of Fig. 77 shows that a part of the pressure wave arriving at the point (r,#) at the time t is that due to the acceleration of the strip of length d = 2 \/a 2 — y 2 and width dy a time (l/c)(r — y sin#) VII.28] RADIATION FROM A PISTON IN A PLANE WALL 345 earlier (if r is much larger than a). The amplitude of this component of the pressure pulse is, from Eq. (27.1), dp ~^r a/o ' ~ y 2 A\t - -(r - ysm&)\dy The total pressure is obtained by integrating this over the surface .d1 Fig. 77. — Radiat ion of a transient from a piston. Sound reaching point P came from strip d = 2 \/a 2 - V 2 a time (l/c)(r - 2/ sin #) earlier. Lower curve shows pressure fluctuation at P due to a velocity pulse U = d(t) of the piston. of the piston, from -a to +a. Change of integration variable and integration by parts yields the general formulas pc p = H 2irr sin 2 # _ ,2 Cf _ pc l J Vo 2 sin 2 ?? - (ct + r - ct) 2 A(r) dr (r + ct — ct) U{r) dr 2irr sin 2 # J e Va 2 sin 2 ?? - (r + ct — ct) 2 (28.12) where the limits of integration are e = t - (l/c)(r + a sin/?) and f = t - (l/c)(r - asintf). This is a very interesting formula, for it shows, perhaps more clearly than Eq. (28.1), the dependence of the pressure wave on the 346 THE RADIATION AND SCATTERING OF SOUND [VII.29 angle #. Directly ahead (# = 0) the pressure wave reproduces the piston acceleration exactly: »=te>H> ( *= o) The force on a microphone diaphragm directly ahead of the piston is therefore proportional to the piston acceleration; and if the diaphragm is mass controlled, its acceleration is proportional to the force, so that the accelerations, velocities, and displacements of piston and diaphragm are proportional. As & is increased, however, the integral covers a larger and larger interval of time t, so that more and more of the piston motion gets blurred together in the pressure wave arriving at P. A simple example of this is for the case when the piston suddenly moves outward a distance A. In this case the displacement of the piston is Au(t), where u{t) is the step function defined in Eq. (6.9); the velocity is A 8(0, proportional to the impulse function; and the acceleration A8'(t) is formally proportional to the derivative of the impulse function, a "pathological" function going first to plus infinity and then to minus infinity in an infinitesimal period of time. Its inte- gral properties are V ^ h'{r - a)j{j) dr = ~[(df(a)/da)u(t - a)]. The resulting pressure at (r,&) is !0 (ct < r — a sin#) pc2A r - ct , _ asiiit? <ct<r + asm #) 27rrsm 2 # v 7 a 2 sm 2# _ ( r _ ct y (c/ > r '+ a sin #) This pulse, shown in Fig. 77, is a stretched out version of the 8' func- tion; the greater the angle &, the greater the stretch. Only directly ahead of the piston, at # = 0, is the pressure pulse as instantaneous as the piston pulse. 29. THE SCATTERING OF SOUND f When a sound wave encounters an obstacle, some of the wave is deflected from its original course. It is usual to define the difference between the actual wave and the undisturbed wave, which would be present if the obstacle were not there, as the scattered wave. When a plane wave, for instance, strikes a body in its path, in addition to the undisturbed plane wave there is a scattered wave, spreading out from the obstacle in all directions, distorting and interfering with the plane VH.29] THE SCATTERING OF SOUND 347 wave. If the obstacle is very large compared with the wavelength (as it usually is for light waves and very seldom is for sound), half of this scattered wave spreads out more or less uniformly in all directions from the scatterer, and the other half is concentrated behind the obstacle in such a manner as to interfere destructively with the unchanged plane wave behind the obstacle, creating a sharp-edged shadow there. This is the case of geometrical optics; in this case the half of the scattered wave spreading out uniformly is called the reflected wave, and the half responsible for the shadow is called the inter- fering wave. If the obstacle is very small compared with the wave- length (as it often is for sound waves), then all the scattered wave is sent out uniformly in all directions, and there is no sharp-edged shadow. In the intermediate cases, where the obstacle is about the same size as the wavelength, a variety of curious interference phe- nomena can occur. In the present chapter, since we are studying sound waves, we shall be interested in the second and third cases, where the wavelength is longer or at least the same' size as the obstacle. We shall not encounter or discuss sharply defined shadows. So much of the scattered wave will travel in a different direction from the plane wave that destructive interference will be unimportant, and we shall, be, able to separate all the scattered wave from the undisturbed plane wave. We shall be interested in the total amount of the wave that is scattered, in the distribution in angle of this wave, and in the effect of this scattered wave on the pressure at various points on the surface of the obstacle. Scattering from a Cylinder. — Let us first compute the scattering, by a cylinder of radius a, of a plane wave traveling in a direction per- pendicular to the cylinder's axis. If the plane wave has intensity To, the pressure wave, if the cylinder were not present, would be Vv = P e ik ^~^ = p„e*(«- «»*-««>, p = V2pcT^ k = — ; A where the direction of the plane wave has been taken along the positive z-axis. In Eq. (19.13) we expressed this plane wave in terms of cylindrical waves: Vv = Poe*< ro "*- rf > = P [J (kr) + 2 j? i™ cos (m0)J«(Jiy)]r w '» (29.1) m = \ 348 THE RADIATION AND SCATTERING OF SOUND [VTL29 The radial velocity corresponding to this wave is w«r — " (?j\ iiJ^kr) + 2 i m+1 [Jm+i(kr) - Jailer)] cos(mtf>)l , — 2irivt When the cylinder is present with its axis at r = 0, the wave cannot have the form given by the above series, for the cylinder dis- torts the wave. There is present, in addition to the plane wave, a scattered outgoing wave of such a size and shape as to make the radial velocity of the combination zero at r = a the surface of the cylinder. We shall choose the form of this outgoing wave to be the general series CO p, = 2^ cos (m4>)[J m (M + iN m (kr)]er™" m = u„ = (L\ iiAo[Ji(Jcr) + iNxikr)] + 1 ^ A ™ cos (m<j>)[J m +i(hr) - J m -i(kr) + iN m+1 (kr) - iN m -!(kr)]\ e- 2 ""' The combination J + iN has been chosen because it ensures that all the scattered wave is outgoing. Our first task is to find the values of the coefficients A which make the combination u pr + u sr equal zero at r = a. Equating u„ to — Upr at r = a term by term, we obtain A m = -e^Po^+^-^sinCTm); Po = V^pcjo ) Ji(fco) . _ Jm-x(ka) - J m +x(ka) > (29.2) tanTo = "" mM)'' tanTm ~ N m+1 (ka) - N m ^(ka) ) where eo = 1 and e m = 2 for all values of m larger than unity. These phase angles y m have already been defined in Eq. (26.6), in connection with the radiation of sound from a cylinder. Values of some of them are given in Table X at the back of the book. The behavior of these phase angles completely determines the behavior of the scattered wave. It is interesting to notice the close connection between the waves scattered by a cylinder and the waves radiated by the same cylinder when it is vibrating. The quantities needed to compute one are also needed to compute the other. The pressure and radial velocity of the scattered wave at large distances from the cylinder are vn.29] THE SCATTERING OF SOUND 349 4pcT a , , N .,/ « Pa H — — $ s (4>)e tk(r - c » ; u s ~ — TV pC ^.(0) = —7= 2 Cm sin (Tm)^- 1 ' 7 " 1 cos (m^) TO=0 The intensity of the scattered part, at the point (r,<f>)(kr^> 1), is, therefore, \U<f>)\'< (29.3) A=|ttoi- 00 |^«| 2 = j— ^j e m e„ sin7 TO sin7„ cos (y m — y n ) cos (m^) cos (n<f>) m,n = where e = 1, e m = 2 (m > 0). This intensity is plotted as a function of ^ on a polar plot in Fig. 78, for different values of p = (2rva/c) = (2iroA). It is interesting to notice the change in directionality of the scattered wave as the wavelength is changed. For very long wave- lengths (n small) but little is scat- tered, and this is scattered almost uniformly in all the backward directions. As the frequency is increased, the distribution in angle becomes more and more complicated, diffraction peaks appearing and moving forward, until for very short wavelengths (much shorter than those shown in Fig. 78) one-half of the scat- tered wave is concentrated straight forward (the interfering beam), and the other half is spread more or less uniformly over all the other directions, giving a polar plot which is a cardioid, inter- rupted by a sharp very high peak Fig. 78. — The scattering of sound waves from a rigid cylinder of radius a. Polar diagrams show the distribution in angle of the intensity of the scattered wave, and the lower graph shows the dependence of the total scattered intensity on fi = 2ira/X. in the forward direction, as will be shown in Eq. (29.4). For very long wavelengths only the two cylindrical waves corre- 350 THE RADIATION AND SCATTERING OF SOUND [VTI.29 sponding to m = and m = 1 are important in the scattered wave. As shown in Eq. (26.6), (^) To ^ — 7i ~ I ~~ 4~ J ^ ~ 72 ' —, V3 ' ' ' > wnen ^« — > The first-order approximation for the scattered intensity at long wave- lengths is, therefore, ~ \ 8c 3 r / T (l - 2cos<f>) 2 (coa «c) Short Wavelength Limit. — For wavelengths very small compared with the circumference of the cylinder, the relatively simple approxi- mation of "geometrical acoustics" is valid, with the "scattered" wave dividing into two parts, the "reflected" and the "shadow- forming" waves. However, the process of demonstrating that the series of Eqs. (29.3) really does behave in a simple manner involves mathematical manipulation of considerable intricacy. In the first place the series of Eqs. (29.3), which is in a form useful for calculating scattering at longer wavelengths, is not particularly suitable for showing in detail the interference between part of the "scattered" wave and the primary wave to form the shadow. In optics, one differentiates between "Fraunhofer diffraction," where intensities are measured at distances so large that the angle subtended by the diffracting object is small compared with the ratio (X/2ira) = (1/ka), and "Fresnel diffraction," for distances large compared with the wavelength but not extremely large compared with {%ra). The Fresnel-diffraction formulas show the shadow with its related diffrac- tion bands, but at the great distances involved in the Fraunhofer- diffraction formulas, the shadow has become blurred out again. Series (29.3) is for distances corresponding to Fraunhofer diffraction, so that what can be demonstrated is the separation of the "scattered wave" into a "reflected wave" and a "shadow-forming wave" but not the details of the interference with the incident wave, which are characteristic of the Fresnel formulas. In the second place, the simplicity of the formulas for very short wavelengths only appears as an average; the scattered intensity varies rapidly with angle in a complicated sort of way, and only the average intensity per degree (or per minute) varies smoothly. This rapid fluctuation is seldom measured, however, for any small change of frequency or of position of the cylinder will blur it out, leaving only the average intensity. Consequently, our calculations should separate VII.29] THE SCATTERING OF SOUND 351 the rapid fluctuations of intensity from the average behavior of the "reflected wave." When all the necessary manipulations are made, the expression for the scattered intensity at short wavelengths is T ' ~ Tr sin (l) + SR COt! (l) Sin2 {k " Sin *> + (rapidly fluctuating terms), kr^>ka^>l (29.4) The first term of this expression constitutes the "reflected" intensity, which, for a cylinder, reflects more in the backward direction (# ~ 7r) than in a forward direction (<^ ~ 0). The second term is the "shadow-forming" beam, concentrated in the forward direction within an angle {j/ka) = (X/2o) which is smaller the smaller X is compared with a. The third term contains rapidly fluctuating quantities that average to zero, and so may be neglected. Total Scattered Power. — The total power scattered by the cylinder per unit length is obtained by multiplying T by r and integrating over <t> from to 2ir. The cross terms in the sum of Eq. (29.3) disappear, owing to the integral properties of the characteristic functions cos (ra0), leaving the result x / m = / (67r 5 a 4 A 3 ) To (X » 2xa) l4aT (X«27ra) (29.5) The limiting value for total scattered power for very short wave- lengths is the power contained in a beam twice as wide as the cylinder, (4a). This is due to the fact, discussed above, that the scattered wave includes both the reflected and the shadow-forming waves, the first and second terms of Eq. (29.4). The integral of r times the first term is the total reflected power, which is just 2a T ; and the integral of r times the second term is also 2a T , showing that the shadow- forming wave has enough power just to cancel the primary wave behind the cylinder. The quantity (n s /4aT ) is plotted in Fig. 78 as a function of ka = (2ra/\). Notice that in spite of the various peculiarities of the distribution in angle of the intensity, the total scattered intensity turns out to be a fairly smooth function of ka. We also should note that II S is not usually measured experimentally, because of the diffi- culty of separating primary from scattered waves at small angles of scattering. What is usually measured is more nearly the quantity 352 THE RADIATION AND SCATTERING OF SOUND [VII.29 n ezp ~ 2 jT' r ~ A % r d<f> where A is a small angle, if the experimental conditions are good, but it is never zero. It turns out that n exp is very nearly equal to II, for the longer wavelengths; but as the wavelength is made smaller, the shadow-forming beam (the second term in Eq. 29.4) is less and less included in the integral; until for very short wavelengths n exp is equal to in 8 . The transition from n s to iU s comes at values of ka near (ir/2A), at wave lengths X near 4aA. The Force on the Cylinder. — Returning now to the expression for the total pressure due to both the undisturbed plane wave and the scattered wave, we find, after some involved juggling of terms [the use, for instance, of the last of Eqs. (20.1)], that the total pressure at the surface of the cylinder at an angle from the x-axis is P. = (p, + *)„ = g£) «-«- 2 2^ «' [ " T " +?] (29.6) m = where the quantities C m are defined in Eq. (26.6). This expression is proportional to the expression in Eq. (26.7), giving the pressure at some distance from the cylinder due to a vibrating line element on the cylinder, when we make the necessary change from <j> to tt — <f> (since now = is in the direction opposite to the source). This is an example of the principle of reciprocity. The pressure at a point A due to a source at point B is equal to the pressure at B due to a source at A, everything else being equal. Therefore the polar curves of Fig. 65 show the distribution of intensity about a cylinder having a line source, and they also show the distribution of the square of the pressure on the surface of the cylinder due to a line source at very large distances away from the cylinder (distances so large that the wave has become a plane wave by the time that the wave strikes the cylinder). When n is very small (X » a) the expression for the pressure at r = a reduces to p — ► P (l + 2ika cos 4>)e~ 2Tivt M->0 which approaches in value the pressure P e- 2Tivt of the plane wave alone as n goes to zero. The net force on the cylinder per unit length is in the direction of the plane wave and is Vn.28] THE SCATTERING OF SOUND 353 -f (29.7) f = a p cos <f>d(f> = 4aP ( j—jr ) e^* - *™***/ 2 ' zco(47T 2 a 2 /c)Poe- iat [« « (c/o)] / 4aX P e i( » /C > («-f')+ix/4 ( X « 27ra) This force lags behind the pressure, which the plane wave would have at r = if the cylinder were not there, by an angle — 71 + (ir/2). The limiting* formula, for co small, has been used on page 149 in dis- cussing the forced motion of a string (see Prob. 14). The quantity F/P is the net force on the cylinder per unit length per unit pressure of the plane wave. This quantity, divided by 2ra, is plotted in Fig. 79 as a function of ka = ix. We note that for small 0.8 r Fig. 79. — Amplitude of sideward force per unit length F on a, cylinder of radius o due to the passage of a plane wave of pressure amplitude P„, plotted as a function of ft = 2ira/\. frequencies the force is proportional to the frequency (i.e., to ft) but that when /* becomes larger than unity (i.e., when the wavelength becomes smaller than the circumference of the cylinder) the linear relation breaks down and the force diminishes with increasing fre- quency thenceforth. This result is of interest in connection with the so-called velocity-ribbon microphone, which consists of a light metal strip more or less open to the atmosphere, pushed to and fro by the sound wave. The ribbon is in a transverse magnetic field, so that the motion induces an emf along the ribbon, which actuates an amplifier. The net force on the strip is, of course, not exactly the same function of ft as that given in Eq. (29.7) for -the cylinder, but the behavior will be the same in general. The force on the strip will be proportional 354 THE RADIATION AND SCATTERING OF SOUND [VII.29 to the frequency for small frequencies, but this linear dependence will break down when the wavelength becomes smaller than twice the width of the strip. There is an approximate method of finding the net force on the cylinder which gives the correct result for wavelengths longer than the circumference of the cylinder. If the pressure in the plane wave is P e ik(Xr - ct) , the pressure at the surface of the cylinder due* to the plane wave is P^^^-^ ivt . If ^ is small, this can be expanded into p p ~ P (l + i/x cos <}>)e- 2irivt . This is the pressure due to the plane wave; there is also a scattered wave, enough of a wave to make the radial velocity at the surface come out to be zero. This scattered wave con- tributes a term ifiPo cos <$> e~ 2 * ivt to this approximation to the pressure, so that the net pressure is that given in Eq. (29.6), and the net force is the limiting value given in (29.7). We thus see that even for very long waves the distortion of the plane wave due to the presence of the cylinder contributes a factor 2 to the net force on the cylinder. Scattering from a Sphere. — The analysis of the scattering of waves from a spherical obstacle follows exactly the same lines as that for the cylinder. The expression for a plane wave traveling to the right along the polar axis is oo p p = p oe «(rcostf- c t) = p ^ (2m + \)i m P m {cosd-)j m {kr)e~^ ivt (29.8) m = where P = \/2p c T , and where the factors P m and j m are defined in Eqs. (27.6) and (27.10), respectively. The expression for the wave scattered from a sphere of radius a whose center is the polar origin is oo Vt = -p ^ (2w + l)i m+1 e- iS » sin 5 TO P TO (cost?). m = • [j m (kr) + m OT (/cr)]e- 2 ™< (29.9) where the angles 8 m have been defined in Eqs. (27.17) in connection with the radiation from a sphere. The values of some of them are given in Table XI at the back of the book. The intensity of the scattered wave and the total power scattered are a 2 To 00 1_ ^ (2m+l)(2n + l) r 2 k 2 c m,n = • sin 8 m sin 8 n cos (8 m - 8 n )P m (cos#)P n (cos#) ( (29.10) ( (16ir 4 i' 4 a 6 To/9c 4 r 2 )(l - 3 cos#) 2 (ka « 1) \ (a 2 /4r 2 ) + (a 2 /4r 2 ) cot 2 (&/2)J\(ka sin &) {ka » 1) VII.29] THE SCATTERING OF SOUND 355 (29.11) n, = 27ro 2 To(2/fc 2 a 2 ) J; (2m + 1) sin 2 5 m m = ( (2567r 5 a 6 /9X 4 ) T • (X » %ra) (27ra 2 To (X « 2wa) The discussion concerning the short wavelength limit for the scattering from a sphere is similar to that preceding and following Eq. (29.4) for the cylindrical case. The total power scattered is that X=2jtci n/^2 12 3 4- Fig. 80. — Distribution in angle of intensity scattered from a sphere of radius a and total power scattered II, per unit incident intensity. contained in an area of primary beam equal to twice the cross section ira 2 of the sphere. Half of this is reflected equally in all directions from the sphere (the first term in the last expression of Eq. 29.10); and the other half is concentrated into a narrow beam which tends to interfere with the primary beam and cause the shadow (the second term in the last expression). If the experimentally measured "total scattered power" includes everything from # = A to & = ir — A, n exp will equal II 8 for wavelengths longer than (4a/ A) and will approach ill s for wavelengths much shorter than (4a/ A). Figure 80 shows polar curves of the scattered intensity as a function 356 THE RADIATION AND SCATTERING OF SOUND [VH.29 of the angle of scattering #, for different values of n = ka, and a curve n s as function of ix. As with the cylinder, the directionality of the scattered wave increases as the frequency increases. The Force on the Sphere. — The total pressure at a point on the sphere an angle # from the polar axis (note that the point # = is the point farthest away from the source of the sound) turns out to be oo p a = iVr*"«(l/A;a) 2 ^5 ??Ltl p m (cos #)*-*<«■*-»«»> m = ~ (1 + $ika cosifyPoer*-* (ka « 1) (29.12) As with the cylinder, this expression is proportional to that for the pressure, at large distances, due to a simple source set in the sphere. The curves of Fig. 66 therefore show the dependence of pi on &. v % 1 8 Fig. 81. — Ratio of pressure amplitude at a point on a sphere to pressure amplitude of the plane wave striking the sphere, plotted as a function of ix = (2ira/\). Solid line is the pressure at a point facing the incident wave; dotted line is the average pressure for a circular area around this point with an angular radius of 30 deg. The amplitude of the pressure at the point nearest the source of sound (# = ir), for a plane wave of unit pressure amplitude, is plotted in Fig. 81 as a function of /*. We notice that for wavelengths long compared with the circumference of the sphere the pressure at # = t equals the pressure of the plane wave but that for shorter wavelengths the distortion of the wave due to the presence of the sphere makes p a differ from P . This general fact will be true for obstacles of other than spherical shape, even though the dependence of p a on n will be some- what different from that given in Eq. (29.12). Therefore, a micro- phone measures the pressure of the wave striking it only as long as its circumference is smaller than the wavelength of the wave. For VII.29] THE SCATTERING OF SOUND 357 smaller wavelengths a correction must be made for the distortion due to the presence of the microphone. By using Eqs. (27.7) we can find the average value of the pressure on that portion of the surface of the sphere contained between the angles # = t and # = -k — # : Pav = j r* cos# J*~» V ' sinddd = Poe-^ 1 >b i m e- i5m pm-i (cosflp) - P m +i (cos^o) l ^3 k 2 a 2 D m I P_ x (cos# ) - Pi (cos #„) iPoe-^l + £ifca(l + cos do)] (ha « V (29.13) where P_i(cos# ) = 1. The dotted line in Fig. 81 gives the values of (pav/Poe - ^) as function of n = ka for the case # = 30 deg. This curve will be useful in discussing the behavior of microphones later in this chapter. Design of a Condenser Microphone. — As an example of the use of the scattering formulas derived above, let us discuss the behavior of a condenser microphone in a spherical housing. We shall have to make a number of simplifications that do not correspond to actuality in order to avoid confusing complications. We shall try to include, however, enough of the complications encountered in actual practice to show the difficulties involved and the relative effect of the various complicating factors. The first simplification that we shall make concerns the container for the microphone. We shall assume it to be spherical in shape, because we have analyzed the effect of a sphere in distorting a plane wave, and we have not so analyzed the effect of a cube, for instance. The effect of the sphere will be enough like the effect of any other object of about the same size so that the behavior that we compute will be typical. The microphone that we shall use will be a resonator-coupled condenser microphone, arranged as shown in Fig. 82. The radius of the sphere will be 5 cm and that of the diaphragm will be 2.5 cm, so that the angle of the circular opening in the sphere is 30 deg = # . Fig. 82. — Simplified cross section of the condenser microphone whose constants are given in Eqs. (29.14). 358 THE RADIATION AND SCATTERING OF SOUND [VH.29 The diaphragm is set back 1.2 cm from the opening; any larger dis- tance would make the first cavity resonance come at too low a fre- quency. The spacing between the condenser plates will be 0.02 cm. We wish the microphone to respond up to 8,000 cps, and, referring to Fig. 44, we see that the lowest resonance frequency of the diaphragm cannot be less than 4,000. If the diaphragm is aluminum, 0.0015 cm thick, the membrane density <r is 0.005 g per sq cm, and the tension must be 3 X 10 6 dynes per cm for the frequency v \ to be 4,000 [see Eq. (20.9) and the paragraph preceding]. If we design the air space in back of the diaphragm properly, we can make the damping constant 6d [see Eq. (20.8) come out to be 3, by using damping material behind the plates. The constants of the system are therefore a = 5; b = 2.5; # = 30°; 6 d = 3; I = 1.2 a = 0.005; A = 0.02; v i = 3,800 ha = (j^) ~ (V1,000); n = (~\ ~ OMa \ (29.14) ©-0.06^ ■ »-(£*) -*. where c a is the velocity of sound in air and Ca that of transverse waves in the membrane. Behavior of the Microphone. — Our first task is to determine the average pressure at the surface of the opening in the sphere. This has been done in Fig. 81, for the case when the opening is pointed straight at the oncoming plane wave. There will be, of course, a different curve if the microphone is pointed in some other direction. Next, we must determine the pressure at the diaphragm, which is set at the back of' a tube of length I and radius 6. This quantity is given by Eq. (23.28) for cavity response and is shown in Fig. 57. Equation (23.28) must be modified so as to use expression (27.20) for the impedance of a hole in a sphere, instead of expression (28.6) for the impedance of a hole in a plane. The modification is not large. The ratio of the average pressure at the diaphragm to the pressure of the undistorted plane wave is shown in Fig. 83 as a function of frequency. The slow rise of the curve is due to the increasingly important distorting effect of the sphere on the wave as v is increased; the large peak at about 4,000 cps is due to cavity resonance. Finally, having obtained the pressure at the diaphragm, we can use Eq. (20.12) to determine the electrical response of the microphone. This response can be expressed as follows: If the intensity of the sound VH.29J THE SCATTERING OF SOUND 359 in the plane wave is T , and if the microphone plate is charged to a potential of E volts, then the amplitude of the emf delivered to the 10,000 Fig. 83. — Ratio of pressure amplitude at the diaphragm of the microphone shown in Fig. 82 to the pressure amplitude of the incident plane wave. 2.0r 500 1000 Frequency, -v Fig. 84. — Response curve for the microphone shown in Fig. 82. is given in terms of Hiy) by Eq. (29.15). amplifier is E = 0.93 X 10- 3 VTo EoH(v) 5000 10,000 The output voltage (29.15) where H(v) is plotted against v in Fig. 84. A resonance peak for the diaphragm-resonator system has merged with the one for cavity resonance at the upper part of the useful range. The sudden drop in response above this is due to the fact that the waves on the dia- 360 THE RADIATION AND SCATTERING OF SOUND [VII.30 phragm become short enough so that the motion of one portion cancels that of another, as was discussed on page 202. Compare this with the second curve of Fig. 44. The response could be made larger by decreasing the resonance frequency of the diaphragm, by increasing E , by decreasing the spacing A, or by reducing the diaphragm density a. The diaphragm has already been made as thin as it can reasonably be, so that <x cannot be much reduced. Presumably, E is made as large as it can be for the spacing A. Decreasing the resonance frequency voi would mean reducing the useful range of frequencies of the microphone, which is undesirable. The diaphragm cannot be set any farther back in the sphere, for then the cavity resonance frequency would intrude on the useful range. The sphere should not be made any larger, for it would then distort the plane wave to a greater extent. Perhaps a slight advan- tage would be gained by reducing the values of a and I below those chosen. 30. THE ABSORPTION OF SOUND AT A SURFACE In the previous discussions of scattering we have assumed that the scattering surfaces were rigid, an approximation that is not always good enough. It is now time for us to take up the task of discussing just what a surface does when it is acted on by a sound wave and of computing what effect this behavior has on the sound wave. Surface Impedance. — Of course different sorts of materials react differently to sound, depending on their structure. Nonporous mate- rials yield slightly, since they are never perfectly rigid, and porous materials also allow some air to penetrate below the surface, producing an additional effective motion of the surface. In any of these cases we can express the reaction of the surface in terms of a specific acoustic impedance, a ratio between pressure at the surface and the normal velocity of the surface. This impedance in general depends on the nature of the surface material, on the frequency of the wave, and on its angle of incidence (we at first consider only plane simple harmonic waves; later we shall build up more complex waves out of these). Just as occurred with the problem of the reaction of the air on a membrane, discussed on page 335, we find that the amount of depend- ence of the impedance on the angle of incidence of the wave depends on how well wave motion can travel, in the surface material, parallel to the surface. If such wave motion is rapidly attenuated or is con- siderably slower than that of sound in air, then the impedance of the VII.30] THE ABSORPTION OF SOUND AT A SURFACE 361 surface is nearly independent of angle of incidence, for in this case one part of the surface is not aware of the motion of another part, and the reaction of one part of the surface is proportional to the local pressure at that point. Such surfaces can be called locally reacting surfaces. On the other hand if the wave motion in the surface is not attenuated, and is as fast as or faster than that in the air, then the reaction of one part of the surface will depend on the motion of other parts of the surface and the surface impedance will depend on the angle of incidence of the wave. Such surfaces can be called surfaces of extended reaction. We shall consider a few typical cases in order to illustrate these general statements. Unsupported Panel.— One of the simplest sorts of surface is an impervious plane panel of dimensions that are large compared with the wavelength in air. We assume that its outer surface at equilibrium is in the (?/,2)-plane and that the panel supports are far enough apart so that they do not play a part in the reaction between panel and air. According to Eq. (21.1) the equation of transverse motion of the panel is «.£--<">, + 9, m ;:^t /3(1 _ s2)] CO.!) where p is the pressure of the air, Q and p the modulus of elasticity and density of the material, 2h its thickness, and v its displacement away from the air (in the negative ^-direction). The constant g is called the flexural rigidity of the panel, and m s is its mass per unit area. The velocity of simple harmonic transverse waves in the panel is c a = Vco (g/m.)*. Now suppose that the wave in air has an angle of incidence $, so that the pressure has the form V = 2Poe*<«"> <»■•»*-*> cos [7— ) cos* + *] The air velocity perpendicular to the surface is u x = i —^ c *(./e)c»-«^ef) sin \h£\ cos* + *1 The value of the phase angle ^ is determined by solving Eq. (30.1) for n and then setting - (dv/dt) equal to u x at x = 0. The solution for — (dij/dt) is 362 THE RADIATION AND SCATTERING OF SOUND [VII.30 _dy _ / 2Po cos * \ c i(a/c) (^^^ dt \ z* ) z s = ^ = -ia>m s + ig (^-)sin 4 $ and the solution for the phase angle is tan 1 ^ = il — J = *'(«• ~ **») (30.2) where z s is the specific acoustic impedance of the panel for the angle of incidence <S>, and k 3 and a s are its acoustic conductance and sus- ceptance ratios. The equation giving z s in terms of a, m„ g, and * shows that this impedance is purely reactive and that at high frequencies it depends very strongly on the angle of incidence. At frequencies low enough so that c s , the speed of transverse waves in the panel, is considerably smaller than c, the speed of sound in air, the specific acoustic imped- ance of the panel becomes z s ~ -i(am s (« « c 2 y/m,/g) a pure mass reactance, due to the mass of the panel. This illustrates the general statement made at the beginning, that if the speed of wave motion in the surface is much slower than that of sound in air the surface has local reaction and the impedance is independent of angle of incidence. When the impervious panel is quite thin (as with the fabric lining of an airplane cabin, for instance), the flexural rigidity is extremely small and z s c^ -i<am 8 over the whole of the useful fre- quency range. For some laminated panels the flexural rigidity factor is complex, g = g s — ir s , so that the impedance has a real part = -wrn, + (ig. + r.) I ^-jsin 4 * and energy is lost by motion of the surface. The amount of energy lost is strongly dependent on frequency and on angle of incidence, however. For such a panel, with very little support, the impedance becomes very small at very low frequencies. Supported Panel.— This first example, which has just been dis- cussed, is one seldom encountered in practice; it is more usual to have the surface material supported by a heavier structure, with distances VII.30] THE ABSORPTION OF SOUND AT A SURFACE 363 between the points of support of the same size or smaller than a wave- length. In this case, transverse waves cannot be propagated far along the surface without being stopped by the supporting structure, so that the panel is, on the average, a locally reacting surface. Each portion of panel between supports is a plate with effective specific admittance (l/s P ) equal to the ratio of average velocity u x , averaged over the portion of panel, to the driving pressure. At the higher frequencies this impedance will show a mass reactance z s ~ -t'ww B , but at low frequencies the stiffness due to the support structures becomes important and z s ~ iK s /a>. There is usually a resistive term. Consequently, the specific acoustic impedance of a supported, impervious panel can be represented by an equivalent circuit of inductance ra s , resistance R s , and capacitance C s = 1/K S in series. Porous Material.— A case often encountered is that of a surface which is porous enough so that the air motion normal to the surface is due more to air motion into and out of the pores than to motion of the panel as a whole. Here we must take into account the wave motion of the air in the pores. The properties of the pore system can be expressed in terms of three quantities: flow resistivity r p , porosity P p , and effective air density pm p . Under steady-state conditions a pressure drop (Ap) across a thick- ness d of the material will force a flow of air through the pores of u x = r p d(Ap) cc per sec per sq cm of surface of the material. This defines the flow resistivity r p . If the inner side of the material is made impervious, then forcing air into the material will raise the pressure of the air in the pores. If the air were compressed adiabatically, as would happen in the open, the rise in pressure would be (pc*/P p d)% [see (Eq. 23.3)] where £ is the displacement of the air into the outer surface (in cubic centi- meters per square centimeter of surface) andP p is the fraction of the volume of the material that is available for the air to flow into (this defines the porosity P p ). In many cases the pores are so small that the air loses its heat to the pore walls, so that the expansion is iso- thermal instead of adiabatic. Then the rise in pressure would be (pc 2 /ycPpd)%, where y c = 1.4 for air. Finally, the air in the pores may have an effective density greater than that in free space because some filaments of the pore material may move with the air, so that m p may be greater than unity. The equation of motion of the air in the pores is therefore equiva- lent to that for electric current in a transmission line, with line resist- ance r p per unit length, line inductance P m p per unit length, and shunt 364 THE RADIATION AND SCATTERING OF SOUND [VII.30 capacitance (ycP p /pc 2 ) per unit length. The equations of motion of the air in the pores, for simple harmonic waves, are —iupnipU + r p u = — grad p | „ , .ycP pP o>, . , n > (30.3) V 2 p + i - — ~- (r p — ia)m p p)p = I pc ) and the characteristic impedance of the material is = P Cyj m p + i(r p /po) 7cP P which is a complex quantity, indicating attenuation of the wave. . Equivalent Circuits for Thin Structures. — If the flow resistance r p is fairly large, wave motion will attenuate rapidly in the material and the pores will exhibit local reaction to the sound waves. This is the case usually encountered in practice, although for some materials (such as hair felt) extended reaction is not negligible. When local reaction predominates, so that the impedance is independent of angle of incidence, and when the thickness of the porous material is small compared with the wavelength, then the specific acoustic impedance of the porous material can be expressed in terms of equivalent circuits for different structures. Figure 85 shows some equivalent circuits for a few structures. In the first case the porous material is mounted on a relatively rigid and impervious wall, so that the air motion in the pores is responsible for the acoustic impedance. The important part of the impedance is the capacitative reactance of the pore volume. The effective resist- ance and inertia are reduced by a factor i, because the air motion in the pores nearest the backing wall is constrained to move less than the air in the pores near the outside. In the second example shown in Fig. 85 the panel is held away from the impervious backing wall by a framework, which prevents flexural waves from traveling along the panel, and which also dis- courages lateral wave motion in the air space behind the panel. Here the panel can move by flexure, as well as allowing air to move through the pores, so that the equivalent circuit shows three parallel paths, one for flexure, one for transmission through the pores, and the third due to the stiffness of the air in the pores. The third example shows that laminated structures are equivalent to filter networks and that the equivalent circuit can be used to calcu- VH.30] THE ABSORPTION OF SOUND AT A SURFACE 365 late transmission of sound through wall structures as well as absorp- tion of sound by the structure. Porous Rigid material { backing o°<y °oo° ° ° ' O oo 0< --d V//; J -AWVAAr- YcPpd pc* mppd ^S S^ o Impervious septum a mass ma ■I- i Impervious septum b mass m/, r p d m p pd jL, pc 2 ' ^AAAAArJpTinT 1 -' B m b "pc 2 r Fig. 85. — Equivalent electrical circuits giving specific acoustical impedance of various wall structures. Equivalence is valid as long as wavelength is long compared to t, I or d. Formulas for Thick Panels.— When the thickness of porous mate- rial is greater than a wavelength of sound in the material, the analysis becomes more difficult. The wave equation for motion of air in the pores can be written in the form te . + <» + *>'(!) , i»-o d 2 p (n + i q y = y J> p [m p + i(^] 366 TEE RADIATION AND SCATTERING OF SOUND [VII.30 and the solution, for the case of material of thickness d f backed by a wall (or air space) of characteristic impedance z b (at x = — d) p = A cosh I- J {q - in)(x + d) + ^ - - feL - * n -^f *-*[© ( * - in)d + *] (m4) ^ = tanh-^To^^/pc^ + n)\ When the frequency is high enough, or the thickness of the material is great enough, the wave effectively dies out before it is reflected from the back of the material, the hyperbolic tangent is approxi- mately equal to unity, and the wall impedance becomes the character- istic impedance of the material, z pc . For lower frequencies, Plates I and II can be used to compute the impedance. The quantity n can be called the index of refraction of the material for sound waves, and q can be called its attenuation index. When n is larger than unity, the wave travels more slowly in the medium than in air; and when q is large, the wave attenuates rapidly. In either case the material would be (more or less) locally reactive, and the wall impedance would be (approximately) independent of angle of incidence. Reflection of Plane Wave from Absorbing Wall.— We must next investigate the behavior of sound waves that strike absorbing surfaces. We have seen from the preceding discussion that in a great many cases the surface reacts locally, so that its specific acoustic impedance is practically independent of angle of incidence. We shall make our calculations for this case, because the results will be valid for most types of surface material, and also because the calculations are easier. In the rest of this chapter and in the next chapter, therefore, we assume that the acoustic properties of a surface are given by its specific acoustic impedance, the ratio between the pressure at a point on the surface and the normal velocity, into the wall, of the air at the surface, pet = pcir!*-** The ratio of the impedance z to the characteristic impedance of air pc will be called the acoustic impedance ratio of the surface, and its reciprocal will be called the acoustic admittance ratio of the surface. fe)-' — * (?) v = K — ia (30.5) VII.30] THE ABSORPTION OF SOUND AT A SURFACE 367 where 0, x, *, and o" are the acoustic resistance, reactance, conductance, and susceptance ratios of the material, respectively. A thin mem- brane has a specific reactance coM and susceptance — (1/coilf), where M is the mass of the membrane per unit area; a stiff supported panel of small mass has specific reactance —(K/u>) and susceptance {<u/K), and so on. Now suppose again that the surface is at the (?/,z)-plane, with air on the positive x side. An incident wave comes in from the right at an angle of incidence * and is reflected from the wall. The equation for the pressure wave is sq = p . (>i(fi>/c) (y sin*— x cos*— ct) p g— i(w/c)(y s in*+a; cos *— ct) where the ratio between the reflected pressure and the incident pres- sure is e -2*(«-*»(p r = p t . e -2 1 r(a-i^)) an( j th e ra ^ between the reflected and incident intensity is e -4 ™. The pressure and air velocity can be expressed in terms of hyperbolic functions : u x — p = -2Poe~ ,r(a_ ^ ) e i(a,/c)(j ' 8in *~ c0 sinh<ir (-?) cos* - a + \> 2 e -ir(a-*0) e »(<o/c) („ sin *- ct ) cog $ g^ J ^ U ^ J CQ g $ _ a _|_ ^ K The values of a and are determined by the impedance of the surface P — P tanh[7r(a - iff)] = * r = f cos* = (0 - i%) cos* (30.6) ±i ~~\~ ± r so that, if the specific acoustic resistance and reactance of the surface are known, the values of a and /3 for a given angle of incidence * can be computed from Plates I or II at the back of the book. The ratio of reflected to incident energy is — o — iva = 1 — f COS* 1 + £ cos* 2 = (1 — 0cos*) 2 + x 2 cos 2 * f „ (1 + cos*) 2 + x 2 cos 2 * ^ ' Curves giving values of this ratio, the fraction of incident intensity which is reflected, as function of |f| cos*, for different phase angles <p of the wall impedance, are shown in Fig. 86. We note that the amount reflected has a minimum value (and therefore the fraction of incident energy absorbed has a maximum) when |f| cos* = 1. The larger the power factor cos <p for the absorbing surface, the greater the frac- tion of energy absorbed, or the smaller the fraction reflected. When the power factor is zero (<p = ±90 deg) , no energy is absorbed. 368 THE RADIATION AND SCATTERING OF SOUND [VH.30 The curves also show that the fraction of energy reflected approaches unity (fraction absorbed approaches zero) as |f| cos<I> approaches zero, i.e., as the angle of incidence approaches 90 deg. In fact Eq. (30.7) indicates that the surface would not absorb any energy from a wave traveling parallel to the surface, no matter what the value of z. This seems to be a contradiction of terms, for the pressure fluctuations in a wave parallel to the surface would cause motion of the surface in a direction perpendicular to the assumed direction of the wave. The fact of the matter is that a plane wave 5 ISkos$ Fig. 86. — Reflection coefficient for plane waves incident at angle of incidence $ on a plane surface of acoustic impedance 2 = pc|f \e — iip. cannot travel parallel to an infinite plane surface of noninfinite acoustic impedance. This will be shown in more detail' in the next section. It can also be seen from the equations for p and u x . We notice that both of these go to zero when $ = 0, so that a solution in terms of plane waves alone is not possible. 31. SOUND TRANSMISSION THROUGH DUCTS Before we discuss the behavior of sound waves in rooms, we should return once more to the problem of the transmission of sound through ducts, in order to investigate the effect of the impedance of the duct wall on the sound wave. This investigation was begun in Sec. 26, where Eq. (26.14) gave an approximate expression for the transmis- sion of sound in a cylindrical duct. Now we wish to carry the analysis further. In recent years the problem of sound transmission in venti- lation ducts has become important, so it will be useful to show how to compute the relation between sound attenuation and duct wall imped- ance, in order to be able to design duct linings to absorb undesirable VII.31J SOUND TRANSMISSION THROUGH DUCTS 369 sound. Since most ventilating ducts are rectangular in cross section, we shall confine ourselves to this simple type in the present section.' Boundary Conditions.— We assume that the duct cross section is rectangular, of width l v in the ^-direction and I, in the z-direction. We also assume that the impedance of the duct walls is uniform, the acoustic impedance ratio being f for each wall. The pressure, for a wave traveling to the right along the duct, is given by the equation cosh /2^\ cosh/2*^ x _ ct) smh \ ly / smh \ h / where r 2 = 1 + (27rc/a>) 2 [(&A)2 + fo./Z,) 2 ], and where the ^'s are determined by the boundary conditions at y = ± (l v /2) and z = ± (l z /2) (we are taking the central axis of the duct to be the origin of the lateral coordinates y and z). The air velocity into the duct wall at y = +(l y /2) must be equal to the specific acoustic admittance (77/pc) = (1/pcf) times the pres- sure at the wall. ,, _ (2*yv\ a sinft / \ COSh/^TIYkzX ., . ,, U " ~ WJ A c 0S h <■*> sink (,-177 6 (r}\ . cosh, cosh (%cqj\ .. . ., This produces the equation for determining g y in terms of r\ : Toth ^> =4„)ft) ; X = (WU) (3L1 ' where the hyperbolic tangent is used for the waves that are symmetric about the center plane y = . (p proportional to cosh(2jr^ tf y/^», and the tangent is used for the antisymmetric modes (p proportional to the sinh). This equation also ensures that the boundary condition at y = — (lv/2) is satisfied, for everything is symmetrical (or antisymmetrical) about the center plane. The boundary condition for the surfaces z = ± (l z /z) is a similar equation tanh coth to*) = *-(^(k); x = (2Wco) The exact solution of these equations is a complicated process, so we shall first obtain an approximate solution, valid for values of wall admittance ratio 17 small compared with (\/l) (large wall impedance). 370 THE RADIATION AND SCATTERING OF SOUND [VII.31 Approximate Solution. — When |i?|(^/X) is small compared with unity, Eq. (31.1) can be solved approximately. There are a large number of solutions, each with a characteristic value of g. They can be put in order of increasing size and labeled with an index number n v = 0,1,2 • • • . For reasons that will be apparent shortly, the solutions using the hyperbolic tangent will be labeled with the even values of %, 0,2,4 . . . , and the solutions using the cotangent will be labeled with the odd values, 1,3,5 .... The properties of the functions tanh and coth which are used to obtain the approximate solutions are tanh (w + iirn) ~ w; coth (w + iirn + iwr) c^ w, w « 1 Using these expressions, we find that the approximate solutions for the characteristic values of g„ and g e are ***-+(^)-*^ +0 '- < ' ) fe) ( "-" 1A8 ' ■ ' with the even values of the n's going with the cosh factor in the expres- sion for the pressure and the odd values going with the sinh factor. The solution for n y = n z = is the "plane wave" or principal wave, which usually has the smallest attenuation and carries the greatest part of the power. The equation for the transmission coefiicient t gives, to the first approximation in the small quantity rj, t(0,0) c* 1 + ^g (<r + in), n v = n e = n* = (31.2) r(n y ,0) cr (n y ,0) + ^fcfi^th + * 1 ^ + «)■ r(0,n.) ^ r (0,n 2 ) + ^y ^ (h + 2l y ){a + «), n y = ^ ^^ r{n v ,n z ) ^ro(n y ,n g ) + ^-^ ~ (<r + «) (ny > 0, n* > 0) ro(n,,n.) = [l - Qg) ~ (^) J VII.31] SOUND TRANSMISSION THROUGH DUVTS 371 where S = l y l z is the cross-sectional area and L = 2(l y + l z ) is. the perimeter of the duct cross section. We note that when n u = the duct walls perpendicular to the y-axis (l z ) are half as effective in chang- ing r as is the case for n y > 0, and similarly for n z and l y . Since the case n y = corresponds to waves traveling "parallel" to the sides l a and the cases n y > correspond to waves being reflected back and forth between these walls, we can reword our remark to say that the walls have (to this approximation) half as much effect on waves travel- ing "parallel" to them as they do on waves that are obliquely incident. The reason we have put quotation marks around the word parallel is that the case n y = does not correspond to waves exactly parallel to the I, walls. In Eq. (30.7) we showed that, if a plane wave could move parallel to a plane wall, the wall impedance would have no effect on the wave. We also showed that this was impossible, and the present results reinforce this, for we find that when the wave is as "parallel" to the wall as we can make it (n„ = 0) the wall impedance still makes itself felt; only half as much as for oblique waves, but not zero times as much. Principal Wave. — This property of waves and nonrigid walls can be clarified to some extent by studying the behavior of the principal wave (n y = n z = 0). Remembering that rj is small and that the expansion of cosh (u) for u small is 1 + %u 2 , we have V ^ A [l + ijL (a + iK)y^ \l +q(? + i*> 2 . e <V2S)(t<r-«)* e i( M /e)(*-rf) > (jl y = Uz = Q) (31.4) In the first place, we see that the wave is not a perfect plane wave, for the pressure amplitude depends on y and z and therefore there is some air motion perpendicular to the #-axis (just enough to take into account the effect due to the sideward motion of the duct walls). In addition the wave is damped, as indicated by the factor e-(^««/ss) j by an amount proportional to the acoustic conductance ratio k of the walls, proportional to the circumference of the duct, and inversely proportional to its cross-sectional area S. The attenuation of the principal wave in decibels per centimeter is therefore approximately (4.34Lk/£) (if L is measured in feet and S in square feet, the attenuation will be 4.34Lk/£ db per ft). We note that the greater the wall area is, compared with the duct volume (L is, compared with S), the greater the attenuation. The attenuation of the higher modes is larger than this, being approximately (SMLK/Sroiny,^)) for n y > 0, n z > 0; and being 372 THE RADIATION AND SCATTERING OF SOUND [VTI-31 (4.34/c//STo(n„,0)) (2l y + l z ) for n g = 0, etc., as long as n y , n z are small enough so that to is real. For all modes above certain values of n y , n z , to is imaginary for a given frequency, and no true wave motion can occur; the attenuation for all higher modes being very much greater than for the ones below the cutoff (see page 308). The characteristic acoustic impedance of the principal wave is u x i(a)/c) — (Lrj/2s) (i _ _£_r \pc 2icoSzp/ where z v is the specific acoustic impedance of the duct walls. There- fore the characteristic wave impedance of the principal wave is anal- ogous to a circuit with resistance pc in parallel with an impedance —ica(2Sz p /Lc). For instance, if the duct walls are covered with porous material of thickness d, flow resistivity r p , porosity P p , and negligible density m p , the equivalent circuit giving the characteristic impedance of the principal wave has one parallel arm a pure resistance pc and the other parallel arm a resistance pc(2S/jcPpLd) in series with an inductance (2Sr p d/3Lc). Transient Waves. — We can use the results of the preceding discus- sion to obtain a first approximation to the transient behavior of sound in a long duct. Suppose that the average air velocity at the input end is the arbitrary function u x0 (t). Then the pressure in the principal wave, a distance x along the tube (assuming no reflection from the far end), is *»H£j>( i+ 5' ,+ *' , X i -&r t* 00 . e -(L V /2s)x+i(w/c) X -iwt j w a . (r)e iuT dr In several cases, where the wall admittance 77 has a simple behavior, this integral can be evaluated. For instance, if the side walls are stiffness controlled, 17 = — iwe where e is the specific elastance of the walls, we use the equations, derived from Eqs. (2.19) and (2.20), .. /» * /» 00 f(t — a) = -=- I e iaa ~ iat du I f(T)e iaT dT 2ir J — 00 J — * 1 f°° f °° d 2 J- I ^ w 2 e--* dco j_J(r)e^dr = - -^f{t) VH.31] SOUND TRANSMISSION THROUGH DUCTS 373 (to be used with due regard for convergence;) to obtain P(t) ~ 1 + (Lce/2S) f'° [' ~ X (~c + M)] ' -;8+d£-['-6 + S)]} It is not very difficult to show that if the wall impedance is a pure resistance, independent of 00(77 = k, a constant), then p(t) ~pcj- t <~ I e-wwuxo (t -~)dw -K£ + d^"*-M)} (31 - 5) The proof of this formula will be left to the reader. The transient waves in the higher modes can be computed in a similar manner, though the calculations are more difficult. The Exact Solution. — In a number of cases, the specific acoustic impedance ratio of the wall, f , is small enough so that the approxi- mations used in Eqs. (31.2) to (31.4) are not valid. Since these cases are also of practical importance, we must set about finding an exact solution of the equations O*)-**; ,-«+>; k-& -^ (31 . 6 ) where z p = Izpler** = (pc/rj) is the specific acoustic impedance of the duct wall. In the approximate solutions, n turned out to be approxi- mately half an integer and | was small; we cannot expect this to be true for the general solution. We shall find a series of solutions, however, with a series of allowed values for n, which can be arranged in order of increasing size, and labeled /i , in, m • • • . Corresponding to each ju„ is a solution for £„, and therefore a g(n) = £„ + ifi n , which is the nth characteristic value. The corresponding characteristic posh function is $ n (y) = ^^ l^rg y (n y )y/l v ] where the cosh function is used for the even values of n y [which are obtained by solving Eq. (31.6) with tanh] and where the sinh is used for odd values of n y [for which coth is used in Eq. (31.6)]. The easiest way to solve Eqs. (31.6) is by graphical methods. The solutions are displayed on Plate V at the back of the book, in terms of contours for constant values of h and <p drawn on the (£,/x)- plane. We see that the solutions approach the integer and half- 374 THE RADIATION AND SCATTERING OF SOUND [VH.31 integer limits on the imaginary axis, as h gets small, as demanded by the approximate solution. The heavy dashed lines separate the modes one from the other. For each pair of values of h and <p there is a point within each modal region. The point in the lowermost region in the "tanh" chart corresponds to go (cosh is used in the expression for p); the point in the lowest region in the "coth" chart corresponds to gi (sinh is used for p) ; the point between the first and second dashed lines (second sheet) of the coth chart corresponds to g 2 (cosh is used for p) ; and so on. The dependence on x is by means of the exponential e i( - WTX/c) , where = 1 + WW] +m) (v)"0 = 1 _ ( ^ j I My(^y) - kl(n y ) n\(n,) - £(n y ) 2j M»(Wy) €»(%) _ 2l - /*«(»«) €«(»». '] The real and imaginary parts of tn (N stands for the pair of numbers n Vf n e ) correspond to the "index of refraction" and the damping index for each wave in the sequence. This equation also cannot be solved by approximate means when r is large. The final solutions = cosh\ 2Tg v (n y )y ~\ cosh f 2irflr«(n,)g l g(Wc)(7VC _ rf) (31 7) Y sinh L l v J sinh \_ U \ are characteristic functions that can be used to fit initial conditions and to compute the effect of perturbations, as was discussed in Sees. 11 and 12. It is obvious that the behavior of the sound waves, even of the principal wave, is quite complicated when the admittance ratio of the walls is not small compared with unity. If one adds to this the fact that many porous acoustic materials have impedances that vary considerably with change in frequency, it becomes apparent that very few sweeping generalities can be made concerning the behavior of sound in ducts with highly absorbent walls. An examination of Plate V shows that for negative phase angles (stiffness reactance) and for large values of h (small values of impedance and/or high fre- quencies) the value of £ can become quite large, and therefore even the principal wave can be highly damped. Further examination shows that in these cases the principal wave is far from a uniform plane wave, the negative reactance of the walls having in some way pulled most of the energy of the wave away from the center of the VH.31] SOUND TRANSMISSION THROUGH DUCTS 375 duct to the periphery, where it is more quickly absorbed as it travels along. A positive phase angle (mass reactance) has the opposite effect. The change of acoustic behavior with frequency is greatest Frequency Parameter, (1/a) Frequency Para meter, (l/a) Fig. 87. — Transmission of sound through square duct of width I covered with acoustic material of acoustic impedance pc(6 — i%), for principal (0,0) and a higher (1,1) model wave. Quantities proportional to attenuation per unit length and phase velocity are plotted against frequency. Sudden rise of quantities for (1,1) mode illus- trates fact that higher modes cannot be transmitted at low frequencies. when h and # are close to one of the "branch points" of Eq. (31.6), shown as circles on the dashed lines separating the modes. An Example. — As an example, we show in Fig. 87 transmission data for a case of a porous wall material with specific acoustic impedance 376 THE RADIATION AND SCATTERING OF SOUND [VII.31 equivalent to a resistance and capacitance in series. To simplify the problem we set l y = l z = I and choose for a frequency parameter (l/\) = (cdZ/27rc). We choose the constants of our acoustic material so that f = 1.5 -f 1.5i(\/l), which is stiffness controlled at very low frequencies, but is fairly ""soft" at (X/Z) = 1. The first curves in Fig. 87 show the variation of acoustic impedance and admittance ratios with frequency. They are fairly typical curves. The next curves show the variation of the characteristic values /io, m, £o, £i for the principal wave and the first higher mode. Finally we show curves for the attenuation of two actual waves [the principal wave (0,0) and the (1,1) wave] in decibels per centimeter and the ratio between their phase velocity and that of sound in the open, l/(real part of t). We note that the attenuation of the principal mode is less, than the higher mode for all frequencies shown (this occurs in nearly every case encountered in practice). We note also the very sudden and large increase in attenuation of the (1,1) mode as the frequency is lowered below its cutoff frequency. Few other generalizations can be drawn from these curves. Slight changes in impedance and in size of duct can produce very considerable changes in the shape of the curves of £ and n and of all the quantities derived from them. Problems 1. A flexible wire 50 cm long is stretched at a tension of 10 9 dynes between rigid supports. The wire is cylindrical, having a radius of 0.1 cm, and weighs 0.4 g per cm length. The mid-point of the wire is pulled aside 0.5 cm and then let go at t = 0. What is the expression for the total power radiated by the string in the form of sound? What is its value at t = and at t = 1? What is the ratio between the intensity of the fundamental and that of the first harmonic at t = and at t = 1? 2. A long cylindrical tube of radius 10 cm has a long slit 1 mm wide in its wall parallel to the tube axis. Air is forced back and forth through the slit at a velocity of 10 e' 2 ™. Plot the total power radiated by the slit cylinder per centi- meter of its length, as a function of v from v = to v = 1,000. Plot on a polar diagram the distribution in angle of the radiated intensity for v = 547.5, v = 876. 3. The portion of the surface of a long cylinder of radius a, which is between —4>a and +<£o, vibrates with velocity normal to the surface, while the rest of the cylinder is rigid. Obtain a series analogous to Eqs. (27.20) for the average acoustic impedance over the vibrating surface. 4. Compute the distribution in angle of the pressure ]^(<£)j radiated by a line source on a cylinder of radius a, for a wavelength X = ira [see Eqs. (26.7)]. 5. A hollow cylindrical tube of inner radius 54 cm is lined with acoustic material of specific acoustic impedance pc[10 + (2,000/o>)i]. Compute the cutoff frequencies and plot the characteristic impedances as function of w for « from to 10,000, for the first three symmetric modes of wave transmission along the tube, VU.31] SOUND TRANSMISSION THROUGH DUCTS 377 6. A hollow cylindrical tube of inner radius a, lined with material of specific acoustic impedance z, is closed at x = I with a rigid plane plate. The other end of the tube, x = 0, is driven by a plane piston. Derive the formulas for the pres- sure amplitude at any point x along the tube and the formula for the radiation impedance on the driving piston, including the effects of the impedance z to the first order of approximation in (pc/z). Show how it is possible to measure the wall resistance and reactance by measuring the standing wave. What measure- ments must be made, and what formulas must be used? 7. The surface of a sphere of radius a vibrates in such a manner that the radial component of velocity at the surface is u a = iC/" (3cos2i> + l) e -* ri '* Show that when X is large compared with a the radiated intensity and power radiated are T = Hpo -^- t/*[P 2 (cos#)p, n = iUpo—Jr Ul Such a source of sound is called a quadrupole source. Plot the distribution in angle of the intensity on a polar plot. 8. A piston of radius 10 cm is set in the surface of a sphere 20 cm in radius. It is vibrating with a velocity \Qe~ 2irivt . Plot the total power radiated as a func- tion of v from v = to v = 1,000. Plot on a polar diagram the distribution in angle of the radiated sound intensity at v = 164.2 and v = 438. 9. The mechanical constants of the piston in Prob. 8 are to = 10, R - 1,000, K = 1,000. Plot the total mechanical impedance of the piston as a function of v from v = 100 to v = 1,000. 10. The piston of Prob. 9 is driven by a force of lOjOOOe -211 "^' dynes. Plot the total power radiated by the piston-sphere system as a function of v from v = 100 to v = 1,000. 11. A dynamic loud-speaker of radius 27.4 cm is set flush in a large flat wall. Plot the transmission coefficient of the speaker as a function of driving frequency from v = to v = 1,000. When the speaker is oscillating with a certain velocity amplitude at a frequency of 400 cps, the sound intensity at a point 500 cm away from the wall, straight out from the center of the piston, is 100 ergs per sec per sq cm. What is the intensity at a point close to the wall, 500 cm from the center of the piston? If the diaphragm vibrates with the same velocity amplitude as before but at a frequency of 200 cps, what will be the intensity of the sound at these two points? At 800 cps? 12. A cylindrical tube of radius 5.47 cm and length 34.4 cm has its open end set flush in a large plane wall (the tube is inside the wall with its axis perpendicular to the surface) . The other end of the tube is provided with a piston that oscillates with a velocity of lOOe-^"'. Plot the total energy radiated out of the tube as a function of v from v = to v = 5,000, and plot the distribution in angle of the radiated intensity at v = 500, 1,000, 2,000. 13. A piston of radius 5.47 cm, set flush in a large plane wall, is vibrating with a velocity lOOe- 2 ™'"'. What are the total power radiated and the distribution in angle of the intensity at v = 200, 500, 1,000, 2,000? If the same piston is set in the small end of an exponential horn of constant x„ = 34.4 cm, whose open end 378 THE RADIATION AND SCATTERING OF SOUND [VII.31 has a radius of 54.7 cm and is set flush in the wall, what are the total power radi- ated and the distribution in angle of the intensity at v = 200, 500, 1,000, 2,000? Assume that the open end is large enough so that all the energy is radiated out of it. What do these results indicate about the directional properties of horns? 14. Suppose that the pressure at the mouthpiece end of a wind instrument is given approximately by the Fourier series po = Po 2* cos [2(2n + l)ir Vl t) n = where the frequency of the driving mechanism has adjusted itself so that the impedance at the fundamental vi is maximum [see Eq. (23.17)]. Compute the air velocity u e at the open end of the instrument. Assuming that this open end acts as a simple source of sound, compute the sound pressure p r a large distance r from the open end. Assuming reasonable values for the constants involved, plot u e and p r as functions of time. 15. A velocity-ribbon microphone consists of a conducting ribbon of mass m per unit length and of width 2a, suspended so that it moves freely at right angles to a magnetic field H and at right angles to its surface {i.e., the mechanical imped- ance of the ribbon per unit length is approximately 2wivm over the useful range of v). The motion of the ribbon induces an emf in it of magnitude Hlu X 10— volt. Assuming that the force on the ribbon due to a plane wave of sound falling vertically on it is equal to the corresponding force given in Eq. (29.7) for a cylinder, show that the emf induced in the ribbon is 4a%m -p. x 10-— > 2 -^*!p x io- mcy?C\ ft-*o mc where P is the pressure amplitude of the incident plane wave. 16. Using the results of Prob. 15, design a velocity-ribbon microphone and plot its response curve. The pole pieces of the magnet are 5 cm long and 2 cm apart, and a field of 1,000 gauss can be maintained between them. An aluminum (density 2.7) ribbon can be used, of thickness 0.001 cm. What width ribbon should be used to have a response constant to within 10 per cent for frequencies below 5,000 cps. Plot this response for the range < v < 10,000. 17. A circular membrane of radius a is set under tension, in a plane wall that is otherwise rigid. It is set into vibration at its fundamental frequency v a i. Plot the distribution in angle of the radiated pressure, and compute the total power radiated for (2iravoi/c) = 10 and for suitable values of the other constants (c here is the speed of sound in air, not the speed of waves on the membrane). Use the curves in Fig. 71 to compute the angle distribution. 18. The distribution in angle of radiated pressure given by the following equation: -(*) e iHr-ct) [$„ _ 0.45*1 - 0.1*2] [where the functions *„ are defined in Eq. (28.4) and plotted in Fig. 71] has very small "side lobes" for radiation at frequencies such that ka = 10. Plot the dis- VH.S1] SOUND TRANSMISSION THROUGH DUCTS 379 tribution in angle of the radiated intensity as function of &, and plot the velocity amplitude of the piston, as function of y, which will produce this radiation. 19. Use the curves of Fig. 72 and the formula of Eq. (28.8) to compute the pressure on a flat piston of radius a, set in a rigid plane, vibrating with frequency v. Plot this pressure as function of r for (2wav/c) = ka = 0.2, 1, 4. 20. Suppose that the condenser microphone whose constants are given in Eqs. (29.14) were set in a sphere of twice the radius, a = 10. Plot the altered response curve of the microphone from v = to v = 10,000, and compare the curve with that of Fig. 84. 21. A ribbon of width 0.5 cm and length 5 cm is set in the side of a cylinder 2.19 cm in diameter, with its length parallel to the axis of the tube. The interior of the tube is so designed that the effective mechanical constants of the ribbon per unit length are w = 0.01, R = 20, K = 50. The ribbon is in a magnetic field of 1,000 gauss, perpendicular to the axis of the cylinder and parallel to the face of the ribbon. Plot the response curve of this "pressure-ribbon microphone" for the frequency range v = to v — 10,000. 22. Discuss the directional properties of the condenser microphone whose constants are given in Eqs. (29.14). Plot the response curve for the microphone for the axis of the diaphragm pointing at right angles to the direction of incident sound and the curve for the axis pointing away from the source of sound. Com- pare these curves with that of Fig. 82. 23. A plane wave of sound of the form [15 sin (z) - 10 sin (3z) + 3 sin (5z)] where z = l,000n- It 1» strikes the microphone whose constants are given in Eqs. (29.14). Plot the emf output of the microphone for one cycle, and compare it with the pressure wave in free space. 24. The pressure variation from a "warble-tone" generator corresponds to the expression p s y|g— 2t»j> «— t(Ai'/j'„) cos (2xj>„0 By using the last of Eqs. (19.4) show that this expression is equal to the Fourier series p - ie- 2T<v o' V J n l — ) g-2Tt>„»i-<(»v*) X '•(£) Compute the values of the amplitudes for the component frequencies, for A = 10, v = 250, Av = 50, v„ = 10. Plot the curve for the real part of p for the time range t — to t = 0.1. 25. Obtain the formula for the ratio of reflected to incident sound energy as function of angle of incidence, for a plane wave falling on a plexiglass window of density m 8 g per sq cm and of flexural rigidity g s - ir„. Discuss the difference between these results and those shown in Fig. 86. 26. The flow resistivity of a certain acoustic material is r p = 50 cc per dyne-sec cm. Its porosity is P p = 0.7, its effective mass is m p = 5, and its thickness is 2 cm. Plot the specific acoustic impedance of the material when its back surface 380 THE RADIATION AND SCATTERING OF SOUND [VII.31 is glued to a rigid wall and also when the panel is spaced out 4 cm from the rigid wall (assume that the effect of panel flexure is negligible). Plot resistance and reactance for both cases as a function of frequency v from v = to v = 2,000 cps. Assume that the equivalent circuit is valid over this range. 27. A plane wave falls normally on the acoustic material of Prob. 26. Plot, as a function of p.(0 < v < 2,000), the fraction of incident power absorbed by the material for the two cases mentioned. 28. A ventilating duct 100 cm square in cross section is to be lined with mate- rial to attenuate sound of 1,000 cps most effectively. The acoustic material described in Prob. 26 is to be used with the material fastened directly to the duct walls (which are assumed rigid) and with the thickness d of material to be chosen for optimum results. How thick should the material be to give the greatest attenuation per foot for 1,000 cps sound? 29. The outlet for a propellor test stand is a honeycomb of ducts, each of 6 in square cross section, lined with material of resistivity 84, porosity 0.7, effective mass m p = 2, and thickness 5 cm. Compute the attenuation of the principal wave for 200 and 1,000 cps by the approximate formula, and also by use of Plate V. How long must the duct be to attenuate the 200-cps sound by 60 db? How much will this length attenuate 1,000-cps sound? 30. Derive Eq. (31.5). Plot curves illustrating the properties of the equation. 31. A duct has a square cross section 34.4 cm on a side. Use Plate V to calculate the optimum wall impedance to give maximum attenuation per length of duct at 400 cps for the least attenuated mode. Repeat the calculations for 1,000 cps. 32. A dynamic loudspeaker has a conical diaphragm with outer radius a, set in a large plane baffle. When the diaphragm is vibrating with frequency (u/2ir) and velocity amplitude V , the air in the plane of the baffle has a normal velocity approximately equal to F [*o(rO +*( f f)*i( r )] (r<a) where 6 is a constant related to the "height" of the cone and where the functions ¥ are given in Eq. (28.3). Show that the distribution in angle of the radiated intensity at great distances from the diaphragm is T ^ - pcuVln 2 ' * + (?H where n = (2xo/X) and where the functions * are given in Eq. (28.4). Plot this distribution as a function of # f or n = 10 and (2irb/X) = 2. CHAPTER VIII STANDING WAVES OF SOUND 32. NORMAL MODES OF VIBRATION In the preceding chapter we studied sound that is radiated into open space, tacitly assuming that there were no obstacles opposing its free flow outward. In most cases, however, sound generators are in rooms of size small enough so that the waves produced are reflected back and forth many times a second. When this occurs we cannot say that the waves generated are all radiating outward from the source; rather we must say that the source sets into motion one or more of the normal modes of vibration of the air in the room. This consideration makes it necessary to alter considerably our picture of the distribution of intensity about a source of sound. We cannot expect, for instance, that the intensity will vary inversely as the square of the distance from the source; in some rooms the intensity at some point far from the source may be considerably greater than at intermediate points. Nor can we expect that the intensity of sound in a room is simply related to the power radiated by the sound generator. The mechanical behavior of a loudspeaker, its mechanical and electrical impedance and total power radiated, will be practically unaltered by the properties of the room, but the intensity of the sound produced, and the distribution of this intensity, will be greatly altered. Room Resonance. — Our point of view with respect to the problem of sound in a room can be stated as follows: We look on the air in the room as an assemblage of resonators, standing waves that can be set into vibration by a source and that will die out exponentially when the source is stopped. When the source is started there will be set up a steady-state vibration, having the frequency of the source, and a transient free vibration, having the frequencies of the normal modes, which will die out. The steady-state vibration may be considered to be made up of a large number of the standing waves (just as the forced motion of a string can be built up out of a Fourier series) whose amplitudes depend on the frequency of the source, the "impedance" of the standing wave in question, and the position of the source in the room. The transient vibration will have the form necessary to satisfy 381 382 STANDING WAVES OF SOUND' [VHI.32 the initial conditions in the room when the source is started and will therefore also be made up of many standing waves; but each normal mode of the transient vibration will vibrate with its own natural fre- quency. We shall study these frequencies in the present section. After the transient has died out, the steady-state vibration remain- ing will have only the frequency of the source. We shall study the shape of this vibration, how it differs from the free radiation from a source in the open, and how it can be built up out of the standing waves, in Sec. 34. When the source is turned off, these standing waves remain, only now they have their own natural frequencies, damping out exponentially according to their free vibration properties, and perhaps interfering with each other (making beat notes) as they do so. The damping of these free vibrations, which is called the reverberation, will be discussed in Sec. 33. From another point of view the room is a transmitter of sound from the speaker to the listener; a generalized horn, so to speak. As such, it ought to have more the qualities of the horn than of a musical instrument. In other words, it ought to transmit all frequencies equally well, and its transient characteristics must not distort the sound wave noticeably. This of course could be done by arranging to have the walls almost perfect absorbers of sound, so that they would not reflect sound back into the room. If this is done, however, we lose the effect of the walls in enhancing the sound level by reflection and require considerably greater output power in order to be heard throughout the room. In the open, sound intensity from a simple source diminishes as the square of the distance. In a well-designed room, however, the sound level 100 ft away from the speaker may be only 5 db below the level 10 ft away, in contrast to the 20-db drop which would occur in the open. The engineering problem in room acoustics is to design the shape and acoustic impedance of the walls so that the room is as uniform a sound transmission system as possible, without losing completely the reinforcing effect of the wall reflections. Statistical Analysis for High Frequencies. — We shall see later in this section that at low frequencies there are few resonances and at high frequencies there are many resonances in any given band width. Consequently, a sound of average wavelength about the same size as the room will excite only a few standing waves in the room; whereas a high-frequency sound, with average wavelength small compared with the room dimensions, will excite hundreds of standing waves. Obviously, the methods of calculation that are easy to use for the low-frequency case will be difficult for the high-frequency case, and Vin.32] NORMAL MODES OF VIBRATION 383 vice versa. The situation is analogous to the difference between the methods of statistical mechanics, which deal with the average behavior of a large number of bodies, and those of ordinary mechanics, which deal with the detailed motions of one or two bodies. The statistical case, appropriate for high frequencies, is the simplest to analyze and is the one usually most useful in practical problems. Here hundreds of normal modes of the room are excited by the source; the sound is usually fairly uniformly distributed throughout the room and is traveling in all directions. When the steady state is reached, the sound at any place in the room can be represented as an assem- blage of a large number of plane waves, each with the frequency of the driver but going in all different directions. p(x,y,z) = f 2ir d<p fj Afot^e*-*-*"' sin# d& (32.1) where k is a vector of magnitude k = (co/c) pointing in the direction of the wave, given by the spherical angles (&,<p); vector r connects the origin of coordinates with the point (x,y,z) where the pressure is measured; and kr is their scalar product (k times r times the cosine of the angle & between them). The quantity A gives the amplitude and phase of the component plane pressure wave in the <p,#-direction at the point (x,y,z). In general this is a function of the coordinates (x,y,z) and of the direction (<p,&) of the component wave, corresponding to the fact that the sound is not usually completely uniformly distributed in position or direction. The average sound energy density at (x,y,z) is propor- tional to the square of A, averaged over all directions of k [for the energy density in a plane wave is (|A| 2 /2pc 2 )] so that w{x,y,z) = — 2 J dip J | A (*>,#) | 2 sin #d# The sound intensity in this case is denned as the net flow of energy per second into a square centimeter of area (it is to be noted that this is not identical with our definition of intensity for plane waves). Suppose that 3> is the angle of incidence of the plane wave Ae^ T on the square centimeter in question. The flow of energy into a square centimeter perpendicular to k would be (l/2pc) | A I 2 , and that flowing on the square centimeter with angle of incidence <S> is (l/2pc)|A| 2 cos3>, so that the net intensity is 1{x,y,z) = j- c I dtp \ |^(^#)| 2 cos<S>sin<I><f<S> 384 STANDING WAVES OF SOUND [VHI.32 which is, in general, dependent on the orientation of the square centi- meter in question (since A is a function of direction). The criterion for a room that is satisfactory for hearing is usually that A be as independent of <p, &, x, y, and z as possible. A non- uniform A produces annoying irregularities in sound intensity, both as a function of frequency and also of position in the room. It turns out that irregularity of wall shape tends to make A more uniform. This is because irregularities tend to scatter sound waves in all direc- tions, particularly if they are about the size of a wavelength. Smooth concave walls produce nonuniform sound distributions, on the other hand, for they tend to focus sound and give rise to localized regions of high energy density and marked directionality. Even smooth flat walls are undesirable, as we shall see later in the chapter. Limiting Case of Uniform Distribution. — The simplest possible case to analyze is that where the sound is uniform in density and intensity throughout the room. This is a case seldom attained in practice but one which it is desirable to approximate, as we have indi- cated above. We assume that the room is large enough so that the sound from the speaker excites many natural modes of oscillation of the room, that the wall shape is irregular enough so that sound is scattered in every direction throughout the room, and that there is enough absorbing material on the walls- so that the sound does not take an unreasonable length of time to die out after the power source is shut off (how long a time is "unreasonable" will be discussed later). In this case the amplitude factor A of the component plane waves has a magnitude that is independent of (x,y,z) and of (<p,&), so that the energy density is w = — Li ergs/cm 3 (32.2) which is assumed to be independent of (x,y,z) even in the transient state. Likewise, the intensity is T = M^J- 2 = fe?\ ergs/cm* sec (32.3) independent of direction. We now assume that the energy density and intensity stay uni- form even during the transient stage. This is very rarely true in actual rooms, but it is a condition that is approached in acoustically satisfactory rooms, and it allows a great simplification to be made VIH.32] NORMAL MODES OF VIBRATION 385 in calculating the intensity of sound in the room. For it means that we do not need to calculate the behavior of the amplitudes A(<p,&), but can concentrate our attention on the average quantities T and w. Our assumption corresponds to the requirement that w always equals (4T/c), even when both are functions of time. To determine how w and T depend on time and on the power output of any speaker in the room in this simplified case, we set up the equation for energy balance in the room. The power input into the room is, of course, the power output II (t) of the speaker, which may vary with time. Energy is lost by conversion of sound into heat in the air and at the walls. At high frequencies (above about 6,000 cps) the air can absorb a fairly large amount of energy, particularly if the humidity of the air is high. But below about 2,500 cps the great majority of the energy lost is lost at the walls, and we can neglect the absorption in the air. Each portion of the wall will absorb a certain fraction of the energy incident upon it ; and since we have assumed that the intensity is uniform, we can conclude that the power lost to the walls is proportional to the area of wall surface and to the instantaneous intensity T(t). Absorption Coefficient. — The fraction of incident energy that is absorbed by a portion of the wall depends on the physical character- istics of the wall (e.g., its acoustic impedance) and on the distribution of sound in the room (e.g., the dependence of A on <p and #). When A is independent of <p and #, as we have assumed here, the fraction of power lost to the wall depends only on the wall and is called the absorption coefficient a of the material. The relation between a and the specific acoustic impedance of the material will be discussed shortly, after we have set up the energy-balance equation. Values of a for different materials and different frequencies are given in Table XIII at the back of this book. The sum of the products of the absorption coefficients a 8 of each material composing the walls, floor, and ceiling of the room times the exposed areas A 8 of each is called the absorption a of the room a = £ <x a A a (32.4) s It is not hard to see that the total power lost to the walls (when. the sound is uniformly distributed) is Ta. The total energy in the room at any instant is equal to the volume of the room V times the energy density (w = (4T/c)). Consequently, the energy-balance equation (for uniformly distributed sound below 5,000 cps) is 386 STANDING WAVES OF SOUND [Vin.32 (32.5) Reverberation. — The solution of this equation is T = (^p. J e~ act/iV I e a °" iV H(t) dr (32.6) indicating that the intensity at a given instant depends on the power output n for the previous (4F/ac) sec, but depends very little on the power output before that time (due to the exponential inside the integral). If the power n fluctuates slowly, changing markedly in a time long compared with (4V/ac), then the intensity T will be roughly proportional to II, and Eq. (32.6) reduces to Intensity level ~ 10 log ( — ) -f 90 db if II is in ergs per second and a in square centimeters. If n is in watts and a in square feet, the equation is © Intensity level ~ 10 log I -J +. 130 db This result is easily seen from Eq. (32.5), for if (dn/dt) is small then d(4:VT/c)/dt can be neglected and oT~H. The intensity is thus inversely proportional to the room absorption a, so that for steady- state intensity to be large, a should be small. On the other hand, if II varies widely in a time short compared with (4F/ac), then the intensity will not follow the fluctuations of II and the resulting sound will be "blurred." If the sound is shut off suddenly at t. = 0, for instance, the subsequent intensity will be T = %e- actMV (32.8) Intensity level = 10 log T + 90 - 4.34 ( |^ J db The "blurring" of rapid fluctuations of speaker power is known as reverberation. It is related to the fact that the intensity level in the room does not immediately drop to zero when the power is shut off, but drops off linearly, with a slope — 4.34(ac/4F) db per sec. This linear dependence of intensity level on time is typical of rooms with Vm.32] NORMAL MODES OF VIBRATION 387 uniform sound distribution. We shall discuss cases, later in the chapter, that have more complex behavior. Reverberation Time. — The slope of the decay curve (the intensity level plotted against time after the power is shut off) indicates the degree of fidelity with which the room follows transient fluctuations in speaker output. The length of time for the level to drop 60 db is used as a measure of this slope and is called the reverberation time T. If lengths are measured in centimeters, this time is = 60 \§i ac ) sec When lengths are measured in feet, and for air at normal conditions of pressure and temperature, the reverberation time is T- 0.049 g) = <^ sec (32 . 6) When the speaker output changes slowly compared with T, then the intensity follows the output; but when the speaker output changes markedly in a time less than one-tenth of the reverberation time, then the fluctuations will not be followed. Therefore in order that the room transmit transient sound faith- fully, the reverberation time should not be large. For this require- ment a should be large, in contradiction to the requirement that a be kept small to keep the steady-state intensity large. A compromise must be worked out between these opposing requirements, a com- promise that varies with the size of the room. For a small room (V ~ 10,000 cu ft) T can be as small as 1 sec and the average intensity will still be satisfactorily high; but for a large room (V c^ 1,000,000 cu ft) T may need to be as large as 2 sec for the intensity to be high enough throughout the room. If the room is used primarily for speech, which fluctuates rapidly, the reverberation time should be about two- thirds of this, for if the hall is large the intensity can be increased by a public-address system. If the room is used chiefly for music, we can allow more reverberation without detriment (in fact the music does not sound "natural" unless there is a certain amount of reverberation). Thus an analysis of an extremely simplified example of sound in a room indicates the sort of compromise between reinforcement and absorption that must be reached for any sort of room, even if the sound is not uniformly distributed throughout its extent. The analysis has also indicated that a useful criterion to indicate the degree of uniformity of the sound distribution is the shape of the decay 388 STANDING WAVES OF SOUND [VIII.32 curve for the sound after the source is shut off. If this is a straight line (on a decibel scale), then the chances are that the sound is fairly evenly- spread throughout the room; but if it is a curve, then it is certain that the sound is not uniformly distributed, either in space or in direction of propagation or both. Absorption Coefficient and Acoustic Impedance. — Before we finish our discussion of the idealized case of uniform distribution of sound, we must compute the relationship between the specific acoustic imped- ance of the wall material and the absorption coefficient a. As stated above, this quantity is the average fraction of power absorbed by the wall when sound is falling on it equally from all directions. To obtain this average, we go back to our discussion of the ampli- tudes A(<p,&). Suppose that we choose <p and t? so that the polar axis is perpendicular to the wall (assumed plane) and so that t? is the angle of incidence of the wave of amplitude A(<p,&). In the case we are at present considering, A is independent of <p and t?, so that the power falling on a unit area of wall is T = ^- \ d<p |A| 2 cost? sin t?dt? = ^ |A| 2 2pc Jo Jo pc ' ' But from Eq. (30.7) we can show that the fraction of power lost by a wave of angle of incidence t?, on reflection from a plane surface of specific acoustic impedance z (acoustic admittance ratio rj = % ~ *<0, is a(t?) = 1 77 — cost?) 2 4k cost? y] + cost?) (k -+- cost?) 2 + <r 2 Therefore the average value of a, which is to be used in the case of uniform sound distribution, is given in terms of the acoustic con- ductance and susceptance ratios of the wall by the formula a = T I d<p \ «WI^I 2 cost? sin t?rft? (32.10) Values of this quantity, in terms of values of acoustic impedance ratios f = (1/r?) = 6 — ix = \^\e~ i<p = (z/pc), can be obtained from the contour plot in Plate VI at the back of the book. This plot shows that the maximum value of absorption coefficient (a = 0.96) comes Vin.32] NORMAL MODES OF VIBRATION 389 when the specific impedance is a pure resistance, a little bit larger than pc{$ ^ 1.25). As the impedance is increased or decreased from this value, the absorption coefficient diminishes, and at very large values of f, a is approximately equal to 8k = 86/(6* + x 2 )- There is no set of values of k and <r for which a is unity. A plane wave can be completely absorbed by material of proper impedance; a mixture of plane waves can never be completely absorbed. Therefore when the room is so designed (irregular in shape, high frequencies, sufficient absorption) that the sound in it is fairly uni- formly distributed throughout its volume, then the acoustical charac- teristics of the room are given by Eqs. (32.6) to (32.8) and the slowness of response to transient sounds is measured by the reverberation time; the absorption coefficient entering into these equations is given in terms of the physical properties of the wall material by Eq. (32.10). If, however, the sound is not uniformly distributed, then Eqs. (32.6) to (32.8) will not be valid, and Eq. (32.10) for the absorption coefficient will have no application; in fact the term "absorption coefficient" will have no application. To analyze this less idealized (and often encountered) case, we must return to our study of the individual standing waves in a room. Standing Waves in a Rectangular Room. — To commence the study we consider a limiting case of another sort, a room with perfectly smooth, rigid walls. Here we are sure that the sound will not be uniformly distributed. We choose a rectangular room with sides l x , ly, hy for simplicity, and note that the boundary condition for a rigid wall is that the air velocity perpendicular to the wall is zero at the wall. The wave equation in rectangular coordinates is d 2 p d 2 p , d 2 p _ 1 d 2 p dx 2 + 7Jy~ 2 + ~dz 2 ~ ~c-~dt? If we choose the origin at the mid-point of the room, the standing waves must have a symmetry about the origin. The solution of the wave equation is cos/ x' p= . Itfa;- sm\ c, ^ x „ , ^ x „, ^ (32U) v = 2^ \/o)l + u>l + <a\ where either the cosine or the sine can be used. In either case the pressure wave behaves in the same manner at x = l x /2 as at # = — l x /2, etc. °) eos L«) cos L*) :/ sm\ c/ sm\ c/ 390 STANDING WAVES OF SOUND [VIIL32 The velocity in the x-direction is -am/ *W jAcmAA cos\ c/ sm\ "c/sm\ c/ This must be zero at a; = ± Z x /2. For the sine function, corresponding to the cosine function in the pressure wave, to be zero at x = +l x /2, 03 x must have the following values: 03 X = j n x T (n x = 0, 2, 4, 6 • • • ) In this case u will also be zero at x = —l x /2. For the cosine function, corresponding to the sine function in the pressure wave, to be zero at x = ± l x /2, <*x = f n x ir {n x = 1, 3, 5 • • • ) i x Therefore, the characteristic functions for the rectangular room are the functions given in Eqs. (32.11) with the following characteristic values of the co's and of the frequency: (32.12) When n x is an even number, cos (co^/c) is used in the expression for p; when n x is odd, sin (co*x/c) is used; and similarly for n y and n 2 . The normal mode corresponding to any particular set of values of n x , n y , and n 2 can be produced by starting a plane wave in the direction given by the direction cosines u x /u, «„/«, and co 2 /co and letting it be reflected from the various walls until it becomes a standing wave. If the values of w x , <a y , co z , and co = 2ttv are related to n x , n y , and n z in the manner specified by Eqs. (32.12), then the reflected parts of the wave will combine in such a manner that the resulting motion oscillates with simple harmonic motion. If we start any other sort of wave, its reflected parts will interfere with each other, and the motion will not be periodic, so it will not correspond to a normal mode of vibration. Distribution in Frequency of the Normal Modes. — The last expres- sion in Eqs. (32.11) suggests that v be considered as a vector with components (« x /2jt), (ay/tor), and (»,/2jr). The direction of the vector gives the direction of the wave producing the standing wave, and the length of the vector the frequency. A normal mode of oscil- co x _ OK "X CTT <t) y — -y- tl y , h 0) z C7T = j- n z "Z (n x , n y , n z = 0, 1, 2,3 • • • ) V CO m+( 'nA 2 <m VIII.32] NORMAL MODES OF VIBRATION 391 lation can therefore be considered as a point in "frequency space," whose x component is an integral number of unit lengths (c/2l x ), whose y component is an integer times (c/2l y ), etc. The length of the line joining this point and the origin is the frequency of the normal mode, and the direction of this line is the direction of the wave that can be used to generate the standing wave. Some of these "characteristic points" are shown in Fig. 88, and it can be seen that they correspond to the intersections of a rectangular lattice with x, y, and z spacings equal to (c/2l x ), (c/2l y ), (c/2l z ), respectively. It can also be seen that all the normal modes are included among the points in the octant of space between the positive v x , v y , and v z axes, for any of the waves of frequency v with directions corresponding to (cn x /2l x , cn y /2ly,cn z /2lz), ( — cn x /2l x , cn y /2l y , cn z /2l z ), (cn x /2l x , —cn y /2l y , cn z /2l z ), etc., will generate, by reflection, the same standing wave. This picture of a lattice of characteristic points in frequency space is extremely useful in dis- cussing the number and type of normal modes having frequencies within a given frequency range. For instance, since there are (8F/c 3 ) lattice cells per unit vol- ume of frequency space (V = ld y l z ), there will be, on the average, (8V/c s )(irv 3 /6) normal modes having frequency equal to or less than v (the factor irv 3 /6 being the volume of an eighth of a sphere of radius v). The actual number of modes having frequency less than v varies in an irregular manner as v increases, being zero until v equals the smallest of the three quantities (c/2l x ), (c/2l y ), (c/2l z ), when it suddenly jumps to unity, and so on. Axial, Tangential, and Oblique Waves. — Referring to Sec. 31, and anticipating the results of the next section a little, we note that waves traveling "parallel" to a wall are affected by the wall (are absorbed by it, for instance) to a lesser extent than waves having oblique inci- dence. Therefore we separate our standing waves into three cate- gories and seven classes: Fig. 88. — Distribution of allowed fre- quencies in "frequency space" for a rec- tangular room of sides l x , l v , and l z . The length of the vector from the origin to one of the lattice points is an allowed frequency, and the direction of the vector gives the direction cosines of the corresponding standing wave in the room. 392 STANDING WAVES OF SOUND [VUL32 Axial waves (for which two n's are zero) x-axial waves, parallel to the x-axis (n y , n z = 0) ?/-axial waves, parallel to the y-axis (n x , n z = 0) 2-axial waves, parallel to the z-axis (n x , n y = 0) Tangential waves (for which one n is zero) y, z-tangential waves, parallel to the y, 2-plane (n x = 0) x, 2-tangential waves, parallel to the x, 2-plane (n y = 0) x, ^-tangential waves, parallel to the x, y-plane (n z = 0) Oblique waves (for which no n is zero) It will turn out that, even in the first approximation, waves of different classes have different reverberation times and, to the first 30,- 2 "o •c E es CO .£ 1 20 10 L=220, A=l80O,V=450O Bond Width 10 cps o jr-rprf 150 200 100 Frequency, v Fig. 89. — Number of standing waves with frequencies between v — 5 and v + 5 in a room 10 by 15 by 30 ft. Irregular solid line gives exact values; dashed smooth curve is plot of Eq. (32.14), giving approximate values of dN. approximation, waves of the same class (with v's approximately- equal) have the same reverberation time. Consequently, it will be quite important to count the number of standing waves of a given class having frequency less than v. The representation in a lattice system is again useful here; for the axial Vm.32] NORMAL MODES OF VIBRATION 393 waves have their lattice points on the corresponding axis in "frequency space" and the tangential waves have their points in the corresponding coordinate planes. Again the number of lattice points can be counted, or a "smoothed-out" average number can be computed. To take an example of practical interest: suppose that the source sends out a pulse of sound of frequency v and duration At. According to Eq. (22.18), if this pulse is to be transmitted in the room without serious distortion of shape there must be a sufficient number (it turns out that a "sufficient number" is more than 10) of standing waves with frequencies within a frequency band between v — (Aj//2) and pq + (Av/2), where Av = (l/At), in order to "carry" the sound. If, for instance, we should wish to have the room transmit adequately a pulse of length tV sec, we would be interested in counting the number of resonance frequencies of the room between *» — 5 and v + 5. If this number is less than 10 for a certain value of v , then a pulse of frequency v and of duration xtr sec would not be transmitted with fidelity in the room. If the number is larger than about 10, and if in addition the reverberation time of each the standing waves involved is less than about a second, then the pulse will be transmitted with reasonable fidelity. Figure 89 shows a curve (solid irregular line) representing a count of this sort for a room 10 by 15 by 30 ft. It indicates that a pulse of to sec duration would not be reproduced adequately unless its frequency were larger than about 150 cps. Average Formulas for Numbers of Allowed Frequencies. — It is quite tedious to count the individual allowed frequencies less than a given frequency or in a given frequency band, so that it is useful to obtain "smoothed-out" formulas for average values of the counts. This can be done by considering that each lattice point "occupies" a rectangular block of dimensions (c/2l x ), (c/2l y ), (c/2l g ) in frequency space, with the actual lattice point at the center of the block. Then the average number of points can be obtained by dividing the volume of frequency space considered by the volume (c 3 /87)(7 = l x l y h) of each block. As an example we can count up the numbers of different classes of waves having frequencies less than v. The average number of x-axial waves is just v divided by the lattice spacing in the v x direc- tion, (2vl x /c) [i.e., it is the number of blocks in a rod of cross section (c 2 /4l y l z ) and length v] and the average number of all axial waves with frequencies less than v is ZV ax © 394 STANDING WAVES OF SOUND [VIII.32 where L = 4(^ -f- l v + h) is the sum of the lengths of all the edges of the room. The average number of y, z-tangential waves is the number of blocks in a quarter of a disk, of thickness (c/2l x ) and of radius v, minus a correction to allow for the axial waves, which have been counted separately. This correction in volume is one half the space "occupied" by the y and the z axial lattice points, viz., {v C y$>v)(i y + u) The factor one-half comes in because only one-half the volume "occu- pied" by the axial lattice points is inside the angular sector formed between the y- and z-axes, which bounds the quarter disk. Therefore the average number of y, z-tangential waves having frequencies less than v is and the average number of all tangential waves with frequencies less than v is N --(cw)-'(€) where A = 2(1 J y + l x l z + l y l z ) is the total wall area. We are neglect- ing the corrections for the overlapping regions at the origin, v = 0, for they are independent of v and are small in magnitude. The volume "occupied" by the lattice points for the oblique waves of frequency less than v is the volume of one-eighth sphere minus the volume already counted for the other classes of wave : where V = l x l y L is the volume of air in the room. Therefore the total number of standing waves of all classes which have frequencies less than v is The correct value for JV fluctuates above and below this average value but is seldom more than one or two units away, unless the room is too symmetrical; this will be discussed later. VIII.32] NORMAL MODES OF VIBRATION 395 Average Number of Frequencies in Band. — The number of stand- ing waves with frequencies in a band of width dv is obtained by differ- entiating the formulas given above: dN ax ,x — (2^/c) dv, etc. dN ax ~ (L/2c) dv dN ta , yz ~ [(2tv/c 2 )I v I z - (l/c)(ly + I,)] dv, etc. dN ia ~ [(ttM/c 2 ) - (L/2c)] dv } (32.14) dN ob ~ [(4™ 2 7/c 3 ) -■ (7rM/2c 2 ) + (L/8c)] dv dN ~ [(4rv 2 V/c») + (7r^/2c 2 ) + (L/8c)] dv £ = 4(Z a + ^ + Z,); A = 2(l x l u + ZJ, + l y l z ); V = l x l y l z The value of dN, obtained from this formula, for dv = 10 and for appropriate values of the other constants is shown as the dashed line in Fig. 89. It is seen that this curve is a good " smoothed-out " approximation to the correct step curve. At very high frequencies just the term proportional to v 2 is important. We notice that the average number of allowed frequencies in a band increases with the square of the frequency at the higher frequencies. If we assume that the average intensity of sound in a room (for a con- stant output source) is proportional to the number of standing waves that carry the sound (i.e., the number with frequency inside the band characteristic of the driver), then the intensity in the room increases as the square of the frequency, for high frequencies, according to Eqs. (32.14). This is very interesting, because the power output into free space from a simple source is proportional to v 2 , according to Eq. (27.4). Therefore the power transmitted from source to receiver in a room varies, on the average, with frequency as it does in the open; but superimposed on the smooth rise are fluctuations (as shown in Fig. 89) due to the fluctuations of the number of standing waves in the frequency band of the driver. These irregularities of response are more pronounced, the more symmetric is the shape of the room, or the narrower is the frequency band of the sound source. When dN, as given by Eqs. (32.14), becomes two or less, then the fluctuations become so large that they appreciably reduce the fidelity of transmission. For the room referred to in Fig. 89 this lower limit, for a band width of 10 cps, is about 50 cps; for a band width of 5 cps, it is about 100 cps; etc. The Effect of Room Symmetry. — We have mentioned several times in the preceding pages that the response curve of a room, as evidenced by the exact curves for dN, is more irregular when the room is more symmetrical. This is due to the increase in the number of degenerate 396 STANDING WAVES OF SOUND [VIII.32 modes, standing waves with different n's that have the same frequency. As an example of the effect of degeneracies we can consider the sequence of rooms of dimensions l x , l y = ql x , l z = L/q, so chosen that the volume Table 4. — Characteristic Frequency Parameters (2lv/c) and Corresponding Quantum Numbers for Standing Waves in a Cubical Room of Side I and in a Room of Dimensions I, I \/2> Z/a/2 Cubical room Non cubical room (2lv/c) n x n y n, (2lv/c) n x n y 71* fl 0.707 1 1.000 . ° 1 1.000 1 k 1 1.225 1 1 1.414 (i 1 1 1.414 {o 2 1 lo 1 ' 1 1.581 1 1 1.732 1 1 1 1.732 {\ 1 f 2 2 2.000 2 1.871 1 1 1 lo 2 lo /2 1 2.000 2 1 ll 2 {I 1 1 2.236 ) 2 )l 1 2 2.121 2.236 3 2 1 1° 2 1 2.345 1 3 \0 1 2 {;. 1 f 2 1 1 2.449 2 2.449 1 2 1 {o 1 1 u 1 2 2.550 3 1 f 2 2 2.739 i 3 1 2.828 2 2 ii 2 lo 2 2 2.828 4 /3 \i 1 2 lo 3 2.915 3 3.000 Jo )2 2 3 1 3.000 P 2 I 2 1 2 li 4 \1 2 2 V = (l x ) z remains the same but the relative dimensions change as we change q. The natural frequencies are -&)Mf) + q 2 n 2 t VUI.32] NORMAL MODES OF VIBRATION 397 which change in spacing as we change q. We shall write down the lowest allowed frequencies for two rooms of this sequence: one for q 2 = 1, a cubical room, the most symmetric; and one for q 2 = 2, which is not symmetric. Table 4 gives the allowed frequencies and the combinations of integers (n x ,n y ,n z ) which label the corresponding characteristic func- tions for these two cases. We notice immediately the tendency of all the characteristic frequencies to "clump together" in the cubical- room case. Threefold and even sixfold degeneracies are common even at these low frequencies (for instance 2.236 and 3.000). These result in large ranges of frequency within which there is no character- istic value, so that the response is very irregular. In contrast, the case of q 2 = 2 never gives more than twofold degeneracies in the frequency range considered, and the allowed frequencies are therefore more evenly spaced along the scale. We note that, because the room volumes are equal, there are approximately the same number of frequencies equal to (3c/2Z) or less (28 in one case, 27 in the other) but the particular values of (n x ,n y ,n z ) included, and their order on the frequency scale differs. If we had picked an incommensurate value for q 2 (the cube root of 5 for instance), we would have had no degener- acies at all, and the allowed frequencies would have been still more evenly spaced along the scale. We can never get absolutely uniform spacing with a rectangular room, of course, because the lattice in frequency space is always rectangular. A room with irregular walls would correspond to a more random arrangement of lattice points in frequency space and, perhaps, to a more uniform response. Nonrectangular Rooms. — Our analysis of standing waves has depended to some extent on the fact that we have chosen to study rectangular rooms. This is not a serious limitation, for most rooms approximate a rectangular form. Nevertheless, it would be more satisfactory if it could be shown that Eqs. (32.13) and (32.14) hold for all room shapes. This cannot be done for several reasons. In the first place, although it is not difficult to generalize the quantities V (room volume) and A (area of walls) to rooms of other shape, the quantity L (total length of edge) becomes a problem. (For example, if L for a cylindrical room is just 4irR, what is L for a room of octagonal floor plan, or with a floor plan that is a polygon of a large number of sides — approaching a circular form — and what is L for a spherical enclosure?) In the second place it becomes progressively more difficult to define axial and tangential waves as the room shape is made more com- 398 STANDING WAVES OF SOUND [VHI.32 plex (this of course is another aspect of the reciprocal relationship between uniformity of room shape and uniformity of wave behaviors- all waves are oblique waves if the room is irregular enough). As an example of how these questions can be answered in one other case, we shall outline the solution for a cylindrical room, with circular floor plan of radius a and of height I. The solution of the wave equation in cylindrical coordinates is cos <t)'-(?) V = gin (m<p) cos^j J»yf) <r M «, " = ^ Va>i + «? (32.15) where z = is at one of the end "walls," the floor. The properties of the Bessel functions J m have been given in Eqs. (19.4) and (19.5). In order to have the z component of the particle velocity zero at z = and z = I, we must have the derivative of p with respect to z zero at these points. The derivative is zero at z = 0, because we have already chosen the cosine function. For it to be zero at z = I, we must have (wj/c) = n z ir, (n z = 0,1,2,3 • • • )• I n order to have the radial particle velocity zero at the cylinder walls, we must have (dJ m /dr) = at r — a. For this to be true, we must have ((o r a/c) = Tra mn where a mn is a solution of the equation [dJ ' m {ira) / 'da] = 0. The characteristic functions for the cylindrical room are therefore those given in Eq. (32.15), and the characteristic values are -j- J (n, = 0,1,2 • • • ) co r = I — ^_ J (where a mn is given in Table 5) c " = 2 V(t)' + ( s S b )' ™ The quantities <x mn have already been mentioned in Eqs. (26.12) and (26.15), in connection with the cylindrical pipe. If I is not longer than 1.71a, the mode with the lowest frequency is a transverse one, with one diametric mode, the air "sloshing" back and forth across the cylinder. The next problem is to label the "axial" and "tangential" waves. The 2-axial waves are obvious ones; they correspond to those for which m = n = 0, where the motion is parallel to the z-axis. Similarly, the waves for which n z = are obviously parallel to the floor and ceiling and should, probably, be called <p, r-tangential waves, by analogy with the rectangular case. The waves for which motion is VIIL32] NORMAL MODES OF VIBRATION 399 entirely radial (parallel to r) for which n z = m — are those which focus the sound along the cylinder axis. They can perhaps be called r-axial waves, though they are not parallel to any wall. The waves for which n z = n = are those which travel close to the curved walls and have little motion near the cylindrical axis. These can be called <p-axial waves, though the air motion is not entirely in the ^-direction (perpendicular to r and to z) . We shall see in the next section that the r-axial waves are least absorbed by acoustic material on the curved walls and the ^>-axial waves are very strongly absorbed. Table 5. — Characteristic Values a mn for the Cylindrical Room Solutions OP dJ m (ira) /da = 1 2 3 4 0.0000 1.2197 2.2331 3.2383 4.2411 1 0.5861 1.6970 2.7140 3.7261 4.7312 2 0.9722 2.1346 3.1734 4.1923 5.2036 3 1.3373 2.5513 3.6115 4.6428 5.6624 4 1.6926 2.9547 4.0368 5.0815 6.1103 5 2.0421 3.3486 4.4523 5.5108 6.5494 6 2.3877 3.7353 4.8600 5.9325 6.9811 7 2.7304 4.1165 5.2615 6.3477 7.4065 8 3.0709 4.4931 5.6576 6.7574 7.8264 «m0 — (w/tt) OCmn ^ n + \m + \ (m » 1) {n ^> 1, n > m) Except for m = 0, each mode is doubly degenerate, corresponding cos to the duality . (m<p) in the characteristic function. J sin v This can be represented formally by allowing m to take on negative as well as positive values, with negative m corresponding to the cos(m^>) case and positive m to the sin(m<p) case (where, of course, a-^ n = a mn ). Frequency Distribution for Cylindrical Room. — A certain amount of arbitrariness comes in when we set up the lattice in frequency space for computing allowed frequencies. Here the standing wave is made up of waves with components in all directions perpendicular to the z-axis, so that there is no obvious direction that corresponds to the v m , n ,n z vector, as there was for the rectangular room. To be strictly accurate, we should represent each wave by a portion of a circle in frequency space, corresponding to the fact that a cylin- drical wave is made up of a distribution of plane waves in a variety of directions. Formally, however, we can set up arbitrarily a sym- 400 STANDING WAVES OF SOUND [VIII.32 metrical grid of points on the (v x ,v y ) plane, each the proper distance from the origin, which can represent all the allowed frequencies. A possible representation is given in Fig. 90, where the r-axial waves (w = 0) lie along the main diagonal and the doubly degenerate ^-axial waves (n = 0) lie along the v x and i^-axes. This type of representation is useful for, as we shall see later, the ^>-axial waves depart most radically from the average reverberation behav- ior, being much more rapidly absorbed than the rest. In fact it is not unreasonable to expect waves tangential to any concave curved wall to be absorbed more rapidly than oblique waves, and waves tangential to a flat or a convex curved wall to be ab- sorbed less rapidly than the rest. The third dimension of the lattice (the ^-direction) is ob- tained by adding similar two- dimensional arrays, one above the other, spaced a distance (c/2l) apart. The process of counting the number of standing waves with frequencies less than v is now carried out as before. The "smoothed-out" formula is somewhat more difficult to calculate, for the asymptotic formulas for the a's for large m and n must be utilized. The equations analogous to Eqs. (32.14) are (c/a) (Zc/a) Fig. 90. — Formal representation of allowed frequencies in a cylindrical room. Allowed frequencies given by radial dis- tance from origin to one of the small circles. Lattice is made up of a parallel sequence of similar planes, spaced (c/2h) apart. dN aXyZ ~ (22/c) dv dNax, v ^ (2xa/c) dv dN ta ,r<p^[2irv/c 2 )(ira 2 ) - (7ra/c)]dv dN ta , z<p ~[(2rv/c 2 )(2T<il) - (2va/c) - (2l/c)]dv dN ~ [(±tv 2 V/c s ) + (tvA/2c 2 ) + (L/8c)] dv V = iraH; A = 27ra 2 + 2-wal; L = lira + U (32.17) This corresponds to our definitions of volume V and area A for the earlier case; but L has a form that could not easily have been predicted. It appears likely that the equation for dN is a general one as far as the terms in v 2 and v go, but that the constant term (if it is needed) must be worked out for each case. It is seldom that this constant Vm.33) DAMPED VIBRATIONS, REVERBERATION 401 term in the series is needed, however. Indeed, in many cases only the first term (4.irv 2 V/c z ) dvis sufficiently accurate. It can be proved that this term has the correct form for a room of volume V of any shape whatsoever. The curve for dN from Eq. (32.17) is roughly similar to Fig. 89, but the allowed frequencies in this case are not so evenly spaced as with the rectangular room, so that the actual value of dN does not approach the average curve for so low a frequency as in the rectangular case. This is due to the fact that at the higher frequencies there are many modes having nearly the same frequency, the asymptotic form of a showing that a m -i t n+\ ^ <w, etc. This "bunching together" of the allowed frequencies is essentially due to the symmetry of the enclosure about the cylindrical axis, so that a number of standing waves can be set up, having different directions but having about the same frequency. The distribution in frequency of the standing waves in a spherical enclosure is even less regular than that in a cylinder, and still higher frequencies must be u3ed before the actual curve for dN is smoothed out and approaches the average curve. Such an enclosure would not be satisfactory for use as a room, because of the fluctuation in its resonating characteristics as the frequency is changed. In fact, we can state as a general rule that the more symmetrical an enclosure is, the larger will be the range of frequency over which the resonance properties fluctuate, and the less desirable will it be for use as an auditorium. The curve for dN for a room of the same volume as that used for Fig. 89, but having very irregular walls, will approach the smooth average curve still more rapidly than the curve shown does. Irregular walls also serve to spread out the sound energy more or less uniformly over the room. Most of the high-frequency standing waves in a rectangular enclosure have an average amplitude that is nearly the same everywhere in the room, but many of the standing waves in a spherical room have larger amplitudes near the center than near the wall. In rooms having smooth concave surfaces, focal points of considerable excess intensity may occur to render the room undesir- able as an auditorium. 33- DAMPED VIBRATIONS, REVERBERATION The standing waves discussed in the last section do not continue in the enclosure forever; they lose energy and damp out. Over the range of frequencies useful in acoustics, most of this energy is lost at the walls, some of it being transmitted through to the outside, and some going to heat the wall. At very high frequencies an appreciable 402 STANDING WAVES OF SOUND [VIII.33 portion of the energy is lost in heating the gas itself, but for the fre- quencies in which we are interested we can consider that this loss is negligible compared with that lost at the walls. We have already computed [see Eq. (32.6)] the reverberation in a room with sufficiently irregular walls so that the sound is uniformly distributed throughout its volume. In this section, we shall discuss the reverberation of individual standing waves in a room. Pursuing the analogy mentioned at the beginning of Sec. 32, the analysis of this section is more analogous to the detailed analysis of ordinary mechanics than to the calculations of statistical mechanics. Rectangular Room, Approximate Solution. — We start first with the rectangular room of sides l x , l v , l z , with walls (and floor and ceiling) that are fairly hard (specific acoustic impedance large compared with pc). In this case the standing waves are not much different in form from those given in Eq. (32.11) for a rectangular room with rigid walls: _ cos /im x x\ cos f -Kn y y \ cos / irn z z \ * N ~" sin\ l x ) sin\ l y ) sin\ l t ) — «vfc) ,+ fey +&)■-*"•' (33.1) where the origin of coordinates is at the center of the room. Even values of n correspond to the use of the cosine in \f/ N , odd values of n correspond to the use of the sine. If the wall impedance has a resis- tive term, energy is lost and the free vibration of each standing wave must contain an exponential factor e~ kt , corresponding to the loss of energy. In Eq. (4.8) we related the damping factor k for a simple oscillator to the fractional loss in energy of free vibration per second. This relationship holds for any oscillation and can be used here to obtain a first approximation to the damping constant for a standing wave of sound. The total energy of the standing wave corresponding to the trio of numbers N = n x , n„, n z is [using Eqs. (33.1) and (25.3)] ^' = 5 J J/ k + ^ "*]*""*! (33 . 2) VA-x * _ /i / \ \ = K — 2» An ~~ W € »* e »» e »J j where e = 1 but d = e 2 = e 3 = • • • =2, and where V = U y l z . We notice that the axial waves have more energy content than the VIII.33] DAMPED VIBRATIONS, REVERBERATION 403 tangential, and the tangential have more than the oblique. This is because each cosine or sine factor has mean-squared value %, whereas a constant factor (n = 0) has average value 1. If there are nodal planes in all three directions (oblique waves), the average value will be least. The energy lost per second at a square centimeter of wall surface is the average value of the pressure p times the velocity normal to the wall; this equals (/cp 2 /2pc), where k is the acoustic conductance ratio of the wall material. The rate of total power loss at the walls is therefore = WcSf K (S)p 2 (S) dS where the integration is over every element of all area for which k is not zero. Referring to Eq. (4.8), we see that the damping constant k is equal to i times the ratio between power loss and total energy of vibration. In the present case, we can write the equation for k as follows: k ^ Ac [average value of tap* over all walls] N 2V [average value of ^* over volume of room] ^ ^ This expression is correct to the first order in k for standing waves in rooms of any shape. A is the wall area and V is the room volume. The "Q of the room" for the iVth wave [see Eq. (4.4)] is (o> N /2k N ). Wall Coefficients and Wall Absorption.— The exponential decay factor for energy or intensity of the standing wave will therefore have a term e~ 2k ^ (because energy or intensity is proportional to the square of p). Comparison of this with Eq. (32.8) for the statistical case of uniformly distributed sound shows that the quantity that takes the place of the average absorption coefficient a in the case of a single standing wave is the average wall coefficient a N , where 5 ^ average value of 4k^ over all walls average value of ^ over room volume (33.4) to the first approximation in the (assumed small) quantity k. In the statistical case, a was independent of the sound; in the present case, a N depends on the standing wave as well as on the wall materials. These formulas show that absorbing material (material with a nonzero value of k) should be placed at those parts of the wall where the pressure is largest if it is wished to absorb the standing wave most 404 STANDING WAVES OF SOUND [VTII.33 rapidly. Material placed at the corners of the room is twice as, effec- tive, on the average, as if it were placed at other locations, because all standing waves have pressure maxima there. One might be tempted to place strips of absorbing material in a regular pattern on the wall, corresponding to the maxima of a particu- lar \f/ N , in order to damp out the wave having a particular trio of values of n x , n y , n z . This would *absorb that particular standing wave rapidly, but other waves, with different values of the w's, would not have their maxima coincide with the strips of absorbing material, and so would be much less rapidly damped. Whenever there is a wide variation in the reverberation times of different standing waves, the acoustical conditions in the room turn out to be unsatisfactorily nonuniform; so any regular pattern of patches of absorbing material will usually lead to unsatisfactory acoustics in one or more frequency ranges. In order to maintain as much acoustic uniformity as possible, it is best to cover several walls completely with material, or else to distribute patches of material in a nonuniform manner over all the walls. A random arrangement of patches has the additional advan- tage of scattering sound and tending to distribute it more uniformly throughout the room. This scattering effect is not calculable by first- order approximations, but will be discussed briefly later in this section. If each wall has a uniform acoustic impedance, or if its impedance varies in a completely random manner, then the maxima and minima of each standing wave fall with equal likelihood on the most absorbing parts of the wall. In this case the average value of (4k^) is equal to the product of the average value of (4k) times the average value of (^), each averaged separately over the wall in question. In this case Eqs. (33.3) and (33.4) can be further simplified. Suppose that we multiply the acoustic conductance ratio of each wall material by eight times its exposed area and sum over all the area of the two walls perpendicular to the z-axis, calling this quantity a*~ Y (8k s )A* (33.5) x walls the absorption (first-order approximation) of the x walls. Comparison with Eq. (32.4) shows that, to the first order in k, the quantities (8/c) for the rectangular room correspond to the absorption coefficients a in irregularly shaped rooms with uniformly distributed sound. The quantities a y and a z can similarly be defined. Finally, utilizing the properties of the squares of sine and cosine to obtain average values for \[/%, we obtain approximate formulas for the Vra.33] DAMPED VIBRATIONS, REVERBERATION 405 damping constant kn and corresponding reverberation time for the standing wave specified by N = (n x , n y , n e ): 0.0497 ^ (33,6) T N \^n x dx + -$*n Jly + Tt^nflz) where Tn is given for dimensions in feet and for air at standard condi- tions. These formulas are good only to the first order in the (sup- posedly) small quantities k; more accurate formulas will be discussed later in the section. The factors ^e„ (= ^ for n = 0; = 1 for n > 0) constitute the difference between this formula and that of Eq. (32.9) for the room with uniform distribution of sound. When n x (for instance) is zero, the x walls contribute one-half as much (to this approximation) to the absorption of the wave as they do for waves with n x > 0. This is due to the properties of the functions \f/ N ; the ratio between the average energy in the room and the average intensity on the x walls for n x = (waves parallel to the x walls) is one-half what it is for the waves that reflect from the x walls (n x > 0), so that for the parallel waves the fraction of total energy absorbed by the x walls is one-half the usual fraction. Reverberation Times for Oblique, Tangential, and Axial Waves. — These formulas illustrate some of the effects of room regularity on sound absorption. A rectangular room has walls smooth enough so that some standing waves move parallel to two or more walls. For these standing waves the walls parallel to the motion have less effect on the damping (one-half as much, to the first approximation) as they do for oblique waves. This behavior is, in general, true for smooth plane walls and smooth convex walls, in rooms of other than rectangu- lar shape (by "smooth" we mean that the radius of curvature of any part of the Wall is considerably larger than a wavelength). For smooth concave walls it turns out that the waves "parallel" to its surface are more strongly absorbed than are oblique waves. In either case, however, the waves moving tangential to one or more walls are absorbed at a different rate from the rest. As an example, it can be mentioned that the standing waves in a room with a floor plan that is an isosceles right triangle also has three classes of tangential waves: those tangential to the plane-parallel floor and ceiling (for which the floor and ceiling absorptions are multiplied by one-half), those "tangential" to the two walls at right 406 STANDING WAVES OF SOUND [VHI.33 angles (for which the absorption for these walls is multiplied by three-quarters), and those tangential to the diagonal wall (the factor for this wall is also three-quarters) . Another example is the cylindrical room, which will be worked out later. In a rectangular room (or indeed in any room with "smooth" walls), if a sound source exciting a number of standing waves is sud- denly turned off, the various standing waves will take different lengths of time to die out. The oblique waves will damp out most quickly, then the tangential waves (if the walls are not concave), and the axial wave parallel to the most absorbent walls will linger nearly twice as long as the oblique waves. Therefore, the distribution of sound in the room will change as it dies out, being at first fairly random, but eventually being mostly parallel to the absorbent walls. Experience shows that this behavior, if the differences of reverberation time are pronounced, is acoustically objectionable. Therefore, if a rectangular room cannot be changed in shape by introduction of irregularities, and if it is to be used for purposes of speech or music, requiring good acoustics, the absorbing materials should be placed about equally on all walls, so that a x , a y , and a z are nearly equal. This will produce the least difference between rever- beration times for a given value of total absorption. Irregularities of wall shape, to scatter the axial and tangential waves, would improve the acoustical conditions considerably, however, and these should be introduced whenever possible. Decay Curve for Rectangular Room. — If the source in the room emits sound in a frequency band dv (between v — \dv and v + \dv), there will be excited, on the average, dN " & oblique waves, dN ta tan- gential waves, and dN ax axial waves, where the formulas for the quantities dN are given in Eqs. (32.14). If each of these wave groups is excited and contributes to the resulting average intensity by an amount proportional to the dN's, we can write down a formula for the average attenuation of sound in a rectangular room T ~ ? D x (t)D v (t)D.(t) where a x , a y , a z are defined in Eq. (33.5). To this approximation, therefore, the sound-decay curve (the intensity level vs. time) for a rectangular room can be expressed in terms of a sum of individual terms, 10 log(D x ), etc. Each of these terms starts at t = as a straight line with negative slope proportional to a x . A time (8V/ca x ) In (c/2vl x ) Vm.33] DAMPED VIBRATIONS, REVERBERATION 407 later the curve has a break, ending up, beyond this, as a straight line with slope proportional to \a x , half the initial slope. Since each of the additive terms has it's "break" at a different time (unless the room is completely symmetrical), the resulting decay curve is quite far from being the straight line that was indicated in Eq. (32.8) for a uniform distribution of sound in an irregular room.. At high enough frequencies, the "breaks" come late enough so that the first 20 or 30 db of the curve is nearly straight, with a slope and indicated "reverberation time" corresponding to the oblique waves. If, by chance, this result were assumed to correspond to that given in Eq. (32.8) for an irregular room with diffuse sound, then the quantities (8k) (where k is the wall-conductivity ratio) would be presumed to equal the absorption coefficient a. We have seen, how- ever, that this is an inaccurate correlation, which may work fairly well for very stiff walls [k very small; see the comments on Eq. (32.10)] but which fails for more absorptive walls, when the break in slope of the decay curve is more pronounced. If most of the absorbing material is concentrated on the two opposite walls of a room, then those standing waves that do not reflect from the absorbing walls will take about twice as long to die out as do all other waves. When a sound with a "spread" of fre- quency is used to excite a number of standing waves at the same time, the dying out of these waves after the source is shut off is a rather complicated phenomenon. When only two or three standing waves are excited, these waves as they die out may alternately reinforce and interfere with each other, owing to the fact that they are of slightly different frequency. The intensity will then fluctuate instead of decreasing uniformly, the sort of fluctuations obtained depending on the position of source and microphone in the room and on the manner of starting the sound. If more than three standing waves have been excited, these fluctuations will be more or less averaged out, and the resulting intensity will first diminish uniformly at a rate dependent on the reverberation time of the standing waves which are reflected from all six walls. After these waves have died out, the rest of the sound, due to waves not striking the most absorbent walls, remains and dies out more slowly. The intensity level as a function of time approximates a broken line, the steeper initial part corresponding to the most of the standing waves, and the less steep later part due to the waves that do not strike some walls. In such cases the term "reverberation time " has a specific meaning 408 STANDING WAVES OF SOUND [VHI.33 only in connection with the damping out of single normal modes, i.e., in connection with the slopes of the two portions of the broken line of intensity level against time. The actual length of time that it takes for the intensity level to drop 60 db will depend on the relative amounts of energy possessed by the rapidly and the slowly damped standing waves. Figure 91 illustrates these points; it shows curves of intensity level in a room 10 by 15 by 30 ft as a function of the length of time after the sound is shut off. The two smallest walls are supposed to be much more absorbent than the other four. The solid line is the curve for Time- Fig. 91. — The decay of sound in a room with two opposite walls more absorbent than the rest. Solid curve shows the average decay of a large number of normal modes; dotted line shows the interference effects possible when only two normal modes have been excited. intensity when the room has been excited by a tone with wide enough frequency "spread" to excite 10 standing waves about equally, so that a smooth decay results, and the "break" in the curve is apparent. The dotted line is the curve when only two standing waves are excited, the resulting fluctuation, due to interference, masking the exponential decay. These interference oscillations and breaks in the curve for decay of intensity level are present even when all the walls are about equally absorbent, but they are less pronounced. Cylindrical Room. — We can use the characteristic functions given in Eqs. (32.15) and (32.16) to compute the damping constants for the different standing waves in a cylindrical room of radius a and height I. If the absorbing material is distributed either at random or com- pletely uniformly over each wall, the quantity k can be averaged Vm.33] DAMPED VIBRATIONS, REVERBERATION 409 separately as before. The wall behavior is then expressed in terms of the absorptions. a z = X(Sk s )A s (over flat floor and ceiling) a r = 2(8k s )A s (over cylindrical surface) The wave properties are evidenced through the averaged values of p 2 over the walls and throughout the volume. The mean-square value of the Bessel function is Therefore the damping constant for the iVth wave (N = m,n,n z ) in the cylindrical room is *'^[***+ l- ( i/„^, ] (33-7) which is to be compared with Eq. (33.6). Values of a mn are given on page 399. The r, ^-tangential waves (n z = 0) are absorbed half as much by the flat floor and ceiling as are the waves for n z > 0, as is the case for the flat walls in a rectangular room. The curved cylindrical walls absorb most waves less effectively than do flat walls; for the factor l/[2 — 2(m/7raw) 2 ] is smaller than unity unless n is small. Most waves in a cylindrical room are focused toward the center, away from the curved wall, and so are not affected as much by this wall as they would if it were flat. In contrast, the standing waves for which n = (the ^,2-tangential waves) are much more strongly absorbed than if the wall were flat; for these waves l/[2 - 2(m/Tra mn ) 2 ] is much larger than unity. These (p, z-tangential waves move "parallel" to the curved wall and have large amplitude only near this wall, so it is natural that it will absorb them strongly. When a sound source in a cylindrical room is shut off, the <p,z- tangential waves damp out most rapidly and the r-axial waves least. Therefore the sound nearest the curved walls vanishes first, eventually leaving the radial wave motion, which focuses the sound along the axis and which attenuates most slowly. This effect was mentioned on page 399, where we discussed the classes and numbers of standing waves in a cylindrical room. Second-order Approximation.— In the foregoing calculations, we took into approximate account the effect of the flow of air into and out 410 STANDING WAVES OF SOUND [VIII.33 of the wall surface on the power absorption at the surface, assuming that the standing wave shape is not changed by the yielding of the walls. Actually, of course, the fact that the wall impedance is not infinite does affect the shape of the standing waves, and the second approximation to the solution must take this into account. How this correction is to be calculated can be seen most easily by assuming that we gradually increase the admittance of the walls, starting from zero and ending at the actual distribution of admittance which is under consideration. When the walls are all rigid, the stand- ing waves are given by the characteristic functions ^n given in Eq. (33.1) or (32.15), no energy is absorbed by the walls, and the natural frequencies are (o)n/2tt), with the characteristic values ca N given in Eqs. (32.12) or (32.16). As the wall admittance is increased slightly the shape of the characteristic functions is not changed very much at first; there is a slight air velocity into the wall, of an amount equal to the specific acoustic admittance of the wall (still small) times the pressure at the wall surface. This motion of air produces radiation out into the room; in other words the original standing wave is modified by the scattered waves produced by the reaction of the wall to the standing wave. Suppose that we start with the standing wave \f/N(P)e~ iaNt [where P stands for the point (x,y,z)]. When the wall admittance is not zero, the air velocity out from the wall at the point Q = (X, Y,Z) on the wall surface is — (l/pc)^w(Q)i?(Q), where rj = (pc/z) is the acoustic admittance ratio (small) of the wall at the point Q. The element of wall area dS Q at Q therefore acts like a simple source of sound, modify- ing the pressure at P by an amount [ivN^ N (Q)ii(Q) dS Q /ch]e i < u « /c '><* r -. et \ where h is the distance between P and Q [see the discussion of (Eq. 28.1)]. Consequently, to the first order in the small quantity rj, the modified characteristic function for the pressure at P = (x,y,z) is walls ¥ W (P) ~ MP) + {£%) J J [ Y J ««•"•>»-"> dS Q (33.8) walls This modified standing wave, vibrating within nonrigid walls, does not vibrate with quite the same frequency as does the wave ip N inside rigid walls. To find the new natural frequency we set the expression of Eq. (33.8) into the wave equation. After a great deal of involved integration, which does not need to be gone into here, we find that the square of the corrected characteristic value is given to the second order in the small quantity 17 by the expression Vm.33] DAMPED VIBRATIONS, REVERBERATION 411 2^ 2 L ■ _ (ic\ average of [r)(P)ty N (P)\f/ N (P)] over all walls \ N \ \u>nJ average of [^(P)] over room volume ) (33.9) where Vn(P) is the function given in Eq. (33.8). To put these formulas into a form that can be calculated with reasonable ease, we assume that the characteristic functions yj/ N have the usual properties of orthogonality fff+»(P)MP) dV p = { ° [* I $ ohuo) room s ' where V is the volume of the room, N and M are trios of numbers labeling different standing waves, and Aw is some dimensionless con- stant depending on the shape of the function $ N . In terms of these functions, the outgoing wave in Eq. (33.8) is given in terms of a series [a simplified form of Eq. (34.4)] where h is the distance between the point P (which can be anywhere in the room) and the point Q (which is somewhere on the wall). The function ^ N (P) can also be expanded into a series of characteristic functions. As long as 97 is everywhere small, the largest term in the series will be i[/ N , the rest being correction terms: where the functions Y MN = Gmn — iBmn, 7] = K — l(T Gmn ' k(Q) = / / ^ (Q) *"( Q) ds<1 Bmn walls <r(Q) are the transfer admittances of the walls, coupling the Nth and Mth standing waves. They have the dimensions of an area, as do the absorption constants a x , given in Eq. (33.5). The prime on the sum- mation symbol 2 indicates that the sum does not include the term M = N, for this term has been written down separately as the first and largest term. 412 STANDING WAVES OF SOUND [Vm.33 Inserting Eq. (33.11) into Eq. (33.9) gives us the natural fre- quency of the iVth wave. The time dependence of the free vibration of the Nth wave is given, to the second order of approximation, by the exponential exp — icon — cY NN . c 2 Y* NN 2FAtf SV 2 usA% "^|AS#^jJ f (33 ' 13) The real part of the exponent gives the damping constant kx, to the second order in the small quantities Y. , cGnN , C 2 COjV GnnBnN , ^0' AGmnBmN /„q -.is The first term in this, of course, corresponds to the first-order expression given in Eq. (33.3). The second term is the correction due to the distortion of the standing wave by the wall admittance. The imaginary part of the exponent also carries a correction to the unperturbed angular frequency un. The first-order correction is — (cBnn/2VAn). While the effect of the wall conductance (to the first approximation) is to attenuate the free vibrations, the effect of the wall susceptance is to change the frequency of free vibration. The second-order correction is of the same general form as that in Eq. (33.14) for k N , except that the quantities i((? 2 — J5 2 ) appear instead oiQB. Scattering Effect of Absorbing Patches. — We can now begin to see the effect of wall irregularity on the distribution of each standing wave. As long as the admittances Ymn in Eq. (33.12) are small, the standing wave ^n differs only slightly from the symmetrical wave ^r for smooth rigid walls. But as the walls are made softer some of the F's increase in size; and if the absorbing material is properly placed so that many of the F's get large, each standing wave becomes a more or less random mixture of a large number of ^'s, corresponding to wave motion in many directions. If enough F's are large enough, each standing wave is a random mixture of waves, all waves of about the same frequency have about the same value of damping constant, and we arrive at the uniform sound-distribution case discussed at the beginning of this chapter. Thus a placement of absorbing material so that as many Ymn's as possible are as large as possible will cause the acoustic conditions to Vin.34] FORCED VIBRATIONS 413 approach the uniform acoustic-distribution case as closely as possible. Uniform distribution of absorbing material over any wall will not pro- duce this effect, for if t)(Q) is constant over any wall (the yz wall for instance) a part of Ymn will equal r\ | I ^m(Q)^n(Q) dy dz, which is x = zero unless m y = n y and m z = n z , so that in this case many of the F's would be zero. A study of the integrals defining Ymn shows that, for as many of them as possible to differ from zero, the absorbing material should be placed on the walls in patches, of dimensions neither large nor small compared with the wavelength, the patches being placed in a random manner over all walls. In this way, the absorbing material will scatter the waves as effectively as possible. A study of the scattering effect of small modifications of shape of the walls (which will not be computed here) shows that shape irregu- larities, if they are about the size of a wavelength and are irregularly placed, are even more effective in scattering than are irregularities of admittance (though, of course, they do not cause absorption). A combination of "bumps" and patches of absorbing material, irregu- larly placed, produces the best acoustic effect. 34. FORCED VIBRATIONS We are now in a position to calculate the excitation of the standing waves in a room by a sound generator and thus obtain an idea of the effect of the acoustic properties of the room on the quality of the sound produced by the generator. We shall see that in a moderately absorb- ing room the shape of the room will have very little effect on the output of the generated sound at high frequencies. The intensity will be greater and will be more uniformly distributed over the volume than is the case when the source is in free space; but the dependence of this intensity on frequency will be the same, its value being simply a con- stant times the power output of the source for free space. This is due to the fact that at high frequencies the energy is "carried" by a large number of standing waves, all of nearly the same natural frequency, so that the resulting effect is fairly uniform. At low frequencies, however, the majority of the energy will be carried by one or two standing waves, and large fluctuations in the amplitude of these waves can occur as the frequency is changed. The variation of output with frequency will depend more on the character- istics of the room than on those of the generator at low frequencies. 414 STANDING WAVES OF SOUND [VIII.34 Simple Analysis for High Frequencies.— The intensity of high- frequency sound due to a source of output power n can be obtained in a very simple manner. If it is possible to assume that the intensity of the sound T is uniform throughout the room, then the formulas derived at the beginning of this chapter can be used. In particular, if the source output II changes little during a reverberation time, Eq. (32.7) can be used. Close to the source most of the sound is radiating outward from the source, and the intensity varies with the inverse square of the distance, as from a simple source. At larger distances the outward radiation is lost beneath the randomly scattered waves, which have more or less uniform intensity everywhere in the room. If II is the power output of the source in watts, r the distance from the source in feet, a the room absorption in square feet [see Eq. (32.4)], then the intensity close to the source in watts per square centimeter is (II/4,0007rr 2 ), where the factor 1,000 is approximately the number of square centimeters in a square foot. For large values of r, the inten- sity is n/l,000a). Intensity and Mean-square Pressure. — At this point we must call a halt, to point out the difference between our definition of intensity and the way sound "intensity" is usually measured. We have defined intensity as sound power falling on one side of a square centi- meter of area. This can conceivably be measured, but in many cases the result may depend on the orientation of the area. Close to the source the intensity is all flowing outward, so that we must arrange that the square centimeter be placed perpendicular to the radius r, if our measurement is to equal (n/4,000jrr 2 ) ; if it were placed parallel tor the intensity measured would not be at all as large. On the other hand, throughout the rest of the room, according to our assumption, the intensity flows equally in all directions, and the intensity-measur- ing device need not have any special orientation to measure the predicted amount. In actual practice, sound intensity is rarely measured directly; what is measured is mean-square pressure, as was mentioned in con- nection with Eq. (22.15) (p 2 ms = i|pi 2 )- This quantity is simply related to the average energy density w by the relation p 2 ^ = pc 2 w, but it is not simply related to the sound intensity. If the intensity is flowing in only one direction the relation is piL = pcT, but if it is flowing equally in all directions the relation is p 2 ^ = 4pcT, as was shown in Eq. (32.3). Consequently, the quantity to compute, which can be checked directly with measurement, is the mean-square pres- sure, rather than the intensity, or else the pressure level (whieh is 20 log (prms) + 74 = 10 log (w) + 136) rather than the intensity level. VHI.S4] FORCED VIBRATIONS 415 In terms of these quantities the statements made above become C 10 log (n) - 20 log (r) + 49 db (r 2 < a/50) Pressure level c^ < (34.1) ( 10 log (n) - 10 log (a) + 66 db (r 2 > a/50) for the statistical case, where II is source power in watts, r the distance from the source in feet, and a the room absorption in square feet. The criterion for range of validity (r 2 vs. a/50) is obtained by equating the two formulas. If the power II is measured in ergs per second, r in centimeters, and a in square centimeters, the formulas for mean-square pressure are 2 f (pcll/4irr 2 ) (r 2 < a/50) Prms — ( (4 pc n/a) (r 2 > a/50) a = ^a s A s s Close to the source, the first expression is larger and is used; far from the source the second term, representing the random sound, is larger and is valid. If the sound generator is a simple source of strength Q Q , then Eq. (27.4) shows that the power generated will be (po; 2 Q§/8xc). Conse- quently, over most of the volume of the room the mean-square pres- sure due to a simple source is (pWQ$\ \ 2ira J A - - DsfV (34 - 2) This equation is obtained here, for the case of the room with uniform distribution of sound, so as to compare it with the expression we shall obtain for a rectangular room, where the assumption of uniform dis- tribution is not valid. Solution in Series of Characteristic Functions. — The preceding equations are satisfactory for high frequencies, where the resonance frequencies are close enough together for the response to be fairly uni- form, and for rooms of sufficiently irregular shape for the sound to be uniformly spread over the room. It certainly will not be valid at low frequencies, where the response is far from uniform. To determine the range over which Eqs. (34.1) and (34.2) are valid, we must analyze the coupling of the source with the individual standing waves. For a given room, the characteristic functions, being the solutions of the wave equation which satisfy the proper boundary conditions, can be represented by the sequence yp N , where N represents a trio of 416 STANDING WAVES OF SOUND [VHI.34 integers and the ^'s satisfy the orthogonality condition given in Eq. (33.10). The corresponding characteristic values can be written as (o) N — ik N ). If the walls are rigid k N = 0, and if the walls have some simple shape (rectangular, cylindrical, etc.) the values of ca N can be computed, as has been done in Sec. 32. If the walls are not rigid, the values of w N and kn can be obtained to the first approximation by Eqs. (33.1), (33.3), (33.6), or (33.7), to the second approximation by Eqs. (33.13) and (33.14). In any case the functions \f/ N are solutions of the equation 2 Aw + A\vY , VVjv = I ) yp N Any source of sound may be considered to be an assemblage of simple sources, as was indicated in Eqs. (27.2) and (27.3) and as was done in Eq. (28.1) in computing the radiation from a piston. A source distribution qixa^e-™' can be expressed as a series of characteristic functions : Qm = (?b) / // q{x',y',z')+ M (x',y',z') dx' dy' dz' ) (34 ' 3) room The steady-state distribution of sound in the room can likewise be represented as a series XA m ^m, with the series satisfying Eq. (27.3), which becomes V A n [(i0)N + &JV-) 2 + i0 2 ]\pM — lO)pC 2 ^ Qm$M Solving this for the coefficients A, we finally obtain a series solution for the steady-state pressure p — pce ^ 2 ^ An + .^ _ w2) Mx,y 4 z) t<54.4) where we have assumed that ## is small enough so that we can neglect k% compared with a>%. This corresponds to the series of Eqs. (10.16) or (11,6) for the string. The equation shows that the steady-state pressure wave at a point (x,y,z) is the sum of the waves corresponding to the different normal modes of the room, each with amplitude proportional to the values of Vm.34] FORCED VIBRATIONS 417 the standing wave at the source and at (x,y,z) and inversely propor- tional to the "impedance" of the standing wave: (2o) N kif/o>) — i[u — (co^/co)]. Steady-state Response of a Room. — The energy density at any point in the room is (p 2 /2pc 2 ), and the intensity, by Eq. (32.3), is (p 2 /Spc) in those parts of the room where the energy is fairly uniformly distributed (i.e., not too near the source). When k is small the only functions yp n that have large coefficients in expression (34.7) are those having allowed values of oo n nearly equal to 2rv (i.e., those standing waves whose allowed frequency is almost equal to the frequency of the source). Even these coefficients may be small if Jjjqrf/ n dv is small (i.e., if the source of sound is located near & node of the nth standing wave). The only important terms in the expression for the intensity at a point (x,y,z) are those for waves whose natural frequency is close to the frequency of the source and which are so distributed in space that neither the source nor the point (x,y,z) is at a node of the wave. At low frequencies the allowed frequencies are spaced so widely apart that for some frequencies no coefficients A M will be large, and for some others only one may be large ; and this will be large only when the source is put in particular places in the room. Only for large frequencies, when several standing waves have frequencies near enough to that of the source to be excited strongly, will the intensity be distributed more or less uniformly over the room and will the intensity at any point be uniform when the frequency of the source is changed. The intensity close to the source will always be larger than that some distance away, as a calculation by Eq. (34.4) will show; .but at high frequencies the intensity throughout the rest of the room will be fairly uniform. As mentioned earlier the measurable quantity is not energy density or intensity but mean-square pressure. This is obtained by squaring the series for p, integrating over the room volume, and dividing by 2V. In virtue of the integral properties of \J/ N , given in Eq. (33.10), this becomes Pma ~ 2V 2 ZA (2u> N k N /u) 2 + \(col/o) - u,) 2 ld4 - D; N where Ajv is the average value of *//% over the room. If the largest dimension of the sound source is less than a half wavelength, the inte- gral Jjjq^NdV becomes Qo^n(S), where Qo is the equivalent source 418 STANDING WAVES OF SOUND [VIIL34 strength (amplitude of total air outflow from source) and where iPn(S) is the value of -^n at the position S of the source. Rectangular Room. — If the room under consideration is a rec- tangular one of sides l x , l y , U, the characteristic functions are those given in Eq. (32.11). To the first approximation the allowed values of un are given in Eq. (33.1) and those of the damping constant k N are given in Eq. (33.3) or (33.6). The pressure at the point P = (x,y,z) due to a simple source Qoe'™* at S = (x ,yo,Zo) is then _ PC 2 Qo iut ^ 6n,e w ,6n.^y(^)^(P) ,„, fi , P— V e 2j (2coa-W") + *'[K/co) - «1 JV where eo = 1, e 2 = e 3 = • • • =2. The mean-square pressure throughout the room is then Prms— 2 y 2 ^ (2u, N k N /u) 2 + [ K/co) - 0>] 2 ^ ' ' N The source factor E N (S) = e^e^e^i/^^) is unity if the source is placed at random in the room (or is moved about in the room to average out the effect of source position), but is e nx e ny e ni if the source is placed at a corner of the room. Transmission Response. — Figure 92 shows response curves for a room 10 by 15 by 30 ft (the one used for Fig. 89), with a damping constant for oblique waves of 1c n = 10 (assumed independent of frequency) corresponding to a reverberation time of about 1 sec. The source is located at one corner of the room, thereby making \[/n(S) unity for all values of N and making the variation in energy depend only on the impedance functions for the normal modes and on the position of the observer. Curves for the energy density at the center of the room, at the center of an opposite wall, and for the average density are plotted as functions of v. We see that at low frequencies the response is very nonuniform, both as to position and as to fre- quency, but that it becomes more uniform at higher frequencies. These curves should be compared with Fig. 89, which gives the number of allowed frequencies between v — 5 and v + 5 for the same room. In Sec. 32 we did not include the damping of the standing waves, and we had to assume that the frequency distribution of the sound source was "spread out" in order to obtain a uniform response even at high frequencies. For a spread of Av we found that the response would be fairly uniform for frequencies larger than \/c 3 /£tVAv [according to Eq. (32.14)]. In the present section we see that absorption at the walls spreads out the response of each VHI.34] FORCED VIBRATIONS 419 standing wave so that, instead of responding "infinitely well" to just one frequency and only moderately well to others, there will be a range of frequencies over which the response will be about equally large. We can show, from Eq. (34.7), that the range of frequency, over which the response of a given standing wave is larger than one-half its maximum response, is between v n — \/3(A-/27r) and v n + V 3(A-/2r), 50 100 50 y v Fig. 92. — Response curves for a room 15 by 30 by 10 ft with reverberation time 1 sec. Dotted lines correspond to approximate formula (34.12), which the actual curves approach asymptotically. so that the response has a "spread" of Sv = \/3(k/ir) c~ (k/2). This means that if the frequency of the source is high enough so that the average difference between the allowed frequencies of the standing waves that resonate is less than (k/2), then several standing waves will have large amplitude, and the response of the room will be uniform even if the source sends out a single frequency. Utilizing the fact that k ~ (7/T), where T is the reverberation time, and using Eq. (32.14), we see that the general rule for the lower 420 STANDING WAVES OF SOUND [VHI.34 limit of uniform response is the following: If a source is sending out energy over a frequency range from v to v + Av in a room of volume V and reverberation time T, the response of the room will be fairly uniform in frequency and in distribution over the room for all fre- quencies greater than v„n a , where 1 £f__V \4a-7 [A»+ (4/T)]/ iovvT (34 - 8) for V in cubic feet VAv + (4/7 1 ) For a pure tone (Av = 0), and for a reverberation time between 1 and 3 sec, this JWn is approximately equal to 10 4 / \/F. For an auditorium as large as 50 by 100 by 200 ft this lower limit is 10 cps, so that no large resonance fluctuations will occur in the useful frequency range (30 to 10,000 cycles). For a room as small as 10 by 12 by 20 ft, however, the response will fluctuate considerably for frequencies below 200 cycles, unless the spread in frequency of the source is as large as 100 cycles or larger. The preceding analysis, of course, does not take into account the possible focusing effects of the walls. If some of the walls are smooth and curved, a considerable concentration of the energy may occur in some parts of the room, with detrimental effects on the uniformity of response and the reverberation time of the room. The Limiting Case of High Frequencies.— As a last example of the method of standing-wave analysis of sound in a room, we shall show how the standing-wave formula (34.7) for a simple source goes over into the simple formula (34.2) when the frequency is high enough for uniform response. In such a case the allowed frequencies are close enough together so the quantity {(wwA-^/tt^) 2 + [(<a%/%rv) — Zwv] 2 }- 1 , when considered as a function of a) N , does not change much as u N changes from one of its allowed values to the next. When this is true we can change the summation of Eq. (34.7) into an integration over the variable u = (a N . We shall have to separate the axial from the tan- gential from the oblique waves in performing the integration, since they have different values of ku, even to the first order. The number of such waves having values of a N between u and u + du can be obtained from Eqs. (32.14), by letting u = 2rv. All these terms are obtained by modifying the series of Eq. (34.7) from a series over N to a number of integrals over dN, where the dN's for different types of waves are given in Eqs. (32.14). The largest term of all comes from the chief term (AttvW/c 3 ) dv = (u 2 V/2t 2 c s ) du in the expression for dN Q b and is VIII.34] FORCED VIBRATIONS 421 P 2 cQW 4tt 2 E (S) !. u 2 du (2uko) 2 + (u 2 - co 2 ) 2 (CJ >>> ^Vaan) where 2£ (£) is unity if the source is not near a wall; is 2 if it is on a wall, not near an edge; is 4 if it is on an edge, not near a corner; and is 8 if it is near a corner ("near" is less than a wavelength). The quantity k Q is given to the first order, according to Eq. (33.6), by the expression ^° ~ \w)' a ° = a * + ay + a * ~ 2 ^ k ^ a " ^ ' all walls where a is the room absorption for the oblique waves at the frequency v (k may depend on v) and is the expression that comes closest to the absorption a given in Eq. (32.4) for a room with uniform distribution of sound. The integrand written above is large only when u is nearly equal to co (since k is usually much smaller than co). Therefore no great error is made if we extend the lower limit of integration from to minus infinity. If this is done, the integral can be readily performed. /: u 2 du if. dx (2uk ) 2 + (w 2 - a> 2 ) 2 — 4 J_ . k 2 + x 2 Therefore this largest term in the expression for w becomes P 2 co 2 Q 2 7T Wo T^VE Eo{s) 2tOo Eo(s) which is to be compared with the expression for p 2 ^ given in Eqs. (34.2) for the statistical case of uniform distribution of sound. Approximate Formula for Response. — The other terms in the integral, for the oblique, tangential, and axial waves, can be computed in a similar manner. The result is, to the second order in (v mia /v), J P f «H28 Pi. B x = 2rrao B X B V B* ^2L O-x "f* dy 4" 0>z ■%0>x -j- CLy 4" Q>z dx + CLy + a z 2 + — 2i x \j£a x + Oy + a z etc. (source not near an x-wall) (source close to an x-wall) (34.9) — 2 (8K a )A B ; a = a x + a y + a z ; X = (2irc/co) x-walls etc. 422 STANDING WAVES OF SOUND [VIII.34 When the room dimensions are large compared with a wavelength, this reduces to Eq. (34.2) which was derived for the case of uniform sound distribution. To recapitulate: at low frequencies, below jw ^ 10 4 s/T/kV, a room is an irregular transmitter of sound, having alternate bands of good and bad transmission, more or less irregularly placed depending on the values of the resonance frequencies of the various standing waves (which are widely spread on the average at low frequencies). If any of the room dimensions is equal (l x = l y , etc.) or is a simple frac- tion of each other (l, = %l y , etc.), then multiple degeneracy will occur and the irregularity at low frequency will be still more pronounced. At frequencies above jw the individual resonance peaks merge together over most of the frequency range and most of the irregularity disappears. At high enough frequencies the mean-square pressure is practically uniform over the room and equal to the value computed statistically, given in Eq. (34.2). The same sort of analysis 1 can be carried through for the pressure wave itself, given by Eq. (34.6). The analysis is somewhat more difficult than that for p^, owing to the fact that when (x,y,z) is near the source the phase difference. between *[/ N (S) and \f/ N (x,y,z) must be taken into account. The results are that when v is larger than v min the amplitude of the pressure wave at a distance r fro m a sou rce in a rectangular room is (j>vQ /2r) when r is less than V^o/loV and is ap proxima tely constant, equal to 2pvQ y/v/a, when r is larger than ■x/ao/l&ir. (This holds when neither the source nor the point (x,y,z) is very close to a wall.) Close to the source the wave behaves as if the source were not in a room, being radiated outward in a form corre- sponding to that given by Eq. (27.4) for the pressure wave from a simple source in free space. Far from the source the waves are uni- formly distributed and are traveling in all directions, though there is everywhere a slight preponderance of waves coming from the source and a slight diminution in amplitude as r is increased. The more highly damped the room is, the greater will be the portion of the room filled with radially outgoing waves, and the smaller will be the portion having uniform density, as was indicated in Eq. (34.1). Of course, if' the room has focusing walls, these results will be considerably altered. When either source or observer is close to a wall, interference effects will occur at certain wavelengths, owing to the reinforcing 1 See E. Fermi, Quantum Theory of Radiation, Reviews of Modern Physics, 4, pp. lOOjf. (1932), for a discussion of the methods used, details of the calculation of the integrals involved, etc. vm.34] FORCED VIBRATIONS 423 effect of the reflected wave. When the source is sending out sound of frequency v (larger than v min ), the mean-square pressure a small distance 8 from a wall will be proportional to cos 2 (2rv8/c), unless the walls are very highly absorbent. The phenomenon is the analogue of the optical effect utilized in the Lippmann process of color photography. Exact Solution. — All the calculations we have made so far have been by the use of approximate solutions of the boundary condi- tions. In some cases these first- (or second-) order approximations are sufficiently accurate ; but if the wall-admittance ratios get large (magni- tude larger than about 0.2) the approximate solutions lose their validity, and the formulas given in this and the previous section may be several hundred per cent in error. In most cases, when this is true, the problem cannot be computed, but in a few cases, where the wall admittance is constant over each wall, an exact solution can be obtained. As an example of such a case, and of the exact solution that can be obtained, we consider a rectangular room of dimensions l x , l y , l z (origin of coordinates at corner of room) with the wall at x = l x having an acoustic impedance [acoustic-admittance ratio (pc/z) = t\ = k. — ia] constant over the whole wall and with the other five walls perfectly rigid. This is a somewhat idealized version of a small sound room set up to. measure the absorbing properties of acoustic material. The material is sometimes placed on one wall, the other walls being bare concrete. The general method of solution has already been discussed in Sec. 31. We set (34.10) N =? n,n y ,n z The boundary condition at x = l x is that p = zu x , or tanh(7T0„) = iiP^b X = (2rc/co) (34.11) The equation is similar to that of Eq. (31.1) or (31.6) and can be solved graphically by means of one part of Plate V, at the back of the book. Note, however, that in this case (five walls rigid, one wall soft) h = (2|i?|Z/X), an extra factor of 2. The solution is complex, g = £ + i\i- We note that the whole set of functions ^n depends on 424 STANDING WAVES OF SOUND [VIII.34 the driving frequency {w/%r) through the complex quantity g n . Con- sequently, these characteristic functions are useful primarily for expres- sing the steady-state driven motion of air in the room in terms of a series expansion. From the steady-state response we compute the transient behavior by the operational-calculus methods. The equation for the forced motion due to a simple source at S = £o,2/o,Zo is, from Eq. (27.3), where P = x,y,z and An is the average value of \[>% over the room volume. The calculation of the series expansion for p is carried through in the same manner as for Eq. (34.6) . The result is V " V ^J A N (co 2 - un) As we have mentioned, the constant un is now a complex quantity. The real part co% corresponds to the resonance frequency, and the imaginary part k% corresponds to the damping constant. If we can neglect (k° N ) 2 compared with (<4) 2 , as we usually can, the quantity wj, can be written as «) 2 - 2ua%K- But from the definition of w 2 . given in Eq. (34.10), using the notation g N = (fw + iiur (where £ and n can be determined from Plate V), we can write co- - a** - w [(?y + fey + fey - <*^)] or 7TC V I /._ I \ I... I \ i,„ I I (34.12) /^■V2„„t„\ v n w vey+ey+ey o ~ , / V 2 c v,vgA The resulting series for p and for the mean-square pressure (when the source is placed at random away from the walls) is PW ~ V 2± An o) 2 - « ~ **&) 5 , * , \ (34.13) Pmu, — 2F 2 ^ (2<^) 2 + [«) 2 - 0> 2 ] 2 We note that since $n and /in, solutions of Eq. (34.11), are functions Vin.34] FORCED VIBRATIONS 425 of w, all the functions rpx and all the quantities &$ and k% are functions of the driving frequency (w/27r). The expression for pL, is quite similar to that of Eq. (34.7), and the discussion of its behavior would follow the same lines as that given earlier. The difference is in the value of k%, which is now ~w~)-\w)\-~ir-r where \ N = (2irc/w° N ) and A a = l y l z , the area of the wall that is covered with acoustic material. Equation (33.6) would give k^ as (») (4e w /c s J, to the first order in the conductance ratio k s , for the case under consid- eration. Detailed examination of the solutions of Eq. (34.11) shows that the quantity a „ sW = (4*w^. n = (e^ (34 . 14) is approximately equal to (ie„/c s ) when k is small (0.1 or less), but is a much more complicated function of k and of <o than this when k is not small. The Wall Coefficients.— The quantity <x% N is the closest analogue to the absorption coefficient for the acoustic material that one can have in a rectangular room with at least one "soft" wall. It has been called the wall coefficient [see Eq. (33.4)]. We notice that it is not only dependent on the acoustic impedance of the material (as is the a for a room with uniform sound distribution), but also depends on the size of the room and on the particular standing wave that is being damped. For hard walls (k small) the variation with room size becomes negli- gible, and the only dependence on the type of wave is in the difference between tangential and oblique classes. For larger values of k the quantity a N (u>) may change appreciably from one value of N to the next (and it can be shown to depend also on the impedance of the opposite wall). In other words, each standing wave has a different reverberation time; and the decay curve for a sound involving several standing waves may be far from a straight line. At present our analysis is for the steady-state, driven motion of the room, so that we cannot yet say that k% is the damping constant for the reverberation. As we shall see shortly, the damping constant 426 STANDING WAVES OF SOUND [VIII.34 (and the related wall coefficient a) for the free vibration of a standing wave is obtained by setting w equal to < - ik° N in solving for £ y and hk. Since < also depends on co, this means a series of successive approximation calculations to find the correct values of k and a to use for the transient decay rate. Transient Calculations, Impulse Excitation. — To show how the free vibrations of the room can be computed by contour integral evaluation from Eqs. (34.13) we shall take as a first example the response to a sudden explosion of unit strength at the point S = x ,yo,z. This corresponds to a simple source at S of strength 8(t), where 5 is the impulse function defined in Eq. (6.11). By an obvious generaliza- tion of Eqs. (6.16), the resulting pressure distribution in the room is Pi(0 (« < 0) where p(co) is the series given in Eq. (34.13). Finding the locations of the poles of each term in the series for p(u) is rather complicated. Both < and k N vary as co is varied (since £ and n depend on co) ; in fact, k% changes sign when co becomes negative (since £ changes sign). However, in principle, a value of «°' and k% can be obtained for every value of co, even for complex values. The position of a pole is found by varying co over the complex plane until it equals <a% - ik% for that same value of co. The value of co at that point can be called < - ik r N , the corresponding values of $ and v can be called & and » r N , and that of a° xN can be labeled just a xN . The other pole turns out to be at - a r N - ik r N , as would be expected from the symmetry of the equations. The corresponding values of £, fi, and a are just — &, n N , and <x x n. # _ The characteristic functions $», which are complex quantities, are also functions of co. Their values at the poles can be represented by the following symbols: For a, = o> N - iky, mp) = r K (P)*"™; A * = W** For co = -< - ik N ; MP) = MP)e-^ p) ; A* = A^r^ where ¥ N is a real function of x, y, z, the magnitude of the pressure wave; and <p N , the phase lag of the JVth wave, also depends on x, y, z. The average value of $\ also has its magnitude and phase. We can now compute the residues about the poles of the function p(co). The resulting expression for the "impulse wave" is VHI.84] FORCED VIBRATIONS 427 ^.j^&cop^,. (<<0) (34 , 5) \ • cos [tf„t - <p' N (P) - ^(Q) + ^] ( t > 0) The individual standing waves damp out exponentially, each with the damping constant k r N , given in terms of the wall coefficients <x xN by Eq. (34.14), where the parameters n r N , & are computed for the reso- nance frequency « = «£ - iV N , as mentioned above. This series does not converge, corresponding to the fact that the excitation cor- responds to an impulse function. But for a physically possible fluctua- tion of source strength Q(t), the corresponding pressure wave P = jl „ Q(j)p»(t - r) dr (34.16) analogous to Eq. (6.17) is given by a series that does converge. This integral is the counterpart, for the exact solution, of Eq. (32.6) for the statistical solution. Exact Solution for Reverberation.— As a last example we shall compute the response of the rectangular room, with the wall at x = l x covered with material of admittance ratio v = K - ia and the other five walls rigid, for a simple source of strength <? sin (2™*) from t oo to t = 0, and of strength zero from t = to t = oo (j e the source, of frequency v, is shut off at t = 0). This can be computed by the use of Eq. (34.16), or it can be computed by obtaining the Laplace transform for the source function and calculating another contour integral for the pressure. We choose the latter method to provide one further example of its use. ' We first compute the steady-state pressure wave in the room if the source is not shut off at t = 0. We calculate the parameters * and M for the frequency v and from them compute the steady-state functions- t or positive values of %rv : For negative values of 2™ : < - iK = u f N e- ia »° < + ik° N = «/e"fc- MP) - f»(?)^(« MP) = r N (P)e- i ^°^ A* = A» e*** An = A >-i**° Then, using the series for p(„) given in Eq. (34.13), we have that the steady-state wave due to a source Q sin (2wvt) is 428 STANDING WAVES OF SOUND [Vm.U -V 2irV 2i A%G%{2nrv)' N p.(2r,0 = ^ real part of |^w ^ _ (2 ^ )2 _ (< _ ^ )t j cos[2*v* - <p%{P) - <p%(S) + ^ + r^ - (34.17) where [G° N (2irv)f = (ZkvY + K) 4 - 2(2ttv) 2 (co/) 2 cos (2Q&) and -«> = [s^)] sinW) Next we compute the pressure oscillations in the room for a source that is zero for t < and Qo sin (2irv0 for t> 0. To do this we need the Laplace transform [see Eq. (6.19)] for this source function X27TV sin (2rrf)«r* dt = ^^ + g2 Then the resulting pressure is, using Eq. (34.13) again, p u (t) = ^ I *(-iw)p(w) dco oc 2 Qo r - 7 _• * "^ * w $n(PWn(Q) = ""V" ^ ,2l JW[co 2 - (2™) 2 ][co 2 - (co» - t/#)*J J- * AT The residues about the poles a> = ±2irv result in the function p s (2rv) of Eq. (34.17); whereas the residues about the poles w = ± W £ - tA* [by a calculation similar to that for Eq. 34.15)] give — p r (t), where, for t > 0, m _ P c2 £° >? M p )^( S Vrit) ~~ ir 2a A^(2x.) pc 2 Qo ^ + r N ( P)MS) p - kN rt. • cosK* - <P r N (P) - <&(&) '+ *Sr + r r J (34.18) where [G* N (%tv)Y = (2rv)» + K - ik r N \ 4 ~ 2(2rv)*K ~ **&!' cos (2^) and and VHI.34] FORCED VIBRATIONS 429 Finally, the effect due to a source that has been of strength Qo sin (2rrvt) from t = — <*> to t = 0, and is turned off at t = 0, must be the difference between the steady-state response ps(2irv) and the expression p s (?Trv) — Pr(t) which represents the effect if the source is zero before t = and is turned on at t = 0. Consequently, the pres- sure wave in the room due to a source that has been on for a long time and is turned off at t = is / p.(2rv) (t < 0) [see Eq. (34.17)] V \ Pr(t) (t > 0) [see Eq. (34.18)] The steady-state response p 8 and the reverberation p r are similar series. There are differences in the values of the amplitudes of the terms, represented by the superscripts and r, owing to the fact that the driving frequency (2rv) is used in p s for the calculation of £# and hn (and thus w%, k%, \f/%, etc.), whereas the "natural frequency" (<a N — ik r N ) is used in p r for the calculation of £ and p. The change in amplitude is not large for those standing waves having resonances near the driving frequency (a> N nearly equal to 2irv). The major difference between the series is in the exponential e~ klfH and in the terms 2irvt or w N t in the cosine terms. The reverberation series p r damps out in time, whereas the steady-state pressure p 8 has a constant amplitude (as, of course, it must). In addition, the indi- vidual standing waves in the reverberation each oscillate with their own natural frequency co^, whereas every term in the steady-state solution oscillates with the driving frequency (2rv). Since there are many natural frequencies u r N with nearly the same value, complicated "beat-note" effects often occur in the reverberation (see Fig. 91). It is a far cry from auditorium acoustics to second quantization; yet the methods used above to analyze the behavior of sound in rooms are quite similar to the methods Dirac used to predict the existence of the positron three years before its experimental discovery. This similarity is an interesting example of the unifying influence of theo- retical physics. Problems 1. A rectangular office room is 15 ft high, 20 ft wide, and 30 ft long. The walls are of plaster, wood, and glass, with absorption coefficient 0.03. The floor is covered with a carpet (a = 0.2) and the ceiling with acoustic material (5 = 0.4). Ordinarily, six persons are present in the room. What is the reverberation time? If four typewriters are going in the room, each producing 1 erg of sound energy per second, what will be the intensity level in the room? 430 STANDING WAVES OF SOUND [VIII.34 2. An auditorium is 30 ft high, 50 ft wide, and 100 ft long and contains 500 wooden seats (a = 0.15 apiece). The walls, ceiling, and floor have an absorption coefficient of 0.03. What is the reverberation time when the auditorium is empty? When it is full? How much acoustic material (a = 0.4) must be placed on the ceiling and walls in order that the reverberation time may be 2 sec when the audi- torium is empty? What is then the reverberation time when the room is filled? What must then be the power output of a public-address system in order to have the average intensity level in the filled auditorium 90 db? What will be the intensity when the auditorium is empty? 3. Calculate and plot the absorption coefficient, as a function of frequency of the material given in Prob. 23, Chap. VII, for both values of spacing from the wall, for < v < 2,000 cps. 4. A room with cylindrical walls, of radius 5 m, has a flat floor and ceiling, 4 m apart. Plot the number of allowed frequencies in the room between v and v + 5 as a function of v from v = to v = 50. Above what frequency will this curve become fairly uniform? 5. A rectangular corridor is 2 m wide, 3 m high, and 10 m long. Plot the number of allowed frequencies in the enclosure between v and v + 5 as a function of v from v = to v — 100. Above what frequency will this curve be fairly uniform? 6. A cubical room 5 m on a side has an average absorption coefficient for floor and ceiling of 0.2; for the walls, a value of 0.04. What is the reverberation time for those waves which strike floor and ceiling? For those waves which do not strike floor and ceiling? List all the allowed frequencies between zero and 100 cps and give the position of the nodal planes and the reverberation times of each corresponding standing wave. 7. List frequencies, nodes, and reverberation times of the normal modes between v = and v = 100 for the room of Prob. 6 when walls, floor, and ceiling all have an average absorption coefficient of 0.1. 8. The air in the room of Prob. 6 is started into vibration so that all the normal modes between v = 98 and v = 102 are set into motion with equal initial amplitudes. What normal modes are excited? Plot the decay curve of intensity level against time after the source is shut off at the mid-point of the room ; at the mid-point of one wall; at a point 167 cm out from two walls and 250 cm up from the floor. 9. A room has constants l v = l z = 500, l x = 505, a x = a y = 0.02, a z = 0.2. All the allowed frequencies between v = 81 and v = 83 are excited with equal amplitude. Plot the decay of intensity level at the mid-point of the room, at the mid-point of the floor, at a point in one corner of the room halfway between floor and ceiling (assume that at this point the pressure due to the different standing waves are in phase when the sound shuts off). Will the decay curve be the same for the corresponding points in the other three corners? 10. The allowed frequencies of the room, discussed in Prob. 9, between v — 66 and v = 68 are excited with equal amplitude and are in phase at the mid-point of the room at the instant when the source is shut off. Plot the decay curves for the mid-point of the room; for the mid-point of one of the walls perpendicular to the z-axis; for a point in one corner between walls, halfway between floor and ceiling. Will the decay curve be the same for the corresponding points in the other three corners? Vin.34] FORCED VIBRATIONS 431 11. The room of Prob. 6 has a point source of strength Qo ■» 10 located at its mid-point. Plot the average energy density in the room as a function of v from v ■= to v = 100. The source is stopped at the instant that its source function q is a maximum, when its frequency is 100 cps. Plot the decay curve of sound at the mid-po'nt of one wall. 12 The point source of Prob. 11 is relocated at the mid-point of one of the walls. Plot the energy density as a function of v from v = to v = 100 at the mid-point of the room; at the mid-point of the wall opposite the source. 13. A point source of source function q = 15 sin(40rf) - 10 sin (120x0 + 3 sin(200ir*) is placed at the mid-point of one of the walls of the room of Prob. 6. Plot the curves of pressure against time for one cycle of the sound for the mid-point of the room; for the mid-point of the opposite wall; for one of the corners of the room farthest from the source. 14. Discuss the reverberating qualities of a cubical room 5 m on a side, having two opposite walls with an absorption coefficient 0.8, the other walls with a coeffi- cient 0.05. Find the "average reverberation time" from Eq. (33.6), and find the reverberation time for the lowest five standing waves by use of Eqs. (34.12). 15. Calculate the exact values (by Plate V) of the damping constants and the corresponding reverberation times for the lowest five standing waves in a room 8 by 12 by 16 ft with acoustic material on one of the 8 by 12 walls and with the other five walls rigid. The specific acoustic impedance of the "soft" wall is 3pce i(ir/6) (|z| = 3pc, <p =» —30°) for the frequencies of these five standing waves. BIBLIOGRAPHY This is not intended as an exhaustive classification of all the books on vibrations and sound; it is simply a list of the books that the writer has found particularly useful as collateral reading or as reference works giving particular problems in greater detail than can be given in the present volume. A good book on the philosophy of the scientific method, with many examples in mechanics and theory of vibrations, is "Foundations of Physics," by Lindsay and Margenau. A good general text on theoretical physics is Slater and Frank "Introduction to Theoretical Physics." Reference works on theoretical physics are Byerly, "Fourier Series and Spherical Harmonics"; Bateman, "Partial Dif- ferential Equations of Mathematical Physics"; and Webster "Partial Differential Equations of Mathematical Physics." Reference works on the type of mathe- matics used in this book are Whittaker and Watson, "Modern Analysis"; Courant and Hilbert, " Mathematische Physik"; and Watson, "Theory of Bessel Func- tions." A good set of tables of the functions used, more complete than those given below, is in Jahnke-Emde, "Tables of Functions." In the field of vibrations and sound the standard reference works are Rayleigh, "Theory of Sound"; and Helmholtz, "Sensations of Tone." Useful books for collateral reading are Lamb, "Dynamical Theory of Sound"; Richardson, "Sound"; Watson, "Sound"; Miller, "Science of Musical Sound"; Crandall "Vibrating Systems and Sound"; Trendelenberg, "Klange und Gerausche"; Stewart and Lindsay, "Acoustics"; and Bergmann and Hatfield, "Ultrasonics." Especially useful are Olsen and Massa, "Applied Acoustics," for its discussion of recent practical applications; Fletcher, "Speech and Hearing"; and Stevens and Davis, "Hearing, Its Psychology and Physiology," for their discussion of the physicophysiological aspect of acoustics. In the field of architectural acoustics the classical reference is the "Collected Papers" of W. C. Sabine. Books for collateral reading are Knudsen, "Archi- tectural Acoustics"; Knudsen's article in the Reviews of Modern Physics, 6, 1 (1934) ; Watson, "Acoustics of Buildings"; and P. E. Sabine, "Acoustics and Architecture."' Of course, the issues of the Journal of the Acoustical Society of America should be consulted. Two articles in the Reviews of Modern Physics, one by V. O. Knud- sen on page 1, Vol. 6 (1933) and one by Morse and Bolt on page 69, Vol. 16 (1944), have material of interest in architectural acoustics. 433 34 320 GLOSSARY OF SYMBOLS The more commonly used symbols are listed below, together with their usual meanings and a reference to the page where the symbol is denned (if a definition is needed) . Only those symbols used in several sections are listed. SymbGl , Meaning p a 3 Radius of cylinder or circular diaphragm, in cm 298 I Absorption constant of a room, in sq ft or sq cm 3g 5 A 5 Amplitude of oscillation, in cm 9 } Area of room in sq ft 38 _ B \ Mechanical susceptance in sec per g on } Magnetic induction, in gauss 34 c Wave velocity, in cm per sec 73 222 C Capacitance in farads ' C m Radiation amplitude for a cylinder 301 du Piezoelectric constant D Ratio mechanical force to current, in dynes per amp D m Radiation amplitude for a sphere 1 J~ 2 - 71828 * * • , base of natural logarithms A Emf, in volts < General function } Fourier transform of force F Force, in dynes G Mechanical conductance, in sec per g ,. h Half thickness of a plate Jj? H Specific response function for a diaphragm * " V ?F I \ . To corre spond to the usual electrical-engineering nota- tion, the x in this book is equal to minus j ■, , / Current, in amps ^ /« Hyperbolic Bessel function .rjj 3 = ~i- See note on i jm Spherical Bessel function J.Z J m Cylindrical Bessel function ^Jl k (Damping constant (R/2m) „_}!! I Wave number („/c) = (2x/X) = (2™/c) ' "* A Stiffness constant of spring <T I Length of string or tube, in cm ?* ln(x) Natural logarithm of x log (a;) Logarithm to base 10 of x L Inductance, in henrys «C Symbol representing Laplace transform ? Mass, in g bZ 201 M Reactance function for piston in plane wall J 435 436 GLOSSARY OF SYMBOLS Symbol Meaning Page m,n Integers labeling normal modes of vibration 84, 390 n m Spherical Neumann function 317 N m Cylindrical Neumann function 196 p Excess pressure in sound wave, in dynes per sq cm 218 j Power, in ergs per sec 26 I Pressure amplitude, in dynes per sq cm 224 P n Legendre function 315 q Source density, in sec -1 313 != (o} m/R) (The "Q" of a circuit or other system) 25, 403 Modulus of elasticity, in dynes per sq cm 152 Strength of simple source, in cc per sec 313 r Radial distance, in cm 311 S Electrical resistance, in ohms 36 I Mechanical resistance, in g per sec 24 {Elastic modulus of crystal 40 Poisson's ratio 209 S Area of cross section, in sq cm 217 t Time, in sec 2 I Tension, in dynes m 72 ( Reverberation time, in sec ' 387 u Unit step function , 47 u,v,w Components of velocity, in cm per sec U Velocity amplitude, in cm per sec 23 ( Potential energy, in ergs 3 I Volume, in cc or cu ft 394 w Energy density in sound wave, in ergs per cc 383 W Energy, in ergs 2 x,y,z Rectangular coordinates ( Stress in crystal, in dynes per sq cm 39 ] Mechanical reactance, in g per sec 29 Displacement of string 71 Y Mechanical admittance, in sec per g 35 z Specific acoustic impedance (p/u) in g per sq cm per sec 237 ( Electrical impedance, in ohms 36, 38 I Mechanical impedance, in g per sec 29, 38 a Absorption coefficient 385, 403 a,/3 Acoustic impedance parameters for plane wave 139, 240 a^ n ,(3„ Characteristic numbers for Bessel functions 189,210,399 ey\ Wavelength constant for bar and plate 155, 209 & e Ratio of specific heats = 1.4 for air 221 y m Phase shift for scattered cylindrical wave 301 r Electromechanical coupling constant 35 5 Unit impulse function 47 / Change in density of air due to sound wave 218 S m Phase shift for scattered spherical wave 320 A Separation of plates in condenser microphone 204 V 2 Laplace operator, Laplacian 174, 296 s Mass per unit length of string, in g per cm 72 X y v GLOSSARY OF SYMBOLS 437 Symbol Meaning «o =* 1, «i = 62 •= es =» • • • = 2 402 f Acoustic impedance ratio = (z/pc) = — ix 144, 240 t\ (Displacement of membrane, in cm 173 \ Acoustic admittance ratio = (pc/z) = k — ia 144, 240 ( Phase angle 31 I Angle in spherical coordinates 311 B Acoustic resistance ratio 135, 240 ( Dielectric susceptibility 40 } Acoustic conductance ratio 144, 240 X Wavelength, in cm 83 m = (wa/c) = ka 320 v Frequency, in cps 6 j Strain in crystal, in cm per cm length 38 I Displacement of air in sound wave, in cm 218 7t = 3.14159 • • • (Electric polarization 39 ( Power radiated, in ergs per sec or in watts 229 \ Density 154 p / Density of air at standard conditions 218 pc Characteristic impedance of air for sound = 42 g/cm ! sec 222 ! Piezoelectric coupling constant, in ohms per g 40 Mass per unit area, in g per sq cm 173 Acoustic susceptance ratio 144, 240 2 Summation symbol 85 \ Ratio propagation velocity to sound velocity c 307 I Transmission coefficient 273 T Intensity of sound, in ergs per sec per sq cm 223 (Phase angle 31 (Angle in spherical or polar coordinates 296, 311 4> Angle in cylindrical coordinates 296 \ Laplace transform of / 52 ( Angle of incidence 367 X Acoustic reactance ratio 135, 240 ^ Complex reflection phase for plane wave = ir(a — i0) 239 SP Characteristic function <*> = 2irv, angular velocity 21 Q Mechanical phase angle 201 Cii Approximately equal to oo Infinity -+. Approaches as a limit > Is greater than < Is less than J> Contour integral 12 |Z| Magnitude of Z = IP + X 2 10 Capital subscripts for Z,R,X, etc., denote electric impedance in ohms, lower- case subscripts denote mechanical impedance in grams per second, Greek sub- scripts denote analogous impedances (see pages 38 and 237). 438 TABLES OF FUNCTIONS Table I. — Trigonometric and Hyperbolic Functions (See pages 4 and 136) X sin (x) COS (x) tan (a;) sinh (x) cosh (x) tanh {x) e* e~ x 0.0 0.0000 1.0000 0.0000 0.0000 1.0000 0.0000 1.0000 1.0000 0.2 0.1987 0.9801 0.2127 0.2013 1.0201 0.1974 1.2214 0.8187 0.4 0.3894 0.9211 0.4228 0.4018 1.0811 0.3799 1.4918 0.6703 0.6 0.5646 0.8253 0.6841 0.6367 1.1855 0-5370 1.8221 0.5488 0.8 0.7174 0.6967 1 . 0296 0.8881 1.3374 0.6640 2.2255 0.4493 1.0 0.8415 . 5403 1 . 5574 1.1752 1.5431 0.7616 2.7183 0.3679 1.2 0.9320 0.3624 2 . 5722 1 . 5095 1.8106 . 8337 3.3201 0.3012 1.4 0.9854 +0.1700 + 5.7979 1.9043 2.1509 0.8854 4.0552 0.2466 1.6 0.9996 -0.0292 -34.233 2.3756 2.5775 0.9217 4.9530 0.2019 1.8 0.9738 -0.2272 -4.2863 2.9422 3.1075 . 9468 6.0496 0.1553 2.0 0.9093 -0.4161 -2.1850 3.6269 3.7622 0.9640 7.3891 0.1353 2.2 0.8085 -0.5885 -1.3738 4.4571 4.5679 0.9757 9.0250 0.1108 2.4 0.6755 -0.7374 -0.9160 5.4662 5.5569 0.9837 11.023 0.0907 2.6 0.5155 -0.8569 -0.6016 6.6947 6.7690 0.9890 13.464 0.0742 2.8 0.3350 -0.9422 -0.3555 8.1919 8.2527 0.9926 16.445 0.0608 3.0 +0.1411 -0.9900 -0.1425 10.018 10.068 0.9951 20.086 0.0498 3.2 -0.0584 -0.9983 + 0.0585 12 . 246 12 . 287 0.9967 24.533 0.0407 3.4 -0.2555 -0.9668 0.2643 14.965 14.999 0.9978 29.964 0.0333 3.6 -0.4425 -0.8968 0.4935 18.285 . 18.313 0.9985 36 . 598 0.0273 3.8 -0.6119 -0.7910 0.7736 22.339 22.362 0.9990 44.701 0.0223 4.0 -0.7568 -0.6536 1.1578 27.290 27.308 0.9993 54 . 598 0.0183 4.2 -0.8716 -0.4903 1.7778 33.335 33.351 . 9996 66.686 0.0150 4.4 -0.9516 -0.3073 3.0963 40.719 40.732 0.9997 81.451 0.0123 4.6 -0.9937 -0.1122 +8.8602 49.737 49.747 0.9998 99.484 0.0100 4.8 -0.9962 +0.0875 -11.385 60.751 60.759 0.9999 121.51 0.0082 5.0 -0.9589 0.2837 -3.3805 74 . 203 74.210 0.9999 148.41 0.0067 5.2 -0.8835 0.4685 - 1 . 8856 90.633 90.639 0.9999 181.27 0.0055 5.4 -0.7728 0.6347 -1.2175 110.70 110.71 1 . 0000 221.41 0.0045 5.6 -0.6313 0.7756 -0.8139 135.21 135.22 1.0000 270.43 0.0037 5.8 -0.4646 0.8855 -0.5247 165.15 165.15 1 . 0000 330.30 0.0030 6.0 -0.2794 0.9602 -0.2910 201.71 201.71 1 . 0000 403.43 0.0025 6.2 -0.0831 . 9965 -0.0834 246.37 246.37 1 . 0000 492.75 0.0020 6.4 -+0.1165 0.9932 +0.1173 300.92 300.92 1.0000 601 . 85 0.0016 6.6 0.3115 0.9502 0.3279 367.55 367 . 55 1.0000 735.10 0.0013 6.8 0.4941 0.8694 0.5683 448.92 448.92 1 . 0000 897 . 85 0.0011 7.0 0.6570 0.7539 0.8714 548.32 548.32 1 . 0000 1096.6 0.0009 7.2 0.7937 0.6084 1 . 3046 662 . 72 662 . 72 1.0000 1339.4 0.0007 7.4 0.8987 0.4385 2.0493 817.99 817.99 1 . 0000 " 1636.0 0.0006 7-6 0.9679 0.2513 3 . 8523 999.10 999.10 1 . 0000 1998.2 0.0005 7". 8 0.9985 +0.0540 + 18.507 1220.3 1220.3 1.0000 2440.6 0.0004 8.0 0.9894 -0.1455 -6.7997 1490.5 1490.5 1.0000 2981.0 0.0003 TABLES OF FUNCTIONS 439: Table II. — Trigonometric and Hyperbolic Functions (See pages 4 and 136) X sin (xas) cos (-kx) tan (trx) sinh (irx) cosh (irx) tanh (rx) e xx e -rx 0.00 0.0000 1.0000 0.0000 0.0000 1.0000 0.0000 1.0000 1.0000 0.05 0.1564 0.9877 0.1584 0.1577 1.0124 0.1558 1.1701 0.8546 0.10 0.3090 0.9511 0.3249 0.3194 1.0498 0.3042 1.3691 0.7304 0.15 0.4540 0.8910 0.5095 0.4889 1.1131 0.4392 1.6019 0.6242 0.20 0.5878 0.8090 0.7265 0.6705 1.2040 0.5569 1.8745 0.5335 0.25 0.7071 0.7071 1.0000 0.8687 1.3246 0.6558 2.1933 0.4559 0.30 0.8090 0.5878 1.3764 1.0883 1.4780 0.7363 2 . 5663 0.3897 0.35 0.8910 0.4540 1.9626 1.3349 1.6679 0.8003 3.0028 0.3330 0.40 0.9511 0.3090 3.0777 1.6145 1.8991 0.8502 3.5136 0.2846 0.45 0.9877 +0.1564 6.3137 1.9340 2.1772 0.8883 4.1111 0.2432 0.50 1.0000 0.0000 00 2.3013 2.5092 0.9171 4.8105 0.2079 0.55 0.9877 -0.1564 -6.3137 2.7255 2.9032 0.9388 5.6287 0.1777 0.60 0.9511 -0.3090 -3.0777 3.2171 3.3689 0.9549 6.5861 0.1518 0.65 0.8910 -0.4540 -1.9626 3.7883 3.9180 0.9669 7.7062 0.1298 0.70 0.8090 -0.5878 -1.3764 - 4.4531 4.5640 0.9757 9.0170 0.1109 0.75 0.7071 -0.7071 -1.0000 5.2280 5.3228 0.9822 10.551 0.09478 0.80 0.5878 -0.8090 -0.7265 6.1321 6.2131 0.9870 12.345 0.08100 0.85 0.4540 -0.8910 -0.5095 7.1879 7.2572 0.9905 14.437 0.06922 0.90 0.3090 -0.9511 -0.3249 8.4214 8.4806 0.9930 16.902 0.05916 0.95 +0.1564 -0.9877 -0.1584 9.8632 9.8137 0.9949 19.777 0.05056 1.00 0.0000 -1.0000 0.0000 11.549 11.592 0.9962 23.141 0.04321 1.05 -0.1564 -0.9877 0.1584 13.520 13.557 0.9973 27.077 0.03693 1.10 -0.3090 -0.9511 0.3249 15.825 15.857 0.9980 31.682 0.03156 1.15 -0.4540 -0.8910 0.5095 18.522 18.549 0.9985 37.070 0.02697 1.20 -0.5878 -0.8090 0.7265 21.677 21.700 0.9989 43.376 0.02305 1.25 -0.7071 -0.7071 1.0000 25.367 25.387 0.9992 50.753 0.01970 1.30 -0.8090 -0.5878 1.3764 29 . 685 29 . 702 0.9994 59.387 0.01683 1.35 -0.8910 -0.4540 1.9626 34.737 34.751 0.9996 69.484 0.01438 1.40 -0.9511 -0.3090 3.0777 40.647 40.660 0.9997 81.307 0.01230 1.45 -0.9877 -0.1564 6.3137 47.563 47.573 0.9998 95.137 0.01051 1.50 -1.0000 0.0000 00 55.654 55.663 0.9998 111.32 0.00898 1.55 -0.9877 +0.1564 -6.3137 65.122 65.130 0.9999 130.25 0.00767 1.60 -0.9511 0.3090 -3.0777 76 . 200 76.206 0.9999 152.41 0.00656 1.65 -0.8910 0.4540 -1.9626 89.161 89.167 0.9999 178.33 0.00561 1.70 -0.8090 0.5878 -1.3764 104.32 104.33 1.0000 208.66 0.00479 1.75 -0.7071 0.7071 -1.0000 122.07 122 . 08 1.0000 244.15 0.00409 1.80 -0.5878 0.8090 -0.7265 142.84 142.84 1.0000 285.68 0.00350 1.85 -0.4540 0.8910 -0.5095 167.13 167.13 1.0000 334.27 0.00299 1.90 -0.3090 0.9511 -0.3249 195.56 195.56 1.0000 391.12 0.00256 1.95 -0.1564 0.9877 -0.1584 228.82 228.82 1.0000 457 . 65 0.00219 2.00 0.0000 1 . 0000 0.0000 267.75 267.75 1.0000 535.49 0.00187 MQ TABLES OF FUNCTIONS Table III. — Hyperbolic Tangent of Complex Quantity tanh [x(o - i0)\ - 6 - i x - |f |e~** tanh (ia) e X in <P X in <f> a = D.00 P = 3.05 0.0000 0.00 0.0000 0.0000 0.0000 0-90° 0.0000 0.1584 0.1584 90.00° 0.0159 0.05 0.0500 0.0000 0.0500 0.00 0.0512 0.1580 0.1660 72.03 0.0319 0.10 0.1000 0.0000 0.1000 0.00 0.1025 0.1567 0.1872 56.82 0.0481 0.15 0.1500 0.0000 0.1500 0.00 0.1537 0.1547 0.2180 45.20 0.0645 0.20 0.2000 0.0000 0.2000 0.00 0.2048 0.1519 0.2549 36.56 0.0813 0.25 0.2500 0.0000 0.2500 0.00 0.2558 0.1482 0.2956 30.09 0.0985 0.30 0.3000 0.0000 0.3000 0.00 0.3068 0.1438 0.3388 25.12 0.1163 0.35 0.3500 0.0000 0.3500 0.00 0.3577 0.1386 0.3836 21.18 0.1349 0.40 0.4000 0.0000 0.4000 0.00 0.4084 0.1325 0.4293 17.98 0.1543 0.45 0.4500 0.0000 0.4500 0.00 0.4589 0.1256 0.4758 15.32 0.1748 0.50 0.5000 0.0000 0.5000 0.00 0.5093 0.1181 0.5228 13.05 0.1968 0.55 0.5500 0.0000 . 5500 0.00 0.5596 0.1096 0.5702 11.08 0.2207 0.60 0.6000 0.0000 0.6000 0.00 0.6095 0.1005 0.6177 9.36 0.2468 0.65 0.6500 0.0000 0.6500 0.00 0.6593 0.0905 . 6654 7.82 0.2761 0.70 0.7000 0.0000 0.7000 0.00 0.7088 0.0798 0.7133 6.43 0.3097 0.75 0.7500 0.0000 0.7500 0.00 0.7581 0.0683 0.7612 5.15 0.3497 0.80 . 8000 0.0000 0.8000 0.00 0.8070 0.0561 0.8090 3.97 0.3999 0.85 . 8500 0.0000 0.8500 0.00 0.8558 0.0432 0.8569 2.88 0.4686 0.90 0.9000 0.0000 0.9000 0.00 0.9041 0.0295 0.9047 1.87 0.5831 0.95 0.9500 0.0000 0.9500 0.00 0.9523 0.0151 0.9524 0.91 = 0.10 = 0.15 0.0000 0.00 0.0000 0.3249 0.3249 90.00° 0.0000 0.5095 0.5095 90.00° 0.0159 0.05 0.0553 0.3240 0.3286 80.32 0.0629 . 5079 0.5118 82.93 0.0319 0.10 0.1104 0.3213 0.3398 71.03 0.1256 0.5031 0.5186 75.98 0.0481 0.15 0.1655 0.3169 0.3575 62.43 0.1878 0.4951 0.5296 69.22 0.0645 0.20 0.2202 0.3106 0.3808 54.67 0.2493 0.4841 0.5445 62.75 0.0813 0.25 0.2746 0.3027 0.4087 47.78 0.3099 0.4700 0.5629 56.61 0.0985 0.30 0.3286 0.2929 0.4402 41.72 0.3692 0.4531 . 5845 50.82 0.11,63 0.35 0.3820 0.2815 0.4745 36.39 0.4273 0.4333 0.6085 45.40 0.1349 0.40 0.4349 0.2684 0.5110 31.68 0.4838 0.4110 0.6347 40.18 0.1533 0.45 0.4871 0.2537 0.5492 27.51 0.5385 0.3860 . 6626 35.63 0.1748 0.50 0.5386 0.2374 0.5886 23.79 0.5914 0.3589 0.6917 31.25 . 1968 0.55 0.5893 0.2196 0.6289 20.44 0.6423 0.3295 0.7219 27.16 0.2207 0.60 0.6390 0.2003 0.6697 17.41 0.6911 0.2982 0.7527 23.34 . 2468 0.65 0.6880 0.1796 0.7110 14.63 0.7378 0.2652 0.7840 19.76 0.2761 0.70 0.7358 0.1576 0.7525 12.08 0.7822 0.2305 0.8155 16.43 0.3097 0.75 0.7827 0.1342 0.7941 9.73 0.8243 0.1945 . 8469 13.28 0.3497 0.80 0.8285 0.1096 0.8375 7.54 0.8642 0.1573 . 8783 10.32 0.3999 0.85 0.8731 0.0838 0.8771 5.48 0.9015 0.1191 0.9094 7.52 . 4686 0.90 0.9166 0.0569 0.9184 3.56 0.9367 0.0800 0.9401 4.88 0.5831 0.95 0.9589 0.0289 0.9594 1.73 0.9695 0.0403 . 9704 2.38 TABLES OF FUNCTIONS 441 Table III. — Hyperbolic Tangent op Complex Quantity.— (Continued) tanh (ira) e X in <P e X in V a /3 = 0.20 13 = 0.25 0.0000 0.00 0.0000 0.7265 0.7265 90.00° 0.0000 0.0000 1.0000 90.00° 0.0159 0.05 0.0763 0.7238 0.7278 83.98 0.0998 0.9950 1.0000 84.28 0.0319 0.10 0. 1520 0.7145 0.7315 78.01 0.1980 0.9802 1.0000 78.58 0.0481 0.15 0.2265 0.7019 0.7375 72.12 0.2934 0.9560 1.0000 72.93 0.0645 0.20 0.2993 0.6831 0.7458 66.34 0.3846 0.9230 1.0000 (67.38 0.0813 0.25 0.3698 0.6593 0.7560 60.72 0.4706 0.8824 1.0000 61 .93 0.0985 0.30 0.4376 0.6312 0.7680 55.27 0.5504 0.8348 1.0000 56.60 0.1163 0.35 0.5023 0.5989 0.7816 50.01 0.6236 0.7818 1.0000 51.42 0.1349 0.40 0.5635 0.5627 0.7964 44.96 0.6896 0.7241 1.0000 46.40 0.1543 0.45 0.6212 0.5235 0.8123 40.12 0.7484 0.6632 1.0000 41.55 0.1748 0.50 0.6749 0.4814 0.8290 35.50 0.8000 0.6000 1.0000 36.87 0.1968 0.55 0.7247 0.4370 0.8462 31.09 0.8446 0.5355 1.0000 32.38 0.2207 0.60 0.7703 0.3907 0.8637 26.89 0.8824 0.4706 1.0000 28.07 0.2468 0.65 0.8120 0.3430 0.8815 22.91 0.9139 0.4060 1 . 0000 23.95 0.2761 0.70 0.8497 0.2943 0.8992 19.11 0.9395 0.3423 1.0000 20.02 0.3097 0.75 0.8835 0.2451 0.9169 15.51 0.9600 0.2800 1.0000 16.27 0.3497 0.80 0.9136 0.1955 0.9343 12.08 0.9757 0.2195 1.0000 12.68 0.3999 0.85 0.9401 0.1459 0.9514 8.82 0.9869 0.1611 1.0000 9.27 0.4686 0.90 0.9632 0.0967 0.9681 5.73 0.9945 . 1050 1.0000 6.03 0.5831 0.95 0.9831 0.0480 0.9843 2.79 0.9986 0.0512 1.0000 .2.93 - 0.30 /} - 0.35 0.0000 0.00 0.0000 1.3764 1.3764 90.00° 0.0000 1.9626 1.9626 90 . 00° 0.0159 0.05 0.1440 1 . 3664 1.3740 83.98 0.2403 1.9391 1.9539 82.93 0.0319 0.10 0.2841 1 . 3373 1.3671 78.01 0.4670 1.8708 1.9283 75.98 0.0481 0.15 0.4164 1 . 2904 1.3450 72.12 0.6699 1.7653 1.8882 69.22 0.0645 0.20 0.5382 1 . 2282 1.3408 66.34 0.8408 1 . 6326 1.8365 62.75 0.0813 0.25 0.6455 1.1537 1.3228 60.72 0.9776 1.4832 1.7762 56.61 0.0985 0.30 0.7419 1.0701 1.3021 55.27 1.0809 1.3261 1.7109 50.82 0.1163 0.35 0.8223 0.9803 1.2794 50.01 1 . 1540 1.1701 1.6432 45.40 0.1349 0.40 0.8885 0.8873 1 . 2556 44.96 1.2007 1.0140 1.5755 40.18 0.1543 0.45 0.9413 0.7933 1.2311 40.12 0.2267 0.8793 1 . 5092 35.63 0.1748 0.50 0.9820 0.7005 1.2063 35.50 1.2359 0.7500 1.4457 31.25 0.1968 0.55 1.0120 0.6102 1.1819 31.09 1.2324 0.6323 1.3852 27.16 0.2207 0.60 1.0326 0.5336 1.1578 26.89 1.2198 0.5263 1.3284 23.34 0.2468 0.65 1.0449 0.4415 1 . 1344 22.91 1.2002 0.4313 1.2755 19.76 0.2761 0.70 1.0507 0.3640 1.1121 19.11 1.1762 0.3467 1.2262 16.43 0.3097 0.75 1.0509 0.2916 1.0906 15.51 1 . 1493 0.2713 1 . 1808 13.28 0.3497 0.80 1.0465 0.2240 1.0703 12.08 1 . 1202 0.2039 1.1386 10.32 0.3999 0.85 1.0387 0.1612 1.0511 8.82 1.0902 0.1440 1.0996 7.52 0.4686 0.90 1.0278 0.1032 1.0330 5.73 1.0599 0.0905 1.0637 4.88 0.5831 0.95 1.0148 0.0494 1.0160 2.79 1 . 0296 0.0428 1.0305 2.38 442 TABLES OF FUNCTIONS Table III. — Hyperbolic Tangent of Complex Quantity. — {Continued) tanh (ia) e X m V 8 X in <p a = 0.40 = 0.45 0.0000 0.00 0.0000 3.0777 3.0777 90.00° 0.0000 6.3138 6.3138 9p.00° 0.0159 0.05 0.5115 2.9990 3.0423 80.32 1.8580 5.7272 6.0211 72.03 0.0319 0.10 0.9565 2.7833 2.9431 71.03 2.9217 4.4691 5.3394 56.82 0.0481 0.15 1.2948 2.4799 2.7976 62.43 3.2313 3.2535 4.5855 45.20 0.0645 0.20 1.5189 2 . 1427 2.6265 54.67 3.1500 2 . 3362 3.9220 36.56 0.0813 0.25 1.6444 1 . 8124 2.4473 47.78 2.9260 1.6953 3.3820 30.09 0.0985 0.30 1 . 6960 1.5119 2.2720 41.72 2.6722 1.2524 2.9511 25.12 0.1163 0.35 1.6966 1.2501 2 . 1074 36.39 2.4310 0.9417 2.6070 21.18 0.1349 0.40 1.6652 1.0277 1.9569 31.68 2.2154 0.7188 2.3291 17.98 0.1543 0.45 1.6144 0.8408 1 . 8208 27.51 2.0269 0.5550 2.1015 15.32 0.1748 0.50 1 . 5547 0.6853 1 . 6989 23.79 1.8633 0.4319 1.9126 13.05 0.1968 0.55 1.4901 0.5414 1.5901 20.44 1.7210 0.3371 1.7538 11.08 0.2207 0.60 1 . 4247 0.4466 1.4932 17.41 1 . 5972 0.2632 1.6187 9.36 0.2468 0.65 1.3609 0.3553 1.4065 14.63 1.4887 0.2044 1.5026 7.82 0.2761 0.70 1.2994 0.2781 1.3289 12.08 1.3931 0.1568 1.4019 6.43 0.3097 0.75 1.2412 0.2129 1.2593 9.73 1.3083 0.1180 1.3137 5.15 0.3497 0.80 1 . 1862 0.1569 1.1966 7.54 1.2331 0.0857 1.2361 3.97 0.3999 0.85 1 . 1349 0.1090 1 . 1401 5.48 1.1655 0.0587 1.1670 2.88 0.4686 0.90 1.0867 0.0674 1.0888 3.56 1 . 1047 0.0360 1 . 1053 1.87 0.5831 0.95 1.0418 0.0314 1.0423 1.73 1 . 0499 0.0167 1 . 0500 0.91 0-0 475 = 0.50 0.0000 0.00 0.0000 12.706 12.706 90.00° 00 0.0000 00 0-90° 0.0159 0.05 5.780 9.030 10.725 57.35 20.000 0.0000 20 . 000 0.00 0.0319 0.10 6.213 4.811 7.859 37.75 10.000 0.0000 10.000 0.00 0.0481 0.15 5.260 2.682 5.905 27.01 6 . 6667 0.0000 6 . 6667 0.00 0.0645 0.20 4.356 1.636 4.653 20.58 5.0000 0.0000 5.0000 0.00 0.0813 0.25 3.662 1.075 3.817 16.35 4.0000 0.0000 4.0000 0.00 0.0985 0.30 3.138 0.7445 3.225 13.35 3 . 3333 0.0000 3.3333 0.00 0.1163 0.35 2.736 0.5367 2.789 11.10 2.8571 0.0000 2.8571 0.00 0.1349 0.40 2.422 0.3979 2.454 9.33 2 . 5000 0.0000 2 . 5000 0.00 0.1543 0.45 2.170 0.3008 2.190 7.90 2.2222 0.0000 2.2222 0.00 0.1748 0.50 1.9638 0.2304 1.9773 6.70 2.0000 0.0000 2.0000 0.00 0.1968 0.55 1.7927 0.1778 1.8014 5.67 1.8182 0.0000 1.8182 0.00 0.2207 0.60 1.6486 0.1375 1.6543 4.76 1.6667 0.0000 1.6667 0.00 0.2468 0.65 1.5238 0.1060 1 . 5293 3.97 1 . 5385 0.0000 1 . 5385 0.00 0.2761 0.70 1.4194 0.0809 1.4217 3.27 1.4286 0.0000 1.4286 0.00 0.3097 0.75 1.3269 0.0605 1.3284 2.61 1.3333 0.0000 1.3333 0.00 0.3497 0.80 1.2458 0.0438 1.2465 2.02 1.2500 0.0000 1.2500 0.00 0.3999 0.85 1 . 1737 0.0300 1.1741 1.46 1.1765 0.0000 1 . 1765 0.00 0.4686 0.90 1 . 1095 0.0183 1 . 1097 0.94 1.1111 0.0000 1.1111 0.00 0.5831 0.95 1.0518 0.0085 1.0519 0.46 1 . 0526 0.0000 1 . 0526 0.00 TABLES OF FUNCTIONS 443 Table IV. — Inverse Hyperbolic Tangent of Complex Quantity 7r(a - ifi) = tanh -1 (0 - t'x) (See page 137) a a P « a /3 a 6 X = X = 0.2 x = 0.4 X = 0.6 x = 0.8 0.0 0.0000 0.0000 0.0000 0.0628 0.0000 0.1211 .0.0000 0.1720 0.0000 0.2148 0.2 0.0645 0.0000 0.0619 0.0664 0.0552 0.1250 0.0468 0.1762 . 0386 0.2186 0.4 0.1349 0.0000 0. 1281 0.0796 0.1118 0.1379 0.0931 0.1894 0.0760 0.2302 0.6 0.2206 0.0000 0.2041 0.0936 0.1703 0.1640 0.1373 0.2135 0.1103 0.2500 0.8 0.3497 0.0000 0.2955 0.1426 0.2255 0.2110 0.1749 0.2500 0.1386 0.2776 1.0 00 0-0.5 0.3672 0.2659 0.2593 0.2814 0.1985 0.2964 0.1576 0.3106 1.2 0.3816 0.5000 0.3271 0.3894 0.2562 0.3524 0.2041 0.3436 0.1661 0.3445 1.4 0.2852 0.5000 0.2681 0.4394 0.2322 0.4013 0.1962 0.3826 0.1655 0.3750 1.6 0.2334 0.5000 0.2255 0.4610 0.2060 0.4307 0.1823 0.4111 0.1593 0.3999 1.8 0.1994 0.5000 0.1950 0.4723 0.1832 0.4488 0.1678 0.4312 0.1505 0.4198 2.0 0.1748 0.5000 0.1721 0.4792 0.1644 0.4605 0.1536 0.4454 0.1409 0.4341 2.2 0.1561 0.5000 0.1542 0.4841 0.1490 0.4686 0.1411 0.4557 0.1317 0.4454 2.4 0.1412 0.5000 0.1399 0.4868 0.1361 0.4743 0.1302 0.4634 0.1230 0.4541 2.6 0.1291 0.5000 0.1281 0.4890 0.1252 0.4786 0.1208 0.4692 0.1151 0.4610 2.8 0.1188 0.5000 0.1181 0.4908 0.1159 0.4819 0.1124 0.4737 0.1079 0.4665 3.0 0.1103 0.5000 0.1097 0.4921 0.1080 0.4845 0.1052 0.4773 0.1016 0.4709 3.2 0.1029 0.5000 0.1025 0.4931 0.1010 0.4865 0.0988 0.4802 0.0959 0.4744 3.4 0.0965 0.5000 0.0961 0.4940 0.0950 0.4875 0.0931 0.4826 0.0907 0.4774 3.6 0.0908 0.5000 0.0905 0.4947 0.0895 0.4892 0.0880 0.4845 0.0860 0.4799 3.8 0.0858 0.5000 0.0855 0.4953 0.0847 0.4906 0.0834 0.4862 0.0817 0.4820 4.0 0.0813 0.5000 0.0812 0.4958 0.0804 0.4916 0.0793 0.4876 0.0778 0.4838 X = 1.0 X = 1-2 X = 1-4 X = 1-6 X = 2.0 0.0 0.0000 0.2500 0.0000 0.2789 0.0000 0.3026 0.0000 0.3222 0.0000 0.3524 0.2 0.0316 0.2532 0.0259 0.2814 0.0213 0.3046 0.0178 0.3238 0.0127 0.3534 0.4 0.0619 0.2686 0.0506 0.2890 0.0417 0.3106 0.0348 0.3285 0.0249 0.3564 0.6 0.0892 0.2786 0.0729 0.3012 0.0602 0.3185 0.0503 0.3360 0.0362 0.3612 0.8 0.1118 0.2993 0.0916 0.3173 0.0760 0.3326 0.0639 0.3459 0.0464 0.3675 1.0 0.1281 0.3238 0.1058 0.3360 0.0885 0.3472 0.0749 0.3574 0.0552 0.3570 1.2 0.1373 0.3493 0.1150 0.3558 0.0974 0.3628 0.0832 0.3699 0.0623 0.3833 1.4 0.1403 0.3734 0.1197 0.3750 0.1028 0.3783 0.0890 0.3826 0.0679 0.3920 1.6 0.1383 0.3944 0.1207 0.3926 0.1054 0.3930 0.0924 0.3949 0.0719 0.4008 1.8 0.1341 0.4120 0.1190 0.4080 0.1056 0.4064 0.0938 0.4064 0.0745 0.4093 2.0 0.1281 0.4262 0.1157 0.4211 0.1042 0.4182 0.0937 0.4169 0.0760 0.4174 2.2 0.1216 0.4376 0.1114 0.4321 0.1017 0.4284 0.0926 0.4262 0.0766 0.4249 2.4 0.1150 0.4468 0.1067 0.4412 0.0985 0.4372 0.0906 0.4344 0.0764 0.4318 2.6 • 0.1087 0.4542 0.1019 0.4488 0.0950 0.4446 0.0883 0.4375 0.0756 0.4381 2.8 0.1027 0.4602 0.0973 0.4551 0.0913 0.4510 0.0855 0.4478 0.0743 0.4437 3.0 0.0974 0.4652 0.0927 0.4604 0.0878 0.4564 0.0828 0.4531 0.0729 0.4488 3.2 0.0924 0.4693 0.0884 0.4648 0.0843 0.4610 0.0799 0.4579 0.0712 0.4533 3.4 0.0877 0.4727 0.0844 0.4686 0.0808 0.4633 0.0771 0.4619 0.0695 0.4573 3.6 0.0835 0.4756 0.0807 0.4718 0.0776 0.4684 0.0743 0.4655 0.0676 0.4609 3.8 0. 0796 0.4781 0.0773 0.4746 0.0745 0.4714 0.0717 0.4686 0.0657 0.4641 4.0 .0.0760 0.4802 0.0739 0.4789 0.0716 0.4740 0.0691 0.4713 0.0639 0.4668 444 TABLES OF FUNCTIONS Table V. — Bessel Functions for Cylindrical Coordinates Jnix) and N n (x) (see pages 188 and 196) X J»{x) No(x) Ji(x) iVi(x) Jz(x) N 2 (x) 0.0 1.0000 GO 0.0000 — w 0.0000 -CO 0.1 0.9975 -1.5342 0.0499 -6.4590 0.0012 -127.64 0.2 0.9900 -1.0811 0.0995 -3.3238 0.0050 -32.157 0.4 0.9604 -0.6060 . 1960 -1.7809 0.0197 -8.2983 0.6 0.9120 -0.3085 0.2867 -1.2604 0.0437 -3.8928 0.8 0.8463 -0.0868 0.3688 -0.9781 0.0758 -2.3586 1.0 0.7652 +0.0883 0.4401 -0.7812 0.1149 -1.6507 1.2 0.6711 0.2281 0.4983 -0.6211 0.1593 -1.2633 1.4 0. 5669 0.3379 0.5419 -0.4791 0.2074 -1.0224 1.6 0.4554 0.4204 0.5699 -0.3476 0.2570 -0.8549 1.8 0.3400 0.4774 0.5815 -0.2237 0.3061 -0.7259 2.0 0.2239 0.5104 0.5767 -0.1070 0.3528 -0.6174 2.2 0.1104 0.5208 0.5560 +0.0015 0.3951 -0.5194 2.4 +0.0025 0.5104 0.5202 0.1005 0.4310 -0.4267 2.6 -0.0968 0.4813 0.4708 0.1884 0.4590 -0.3364 2.8 -0.1850 0.4359 0.4097 0.2635 0.4777 -0.2477 3.0 -0.2601 0.3768 0.3391 0.3247 0.4861 -0.1604 3.2 -0.3202 0.3071 0.2613 0.3707 0.4835 -0.0754 3.4 -0.3643 0.2296 0.1792 0.4010 0.4697 +0.0063 3.6 -0.3918 0.1477 0.0955 0.4154 0.4448 0.0831 3.8 -0.4026 +0.0645 +0.0128 0.4141 0.4093 0.1535 4.0 -0.3971 -0.0169 -0.0660 0.3979 0.3641 0.2159 4.2 -0.3766 -0.0938 -0.1386 . 3680 0.3105 0.2690 4.4 -0.3423 -0.1633 -0.2028 0.3260 0.2501 0.3115 4.6 -0.2961 -0.2235 -0.2566 0.2737 0.1846 0.3425 4.8 -0.2404 -0.2723 -0.2985 0.2136 0.1161 0.3613 5.0 -0.1776 -0.3085 -0.3276 0.1479 +0.0466 0.3677 5.2 -0.1103 -0.3312 -0.3432 0.0792 -0.0217 0.3617 5.4 -0.0412 -0.3402 -0.3453 +0.0101 -0.0867 0.3429 5.6 +0.0270 -0.3354 -0.3343 -0.0568 -0.1464 0.3152 5.8 0.0917 -0.3177 -0.3110 -0.1192 -0.1989 0.2766 6.0 0.1507 -0.2882 -0.2767 -0.1750 -0.2429 0.2299 6.2 0.2017 -0.2483 -0.2329 -0.2223 -0.2769 0.1766 6.4 0.2433 -0.2000 -0.1816 -0.2596 -0.3001 0.1188 6.6 0.2740 -0.1452 -0.1250 -0.2858 -0.3119 +0.0586 6.8 0.2931 -0.0864 -0.0652 -0.3002 -0.3123 -0.0019 7.0 0.3001 -0.0259 -0.0047 -0.3027 -0.3014 , -0.0605 7.2 0.2951 +0.0339 +0.0543 -0.2934 -0.2800 ' -0.1154 7.4 0.2786 0.0907 0.1096 -0.2731 -0.2487 -0.1652 7.6 0.2516 0.1424 0.1592 -0.2428 -0.2097 -0.2063 7.8 0.2154 0.1872 0.2014 -0.2039 -0.1638 -0.2395 8.0 0.1716 0.2235 0.2346 -0.1581 -0.1130 -0.2630 TABLES OF FUNCTIONS ; *45 Table VI. — Hyperbolic Bessel Functions I m (z) = i~ m J m {iz) (see page 210) z -Tc(z) Ii« ItW 0.0 1.0000 0.0000 0.0000 0.1 1.0025 0.0501 0.0012 0.2 1.0100 0.1005 0.0050 0.4 1.0404 0.2040 0.0203 0.6 1.0921 0.3137 0.0464 0.8 1 . 1665 0.4329 0.0843 1.0 1.2661 0.5652 0.1358 1.2 1 . 3937 0.7147 0.2026 1.4 1.5534 0.8861 0.2876 1.6 1.7500 1.0848 0.3940 1.8 1.9895 1.3172 0.5260 2.0 2.2796 1.5906 0.6890 2.2 2.6292 1.9141 0.8891 2.4 3.0492 2.2981 1.1111 2.6 3.5532 2.7554 1.4338 2.8 4.1574 3.3011 1.7994 3.0 4.8808 3.9534 2.2452 3.2 5.7472 4.7343 2.7884 3.4 6.7848 5.6701 3.4495 3.6 8.0278 6.7926 4.2538 3.8 9.5169 8.1405 5.2323 4.0 11.302 9.7594 6.4224 4.2 13.443 11.705 7.8683 4.4 16.010 14.046 9.6259 4.6 19.097 16.863 11.761 4.8 22.794 20.253 14.355 5.0 27.240 24.335 17.505 5.2 32.584 29.254 21.332 5.4 39.010 35.181 25.980 5.6 46.738 42.327 31.621 5.8 56.039 50.945 38.472 6.0 67.235 61.341 46.788 6.2 80.717 73.888 56.882 6.4 96.963 89.025 69.143 6.6 116.54 107.31 84.021 6.8 140.14 129.38 102.08 7.0 168.59 156.04 124.01 7.2 202.92 188.25 150.63 7.4 244.34 227.17 182.94 7.6 294.33 274.22 222.17 7.8 354.68 331.10 269.79 8.0 427.57 399.87 327.60 446 TABLES OF FUNCTIONS Table VII. — Bessel Functions fob Sphekical, Cooedinates j«(x) = y/tc/2x J n+ \(x), Unix) = -y/ir /2x N n+i (x) (see page 317) X io(x) no(x) h(x) ni(x) h(.x) ws(«) 0.0 1.0000 — X 0.0000 _» 0.0000 — 00 0.1 0.9983 -9.9500 0.0333 - 100 . 50 0.0007 -3005.0 0.2 0.9933 -4.9003 0.0664 -25.495 0.0027 -377.52 0.4 0.9735 -2.3027 0.1312 -6.7302 0.0105 -48.174 0.6 0.9411 -1.3756 0.1929 -3.2337 0.0234 -14.793 0.8 0.8967 -0.8709 0.2500 -1.9853 0.0408 -6.5740 1.0 0.8415 -0.5403 0.3012 -1.3818 0.0620 -3.6050 1.2 0.7767 -0.3020 0.3453 - 1 . 0283 0.0865 -2.2689 1.4 0.7039 -0.1214 0.3814 -0.7906 0.1133 -1.5728 1.6 0.6247 +0.0183 0.4087 -0.6133 0.1416 - 1 . 1682 1.8 0.5410 0.1262 0.4268 -0.4709 0.1703 -0.9111 2.0 0.4546 0.2081 0.4354 -0.3506 0.1985 -0.7340 2.2 0.3675 0.2675 0.4346 -0.2459 0.2251 -0.6028 2.4 0.2814 0.3072 0.4245 -0.1534 0.2492 -0.4990 2.6 0.1983 0.3296 0.4058 -0.0715 0.2700 -0.4121 2.8 0.1196 0.3365 0.3792 +0.0005 0.2867 -0.3359 3.0 +0.0470 0.3300 0.3457 0.0630 0.2986 -0.2670 3.2 -0.0182 0.3120 0.3063 0.1157 0.3084 -0.2035 3.4 -0.0752 0.2844 0.2623 0.1588 0.3066 -0.1442 3.6 -0.1229 0.2491 0.2150 0.1921 0.3021 -0.0890 3.8 -0.1610 0.2082 0.1658 0.2158 0.2919 -0.0378 4.0 -0.1892 0.1634 0.1161 0.2300 0.2763 +0.0091 4.2 -0.2075 0.1167 0.0673 0.2353 0.2556 0.0514 4.4 -0.2163 0.0699 +0.0207 0.2321 0.2304 0.0884 4.6 -0.2160 +0.0244 -0.0226 0.2213 0.2013 0.1200 4.8 -0.2075 -0.0182 -0.0615 0.2037 0.1691 0.1456 5.0 -0.1918 -0.0567 -0.0951 0.1804 0.1347 0.1650 5.2 -0.1699 -0.0901 -0.1228 0.1526 0.0991 0.1871 5.4 -0.1431 -0.1175 -0.1440 0.1213 0.0631 0.1850 5.6 -0.1127 -0.1385 -0.1586 0.0880 +0.0278 0.1856 5.8 -0.0801 -0.1527 -0.1665 0.0538 -0.0060 0.1805 6.0 -0.0466 -0.1600 -0.1678 +0.0199 -0.0373 0.1700 6.2 -0.0134 -0.1607 -0.1629 -0.0124 -0.0654 0.1547 6.4 +0.0182 -0.1552 -0.1523 -0.0425 -0.0896 0.1353 6.6 0.0472 -0.1440 -0.1368 -0.0690 -0.1094 0.1126 6.8 0.0727 -0.1278 -0.1172 -0.0915 -0.1243 0.0875 7.0 0.0939 -0.1077 -0.0943 -0.1029 -0.1343 0.0609 7.2 0.1102 -0.0845 -0.0692 -0.1220 -0.1391 0.0337 7.4 0.1215 -0.0593 -0.0429 -0.1294 -0.1388 +0.0068 7.6 0.1274 -0.0331 -0.0163 -0.1317 -0.1338 -0.0189 7.8 0.1280 -0.0069 +0.0095 -0.1289 -0.1244 -0.0427 8.0 0.1237 +0.0182 0.0336 -0.1214 -0.1111 -0.0637 TABLES OF FUNCTIONS 447 Table VIII. — Impedance Functions for Piston in Infinite Plane Wall 0o - ixo = 1 - (2/w)Ji(w) — iM(w) = tanh[x(« p - ip P )]; w - (4ira/X) (see page 333) to So xo a p fip 0.0 0.0000 0.0000 0.0000 0.0000 0.5 0.0309 0.2087 0.0094 0.0655 1.0 0.1199 0.3969 0.0330 0.1216 1.5 0.2561 0.5471 0.0628 0.1663 2.0 0.4233 0.6468 0.0939 0.2020 2.5 0.6023 0.6905 0.1247 0.2316 3.0 0.7740 0.6801 0.1552 0.2572 3.5 0.9215 0.6238 0.1858 0.2800 4.0 1.0330 0.5349 0.2175 0.3008 4.5 1 . 1027 0.4293 0.2517 0.3194 5.0 1.1310 0.3231 0.2899 0.3353 5.5 1 . 1242 0.2300 0.3344 0.3460 6.0 1.0922 0.1594 0.3868 0.3456 6.5 1.0473 0.1159 0.4450 0.3207 7.0 1.0013 0.0989 0.4788 0.2600 7.5 0.9639 0.1036 0.4594 0.2050 8.0 0.9413 0.1220 0.4241 0.1887 8.5 0.9357 0.1456 0.3980 0.1958 9.0 0.9454 0.1663 0.3839 0.2132 9.5 0.9661 0.1782 0.3799 0.2344 10.0 0.9913 0.1784 0.3845 0.2565 10.5 1.0150 0.1668 0.3964 0.2774 11.0 1.0321 0.1464 0.4153 0.2958 11.5 1 . 0397 0.1216 0.4410 0.3097 12.0 1.0372 0.0973 0.4734 0.3158 12.5 1.0265 0.0779 0.5101 0.3083 13.0 1.0108 0.0662 0.5421 0.2810 13.5 0.9944 0.0631 0.5490 0.2409 14.0 0.9809 0.0676 0.5316 0.2117 14.5 0.9733 0.0770 0.5073 0.2032 15.0 0.9727 0.0881 0.4877 0.2092 15.5 0.9784 0.0973 0.4758 0.2231 16.0 0.9887 0.1021 0.4718 0.2406 16.5 1.0007 0.1013 0.4750 0.2591 17.0 1.0115 0.0948 0.4852 0.2767 17.5 1.0187 t).0843 0.5017 0.2914 18.0 1.0209 0.0719 0.5247 0.3007 18.5 1.0180 0.0602 0.5522 0.3010 19.0 1.0111 0.0515 0.5798 0.2879 19.5 1.0021 0.0470 0.5968 0.2610 20.0 0.9933 0.0473 0.5940 0.2314 448 TABLES OF FUNCTIONS Table IX. — Legendre Functions foe Spherical Coordinates (See page 315) & P-i - Pe Pi(cost?) P2(COS t?) Pt(costf) P4(cos#) 0° 1.0000 1.0000 1.0000 1.0000 1.0000 5 1 . 0000 0.9962 0.9886 0.9773 0.9623 10 1 . 0000 0.9848 0.9548 0.9106 0.8352 15 1 . 0000 0.9659 0.8995 0.8042 0.6847 20 1 . 0000 0.9397 0.8245 0.6649 0.4750 25 1 . 0000 0.9063 0.7321 0.501-6 0.2465 30 1.0000 0.8660 0.6250 0.3248 0.0234 35 1.0000 0.8192 0.5065 0.1454 -0.1714 40 1.0000 0.7660 0.3802 -0.0252 -0.3190 45 1.0000 0.7071 0.2500 -0.1768 -0.4063 50 1 . 0000 0.6428 0.1198 -0.3002 -0.4275 55 1.0000 0.5736 -0.0065 -0.3886 -0.3852 60 1 . 0000 0.5000 -0.1250 -0.4375 -0.2891 65 1.0000 0.4226 -0.2321 -0.4452 -0.1552 70 1 . 0000 0.3420 -0.3245 -0.4130 -0.0038 75 1 . 0000 0.2588 -0.3995 -0.3449 + 0.1434 80 1 . 0000 0.1736 -0.4548 -0.2474 0.2659 85 1.0000 0.0872 -0.4886 -0.1291 0.3468 90 1.0000 0.0000 -0.5000 0.0000 0.3750 t» P 5 (cost?) Ps(cos t?) P7(cos#) P8(cos t5) P»(cost») 0° 1 . 0000 1 . 0000 1.0000 1.0000 1.0000 5 0.9437 0.9216 0.8962 0.8675 0.8358 10 0.7840 0.7045 0.6164 0.5218 0.4228 15 0.5471 0.3983 0.2455 0.0962 -0.0428 20 0.2715 0.0719 -0.1072 -0.2518 -0.3517 25 0.0009 -0.2040 -0.3441 -0.4062 -0.3896 30 -0.2233 -0.3740 -0.4102 -0.3388 -0.1896 35 -0.3691 -0.4114 -0.3096 -0.1154 +0.0965 40 -0.4197 -0.3236 -0.1006 +0.1386 0.2900 45 -0.3757 -0.1484 +0.1271 0.2983 0.2855 50 -0.2545 +0.0564 0.2854 0.2947 0.1041 55 -0.0868 0.2297 0.3191 0.1422 -0.1296 60 +0.0898 0.3232 0.2231 -0.0763 -0.2679 65 0.2381 0.3138 0.0422 -0.2411 -0.2300 70 0.3281 0.2089 -0.1485 -0.2780 -0.0476 75 0.3427 0.0431 -0.2731 -0.1702 +0.1595 80 0.2810 -0.1321 -0.2835 +0.0233 0.2596 85 0.1577 -0.2638 -0.1778 0.2017 0.1913 90 0.0000 -0.3125 0.0000 0.2734 0.0000 TABLES OF FUNCTIONS 449 Table X. — Phase Angles and Amplitudes for Radiation and Scattering from a Cylinder M = ka = (27ra/\) = (wa/c) (see page 301) [Dashes indicate values that can be computed by Eq. (26.6)] ka Co 7C Ci 71 C 2 72 Cz 73 Ct 74 0.0 00 0.00° CO 0.00° 00 0.00° 00 0.00° 00 0.00° 0.1 12.92 .44 63.06 -.45 2546 0.00 — 0.00 — 0.00 0.2 6.651 1.71 15.55 -1.82 318.2 -0.01 9565 0.00 — 0.00 0.4 3.583 6.28 3.875 -6.97 39.71 -0.14 600.7 0.00 — 0.00 0.6 2.585 12.82 1.844 -13.62 11.72 -0.69 119.6 -0.01 1595 0.00 0.8 2.091 20.66 1.199 -18.73 4.922 -2.09 38.20 -0.06 382.9 0.00 1.0 1.793 29.39 0.9283 -20.50 2.529 -4.77 15.81 -0.20 127.3 0.00 1.2 1.593 38.74 0.7884 -18.94 1.503 -8.91 7.712 -0.57 52.03 -0.02 1.4 1.447 48.52 0.7035 -14.80 1.012 -14.06 4.212 -1.35 24.54 -0.06 1.6 1.335 58.62 0.6453 -8.84 0.7627 -19.03 2.504 -2.77 12.85 -0.16 1.8 1.246 68.96 0.6019 -1.61 0.6309 -22.49 1.596 -5.08 7.290 -0.37 2.0 1.173 79.49 0.5676 +6.52 0.5573 -23.69 1.086 -8.44 4.405 -0.79 2.2 1.112 90.15 0.5392 15.31 0.5130 -22.56 0.7898 -12.71 2.801 -1.55 2.4 1.060 100.93 0.5152 24.57 0.4836 -19.45 0.6158 -17.32 1.861* -2.80 2.6 1.014 111.81 0.4944 34.20 0.4624 -14.75 0.5136 -21.41 1.287 -4.73 2.8 0.9743 122.75 0.4760 44.11 0.4457 - 8.84 0.4535 -24.15 0.9265 -7.46 3.0 0.9389 133.76 0.4597 54.24 0.4319 - 1.99 0.4175 -25.09 0.6965 -11.01 3.2 0.9071 144.82 0.4450 64.55 0.4198 + 5.59 0.3952 -24.19 0.5496 -15.13 3.4 0.8785 155.92 0.4317 75.01 0.4090 13.73 0.3804 -21.64 0.4566 -19.29 3.6 0.8524 167.06 0.4195 85.58 0.3992 2.33 0.3698 -17.71 0.3987 -22.81 3.8 0.8286 178.23 0.4084 96.25 0.3901 31.29 0.3617 -12.66 0.3631 -25.11 4.0 C.8067 189.42 0.3980 107.01 0.3816 40.55 0.3549 -6.72 0.3412 -25.90 4.2 0.7865 200.64 0.3885 117.83 0.3737 50.06 0.3489 -0.04 0.3275 -25.14 4.4 0.7678 211.88 0.3796 128.72 0.3662 59.77 . 3434 +7.22 0.3187 -22.95 4.6 0.7503 223.14 0.3713 139.65 0.3592 69.66 0.3383 14.97 0.3126 - 19 .54 4.8 0.7341 234.42 0.3635 150.64 0.3525 79.70 0.3334 23.13 0.3081 -15.10 5.0 0.7188 245.71 0.3562 161.66 0.3462 89.87 0.3287 31.62 0.3044 - 9.81 ka Ci 75 Ce 76 Ci 77 C 8 78 C 9 79 2.0 22.07 -0.04° 130.8 0.00° 903.5 0.00° 7144 0.00 0.00 2.2 12.82 -0.10 68.99 0.00 432.1 0.00 3099 0.00 — 0.00 2.4 7.834 -0.22 38.65 -0.01 221.4 0.00 1452 0.00 — 0.00 2.6 4.999 -0.45 22.78 -0.03 120.2 0.00 725.6 0.00 4941 0.00 2.8 3.309 -0.86 14.03 -0.06 68.58 0.00 383.4 0.00 2418 o.oo- 3.0 2.261 -1.53 8.967 -0.13 40.86 -0.01 212.6 0.00 1248 0.00 3.2 1.590 -2.59 5.922 -0.25 25.27 -0.02 122.9 0.00 674.6 0.00 3.4 1.149 -4.18 4.025 -0.47 16.16 -0.04 73.80 0.00 380.0 0.00 3.6 0.8534 -6.41 2.805 -0.83 10.65 -0.07 45.80 0.00 222.1 0.00 3.8 0.6539 -9.35 2.000 -1.41 7.200 -0.14 29.29 -0.01 134.1 0.00 4.0 0.5190 -12.92 1.456 - 2.30 4.985 -0.26 19.24 -0.02 83.43 0.00 4.2 0.4287 -16.83 1.082 - 3.60 3.526 -0.46 12.95 -0.04 53.32 0.00 4.4 0.3693 -20.62 0.8211 - 5.42 2.542 -0.77 8.907 -0.08 34.92 -0.01 4.6 0.3312 -23.74 0.6374 - 7.85 1.865 -1.26 6.252 -0.14 23.40 -0.01 4.8 0.3071 -25.76 0.5081 -10.88 1.391 -2.00 4.471 -0.25 16.00 -0.02 5.0 0.2921 -26.44 0.4177 -14.40 1.054 -3.06 3.251 -0.42 11.15 -0.04 450 TABLES OF FUNCTIONS Table XI. — Phase Angles and Amplitudes fob Radiation and Scattering from a Sphere n = ka = (2tto/X) = (ua/c) (see page 320) [Dashes indicate values that can be computed by Eq. (27.17)] ka Do So Di Si Di Si D, Si Da. St 0.0 00 0.00° 00 0.00° 00 0.00° 00 0.00° 00 0.00° 0.1 100.5 0.02 2000 -0.01 — 0.00 — 0.00 — 0.00 0.2 25.50 0.15 250.1 -0.08 5637 0.00 — 0.00 — 0.00 0.4 6.731 1.12 31.35 -0.58 354.6 -0.01 5906 0.00 — 0.00 0.6 3.239 3.41 9.408 -1.82 70.73 -0.06 785.5 0.00 — 0.00 0.8 2.001 7.18 4.101 -3.80 22.67 -0.25 188.9 - 0.01 2058 0.00 1.0 1.414 12.30 2.236 -6.14 9.434 -0.70 62.97 - 0.02 547.8 0.00 1.2 1.085 18.56 1.426 -8.11 4.646 -1.59 25.82 - 0.08 186.9 0.00 1.4 0.8778 25.75 1.021 -8.97 2.583 -3.07 12.22 - 0.22 75.74 -0.01 1.6 0.7370 33.68 0.7931 -8.25 1.584 -5.19 6.426 - 0.51 34.84 -0.02 1.8 0.6355 42.19 0.6529 -5.87 1.057 -7.77 3.667 - 1.05 17.66 -0.06 2.0 0.5590 51.16 0.5590 -1.97 0.7629 -10.40 2.236 - 1.97 9.669 -0.15 2.2 0.4993 60.49 0.4918 + 3.21 0.5901 -12.49 1.444 - 3.38 5.635 -0.32 2.4 0.4514 70.13 0.4411 9.44 0.4837 -13.51 0.9823 - 5.34 3.459 -0.64 2.6 0.4121 80.01 0.4011 16.50 0.4148 -13.14 0.7036 - 7.80 2.220 -1.18 2.8 0.3792 90.08 0.3686 24.23 0.3676 -11.31 0.5308 -10.54 1.481 -2.02 3.0 0.3514 100.32 0.3415 32.49 0.3333 -8.11 0.4214 -13.16 1.024 -3.27 3.2 0.3274 110.70 0.3184 41.18 0.3071 -3.73 . 3508 -15.17 0.7334 -5.00 3.4 0.3066 121.20 0.2985 50.23 0.2862 + 1.65 0.3042 -16.17 0.5443 -7.23 3.6 0.2883 131.79 0.2811 59.57 0.2688 7.86 0.2723 -15.94 0.4195 -9.83 3.8 0.2721 142.47 0.2657 69.15 0.2540 14.75 0.2496 -14.41 0.3364 -12.58 4.0 0.2577 153.22 0.2519 78.92 0.2411 22.20 . 2326 -11.67 0.2807 -15.10 4.2 0.2448 164.03 0.2396 88.88 0.2296 30.12 0.2193 - 7.84 0.2432 -17.00 4.4 0.2331 174.91 0.2285 98.97 0.2194 38.44 0.2084 - 3.08 0.2174 -17.95 4.6 0.2225 185.83 0.2184 109.20 0.2101 47.08 0.1992 + 2.47 0.1994 -17.79 4.8 0.2128 196.79 0.2091 119.55 0.2016 56.00 0.1912 8.70 0.1863 -16.47 5.0 0.2040 207.79 0.2006 129.98 0.1939 65.16 0.1840 15.48 . 1764 -14.04 ka Dt, Ss -De 5<s Di 57 Z>8 5s Dt 59 2.0 51.31 - 0.01° 323.7 0.00° 2370 0.00° — 0.00° — 0.00° 2.2 27.14 - 0.02 155.2 0.00 1030 0.00 7790 0.00 — 0.00 2.4 15.25 - 0.04 79.69 0.00 483.5 0.00 3343 0.00 — 0.00 2.6 9.021 - 0.10 43.38 -0.01 242.2 0.00 1541 0.00 — 0.00 2.8 5.573 - 0.20 24.83 -0.01 128.2 0.00 755.6 0.00 5002 0.00 3.0 3.576 - 0.37 14.83 -0.03 71.28 0.00 390.7 0.00 2407 0.00 3.2 2.371 - 0.68 9.206 -0.06 41.34 0.00 211.7 0.00 1219 0.00 3.4 1.620 -1.16 5.907 -0.11 24.88 -0.01 119.5 0.00 645.8 0.00 3.6 1.137 - 1.91 3.904 -0.21 15.49 -0.02 70.01 0.00 356.1 0.00 3.8 0.8183 - 2.99 2.649 -0.38 9.933 -0.03 42.39 0.00 203.6 0.00 4.0 0.6043 - 4.48 1.841 -0.66 6.545 -0.06 26.44 0.00 120.2 0.00 4.2 0.4583 - 6.43 1.308 -1.09 4.418 -0.12 16.94 -0.01 73.09 0.00 4.4 0.3577 - 8.79 0.9486 -1.73 3.050 -0.22 11.13 -0.02 45.66 0.00 4.6 0.2881 -11.45 0.7015 -2.65 2.148 -0.37 7.479 -0.04 29.24 0.00 4.8 0.2399 -14.15 0.5290 -3.92 1.542 -0.61 5, 130 -0.07 19.16 0.00 5.0 0.2065 -16.56 0.4072 -5.58 1.126 -0.98 3.587 -0.12 12.82 -0.01 TABLES OF FUNCTIONS 451 Table XII. — Impedance Functions for Piston in Infinite Plane Wall Jo(irao n r/a) l^n — ^Xn) n = p(r) = pcuoe-** V (e n - »x.) /o(7r "°" r/a) ; M = (2*»/A) (see page 334) M 0o xo 0i XI 02 *2 03 X» 04 X4 0.0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.5 0.1199 0.3969 -0.0014 -0.0480 -0.0004 -0.0180 -0.0002 -0.0096 -0.0001 -0.0062 1.0 0.4233 0.6468 -0.0200 -0.1128 -0.0058 -0.0404 -0.0028 -0.0214 -0.0016 -0.0131 1.5 0.7740 0.6801 -0.0656 -0.1806 -0.0228 -0.0616 -0.0106 -0.0324 -0.0062 -0.0204 2.0 1.0330 0.5349 -0.1966 -0.2138 -0.0496 -0.0750 -0.0228 -0.0387 -0.0132 -0.0236 2.5 1.1310 0.3231 -0.3496 -0.1664 -0.0788 -0.0653 -0.0356 -0.0392 -0.0204 -0.0238 3.0 1.0922 0.1594 -0.4404 -0.0340 -0.0868 -0.0486 -0.0386 -0.0303 -0.0220 -0.0210 3.5 1.0013 0.0989 -0.4458 +0.1502 -0.0776 -0.0408 -0.0344 -0.0286 -0.0196 -0.0188 4.0 0.9413 0.1220 -0.3462 0.3078 -0.0680 -0.0604 -0.0302 -0.0374 -0.0172 -0.0272 4.5 0.9454 0.1663 -0.1830 0.3740 -0.0852 -0.1016 -0.0360 -0.0532 -0.0202 -0.0355 5.0 0.9913 0.1784 -0.0260 0.3420 -0.1480 -0.1348 -0.0554 -0.0644 -0.0302 -0.0390 5.5 1.0321 0.1464 +0.0568 0.2394 -0.2404 -0.1202 -0.0784 -0.0605 -0.0414 -0.0398 6.0 1.0372 0.0973 0.0646 0.1389 -0.3268 -0.0409 -0.0920 -0.0448 -0.0470 -0.0325 3.5 1.0108 0.0662 0.0188 0.0820 -0.3516 +0.0882 -0.0868 -0.0286 -0.0446 -0.0251 7.0 0.9809 0.0676 -0.0260 0.0811 -0.3006 0.2170 -0.0738 -0.0362 -0.0382 -0.0301 7.5 0.9727 0.0881 -0.0366 0.1072 -0.1850 0.2901 -0.0756 -0.0646 -0.0386 -0.0423 8.0 0.9887 0.1021 -0.0140 0.1246 -0.0540 0.2836 -0.1140 -0.0970 -0.0516 -0.0544 8.5 1.0115 0.0948 +0.0143 0.1121 +0.0296 0.2272 -0.1832 -0.0989 -0.0716 -0.0552 9.0 1.0209 0.0719 0.0264 0.0828 0.0538 0.1370 -0.2602 -0.0768 -0.0870 -0.0428 9.5 1.0111 0.0515 0.0130 0.0568 0.0264 0.0795 -0.2962 +0.0487 -0.0864 -0.0270 10.0 0.9933 . 0473 -0.0072 0.0514 -0.0116 0.0666 -0.2710 0.1609 -0.0748 -0.0254 452 TABLES OF FUNCTIONS Table XIII.- — Absorption Coefficients fob Wall Material* (See page 385) Based on Bulletin of Acoustic Materials Association, VII, 1940 Material Values of absorption coefficient a Frequency, cps 128 256 512 1,024 2,048 0.27 0.50 0.88 0.80 0.26 0.79 0.88 0.76 0.40 0.50 0.80 0.55 0.03 0.03 0.04 0.05 0.25 0.37 0.34 0.27 0.27 0.76 0.88 0.60 0.08 0.11 0.25 0.30 0.27 0.50 0.80 0.82 0.01 0.02 0.02 0.02 0.04 0.03 0.03 0.03 0.04 0.03 0.03 0.03 0.04 0.03 0.03 0.02 0.01 0.01 0.02 0.02 0.34 0.74 0.76 0.75 0.02 0.03 0.04 0.04 0.40 0.54 0.52 0.50 0.50 0.40 0.35 0.30 0.07 0.06 0.06 0.06 4,096 Acoustex 40R* Acoustone D* Acoustic plaster Brick wall, unpainted . . Carpet, lined CelotexC3* Curtains, light heavy drapes Floor, concrete wood linoleum Glass Marble or glazed tile. . . Permacoustic* Plaster, smooth on lath Temcoustic F2* Ventilator grill Wood paneling Object 0.16 0.13 0.30 0.02 0.11 0.25 0.04 0.10 0.01 0.05 0.04 0.04 0.01 0.19 0.02 0.33 0.50 0.08 Values of absorption a = aA in sq ft Frequency, cps 128 Adult person, seated in audience Chairs, metal or wood Theater chair, wood veneer Leather upholstered Plush upholstered 1.5 0.15 0.18 1.0 1.5 256 512 1,024 2,048 2.5 3.8 5.0 4.8 0.16 0.17 0.19 0.20 0.20 0.22 0.25 0.25 1.2 1.4 1.6 1.6 2.0 2.5 2.8 3.0 0.70 0.74 0.50 0.05 0.24 0.25 0.30 0.75 0.02 0.03 0.03 0.02 0.01 0.74 0.04 0.42 0.25 0.06 4.5 0.19 0.23 1.5 3.0 * Measurements for acoustic material cemented to plaster or concrete. Other means of mount- ing give other values for a. Acoustical Constants for Air and Water Air at 760 mm mercury, 20°C, cgs units: P = 0.00121, c = 34,400, P c = 42, P c 2 = 1.42 X 10 6 y e = 1.40, Po = 1.013 X 10 3 Water at 20°C, cgs units: P = 1.0, c = 146,000, P c = 1.5 X 10 s , pc* = 2.1 X 10 10 PLATES 453 o \ \ / / A 2 7 b ^ i \ y- v. ./ ' s \< ^2;^ <jX, a^oA. A \ y r- g N / \ c' ■ .^v /P7 > s X / \ 7T^i • k ?v •? r ^ < /^^ \ \ i \ — ^ PK Jl rf p-as — kio\ *PjPk p 5 r )y s V fi®Ks\ J" V£ > \ / Vl\ / y .1 \k$ \f£° ' lb \ ^-es \ ^ < \ Itfr T ' <3\ "Op y jc ^Ct ^ , o^X / ^Pl y^ \^ / — — T v •2 \ s?> \. — — V ^' — — - A; s* N? h \ > -At ^ \ \ ■7, \>X\ \ \ . 1 2 3 Resistance Ratio = (R/pc) Plate 1. — Bipolar plot of wave-reflection parameters a and /3 against impedance ratio f = (z/pc) = — i% = tanh [ir(a — t'/3)j. Standing wave ratio = e~ z * a . See pages 137 and 240. 454 PLATES •*-» 3 = 1.5 0.8 0.6 0.4 0.3 0.2 0.15 => '"^i / \ /£s 5^ *X ^ / II / ^ ^A M^ ***• / $\ fe./ \0\ ^Oeo> SfOAU f=o.arh^ \^ -^n£035 0=0.70 p-0-75 \ 1 1 0=0.30 Kr-i III *3 0=0.25 p-0.80 . g. "0=0.20 p=085V^ J^aTT B^OBO^- j^Jy r^r£— — Vo^ \o.o> ^C TV 1^1 V? n .q6X. / Q ^> \ J k # — ^ "C V -90° -60 € -30° 0° 30° Phase Angle ,ij? 60' 90° Plate II. — Plot of wave-reflection parameters a and /3 against magnitude and phase of .; impedance ratio (z/pc) = f = If \e~ i,p = tan h [ir(a - t/3)]. See pages 137 and 240. PLATES 455 .06 «09- O oe- o * o0£+ o09+ sinh[tr(a-ip)]=pe■ |, '' cp 120° Phase angle of sinh Plate III. — Magnitude and phase angle of sinh [*-(« - «£)] and cosh frr(a - *£)]. See pages 137 and 241. 456 PLATES Maximum Pressure! Minimum Displacemei Maximum Displacement Scale for Standing Wave Ratio | q. I = I A r /A; 0.5 i.O -U-i. I i ' i i ' i ' | ' i ' 1.0 0-5 Corresponding Values of (A m in/Am<w) Plate IV.— Ratio g of reflected to incident plane wave, in phase and amplitude (\q\ = standing wave ratio), for different valves of resistance and reactance ratios B and X . where (z/oc) = r = 6 - i X . Standing wave ratio is distance from center of plot to point 6, x- Angular measure for in units of half-wave-length along tube or fttring. See pages 137 and 241. PLATES 457 sN \ -pni 3\ V °4\ UJ Q ZO UJ2 / a. o UJ £i z V 7 - \ y y $> ^ l «: \^ -H> 1 1 1 u Ou ii %2 ' *7 'v^' ** M \$\ PN ^ ' A "A L ^ ^fr |A sJ&Sf T^ /$S b— 1 c c II £ § II ^ o - s *-< ST3 ••a 5 3 r ^5 © 5 o m s (0 £ >> a s i» ^ ■73 73 >5 £ $ 2" * ftC.£S 03 s Si. R-T3 — a o -a ll ^3 !«d .9 S^-a a o 3«Sfi 03 J? <S rf3 •3- S ft "" vj. ft oj § II 2 -T3 r^ X ® ft » H (j a S °5 ^ b • **■* © d o t> go © £ °^ .sis 83 T3 T) CJ "O 5 § « -c so glft-2^ ISIg'S o *■> a ajR © a 5* » 5 ■^03 <-,. 00 03 © H« ? ? . "3 ^^ ■ -I 3 _ © So" P ft-oS 458 PLATES -90° -60° -30' 0° 30° Phase Angle cp 60° 90° Plate VI. — Contours for absorption coefficient a corresponding to various values of magnitude and phase angle of acoustic impedance ratio (z/pc) = |f|e - **' for wall material. Only to be used for rooms and frequencies such that sound is uniformly dis- tributed in room. See page 388. INDEX Absorption, of sound, 360, 385 wall, 403-405 Absorption coefficient (see Coefficient) Absorption constant, 385, 452 Acoustic admittance ratio, 240 of horn, 285 of surface, 306, 366, 369, 410 Acoustic conductance ratio, 240, 306, 367, 371, 403, 411 Acoustic constants of air and water, 222, 452 Acoustic filters, 235, 290 Acoustic impedance, 237 analogous, 234, 237 of constriction, 234 of horn, 272, 285 of opening, 235 of tank, 235 of catenoidal horn, 282 characteristic, of air, 223, 238, 307, 452 of conical horn, 272 at diaphragm, 200, 333 of exponential horn, 281 measurement of, 242, 456 of piston in sphere, 324 in wall, 332, 447, 451 radiation, 237 of surface, 360, 361, 365, 368, 388 ratio, 240 specific, 234, 237, 239, 287 of surface, 366, 368, 388 Acoustic power, sources of, 228 of voice, 228 Acoustic reactance ratio, 240, 367 Acoustic resistance (see Acoustic imped- ance) Acoustic resistance ratio, 240, 367 Acoustic susceptance ratio, 240, 306, 367, 411 Adiabatic compression, 221, 363 Admittance, acoustic (see Acoustic admittance) mechanical, of oscillator, 35, 37, 50 of string support, 144 transfer, 93, 95, 105 wave, 91 Air, acoustic properties of, 222, 452 load of, on membrane, 310, 335 localized, 198, 361 uniform, 193, 200, 361 Allowed frequencies (see Frequencies, natural) Amplitude, average, of diaphragm, 201 of harmonic, 87, 91 of oscillation, 9, 87, 190, 201 radiation, 301, 320 velocity, 23, 224 of wave, 266 Analytic function, 14 Analogous circuits, 233, 365 Analogous impedance (see Acoustic impedance) Angle, of lag (see Phase angle) Angle-distribution function, 321 Angular distribution, of radiated sound, from cylinder, 304 from dipole, 318 from piston, 323, 328, 331 from point on sphere, 321 of scattered sound, from cylinder, 349 from sphere, 354 Angular momentum, of whirled string, 119 Antinode, 84 Approximate calculation (see Perturba- tion calculations) 459 460 VIBRATION AND SOUND Attenuation index, for porous material, 366 Attenuation, of sound, in ducts, 307, 371 Axial waves, 391, 405 B Bar, 151-170 clamped-clamped, 161, 170 clamped-free, 157, 170 free-free, 162, 170 nonuniform, 164, 171 plucked and struck, 160 whirled, 171 Bel, 225 Bending, of plate, 209 Bending moments, in bar, 153 Bessel functions, 6, 17, 188, 298, 444 spherical, 316, 446 Boundary conditions, 75, 117, 175, 187 for air in duct, 306, 369 for bar, 157, 162 for flexible string, 75, 84, 135 for membrane, 175, 194 at plane surface, 360, 367, 380, 388 for plate, 210 for stiff string, 167 at wall of room, 388, 423 Branch point, 13 Capacitance, of crystal, 41, 69 Catenoidal horn, 267, 271, 281-283, 292 Cavity resonance, 228, 258, 359 Characteristic frequencies, of vowel sounds, 233 Characteristic functions, 107—109 for bar, 159, 162, 171 for membrane, 180-183, 189-191, 195 for plate, 211 for sound in room, 389, 402, 415, 423 in tube, 309, 373 for string, 107, 112, 118, 143, 149 Characteristic impedance, of air, 223, 238, 364, 452 in tube, 308, 372 Characteristic impedance, for string, 93, 126 of water, 452 Characteristic values, 107 for bar, 158, 162 for Bessel functions, 189, 399 for membrane, 180, 189, 195 for plate, 210 for sound in room, 390-401, 411 for string, 107, 113, 117, 149 for wire, 168 Circular membrane, 183-195, 214, 297, 306 Circular waves, 184-187, 297-311 Clamped bar, 157, 161 Clamped capacitance, 41 Clamped impedance, 36, 38 Clarinet, 248, 285 Closed tube, 244, 249, 258-261, 291 Coefficient, absorption, 385, 388, 452 wall, 403-405, 425 Coefficients, of Fourie: series, 87 of power series, 4, 6, 7 of series of characteristic functions, 108, 160, 309, 428 of series for scattered wave, 348, 354 Coil, driving, electromagnetic, 34, 67, 277, 289, 292 Complex exponential, 7 Complex numbers, 8 Complex plane, 8 integrals in, 12-19, 42-44 Compliance, 21, 29 Compressibility, of air, 220, 363 Condenser microphone, 195-208, 211- 215, 357-360, 379 .. Conditions, boundary (see Boundary conditions) initial (see Initial conditions) Conductance, acoustic (see Acoustic conductance) mechanical, 35, 144 Configuration, plane of, 58 of system, 57, 64 Conformal transformation, 136, 373, 453-457 Conical horn, 271-279, 291-292 Constriction, in tube, 234, 247 Continuity, equation of, 218, 294 INDEX 461 Contour integrals, 12-16, 19, 42-46, 100, 132, 144, 206, 263, 288, 428 Convention, for complex numbers, 9 Coordinate systems, 174, 296 Coordinates, normal, 56, 58, 90-91 Cosine, 4, 438-439 Coupled oscillators, 52-66, 68 Coupling, small, 59 of source, with standing waves, 415— 418 Coupling constant, 54 electromagnetic, 35 piezoelectric, 40 Crystal, piezoelectric {see Piezoelec- tric force) Cutoff frequency, 235, 280, 282, 308 Cylinder, force on, 352, 378 radiation from, 297-305, 376 scattering from, 347—352 transmission inside, 305-311, 377 Cylindrical coordinates, 296 Cylindrical room, 398-401, 408-409 D Damped vibrations, 23-27 of air, in room, 386, 401-409, 427-429 in tube, 262-264 of bar, 170 of membrane, 206-208 of string, 106, 132, 145-146 Damped waves, 243, 291, 307, 371, 375, 380 Damping constant, 24, 133, 243, 386, 402, 416, 419, 426-429 Decay, modulus of, 25 Decay curve, for sound, in room, 406 Decibel scale, 225 Decrement, 25 Degeneracy, 181, 190, 395 Delta function, 48, 97 Density of air, 218, 363, 452 of solids, 152 Design, of condenser microphone, 357- 360 of dynamic speaker, 339-344 of horn loud-speaker, 274-279 Dielectric susceptibility, 40 Diffraction, of sound, 329, 350 Dipole source, 318 Dirac, 429 Dirac delta function (see Delta func- tion) Directionality, of sound from piston, 329 (See also Angular distribution) Dispersion, of waves, 154, 209, 307 Displacement, of air in wave, 218 Distribution, of natural frequencies, 390-401 of sound, in room, 383, 401 Divergence, of vector, 294 Driver, 27, 59, 62 Ducts, sound transmission through, 368-376 Dynamic loud-speaker, 34, 67-69, 323- 326, 338-344, 377-379 Dyne, 2 E Ear, response of, 226-228 Echo, flutter, 261-264, 291 Effective length, of tube, 235, 247 Elastic modulus, of bar, 152 of crystal, 40 of plate, 209 Electrical analogue, 233, 275, 365 Electromagnetic drive, for bar, 171 for diaphragm, 341-344, 377 for horn, 276-279, 289, 292 for oscillator, 34—38 for string, 148 Electromechanical driving force, 34 Element, of cylinder, radiation from, 300-303 of sphere, radiation from, 321-323 Energy, 2 kinetic, 2, 89, 163 in normal coordinates, 59, 91 potential, 2, 90, 163 of vibration, 23 of bar, 162 of coupled oscillators, 58 of driven oscillator, 33 of oscillator, 23 of sound wave, 223, 224-226, 402, 414 of string, 89 462 VIBRATION AND SOUND Energy density of sound, 223, 240, 383, 414 Energy loss, 26, 33, 300, 319, 367, 403 Energy transfer, 61 Erg, 2 Exponential function, 7, 438-439 Exponential horn, 279-281, 292 Extended reaction of surface to sound, 200, 361 F Fermi, Enrico, 422 Filters, acoustic, 235, 290 , Flange, on tube, 246, 258, 326, 336- 338 Flexible string (see String) Flow resistivity, 363 Fluctuations of sound in room, 408 Flute, 248 Flutter echo, 261-264, 291 Force, 2 on cylinder, due to wave, 352 reaction of membrane to, 176 on sphere, due to wave, 356 transient, 16 Forced vibration, of air, in tube, 242- 258, 305-311, 368-372 in horn, 271-287 of bar, 166 of coupled oscillators, 62 of membrane, 195-206 of oscillator, 27 of plate, 211-213 of sound, in room, 413-429 of string, 91-100, 119-121, 129-132 138-144 Fourier series, 85, 97 Fourier transform, 16, 43, 51, 94, 229, 288,372 Fraunhofer diffraction, 350 Frequencies, complex, 45, 424 natural, 22, 24, 107, 133 of bar, 158, 162 of coupled oscillators, 55, 68 of kettledrum, 194 of membrane, 180, 189, 195 of plate, 210 of resonating horn, 286 Frequencies, natural, of sound, in room, 390-401, 411 of string, 84, 107, 113, 117, 149 of wire, 168 Frequency, 5 antiresonance, 203, 212 cutoff, 235, 280, 282, 308 fundamental, 84 natural (see Frequencies, natural) resonance, 31, 95, 198, 418-420 Frequency distribution, of normal modes, in room, 390-401 of sounds, 229 Frequency space, 391-400 Frequency spread, of sounds, 229, 419- 420 Fresnel diffraction, 350 Friction, effect of, 23, 27 on air in pores, 363—366 on bar, 166 on membrane, 198-203, 207 on string, 104-106, 130 Functions, analytic, 14 Bessel, 6, 17, 188, 298, 316, 444-446 characteristic (see Characteristic functions) delta, 48, 97 hyperbolic, 18, 136, 438-439 Legendre, 18, 118, 314, 448 Neumann, 7, 196, 444-446 step, 47 trigonometric, 4, 438—439 Fundamental frequency, 84 (See also Frequencies, natural) G Gradient, 295 Gyration, radius of, 152 H Half-breadth, of resonance peak, 68 Half width of pulse, 231 Hanging string, 149 Harmonics, 85 even and odd, 88 in clarinet, 252 in closed tube, 245 in open tube, 248 INDEX 403 Helmholtz resonator (see Resonator) Horn, 265-271 catenoidal, 267, 281-283 conical, 271-279 exponential, 267, 279-281 resonance in, 263—287 shape of, 269-271 transients in, 287-288 transmission coefficient of, 273, 281, 282 waves in, 265-269 Hyperbolic Bessel functions, 210, 445 Hyperbolic functions, 18, 136, 438-439 Hyperbolic tangent, 136, 239-243, 255, 284, 369, 423, 438-443 Imaginary units i and j, 10 Impedance, acoustic (see Acoustic impedance) blocked, of coupled oscillator, 63 clamped, of coil, 36, 38 input, of coupled oscillators, 63 magnetomotive, 37, 38 mechanical (see Mechanical imped- ance) motional, 38 of crystal, 41 mutual, of coupled oscillator, 63 radiation (see Radiation load) of string support, 133, 144 of surface, 360 transfer, of coupled oscillators, 63 for string, 93, 95, 105, 128 of wave, in room, 417 wave, for string, 91, 139 Impulse excitation, 426 Impulse function, 47 Incident and reflected waves, 76, 134, 238, 366, 388 Index of refraction, for porous material, 366 Initial conditions, 21 for bar, 160 for membrane, 183 for oscillator, 21 for string, 74, 86 Integral, contour, 12-16, 19, 42-46, 100, 132, 144, 206 infinite, 14^16 Intensity, 223, 226, 414 of radiation, from cylinder, 297-305, 376 from piston, 328-332, 377-379 from simple source, 312—314, 386, 414-^15 from sphere, 311-324, 377 of reflected sound, 367-368, 388 of scattered sound, from cylinder, 350-351 from sphere, 354—356 of sound, in room, 383, 386, 414 in plane wave, 223, 226 in standing wave, 240 Intensity level, 226 of sound, in room, 386, 415 Interference of waves, 258, 303, 328, 349-351, 355 Interference fluctuations, of sound, 408 Isothermal expansion, 221, 363 K Kettledrum, 193-195 L Laplace transform, 51, 293, 428 Laplacian operator, 174, 295 Legendre functions, 18, 118, 314, 448 Level, intensity, 226, 386, 414 pressure, 226, 415 spectrum, 229 Lippman color photographs, 423 Loaded bar, 171 Loaded membrane, 214 Loaded plate, 216 Local reaction of surface to sound, 198, 336, 361 Loop, 84 Loudness, 226-228 Loud-speaker, dynamic, 34, 67-69, 323-326, 338-344, 377-379 Loud-speaker horn, 265-293 464 VIBRATION AND SOUND M Magnetomotive impedance, 37 Magnitude, of complex number, 10 Mass, 2 effective, for air load, 234, 247, 300, 319 for membrane, 202 for string, 108, 124, 127 nonuniform, of bar, 164 of string, 111-114 Mass-controlled vibrations, 33 Mechanical admittance, 50 Mechanical impedance, 29, 38 of coupled oscillators, 63 of oscillator, 29, 31, 50 Mechanical input and transfer imped- ance, 63 Mechanical reactance, 29 Mechanical resistance, 24, 29 Membrane, 172-208 circular, 183-208 forced motion of, 195-208 forces on, 173, 176 plucked, 185 rectangular, 177-183 struck, 183, 186 transient motion of, 206-208 waves on, 172, 184-187 Microphone, condenser, 195—208, 211- 215, 357-360, 379 ribbon, 353, 378 Modulus, of decay, 25 of elasticity, 40, 152, 209 Moment, bending, 153 Motional impedance, 35, 38 N Natural frequency {see Frequencies, natural) Neumann functions, 7, 196, 444-446 Nodal line, 180-182, 190, 199, 202, 211 Nodal point, 84, 140, 241 Noise, analysis of, 230 Nonuniform bar, 164 Nonuniform string, 107-121, 123-130 Normal coordinates* 56, 58 Normal modes of oscillation, 84, 107- 109 of bar, 156-160, 171 of coupled oscillators, 55 of membrane, 179-183, 189-191, 195 of plate, 210-211 for room, 390, 402, 415, 423 for sound in tube, 309, 373 of strings, 84, 91, 107, 112, 118, 143, 149 effective impedance for, 128 Normalization constant, 108 O Oblique waves, 391, 405 Oboe, 286 Open tube, 246-247, 255-258 small-diameter, 247-253 Operational calculus, 50, 104 Organ pipe, 245, 248 Orthogonality, of characteristic func- tions, 108 Oscillations (see Vibrations) Oscillator, coupled, 52-66 clamped, 23-27 forced, 27-42 simple, 20-23 energy of, 23, 33 Overtones, 84, 158 Panel, reaction of, to sound, 361 Parallel waves, 172 Particle velocity, in sound wave, 222, 224 Period of vibration, 5 Periodic motion, of string, 78, 86 Perturbation calculations, 122—133, 164-165, 402-412, 415, 422 Phase, constant, surfaces of, 266-269 Phase angle, 10 for driven oscillator, 31 Phase shift, scattering, for cylinder, 301, 449 for sphere, 320, 450 Piano string, 102-104 Piezoelectric constant, 40 Piezoelectric force, 38 INDEX 465 Piston, in duct, 309-311 in sphere, 323-326 in plane wall, 326-336, 344-346, 447, 451 Pitch, 226 Plate, vibrations of, 208-213 Point source, 313 Poisson's ratio, 209 Polar coordinates, 174, 187, 398, 409 Polarization, electrical, 39 Pole, of complex function, 13, 45 Porosity, 363 Porous surface, 363-366 Power, absorbed by wall, 367-368, 385, 388 lost to friction, 33, 36 radiated, from cylinder, 299, 302, 305 from horn, 272, 278, 280, 282 from opening, 235 from piston, in plane, 338, 344 from simple source, 313 from sphere, 313, 318, 321, 322 from various sources, 228 scattered, by cylinder, 351 by sphere, 355 Power series, 4 Pressure, of air, 218, 452 mean square of, 414 in tube, 240 maxima and minima of, 241 Pressure level, 226, 415 Principal wave, in duct, 308, 371 Pulse, of sound, frequency distribution of, 230 in room, 126 in tube, 261-264, 291 Q Q of system, 25, 403 Quadrupole source, 377 Quality, tone, of sound, from bar, 158 dependence on intensity, 228 from string, 86 R Radiation amplitudes, for cylinder, 301, 449 for sphere, 320, 450 Radiation of sound, from cylinder, 298-305 from dipole, 318 from piston, in plane, 326-336 in sphere, 323-326 in tube, 309-311 from distributed source, 313 from dynamic speaker, 338—344 from open tube, 246-247, 336-338 from simple source, 313 from sphere, 311—326 from vibrating wire, 299-300 Radiation load, due to medium, 38, 104 on diaphragm, 193, 198, 200 on dipole source, 319 on piston in sphere, 324 in wall, 332, 451 on wire, 300 (See also Acoustic impedance) Radius of gyration, 152 Reactance, acoustic (see Acoustic react- ance) mechanical, 29 Reaction, of surface, to sound, 310, 333 extended and local, 361 Reciprocity, principle of, 63, 352 Rectangular membrane, 177—183, 199 Reed instruments, 248—253 Reed motion, 249 Reflection, of waves, from absorbing surface, 366 from obstacles, 347, 355 on string, 76, 134 Residue of function at pole, 13 Resistance, acoustic (see Acoustic resist- ance) mechanical, 24, 29 Resistance constant, 24 Resistance-controlled vibrations, 33 Resistivity, flow, 363 Resonance, 31 cavity, 228, 258-261 of coupled oscillators, 64 of horn, 283, 286-287 of membrane, 202 of oscillator, 31 of plate, 212 of room, 381, 418-420 466 VIBRATION AND SOUND Resonance, of string, 95, 99 of tank, 235 of tube, 245, 248, 252, 253, 256 Resonance peak, half-breadth of, 68, 419 Resonator, Helmholtz, 235 Response, of microphone, 205, 212, 358 of room, 395, 419, 421 transient (see Transient vibrations) Reverberation, 382, 386, 401 Reverberation time, 387 Ribbon microphone, 353, 378 Rigidity, of panel, 361 Room, cubical, 396 cylindrical, 398-401, 409 nonrectangular, 397-401 rectangular, 389-394, 402-408, 418- 429 response of, 395, 419, 421 reverberation in, 386-388 S Salmon, Vincent, 271 Scattered and reflected sound, 347, 350 Scattering, of sound, 346-347 from absorbing patches, 410, 412 from -cylinder, 347-352 from sphere, 354-357 Scattering shift, of phase, for cylinder, 348, 449 for sphere, 354, 450 Separation of variables, 179 Series, of characteristic functions, 108, 183, 415 Fourier, 85, 97 power, 4 Series coefficients, 87, 108 Shadow, acoustic, 350 Shearing force, in bar, 153 Simple harmonic vibrations, 23, 80, 156 Simple source (see Source) Sine, 4, 438-439 Sound power of various sources, 228 Source, dipole, 318-319 simple, 312-313, 415, 418-421 on sphere, 321-322 Source function, 313 Specific acoustic impedance (see Acous- tic impedance) Specific heats of gas, 220, 452 Specific impedance of surface, 306 Spectrum level, 229 Sphere, radiation from, 311-326 scattering from 354—357 Spherical Bessel functions, 316; 446 Spherical coordinates, 296, 311 Spherical room, 401 Spring supports, for string, 146 Spring, 3 mass on, 3, 20-33 Standing waves, in air in tube, 240 in room, 381, 389 on string, 83, 140 Standing wave ratio, 142, 456 Statistical analysis, of sound in room, 382-385 Steady state, 29, 31 Step function, 47 Stiff string, 166-170 Stiffness constant, 21 Stiffness-controlled vibrations, 33 Strain in crystal, 39 Strength of simple source, 313 Stress, in crystal, 39 String, damped, 104-106 energy of, 89-90 forced motion of, 91-104, 129-132, 138-141 free vibration of, 84-90 hanging, 149 impedance of, 91, 95, 126-129 characteristic, 93 nonuniform, 111-113 plucked and struck, 80, 87 stiff, 166 waves in, 72-80, 134-136 weights on, 68 whirling, 114-121 String support, effect of, 76, 134 impedance of, 133 Surface, acoustic impedance of, 362 porous, impedance of, 363-366 sound absorption at, 360-368, 388 Susceptance, acoustic (see Acoustic susceptance) mechanical, of oscillator, 35 INDEX 467 Susceptaace, mechanical, of string sup- port, 144 Susceptibility, dielectric, 40 Symmetry of room, effect of, 395 Tangential waves, 391, 405 Tank, 235 Temperature change due to sound, 222, 288 Tension, on membrane, 173 on string, 72 variable, 116 Thermodynamic relations for gas, 220- 221 Threshold, of hearing, 227 of pain, 227 Transducer, electromagnetic, 34-38 electromechanical, 38-42 Transfer admittance and impedance, of coupled oscillator, 63 for string, 93, 95, 105 "for walls of room, 411 Transform, Fourier, 16, 43, 93 Laplace, 51, 293, 428 Transformation, conformal, 136 Transient force, 16, 42, 93, 110 Transient vibrations, 42-52 of air, in horn, 287-288, 293 in room, 386, 426-129 in tube, 261-264, 372 of coupled oscillators, 64 of diaphragm, 206-208 of oscillator, 29, 44-52 of radiation, from piston, 344-346 of string, 93, 100-106, 110, 132, 145, 147-148 Transmission, of sound, through ducts, 368-376 inside cylinders, 305-311 inside horns, 265-283 through porous material, 365-366 in room, 418-425 inside tubes, 224, 237-240, 254- 258 Transmission coefficient, 273 of horn, 273, 281, 282 of piston, 338 Transmission line, analogous, 254, 366 Trigonometric functions, 4, 438 Tube, analogy of, with transmission line, 254 cavity resonance in, 258—261 closed, 244-245 constriction in, 234 effective length of, 234 flaring (see Horn) open, 246-253, 255-258 waves in, 222-225, 238-244, 305-311, 368-376 Tuning fork, 158 U Unit impulse function, (see Delta function) Unit step function, (see Step function) Units, physical, 2 Velocity, particle (see Particle velocity) wave (see Wave velocity) Velocity amplitude, 23 Vibrations, damped (3ee Damped vibra- tions) free, 20-23 of air, in room, 390-401 of bar, 156-160, 168 of coupled oscillators, 54-59 energy of, 23, 58, 89, 214, 223, 296 of membrane, 180-183, 187-191 of plate, 208 simple harmonic, 20-23 of string, 84-91, 112-118 forced (see Forced vibrations) Vibrator (see Oscillator) Vowel sounds, 232 W Wall coefficient (see Coefficient) Warble tone, 379 Water, acoustic constants of, 452 Wave, 72-74 axial, tangential and oblique, 391- 400, 405, 409 468 VIBRATION AND SOUND Wave, in bar, 155-156 circular, 184, 191 damped, 243, 307, 371 longitudinal and transverse, 217 in membrane, 172, 184-186, 191 plane, 217, 266-268 in porous material, 365-366 principal, in duct, 308, 371 standing, 83, 140, 240, 381, 389 in string, 71-80, 92-93 in wire, 167 Wave admittance, 91 Wave equation, 81 for air, 221, 294 for horn, 269 for membrane, 174 for porous material, 364 separation of, 179, 268 for string, 81 Wave impedance, of principal wave, in duct, 372 for string, 91, 139 (See also Acoustic impedance) Wave number, 225 Wave velocity, 72 Wave velocity, in air, 222, 452 in bar, 154 in membrane, 177, 187 in* string, 72 in tube, 307, 371-374 in water, 452 Wavelength, 83, 134, 225 Weighted string, 68, 71 Width, of pulse, 231 of resonance peak, 68 Whirled bar, 171 Whirled string, 114 Wind instruments, 253, 285 Wire, radiation from, 299 vibrations of, 166-169 Work, 2 Young's modulus of elasticity, 152 Z Zero response of diaphragm, 202-203, 212-213 hrt Ll -^