RESEARCH DIVISION COLLEGE OF ENGINEERING NEW YORK UNIVERSITY DEPARTMENT OF METEOROLOGY A UNIFIED MATHEMATICAL THEORY FOR THE ANALYSIS, PROPAGATION, AND REFRACTION OF STORM GENERATED OCEAN SURFACE WAVES PART I Prepared for BEACH EROSION BOARD DEPARTMENT OF THE ARMY Contract No. W 49-055-eng-1 OFFICE OF NAVAL RESEARCH DEPARTMENT OF THE NAVY Contract No. N onr-285(03) A UNIFIED MATHEMATICAL THEORY FOR THE ANALYSIS, PROPAGATION, AND REFRACTION OF STORM GENERATED OCEAN SURFACE WAVES Paritee By Willard J. Pierson, Jr. Preliminary Distribution Prepared under contracts sponsored by the Office of Naval Research and the Beach Erosion Board, Washington, D. C. INN ii a) sil New York University College of Engineering Department of Meteorology March 1, 1952 NAT We. 0% ‘Bese FRONTISPIECE ih aon alte ein . is a ma tay eve Preface ————= The following pages represent part one of a book entitled "A Unified Mathematical Theory for the Analysis, Propagation, and Refraction of Storm Generated Ocean Surface Waves." They contain the first ten chapters of this projected book. These first ten chapters are not a logical stopping place in the book. Only part of the unified theory is presented. That part covers the theory of wave propagation (wave forecasting) and the theoretical part of the theory of wave analysis. There are also many references to chapters in the book which are not presented in part one. Part one is presented in this disjointed form with all the apparent loose ends because more of the book could not be com-=- pleted before the publication deadline and because an error was made by the author in estimating the date of completion of the book. There are many important decisions which have to be made soon in connection with the problem of adequate methods of wave analysis and wave recording and it is hoped that the contents of part one will help in these decisions. The remainder of the book will be presented in bi-monthly installments until the book is completed. It is planned to present the mathematical theory of additional properties of waves in deep water, the theory of waves in the transition zone, and the theory of wave refraction in the next chapters. The mathematical theory is complicated, but the practical application is straightforward and easily avplied. After these chapters, the book will consist of examples and applications of the theory, of examples from the work of others which substantiate the theory, and of suggested procedures for further verification. The work presented herein has been sponsored by the Beach Erosion Board and the Office of Naval Research. The Office of Naval Research is supporting the research which applies to the problem of wave forecasting. The Beach Erosion Board is supporting the research which applies to the problem of wave analysis and wave refraction. If the reader wishes, he can select the various parts of each chapter which apply to each of the Sponsors. However difficulties will occur in deciding what parts apply to which sponsor because adequate methods of analysis are a prerequisite for adequate methods of wave forecasting and a firm understanding of basic hydrodynamics is a prerequisite for any part of the theory. One of the most important features of government sponsored research in science is the wide latitude of action permitted the researchers by the sponsoring agencies. This is especially true of the Office of Naval Research and the Beach Erosion Board. The original contracts were thought of as separate entities, and it was planned to present separate reports to each. However, as things worked out, it became possible to unify the entire theory and present the whole subject as an entity. It is hoped that both sponsors will be pleased with the final outcome. March 1, 1952 Willard J. Pierson, Jr. Department of Meteorology New York University bal Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter uve Introduct ion e e e e e e e © e 2s oe e e e e e eo es 2. Resumé of Classical Wave Theory ....... 3. Types of Periodic Functions and Non-Periodic Functions and their Representation by Discrete Gioyol Copdimatio yong Syren) 6 of G6 0 51600 6°56 6 6 6 4 4. The Propagation of a Finite Wave Group in Ibobeabephnoalyy IY CHe) WWeENBCEE Mo Glo KS Ol Ou opwollo oso ol ¢ 5. The Propagation of a Finite Wave Train in IGapealb alain slay MOyeVeyoy eMEHe BG! Gl Olio 6b los SMiow GG. oc 6. Some Model Wave Systems in Which the Crests are IbohtabeieeMay lela 56 6 6 oF G50 6 oO) ola Ga 66 4 0 0 6 7. The Most Realistic Wave Systems with Infinitely JOINS ORROSES 6 6566 eo Oo) oo 6 oo a Gold dio) 6 SamiohortiucrestedmWaverSysiemsivs va We vs. ve eel a ete ee 9. The Mathematical Representation of a Short Crested Sea Surface by a Lebesgue Stieltjes Power Integral and the Problem of Forecasts for a Storm of Finite Width and Finite Duration over a Fetch @he Ue Wi 6) S16) SiG 6 Chapter 10. Methods for the Determination of Power Spectra . Appendix e e e e e e e e eo e e e e e e e e e e e e e e e e e NGISRO VALE CIOMSINOS 5 ANEMG NoMa <6 66) 8) 6 SEG 6G 6 Ge ob) 6 6 Index to Index to Index to Partial the Figures e ° e ° t) e e e e e ) e e ° ° ° ° ° e ° the Plates C) e ° © e e e ° ° ° e ° e ° ° e ° e e e the Tables e e e e e e e e e° e e e e ) e ° e e e aS GiOi ede PEN CESier, ise) veiine. viel Mos vain fall) feuilcedieees ce Citeh tenhivey Wires Rte alalal Page 102 122 173 206 255 311 321 324 326 331 332 aie Ad Py 4 ‘* 4 « * 4 ‘ ° RN re ESR) ch a inl: Sergi, neh eis ital ir Think AP ae iu wah Bar eae a) ae ise alah he aR Se Bets emia mm ah Wn ek ‘ F ee had ; A r ay /.: Sea oe bree er ta Men eitn, wee RE iu a) a eos soiigs Sie ed Ba i bear, Pak yas 103) peae erat | iy Pate Wome ee EGE eve Cito: pi A a aC NCU eyo mm yc ve ditt a, bs ta wy hg | = ay a ins i} SL, fate } ra at . By iy ATES tore Aine oe 7 ia os Se Te ee i a ee | ery . 7 Py « o i eT ae waist! wit fer \- be ye? Yi Lc ’ Wid tegen fA Hh oe ¢ £6 2. hie kes citi Sie Cima Last} He eth a Pekin tnt: Mae aay: Riek iceiondd ‘ais * rss Trea ae eid de) sg By ; . 4s RI Ce NOs lean f A tke yi 4 } : ' a te ah Dae ls : LPL A ee ; : 1's) Bates ete oR zy : f P ' et we i vs ; i , } rua = “es oT Wa Be Te ee i es r : L oS 8) a4 \ & ' ee f MAM PEs ren PD, Gave ak ties nme A UNIFIED MATHEMATICAL THEORY FOR THE ANALYSIS, PROPAGATION, AND REFRACTION OF STORM GENERATED OCEAN SURFACE WAVES SIR HORACE LAMB M.A., LL.D., SC.D., F.R.S. "SINCE THE CONDITIONS ARE UNIFORM WITH RESPECT TO X, THE SIMPLEST SUPPOSITION WE CAN MAKE IS THAT @ IS A SIMPLE HARMONIC FUNCTION OF X; THE MOST GENERAL CASE CONSISTENT WITH THE ABOVE ASSUMPTIONS CAN BE DERIVED BY SUPERPOSITION IN VIRTUE OF FOURIER'S THEOREM." Chapter 1. INTRODUCTION The origin of the remark is lost in antiquity, but many persons claim that ocean waves are just bumps on the water. Cer- tainly, wave records show that the waves pattern is sometimes chaotic, sometimes irregular, and other times smooth. Wave records are not sinusoidal, nor are they obviously periodic. In this paper the supposition that waves are just bumps on the water will be admitted, and then it will be possible to show how waves can be represented by the sum of a number of sinusoidal terms in a way which will preserve many of their observed properties and which will be amenable to theoretical work. The overall theory of wave forecasting is a mixture of various concepts which do not fit together well at the edges. Waves are treated partly as non-conservative waves and partly as classical waves. The significant waves are forecasted over a fetch and they are supposed to represent the average height and period of the one third highest waves. Admittedly, they are not classical a Pos waves, and yet they are forecasted to travel with classical group velocities, and refracted as if they were purely sinusoidal waves of one single period. The Sverdrup-lunk theory [1947, 1949a], as extended in part by Arthur [1948, 1949], depends on the validity of the assumption that two parameters, viz., the significant wave height and period, can adequately describe the sea surface at any time and any place. It will be shown that these parameters are not sufficient to characterize the sea surface and that the inadequacy of these parameters in part can explain the failure of the forecasting method to forecast the significant period as shown by Donn [1949], Isaacs and Saville [1949], and Pierson [1951p]. . Wave records at present are frequently obtained by recording the pressure as a function of time at the bottom as the wave passes overhead. The pressure records are then analyzed for the Significant period and significant pressure amplitude, and the Significant height and period at the free surface are computed from these values. It will be shown that this procedure is in- correct. The use of the significant period in problems in wave re- fraction is also a most doubtful procedure. A method will be developed in this paper which will be far more applicable to actual sea conditions than the present techniques. The overall plan of this paper is to start with the simple and proceed to the complex in the derivation of various models of waves on the sea surface. Models of the sea surface will finally be obtained which will prove to be adequate for a correct de- scription of the sea surface. - Cm After these models have been obtained, they will be dis- cussed in connection with the problem of wave forecasting, wave recording, and wave refraction. It will be shown that they aera a basis for a correct forecasting theory of ocean waves. It will also be shown that the current debate in the literature about whether friction against the atmosphere or eddy viscosity in the moving water causes the decrease of wave height with travel into the decay area is an argument about nothing, because this decrease of height of waves with travel can mostly be explained by classical concepts without the use of any type of friction in the theory. Many of the points which will be discussed in this paper are purely theoretical. Some of the instrumentation and methods of analysis which will have to be devised in order to place the techniques which will be described into practical use have yet to be developed. The data which are obtained at present are in- adequate. Procedures for obtaining data which will adequately characterize waves produced by a storm at sea will be described. The final result of this paper will be to obtain a unified mathematical theory for the representation of ocean surface waves as they are. The behavior of irregular waves will be described completely from the time they leave their source until they enter the breaker zone. Applications to vroblems in beach erosion and ship design and other far-reaching implications will be described. “a Chapter 2. RESUME OF CLASSICAL WAVE THEORY Introduction Classical wave theory has discussed rotational waves such as Gerstner's waves and irrotational waves such as shallow water waves, waves of finite height, deep water waves of infinitesimal height, and solitary waves. This paper will be concerned with the theory of deep water waves of infinitesimal height at first, and later the refraction and diffraction of these waves will be dis- cussed. Before the theory of waves of infinitesimal height is applied to the sea surface, a reason should be given for not using the theory of waves of finite height. The reason is simply that the waves of infinitesimal height combine linearly whereas waves of finite height do not combine linearly. The irregularity of the sea surface is its dominant feature. As such, it can be treated mathematically. The non-linearity of the sea surface cannot be treated mathematically without suppressing the irregularity of the sea surface. These points will be clarified in the resume of the theory of waves which follows. Non-linear equations If irrotational motion is assumed, the problems connected with gravity wave motion on a free surface approximated by a plane despite the curvature of the earth can all be considered to be solutions of equations (2.1), (2.2), (2.3), (2.4), and (2.5) which are shown in Plate I and which for example are given by Lamb [1932]. Irrotational Non-linear Equations for Motion Bounded by a Free Surface and o Bottom of Variable Depth. Potential equation @,,+Oyy+ 22+ 0 (2.1) Bernoulli's equation os $1 - 5 ( + Oy? +027)-92Z (2.2) Boundary conditions: at Z=-h(x,y) 6, = 0 (2.3) at 7a igh n=g Ot-ag (OF + by? + O27) (2.4) at Z=n nye- Oz (2.5) Irrotational Linear Equations for Motion Bounded by a Free Surface and a Bottom of Variable Depth. Potential equation Oxx + Oyy + Ozz = O (2.6) Bernoulli's equation 5 = Ot - 9Z (2.7) Boundary conditions: at Z=-h(x,y) 162 (2.8) ate Z= 0 =9 Ot (2.9) ay Ze© ny=- Oz (2.10) PRAT Es Pak —-5- Equation (2.1) is the potential equation which originally comes from the equation of continuity. Equation (2.2) is Bernoulli's equation where an arbitrary function of time has been neglected. Equation (2.2) need not be considered explicitly in solving wave problems; it simply gives the pressure after 9 has been obtained. Equation (2.3) is a boundary condition equation and states that there is no fluid motion normal to the bottom. Equation (2.4) is the free surface boundary condition for pressure continuity where 7 is the free surface. Finally, equation (2.5) is the kinematic boundary condition at the free surface. It states the condition that a particle at the free surface must remain at the free surface. In this paper, partial derivatives will be denoted by subscripts; for example, ®, means O® At. These equations have never been solved completely. Partial solutions have been obtained only after simplifying the equations. Even the known solutions for waves of finite height are approxi- mations. The difficulty arises in equation (2.4). The term (p,- + Py + 95° )/28 is the cause of the difficulty. It is a non-linear term. Suppose, for example, that Py satisfies equations Wel) Ce ig) Cea) ge 4) ance (2.5) and = that P5 also satisfies the same equations. Then Py plus 95 will not satisfy the equations, and P5 plus P, and 14 plus 75 have no meaning. Thus the original equations for wave motion are non-linear. At this point, then, in the study of wave motions there are two possible ways to proceed. One way to proceed is to concentrate on the non-linear properties of the equations. The second way to proceed is to reduce the equations to a linear form with the smatl~ 260s height assumption so that the property that the sum of two solutions is also a solution will be obtained. If the non-linearity of the problem is of the greatest interest, the work of Stokes [1847] as summarized by Lamb [1932] illustrates the type of results which are obtained. In Chapter IX, Section 250 of Lamb, for example, the problem of waves of finite amplitude in water of infinite depth is treated "as a case of steady motion" under the assumption that the wave is periodic in time. The notation is somewhat different from that which is used here, but equation 4 in Lamb shows that the non-linearity of the free surface and of Bernoulli's equation is considered in the derivation to find the speed of the wave. The solution is approximate because it is in series form. The wave profile is approximated for the first three terms by a trochoid, and the whole wave profile moves forward with the speed c= (2 (1+ ee)? , x = ‘ Davies' Results A recent monograph by Lowell [1950] on gravity waves of finite amplitude describes some results which have been obtained by T.V. Davies [1951] of King's College, University of London. jtorvenlll De summary of his monograph is quoted in full below. Davies' work has unified the previous theories of waves of finite height and has yielded some improved theoretical relationships about the ratio of wave height to wave length. . GRAVITY WAVES OF FINITE AMPLITUDE 'T, V. Davies of King's College, University of London, has discovered a new method for treating tne classical problem of steady gravity waves in an irrotational, incom- pressible fluid. He has been able to solve the problems of (a) periodic waves in a channel of infinite depth, (b) the solitary wave, (c) periodic waves in a channel of finite depth, and (ad) periodic waves at the interface of two streams of finite depth. "The method used by Davies is a development of that of Levi-Civita in his paper of 1925. The first approximate solution contains a variable parameter yp which satisfies Oppo, ( p. being known in each case); the lower range of #» corresponds to the classical waves of small amplitude, the upper limit corresponds to the case in which breaking occurs at the crest. The Stokes result, that the angle of breaking at the crest is 120°, is verified in each case and the problems of wave velocity, energy, form of the free surface, and the drift at the base of the fluid, have in the main been solved. The first approximation is in error by 137 at the extreme case of breaking at the crest, but the error decreases when the crest is horizontal and when the ratio of wave height to wave length is smaller. The higher approximations have been derived in cases (a) and (b).'™ *The / in this quotation has a meaning here which is different from its meaning in the rest of the text. In the theory of waves of finite height, it is not possible to take two solutions, add them, and find a solution for the combined effect of the two profiles. Thus the known solutions for waves of finite height are all purely periodic, and they do not apply to the sea surface if the sea is irregular. Linearized Equations Since the irregularity of the sea surface will be of the greatest interest in this paper, the equations must be linearized if known mathematical techniques are to be applied to the analysis. The assumption can be made that » is so small that the square of pm and its partial derivatives can be neglected compared to the magnitude of m and its partial derivatives. Under this assumption equations (2.1), (2.2), (2.3), (2.4) and (2.5) can be replaced by equations (2.6), (2.7), (2.8), (2.9) and (2.10) which are also given in Plate I. Equations (2.6) through (2.10) are very much simpler than the first set of equations. The non-linear terms have been omitted from equations (2.2) and (2.4), and to the same degree of validity, it is possible to evaluate the free surface boundary conditions at z = O instead of at z =) os These equations are linear. If @, is a solution of equations (2.6), (2.7), (2.8), (2.9), and (2.10)'5 and if > is a solution of the same equations, then 9, + 95 is also a solution. In addition, P, + P5 and 7),+ 7» are defined. Strictly speaking the equations hold exactly only for waves of infinitesimal amplitude. In what follows, they will be applied to waves of finite height with the reservation that S86 the higher the ratio of the wave amplitude to the wave length the more inaccurate the results. This linearized theory is used in nearly all practical wave studies and especially in the theory of wave refraction and diffraction. General solutions The above set of equations is, nevertheless, still compli- cated, and the manifold of possible solutions is extremely large. The number of known solutions is quite small. There are two general types of solutions to those linear- ized equations. One is the periodic solution in time, and the other is the non-periodic solution. A periodic solution in time is a solution such that at any point in the fluid or at the free surface, the same conditions are found one period later aS were found at the time of the initial observation. The conditions must be the same for all time. Thus, the conditions for a periodic solution can be stated as in equation (2.11) in Plate II. From equation (2.11), it follows that the free sur- face is also periodic. The concept of periodicity will be investigated in detail in the next chapter. It should be noted at this point, that the sum of two periodic solutions need not be periodic unless some additional conditions are satisfied. In addition, a whole class of non=-periodic solutions can be obtained from integration by Fourier's Integral Theorem over a continuous spectrum of periods. The quotation at the Start of this paper emphasizes the fact that the way to obtain non- periodic solutions is to build them up mathematically from = Ok i= periodic solutions by the use of Fourier's Integral Theorem, One purpose of this paper is to show what information is needed to carry out this process for the sea surface. It will be found that classical wave theory is not quite general enough to represent the sea surface. The more recent extensions of Fourier Theory to stationary time series have to be employed in order to represent waves from a storm at sea adequately. Periodic solutions To return to the linear equations, then, it becomes necessary to study, the nature of purely periodic solutions. In order to do this, it is possible to split off the periodicity in time by use of the equations (2.12) and (2.13) in Plate II. In equations (2.12) and (2.13), Re is read "the real part of." % and 9 are complex quantities, and some examples will be given later. If equations (2.12) and (2.13) are substituted into the linearized equations, a set of reduced equations is obtained in which 9 and 7)' are not functions of time. However, (2.12) and (2.13) yield the progressive wave solutions. Equation (2.7) will not be used for a while and it will not be given in modi- fied form. The reduced linear equations given in Plate II have been solved exactly for a constant depth and for a linearly sloping beach. They have not been solved for z = - h(x,y) where h(x,y) is an arbitrary function of x and y. There are two solutions for constant depth. One solution yields an infinite train of traveling infinitely long straight parallel wave crests with a free surface which varies sinusoidally ay ayes Form of the Irrotational Linear Equations when the Require - ment that the Solution be Periodic in Time ts Imposed. Definition of periodicity (x,y,z,t)= O(x,y,z,tt+T) (2.11) then @Q(x,y,z,t) = Aatihtanehoe a= (2.12) N(x,y,f) = Ren'(x,y) ei (2.13) and, potential equation Qxx+ yy + zz = 0 (2.14) boundary conditions: at Z=-h(x,y) dn=O (2.15) at Z=0 ni: - tt ad = (2.16) a | 2S) ere (2.17) Known solutions if =-h (h constant) Straight wave crests 0 eu ° g . Ati eile cos® ty sin®)+ 3] ogn 2 (Zz +h) , cosh 22h (ete) L N= Acos [42 (x cos@ + ysinO@) + § - at] (2.19) 2 L 2245-2 tanh 2h. (2.20) Circular wave crests = const Ho(4=") cosh 22 (z +h) | (2.21) joey Ga PE eg alle b= © conatwelaee Neko) Sy 5 Ye) @ cosh 2 (Z +h) (2.22) V (x- xe)? +(y-y,)2 | ne const eos (an ¥(X-X0)* +(y- yo)? -5-271) (2.23) V(x-X0)2 + (y= yo)? PEATE “TE =/2= in one direction. The other solution, given by a Hankel function, yields circular wave crests which radiate from a point source. The first solution is found in the Cartesian coordinate system which has been used so far. g is given by equation (2.18). 6 is an arbitrary phase lag, and A is the amplitude of the wave crest. From equations (2.18), (2.13), and (2.16), it follows that 7) is given by equation (2.19). This representation for the free surface has been chosen in order to point out all the arbi- trary parameters in the solution. The most important one to note is the © which permits the choice of any wave direction, if © varies through 27 radians. The equation for the speed of waves in water of constant depth follows from equations (2.18), (2.16), and (2.17). The fact that the depth is constant per- mits the easy treatment of the problem. The speed of the crests is given by equation (2.20). Equation (2.19) is the only wave with straight crests, which travels with the classical wave velocity of waves with small amplitude. As written, it states that there are an infinite number of crests present, that the wave record will be observed for an infinite time at any point, that the period and wave length are everywhere the same and everywhere constant, and that the heights of all crests are the same. If any single one of these requirements is not satisfied in nature, then the equation is not valid, and a more refined analysis is needed. Sees The second solution is found in cylindrical coordinates, and the solution is given in terms of the distance r from a fixed point, r, = 0. @ is given by equation (2.21) where HS (2rmr/L) is the first Hankel function of order zero (see Sommerfeld [1949]). If r is large and x,,y, are the coordi- nates where r = 0, then 9 is approximated by equation (2.22). The free surface,7, is then given by equation (2.23) under the same assumption that r is large. The same condition for the speed of the wave crests holds that was given in equation (2.20). The point of origin of the circular wave crest is arbitrary. Equations (2.21), (2.22), and (2.23) will not be used in this paper. Mathematical techniques similar to the ones which will be employed in this paper (but more difficult) are appli- cable to problems involving these equations. They are given here in the interest of completeness, and in order to make one very important point. For narrow fetches with very turbulent and extremely vari- able winds, and for wave generating areas such as those found in hurricanes, a detailed study of the sea surface would have to be made with these equations as a starting point. Problems in wave decay in particular must be studied because the form of equation (2.23) provides a means for the decrease of wave height with distance traveled. In view of these considerations, the results of this paper will be based upon the assumption that the elemental unit of.analysis is a wave of the form of equation (2.19). The consequences of this assumption will be Ages discussed in detail in a later chapter. The remaining known solutions to these reduced equations have been obtained by Stoker, [1947], and his co-workers for the problem of a linearly sloping beach with waves parallel to the beach and recently by Peters for waves at an angle to the beach (unpublished). For additional information, see the paper referred to above. No exact solutions for the reduced equations have been obtained under the condition that the depth is an arbitrary function of x and ye Graphical methods of solution based upon the principles of geometrical optics have been given by Sver- drup and Munk [1944], and by Johnson, O'Brien and Isaacs [1948]. Pierson [195la] has discussed these results and formulated the problem which would have to be solved in order to proceed from equations (2.14) through (2.17) to a result which would prove that the principles of geometrical and physical optics are ap- plicable to problems of ocean wave refraction and diffraction. Eckart [1951] has obtained an approximate solution to the com- pletely general problem, accurate everywhere to within a few percent. The solutions which have been discussed so far apply to depths which range from infinite to one or two tenths of the deep water wave length. The solutions do reduce to the shallow water theory, if h is picked smaller and smaller, but a wave of the finite height progressing from deep to shallow water in Stoker's work [1947] becomes infinitely high as it approaches *See References. Sache the shore and thus the linearized theory breaks down. Another class of solutions can be obtained under the assumption that the water is shallow. The shallow water theory of the solitary wave, for example, as obtained by Airy is treat- ed in Lamb [1932]. Recent refinements in the theory have been obtained by Keller [1949]. Lowell [1949a] has studied the propagation of waves in shallow water. Munk [1949] has studied the breaking of solitary waves in shallow water. Stoker [1949] has applied the non-linear shallow water theory to the formation of breakers and bores and to the breaking of waves in shallow water. The results which will be obtained in this paper will hold only up to the shallow water zone. It will be possible to generalize the theory of ocean wave refraction to the dis- turbances studied herein. The breaker zone, however, will not be treated, although an extension of the results obtained by Biesel [1951] may make this possible. Biesel's graphs of waves just before breaking appear to be the most realistic mathemati- cal breakers ever presented. Non-periodic solutions One final important class of solutions which has been ob- tained in classical wave theory remains to be discussed. They are the solutions which have been obtained by the use of Fourier's Integral Theorem for waves in infinitely deep water. The gen- eral procedure is’ to integrate the potential function and the representation of the free surface given in equations (2,18) and (2.19) over a continuous spectrum of angular wave frequencies “a6. (4 = 27r/T) and thus obtain some special case non-periodic solutions. For infinitely deep water (for practical purposes about five hundred feet for waves with periods of ten seconds or less), the potential function, the free surface, the pressure (z de- creases from zero), the wave speed and the wave length are given by equations (2.24), (2.25), (2.26), (2.27) and (2.28) where ft = 2r/T. The equations follow from equations (2.7), (2.9), (Qol2)5 (Bore (aout) eiael (Boils))s In these equations (Plate III), ® is a function of the three space coordinates and time. © also depends upon the para- meters, , 0, A, and 6. If ®, = D 5 (x79 29by oe 304 28784)” is one potential function, and if Dy = @5(xX,y¥,2,t, pp 51959Ao965) is a second potential function, then ® = @, + @, is a third potential function. Moreover, if A and 6 are functions of » and ©, then a double integral of ® over p# and © is also a potential function. A(p ,®) and 5( 4,8) must behave properly in a mathematical sense for largey» . In particular aqne sien (2.29) is a potential function which satisfies equation (2.6) and (for z = -@) equation (2.8). Also7 can be found from equation (2.9) and the pressure can be found from equation (2.7). The condition, (2.10), is satisfied. If one picks some functional form for A(y,6) and b(t 58) and if then the indicated integration can be performed on equa- tion (2.29), the resulting expression for the potential function Pgs ora ® is a function of the time and space variables and one set of fixed values for the parameters. Sas Periodic Wave Solutions For infinitely Deep Water Express- ed in Terms of Angular Wave Frequencies. ast Potential function -A 2 O = Aa ue sin( 4 (x cos® + y sin@)- att ) (2.24) Free surface a2 3 n= Acos(4—(x cos@ + y sine) — ut +6) (2.25) Pressure 2 2 D = pgAe 2/9 cos( 4 (x cos@ +y sin®@)-at+ 5)-gpz (2.26) Wave speed C=— (2.27) Wave length L= ae (2.28) A non periodic potential function results from integration over 4 and @. 00 +1 5 2 2 O = if f eee et 2/9 sin( 4-(x cos + y sin@)-at +6 (u,0))dude -00 -TT (2529) PEAKE? itr = (ies will yield all the usual information about the effects of the disturbance on the free surface. Lamb [1932] summarizes some of the results which have been obtained by the use of the Fourier Integral Theorem. Among the results, the Cauchy Poisson Wave problems are of the great- est interest as far as this paper is concerned. One problem gives the wave system propagated from an initially concentrated elevation of the free surface, and the other problem gives the wave system propagated from an initially concentrated impulse applied to the free surface. The first problem gives the wave system which would result, if at the given time t = 0, an infinitely high, infinitesimally wide, infinitely long column of water were to start falling into the ocean at the point x = 0. The free surfacey7) » is given by equation (2.30) if et-/4x is large. | ml linn iy al At any x as t approaches infinity, 7 oscillates more and more rapidly and approaches infinite values of height. Since the original formulas upon which this solution is based were founded upon the assumption that the height of the initial dis- turbance is small, the physical reality of the problem is ser- iously open to question. The second problem gives the wave system which results from the action of an infinitely intense impulse upon the line pS eee *Note also that the u, Vv, and w components of the fluid velocity can be found from@® . an Gh = r x = 0 at the time t = 0. The free surfiace for large et /4x is found by partial differentiation of equation (2.30) with respect to t and multiplication by 1/gp. Again the physical reality of the problem is seriously open to question. The two problems described above have been used (frequently in a most uncritical way) by many authors in attempts to devise methods for nomecasiamae ocean waves. Until some ship reports an infinitely high, infinitely long, infinitesimally wide colunn of water over the ocean or an infinitely intense local impulse concentrated on a line, it will be necessary to interpret these results "cum grano salis." There is one remaining classical application of the Fourier Integral Theorem which is of great interest in this study. It is the Gaussian wave packet. Coulson [1943] gives a readily available summary of the chief results obtained (see reference; ppe 135-138). The representation for the free surface obtained from the Gaussian wave packet depended upon the integration of equation (2.31) where to transform to the notation of equation (2.29), K would be given by K =p Je and n would be given by n= Kee [been [o0) —O(K-Ke)* i277 (Kx-nt) nlx,t) = /Ae e dK (2.31) For t = 0, the integral can easily be evaluated and the free surface is found to be given by equation (2.32). *Coulson [1943] uses ¢ for the free surface and not the potential function. Li 36) = m2x2 (x0) = SRAVE @ 7 Eemikor (2.32) Equation (2.32) represents a wave as a function of x on the free surface with a wave length, L, = 2n/K. » which is modu- lated by a probability curve envelope. In order to evaluate (2.31) as a function of time also, n was expanded as a function of K in a Taylor series about the point aS “Kez . Only the first two terms of the expansion were used in the integration. The solution thus obtained was an approximation because of the series approximation of n. It showed that the probabi- lity curve envelope advanced with the group velocity appropriate to waves with a wave length, L, = 2r/K, that the envelope flat- tened out with time and decreased in maximum amplitude, and that there was a gradual phase shift of the individual waves under the envelope. The Gaussian wave packet is a far more realistic problem than the Cauchy-Poisson problem because the condition that the height of the waves be small is satisfied everywhere if it is satisfied at the time t = 0. As it stands, however, it is vroba- bly not applicable for moderately large values of o and for large values of time or displacement in the x direction because the effect of dispersion is partially neglected in the series approximation of n. -It should be noted that none of the classical solutions have considered the possible variation in the direction of pro- pagation of the wave crests as indicated in eouation (2.29). eho os This will be done in great detail in this paper when models which describe waves from a storm at sea are obtained. Chapter 3. TYPES OF PERIODIC FUNCTIONS AND NON-PERIODIC FUNCTIONS AND THEIR REPRESENTATION BY DISCRETE AND CONTINUOUS SPECTRA Introduction The surface of the oceans, if represented accurately every- where, would have to be given by a function of latitude, longi- tude, and time. The function would include the effects of tides, piled up water due to wind stress, other things, capillary waves, and gravity waves. This representation for the sea surface would be an extremely complicated function. In fact, it is so complex that it is necessary to restrict the scope of the prob- lem and to study the various effects separately. This study will be restricted to the mathematical analysis of ocean gravity waves with periods ranging from one or two minutes through one half seconds. Even this restriction is not enough. It is also necessary to restrict attention to homogene- ous areas of the ocean over which conditions can be expected to be relatively the same and to line segments on which the waves as they pass are the same in essential character for a relatively long time. A generating area or fetch such as the ones treated in the Sverdrup-Munk Theory [1944a, 1947] might be such an area of study if the waves have reached a steady state condition. It will be shown that the usual measurements of significant height and period are not sufficient to character- ize such a steady state condition. Frequently the character of recorded wave data changes - 23 - slowly over time intervals of the order of four hours. In the Same sense that sin t can be approximated in the neighborhood of t = 0 by t, it will be possible to represent wave records obtained at a certain time by the functions which will be studied. The more rapidly the sea surface characteristics change, the less valid some of the techniques described here will be. The nature of available data Ocean waves are recorded by two methods at the present time. Either the actual height of the free surface is recorded at a fixed point as a function of time, or the pressure at some depth below the free surface and at a fixed point is recorded as a function of time. Neither method is sufficient to determine com- pletely the actual space and time distribution of the free sur- face, the pressure, and the fluid motions. By a sufficient number of simplifying assumptions, it is possible to draw a few con- clusions about the distribution in space and time of the above properties. The actual height of the free surface is frequently measured on the open ocean by an upright graduated pole with a large disk on the bottom to damp out the motions of the pole. On.the end of piers or at fixed installations such as oil drilling structures in the Gulf of Mexico as reported by Glenn [1950], it is possible to use the instrument developed by the Beach Erosion Board and described by Caldwell [1948]. In either case a record is ob- tained of the height of the free surface as a function of time at a fixed point. Or in terms of the equations employed in this paper, 7] = 1(x,,¥,92 = 0,t) is known. Fey We Most wave records at the present time are not actual mea- surements of the free surface. They are measurements of the pressure at some depth below the free surface. In all but one known case, the depth is the bottom at a short distance (rela- tively speaking) from the shore. In this one case, the pressure was recorded by a submarine below the sea surface as reported by Ewing and Press [1949]. In terms of the equations employed herein, P = P(X),y,,z,,t) 1s known where usually z, is equal to - W(X yy), the depth of the water below the (X59Y5) point of installation of the instrument. From either 7 = 7 (X,,¥,,0,t) or P = P(X 6 sV59Z52t)s the problem is to find out what P = P(x - x) ¥Y - Ygs 25 By) 6 : i) =a) (Gs S85 3/ S sean 12) etal (Sane) WS WC = 5.5 = Nig) Bo 8) are like. The problem is not simple. In fact, with the given data, the problem cannot be solved. As a start, though, it is necessary to study what is most accurately known, namely either ) = 7) (X59¥590,t) or P = P(X 59Vo2259t)- The free surface will be used in this part of the discussion although the remarks can be modified so that they apply to the pressure. The question is, "What ways are there to analyze the free surface as a function of time?" Over time intervals of the order of days, 7 = 7) (t), at any fixed point, is not even remotely periodic. The amplitude of 7) may vary from small departures from zero to storm wave heights. The problem, then, is how to analyze 7 under the assump- tion that some property of 7) is preserved for time intervals of the order of twenty minutes or so, with the reservation that - 25 - the situation is still undefined outside of some possibly larger time interval. Consider a wave record, say twenty minutes long. Is it possible to pick some functional representation for7) (t) which will coincide with the wave record for the twenty minutes over which our attention is concentrated? Many functional represent= ations are so obviously inadequate that they will not even be considered, but for other functional representations it is not immediately obvious that they do or do not apply. As a start consider a wave record which is not too irregu- lar.* Such a wave record might appear as sketched in Plate IV. The essential feature of the record for this part of the discussion is that there are groups of high waves and that between the groups of high waves there are time intervals where the amplitude of the disturbance of the free surface is small compared to the ampli- tude near the center of the group. These groups of high waves will simply be referred to as "wave groups." One way to analyze the actual wave record would be by the Significant wave method of analysis as defined by ereranue and ‘tank [1947]. Suppose that the significant height and period are ten feet and eight seconds. Now Sverdrup and ifunk carefully state that the significant wave does not behave like a classical wave, yet in many applications it is tacitly assumed that the free sur- face at a point in relatively deep water can be represented by equation (3.1) of Plate IV where in this case A = 5 and T = 8. *Irregular wave records will be discussed very much later. a eae Types of Periodic Functions and Non Periodic Functions and Their Representation by Discrete and Continuous Spectra Actual wave record n= Malt) a t=0 t=t, t=t> t=ts | | minute | Analysis no. | Purely periodic with one discrete spectral component. At n(t) = A,cos nee =eANICOS it (3.1) | n,(t) 7 n(t+7,) (3.2) By A Portiioni off graphy mori 7) ite 1a Neu 000 Analysis no. 2 Purely periodic with many discrete spectral components. + (3.3) A no(t) = = Ancos(2=mt A py) e = NGOS (rahi S Sp)! ee esi) aca) (3.4) Une 2 cine) en = a (3.5) Portion of graph of 7(t) Analysis no. 3 Not periodic with a continuous spectral representation. Col 1) na(t) = folt) + f,(t-t)) + fo(t-tg) + fa(t-ts) (36) Ga(ne= Lf folt)cosptdu bolu) = tf folt)sinutdy (3.7) folt) = foo(u)cosutdy +f bo(u)sinutdy (3.8) Co(p) = + VGou)® + (Don)*® (3.9) VV pe Graph of 43(t) PLA Je Si This representation implies that the spectrum of the wave ampli- tude is concentrated at one value, T): It also implies that the wave record repeats itself every Ty = 8 seconds and that the wave amplitude is constant and that therefore equation (3.2) holds. Now try to match the graph of 7 ,(%) with the actual wave record. A point for t to be equal to zero can be chosen at a sharply defined crest in the actual wave record. The two records will coincide in apparent phase near t = 0, but they will soon get out of phase. In addition, the heights of the two wave re- cords will rarely coincide. Five sixths of the time the actual wave heights will be lower than the heights in the function which is supposed to represent the wave record. One property which the function which is to represent the actual wave record should have is that that function should re- present the potential energy of the sea surface averaged over time at the point of observation. Thus equation (3.10) should hold where T represents a time interval which is long compared to the length of a wave group but short compared to the rate at which the features of the wave record change (say, twenty minutes). {eh eel == ( 4 (t))Pdt (3.10) ie ial Si a ie Obviously this particular method of representing the sea surface is an overestimate of the potential energy of the sea surface. In addition, many different actual wave records could have the Same significant height and period and completely different values for the potential energy. = BG It is possible to assign a value to A in equation (3.1) so that equation (3.10) will hold. The value of A would not be one half of the significant height. If this were done, equation (3.1) would still not be a good representation for the actual wave record for reasons which will become apparent later. In summary, if the actual wave record is represented by a purely periodic function with one discrete spectral component, there are only two parameters which can be chosen. ‘These two parameters do not adequately describe the actual wave record as a function of time. Many discrete periods A second way to analyze the actual wave record would be to pick out a well defined wave group (if there is one) in the record and assume that that wave group repeated itself every T seconds exactly. Here T is the time interval separating either the re- lative low wave height areas or the relative maxima from wave group to wave group. By a proper choice of the origin of the time axis, and by the assumption that the wave group is repeated per- iodically, it is then possible to analyze that one wave group by a Fourier series. The wave record will then be given by equation (3.3). It must repeat itself every T seconds. The discrete spectral wave periods which determine those component waves which vary sinusoidally are determined by dividing the period of repi- tition by the integers. Suppose such an analysis were carried out on the records shown by some computational method such as the one given by Conrad [1946]. The record would be multiplied by cos 27t/z . The area = 29) = under the curve would be computed with proper regard for positive and negative areas. A similar computation with sin 2rt/T would pe carried out. Then by proper correction factors and by ele- mentary computations, the amplitude and phase of the first har- monic could be found. If this were done for an actual wave record, or for the one sketched, the amplitude of the first harmonic would undoubtedly come out to be negligible. Infact if tT were, say, one hundred seconds, in most records, the amplitude of the harmonic components would not become appreciable until n were equal to five or six. It would become a maximum with n about twelve (if the signifi- cant period was near eight seconds) and die out again as n became higher than 25 or 30. Such a computation would’ be extremely tedious. But it would emphasize the fact that the areas of low wave height are essent- ially caused by the phase cancellation of a great many sinusoidal waves of low amplitude and the fact that the areas of high wave height are essentially caused by the phase reinforcement of the Same sinusoidal waves of low amplitude. The representation thus obtained would be a true representa- tion of the one wave group studied. However, if the representation for the actual wave record were compared to the actual wave record, it would only match up for the one wave group chosen. It would not match the followinz or preceding wave groups because they are not exact duplicates of the chosen wave group. The other wave groups would vary in amplitude and phase, they would not occur at regularly spaced time intervals, and they might possibly have a different apparent overiod and/or frequency spectrum. Also = 30) there might be long stretches of the original record which do not show any groups. If equation (3.10) is applied to7 (t) instead of 17 ,(t), there is a much better chance that the potential energy of the representative wave record will be apvroximately equal to the potential energy of the actual wave record. However, the wave grouo chosen and the time interval,T, might not be representative of the entire wave record. There are several other, not so important, ways in which the actual wave record could be analyzed which would yield a discrete spectrum. For example, it could be assumed that a ten or twenty minute length of record repeats itself periodically every ten or twenty minutes. Such an analysis would be carried out along the lines of the one described above. The harmonics would not become appreciable until n was of the order of forty- five or fifty. The analysis would be even more tedious than the one described above, and the results would not be too amen- able to theoretical work. The portion of the wave record studied would be repeated exactly, but the record and its representation would not agree outside of the time interval studied. It could also conceivably happen that a wave record was com- posed of discrete spectral components which were irrational. For example, 7 (t) = cos 2rt//2 + sin 27t//3 is not periodic. There is no time interval, T , such that 7 (t) = (t+T). Sucha representation for the free surface would be called an almost periodic function. For additional theoretical considerations, reference is made to the book by Bohr [1947] on the subject. = Ail = For additional comment on this point, see Chapter Seven. Continuous spectrum A third possible way to analyze the actual wave record would be to pick out the well defined wave groups and analyze them by means of the Fourier Integral Theorem. For instance, the wave group centered at t = 0 could be defined to be identi- cally zero beyond the arrows which bracket it. The function f(t) could be given by the wave group between the arrows and by the zero outside of the arrows. Then equation (3.7) could be applied to the function and finally, f(t) could be represent- ed by equation (3.8). For conditions on f(t) and for defini- tions of the symbols used, see Sommerfeld [1949]. Similar analyses of f,(t - t,), fo(t - to), and f(t ~ t3) in (3.6) could be carried out. Hach analysis would yield a con- tinuous spectrum of wave frequencies given by the appropriate form of equation (3.7) and the relative importance of various parts of the spectrum would be given by equations of the form of (3.9). There is no known precise procedure with which one could start with the wave record and find the appropriate a; Cu ) and b,(# ), but such a procedure is theoretically possible. Finally, 1 3(t) can represent the wave record exactly over any length of time chosen for analysis. If eguation (3.10) were applied to the actual wave record, the two sides of the equation would be exactly equal. Thus this method of analysis represents exactly the potential energy aver- aged over the wave record as a function of time at a fixed point. Such an analysis would make it possible to represent a wave 8D e record as a function of time as observed at a fixed point. How- ever, it tells us very little about what to expect for times outside of the interval in which the analysis was performed. There is also the difficulty that the wave groups as defined above do not seem to be really persistent phenomenon, that is, there is no mean time, T , which separates the wave groups. Time series analysis A fourth method of analysis, which has not been illustrated in Plate IV, can be carried out with the aid of the more recent concepts of time series analysis. These concepts will be dis=- cussed in Chapter 7 where it will be shown that they form the most realistic method of wave analysis. In this method of analy- sis, equation (3.10), for example, has a most convenient inter- pretation. The problem of variable direction When 7 (t) has been represented as a function of time by one of the methods described above, the problem of representing the short crested appearance of the sea surface is still unsolved, and the representation of 7 (t) is not enough to yield the com- plete solution. More information is needed to solve the short crested wave problem. The exact information needed will be de= scribed from Chapter 8 onward in this paper. Ge doleue mand Ppl ano ib ac ment aphueesehapters The first three methods of analysis are all inadequate. Various simple models which have the properties of the second and third models and which have infinitely long crests will be con- sidered mathematically in Chapters 4, 5, and 6. When these models - 33 - are compared to the realistic models which will be obtained from time series analysis, the reasons for their inadequacy will be- come evident. It will also be evident that the analysis based on method number one is completely inadequate. a RYE Chapter 4. THE PROPAGATION OF A FINITE WAVE GROUP IN INFINITELY DEEP WATER Introduction Some interesting results can be obtained by the application of Fourier Integral Theory to the problem of a finite wave group. In this chapter, a special wave group will be studied in order to show some of the properties of dispersion in infinitely deep water. The particular form of the wave group studied in this chapter will be employed in studies of various not-too-realistic models of the sea surface. This particular form of the wave group is too specific for reality, but if it is imagined that the steps taken with reference to the specific modulation envelope employ=- ed are taken with reference to arbitrary forms for the envelope, then it is possible to see how some of the properties of ocean waves can be studied. There are a great many possible forms for the finite wave groups discussed in the previous chapter. A very special one will be picked for this chapter. Formulation Suppose then that the origin of the space coordinate system is located at the point x = 0, y = 0, z = O on the surface of an ocean of very great depth. At this point the height of the free surface as a function of time is measured and it is found that the equation for the observed free surface is given by equa- tion (4.1) of Plate V. Equation (4.1) has three parameters. The parameter, 4A, determines the amplitude of the disturbance which is greatest - 35 - The Propogation of a Finite Wave Group in Infinitely Deep Wafer. ye (0 ) ere te sin ant +00 242 b(4) = =f eT sin = sin wt dt +00 2H “gia arom (-cos(2% +)t + cos( - u)t) dt -(u + 210/T -(u- 27/7)? -VTA (-@ lu sem/e)” +e 40 ) 2710 - (a+ 2m/t)* n(0,0,1) “=| -e 4q2 sin alt da oc 2m/t)* +A fe 4 sin at du Virb = -(a + 271/T)2 mest) aa 4G sin (FESS & ea) dea ° eo _(a- 271/T)* ee ee 4G 2 sinj( 2 at) da co =- CO ° [ tunee = il f(s) Gai = i f(-u)du VTA 19 = (Ce erems 1)* 2 n (x,t) = OTS =€ 40° sin (Em lin Bist NO AVT -(AL- 2TT/T) se e4 : 2 p(x,z,t) = AS Sen mG Lao sin(4*- ut)du - gpz PAL Vble AVE, (4.1) (4.2) (4.3) (4.4) (4.5) (4.6) (4.7) near t = 0. The parameter, o0 , determines the rate at which the probability curve envelope dies out from t = O as t becomes large either positively or negatively. The parameter, T, deter- mines the period of the sinusoidal term which is modulated by the probability curve. For example if a0 = 1/30 sec7-, A = 3 meters, and t = 10 seconds, in three cycles of the sinusoidal term the amplitude of the disturbance would die down from a peak near 3 meters to a value of about 1.1 meters. In six cycles the amplitude would be .4 meters; in nine cycles it would be 9 db5) meters; and in twelve cycles it would be .055 meters. Thus for these values of the parameters, the disturbance would essentially pass completely in two hundred and forty seconds (four minutes). For this reason, the wave group will be referred to as a finite wave group because it lasts for only a finite length of time at the origin. Two hours later at the point x = 0, y = 0, z = 0, the sea surface is essentially undisturbed. It would be nice to know where the disturbance is at two hours after the time t = 0, and what disturbance of the free surface it is causing wherever it is. It would also be nice to know what pressure disturbance at depths below the free surface is being caused by the passage of the wave group overhead. Method of solution The first step in solving the problem is to find the contin- uous Fourier spectrum of the function given by equation Cami The expression,7) (0,0,t),is an odd function, that is, 7 (0,0,t) equals =m) (0,0,-t), and so only b(y ) must be found as given by =- 37 = equation (3.7). The expression, b(y ),is given in equation (4.2) for the particular problem under study. The integral is evaluated in Bierens de Haan [1867]. It follows that equation (4.3) is just another way to write equation (4.1), and if it were integrated (4.1) would be obtained. As written, equation (4.3) is more informative than equation (4.1) because it is an integral which contains a term which varies sinu- soidally as a function of time, and, from Chapter 2, a great deal is known about how such waves travel. Neither equation (4.1) nor equation (4.3) gives sufficient information to determine the solution of the problem completely. There are many disturbances of the free surface which could have produced the observed variation in time at the point of observa- tion. The various spectral components which combine at the point x = O and y = O to produce the disturbance might have come from many different directions. It will be assumed that most of the disturbance came from the negative x direction and is traveling in the positive x direction. Thus variation in y does not occur and 7 will be a function of x and t alone. This assumption is definitely an approximation to what occurs in nature because it implies that the crests of the disturbance are infinitely long in the y direction. The first term in equation (4.3) contributed only a very small amount to the total integral because, with, positive, the magnitude of the exponential term is small to start with and be- comes smaller as php increases. Let these spectral components travel in the negative x direction. The second term in equation (4.3) contributes a major part - 36 - to the integral because, for « = 27/T, the exponential term is unity. Let these spectral components travel in the positive x direction. Under these conditions, the variation of 7 with x can be expressed, and equation (4.4) follows from equation (4.3). Equation (4.4) reduces to equation (4.3) if x is set equal to zero. In addition, a correct spectral wave length has been assigned to each spectral frequency, - Equation (2.25) applies where 6 is equal to -7r/2 and 9 is zero. In the first term of equation (4.4), the limits of integration can be changed from zero through infinity to minus infinity through zero by the relations given in equation (4.5), and finally 7 (x,t) can be expressed by equation (4.6). Again equation (4.6) reduces to equation (4.3) if x is set equal to zero. If equation (4.6) were integrated at this stage of the deri- vation, an expression for the free surface as a function of x and t would be obtained. It is better to delay the integration and consider the possibility of obtaining some information about the pressure at the depth z below the free surface. The pressure can be found immediately from consideration of Somerset (oo) 4 (2665 C2875 (262), (25225 eucl (2526))5 eon equation (2.9), the value of © is known for z = 0. From equation (2.6), 6 and ®, as a function of z follow, and substitution of @, as a function of x, z, and t into equation (2.7) gives the pressure. Equations (2.9), (2.6), and (2.7) are perfectly general. In particular, if equation (2.25) is the free surface, then equa- tion (2.24) must be the potential function, and equation (2.26) must represent the pressure. Integration over the parameter, p , does not affect these relationships and the pressure as a function * Within the linear approximation. - 39 - of time, distance, and depth is given by equation (4.7). The pressure given by equation (4.7) is evidently a rather complicated function. It is complicated because infinitely deep water is a dispersive medium. The various spectral components of the pressure are attenuated at different rates with depth and the various spectral wave components travel at different speeds along the surface. It is therefore to be expected that the shape of the wave profile as a function of time at different values of x will not be the same as the shape of the wave profile at x equal to zero and that the apparent period of the pressure vari- ation at a depth zg below the surface will not be the same as the apparent period of the disturbance at the surface. Solution The next step is to integrate equation (4.7). The value of the integral is given in Table 269 on page 375 of the table of definite integrals compiled by Bierens de Haan [1867]. After © some algebraic manipulations the result can be put into the form of equation (4.8) of Plate VI. The free surface can be found from equation (4.8), with the use of equation (2.9), by substituting p = 0 on the left, z = 0 into the first term on the right, and z =7 into the second term on the right. Equation (4.9) then gives the free surface. The pressure as a function of time and depth at the point x = 0, is also of interest because the expression is simpler. By substituting x = 0 into equation (4.8) and clearing fractions, the pressure can be found below the original point of observa- tion. The pressure is given by equation (4.10). The derivation given in Plate V and the results obtained SPA (Ol B) (64) (8b) Zo 6 - (zy2r -1)8 X22 | uD} + Ime | UW le 6 5 (z oe -t)t (3 ray ll) Z96 — || us» ——2 - (1'20)d 3 Jby- 6 ape — | ° Tore” = 6 6 z 2 6 2 gX,oar +! wo (QF eooI +1) X22” , U0} T 6 1 216 uIS : = 3 (Vx)yu Ketyoe }22 X2ae ui +1) OV FEC = (Ss 222291 6 ( 26 + = 1s) - 1) rigasa +t Zz>08 '! 6 o/) (BE ZUOU BRO ( tot )d uIS « yz ‘x 16 6 eo ee 216 6 db 2 Tgenze * XT, 50 «jae ~ Bae (qpr¥2z 28 * DL ={))) S V ae v = Bev 126 a [ (e-*)+ (RMI — TP =] as (UOIIN]OS) dnosy BADM 94lul 4 in Plate VI could just as easily have been carried out if the sin 2rt/T in equation (4.1) had been replaced by the cos 2rt/T. All that is needed is a few changes in sign in appropriate parts of the derivation. An arbitrary phase lag, 5, can be inserted into the sinusoidal term of equations (4.8), (4.9) and (4.10) and the equations will still be valid. Evaluation Now that the solutions have been obtained, some graphs and tables will be presented in order to show how the functions vary, why they vary the way they do, and what physically significant conclusions can be drawn from the data assembled. When the para- meters of the solution are varied, the behavior of the solution varies markedly. The behavior of the solution depends most strongly on the parameter, o , which, in equation (4.1) determines the rate at which the envelope of the sinusoidal term goes to zero. For large values of o the duration in time of the original disturb- ance is short. For small values of o0 the duration of the ori- ginal disturbance is long. Spectrum Thus o is an interesting parameter to trace through the re- maining equations. Consider, for example, the effect of o in equation (4.6) in which it determines the nature of the continuous spectrum. The amplitude of the continuous spectrum is given by e( 4) as shown in figure 1 where the minus sign is omitted by virtue of equation (3.9). The graph of the spectrum is a probability curve with a SAD 30 25 20 C (4) oa @ = 1/100 O@ = 1/100 A _ (a 2T/T)* = 4G2 C(u)= Sao e Cc G@ =1/50 G@ = 1/50 Ge 1/30 = 1/30 I. = 1/20 i, Ge: 1/20 AK AR 0.6 0.7 0.8 T=10 sec. an mae 1.4 1.6 AL Fig |. GRAPHS OF THE CONTINUOUS SPECTRA OF FINITE WAVE GROUPS FOR VARIOUS VALUES OF © AND T. a3 maximum amplitude given by A/2/ra when p = 27/T. The larger the value of o the more slowly the probability curve dies down to zero as varies and the lower the peak amplitude. Aso approaches zero, c(p) approaches an infinitely high spike at the point p = 2r/f. Note that as o approaches zero, equation (4.1) approaches 7 (t) = A sin 2rt/T, that equation (4.8) approaches the correct expression for the pressure which would be caused by a purely sinusoidal wave, and that equation (4.9) approaches 7 (x,t) = A sin(27x/L - 2rt/T) where L is the appro- priate wave length for a wave with a period, T, in deep water. The continuous spectrum of the disturbances is graphed in figure 1 for A equal to unity, 7 = 1/20, 1/30, 1/50, and 1/100 sec and T = 5 and 10 sec. The shorter the duration of the wave group, the wider the spread of the wave spectrum, and it should be expected that the more rapidly the shape of the disturbance will change as it travels onward. In the formulation of the problem, part of the spectrum of the wave group at x = O was made to travel in the negative x direction. As a result an integral form of the solution was ob- tained which could be evaluated in closed form. If this had not been done at that time, the solution could only have been obtained in series form and it would have been more difficult to interpret and evaluate. Figure 1 shows that the contribution to the total spectrum of these components is indeed so small that it does not show on the graphs for the values of the para- meters employed. It can be concluded that the effect of this tail of the probability curve which exists for negative values of # will not affect the properties of the solution very much. Bey Properties of the Solution 16 ao tx2 definition D= |+ 5 2 S ( )= —sin 4 12x fet emt 40% |y -l\4ax 4 ee Dt ot Dg see eins peemiier 4iattix wu DT Dg _2rltt+T — 4c (t+ TH? x, Dia gD 40% T*)? (22 Bat x * BT og dan © FR, | ae g my g l Lf f on (THe T gw7D =) ClO MES a= Cath TX “2agD ies ats 4 4+ T 2 8 4, 72 Sel ha. ott'x, fi o*XyT Eee ted 4a! xT g7D g7D g7D 2 Bye mitmils hous Oma tay Srey, contaqena. wait DT gD fe 20 et ____ f Aig Aitix Ae mle XyliG He (igs GoAtxd gmD g7D g7D Te u 4a*t'xT ) 420 g7D (Ge (N11) (4.12) (4.13) (4.14) (4.15) (4.16) (a5 7) (4.18) (4.19) (4.20) (4.21) (4.22) (4.23) The assumption made in the formulation essentially noes the form of 1 (0,t) which can be arbitrarily given in such a prob- lem. The possibility of very low waves traveling in the opposite direction out of the group is indeed a very small price to pay for a closed complete easily evaluated solution. Envelope The free surface given by equation (4.9) is a product of a slowly varying term which determines the envelope of the disturb- ance times a term which is the sine of a complicated function of x and t and which varies rapidly as a function of x and t in order to give the individual waves in the wave group. Consider, first, the envelope of the free surface given by E(9 ) in equation (4,12) where D is defined in equation (4.11). The minus sign can be considered to be part of the phase of the sinusoidal tern. At x = 0, the envelope becomes Ae~ ote Substitute some constant value for x into the equation for the envelope, say x = xy and keep that value. As the time varies, what happens to the amplitude of the disturbance? The disturbance is greatest when t = 4nrx,/gt which shows that the envelope moves in the positive x direction with the group velocity of waves of the period T. The maximum value of the envelope is given by a/(D) 1/4 and so the greatest value of the amplitude of the dis- turbance decreases as the wave group travels in the positive x direction. Letet = 4rx,/er Gia te + t' (Equation (4.13)), so that attention can be concentrated on the times near the time when the wave group passes the point Xz° The exponent of e in the ieee equation for the envelope takes the form -( o7/D)(t)* (equation (4.14)). This shows that it takes longer for the envelope to decrease to 1/e of its maximum value at x = Xy than it does at x = 0. The behavior of the envelope as a function of time at a fixed point is shown by Tables 1 through 4 for the same values of o and T which were employed in graphing figure 1 and for G =) 1720, La= 20)sec. ‘Table shows) the appropriate values for o equal to 1/100 sec™+ and T equal to either 5 seconds or 10 seconds. At x equal to zero and t equal to zero the amplitude of the envelope is one. Thirty-two and four tenths seconds be-=- fore or after t equal to zero the amplitude of the envelope at X equal to zero is nine tenths. On hundred fifty-two seconds before or after t equal to zero the amplitude of the envelope is one tenth. The highest part of the wave group passes the point x equal to zero in three hundred and four seconds (5.07 min). Of course, the wave group never completely passes a given point. For example, it takes five hundred twenty-two seconds (or 8.7 minutes) for the .O1 values of the envelope to pass. When the maximum amplitude of the wave group reaches the point x equal to 17.7 km, the maximum possible value of the enve- lope is 0.90. The maximum amplitude of the wave group passes that point x = 1.77 km at the time indicated by too which in this case is given by 4,560 seconds (or 1.27 hours) if the period of the waves under the envelope is 5 seconds and by 2280 seconds (or -635 hours) if the period of the waves under the envelope is 10 seconds. This shows that the envelope of the 10 second waves travels twice as fast as the envelove of the five second waves. SAG) 3 “ONT BA peZeoTpUT 944 *sanoy UT UeATS 54 *spuooes OL = LZ a0g 54 jo onTeaA su} SeATZ OM4 uum[OD9 °*spuooses G = | togJ 4 Jo onTeA 9y4 SOATS suo UUNMTOD °*x g4utod 9 ay, sessed edoTeaue eyy Jo en~[eA unuTxew oy YOTYM Ye owt eUa ST 4 SIOYeWOTTY UT uoTZeadesgo jo yutod eyq st Ty *oaoqe peqeTngey 4% 19 + 99 = 9 OSTST 0 2°48 (dT cSve LeOr- =csive: © etc Olay. rae) 6v0c S9TT 404 O Bee Wei che OOeT STZ OcS SEE 9) 62S = OabE O°+eL T6L 906 €8€ 992 06T 0 TG°E G4°9 8°66 Coan acute Woce alec L4T 6TT ) Ciacecomy, 0°gé Ocry 8c 98cec 93gT vST OeT 08 0 OG ae stage 9°fr 42E See vgt SST OtT Moye oe) 472 e9) 0 SG a gee 9°62 Zoe ene eeetsik Gite CLs OSS = Wa Ore) 7 ay 0 CEG% a 7igeit GGL Glee acGm GclarOtl mgvaG) wcucG) = O7lA 0.09) scway, vocca © 0 ©) 0 Tose sO! clO. cm Bys0 Ga sOS0n ec Om = STZ? S6-Or vOT 1°49 0 TEES CC Vis 9 Omar, CYA — GAG (OAKS 0 890° Gel aa Gc. mola, GagS cGy 6 °Se. Cec @) 6¥0° (HSO? — GES -PECE LoGt 946 GOs Geecm OGL 0 Tee AYO? eee hoe BOC)KS AOS OPC eke CCE CAL 0 (5110)? GIO? —AGYHO)2 ~ Berit VeOc 20-Gc Beccc: OF CIe CG leer cb asa9 0) Zt0° GeO" 1S0- mies GG 2G LEG GONE = CWE OCie PG GPO, © @) 0 0 0 0 G0 v0 G0 90 LO BO BO OT~*~SS*S (ai Ofa hm 5 = aj OOS Of & ah joe [OR SA, PG S apn ee 28S O¢/T = 9 Joy outy Jo uoTYOUNZ e se squtod pexTs T 4e@ eoejins selj 949 jo edoTeause 244 jo senTe, vy STOPL =- 5] - Put another way, for the same value of o , the amplitude of the envelope decreases twice as fast for 10-second waves as at does for 5-second waves. Forty-two and four tenths seconds be- fore and after the time, t_, the amplitude of the envelope is 0.80. One hundred eighty-three seconds before and after t. the ampitude is 0.10. The above examples show that this method of computation breaks down the dependence on time of the amplitude of the enve- lope into two different parts. The time is given by t = t, ti ie The part, tos evaluates the gross effect of the speed of travel and it is of the order of magnitude of hours in the computations. t, depends only .on the value of Xy considered and the period of the waves under the envelope. The part t', evaluates the time it actually takes the wave group to pass a given point, and it is measured in seconds. t' depends only on the value of X, con- sidered and the value ofo. ‘Computation of the actual time t would require prohibitive accuracy in order to compute the time of passage of the wave group at large values of xy because t! is essentially the difference between two large numbers. Now compare Table 4 with Table 1. In Table 4, 0 equals 1/20 sec7+, The maximum amplitude of the envelope dies down much more rapidly. In fact, Xj need be only four hundredths of the distance given in Table 1 for the amplitude to decrease a cor= responding amount. In Table 1, the envelope could travel 17.7 km before the amplitude would decrease to nine tenths of its original value. In Table 4, the envelope would travel only .71 km and then its amplitude would decrease to nine tenths of its pe original value. Thus the larger the value of ao , the wider the spectrum of the disturbance, and the more rapidly the disturb- ance dies down in amplitude as it travels along. The time required for the wave group to pass the point of interest is simply two tenths of the time required for the wave group to pass the corresponding point of interest in Table l. Thus, the Modi eatioe of the wave group with a large value of 0 occurs much more rapidly than it does for a small value ofc . In summary, the envelope travels in the positive x direction with a speed determined by the group velocity of waves with a period T. The larger the value of T, the more rapidly the group travels in the x direction. Its maximum amplitude decreases as it travels along, and it spreads out over the sea surface more and more the further it gets away from the origin. The larger the value of T and the larger the value of o , the more rapidly the group disperses in time. Apparent local period The rapidly varying sinusoidal term which determines the nature of the waves as modulated by the envelope can now be con- sidered, The sinusoidal term is given by S(7m ) in equation (4.15). The term varies between plus one and minus one as x ane t are varied and it is defined everywhere in the x,t plane. It is not periodic in t since there exists no constant T such that S(ym )[t] = S(m)[t + 7]. In addition, the sinusoidal term is not periodic in x. In fact, the entire solution is not periodic. Consider the term in the brackets in equation (4.15) for a fixed positive value of Xy° As a function of t, it is a parabola ee Fig 2 SOnmim Of . ihe Graph Of “thie “Argument Of ~ Si): 7, cond «are derined’ at (P,. lehe = *higtire” > tsi? note to”! s ciate. 54 which achieves some maximum value when t is negative at the point t = tir as graphed in figure 2. For t>ty an increase in t results in a decrease of the argument of the sine curve. The minus sign at the front of the expression can be put in- side by adding wm to the term in brackets. Since the original problem was, in a sense, an initial value problem, the main point of interest will be in the behavior of S57) for t>O>t,. For this reason, consider the point P,. The terms which are constant for constant x, can be ignored and equation (4.16) can be written. Then if t is increased by the amount , Bas the new constant value will be equal to the old constant value minus 27, and S(7 ) will have the same value as before. Equation (4.18) can then be obtained by subtracting equation (4.17) from equation (4.16). Equation (4.18) trans- forms easily into equation (4.19) with the use of equation (4.13). Finally equation (4.20) can be obtained if equation (4.19) is solved for aes and the reciprocal of the solution is taken. By an exactly similar procedure, T.* can be found from equation (4.16) and equation (4.21). ly? is given by eouation (Aho22))6 The only difference between equation (4.22) and (4.20) is that the second term under the radical is negative in equation (4.22). Thus for certain values of t' near t' = - t. + t,,, the M value of the term under the radical is negative and T> dis imaginary. Such a value of t' (or t) is shown at the point P, * in figure 2. An increase in the value of t by T) results in a decrease in the value of the argument by 217, but there is no S55 = possible way to decrease the value of t by B5* and cause an in- erease in the value of the argument by 27. This is the reason why T,* is imaginary for certain values of t'. The derivation given above applies only to values of t>ty (or t'>- Be + tig) and X,> 0. For the other three possible com-= binations of inequalities, similar derivations can be carried out. One of the two quantities T,* or T5*, will always exist for the entire range of applicability. The other will be imagi- nary only over a very short range. Although S(7) ) is not periodic, it now becomes convenient to define a term which is somewhat analogous to the period of a periodic function. This term will be denoted by T* and it will be defined to be the average value of T,* and T5*e wee be called the apparent local period of S(7 ). It has been shown that the maximum value of the envelope occurs for t' = 0, and therefore T* is most important near t' = 0. For the values ofa , xy and T employed in Tables 1 through 4, 8 ot x40°/enD is always less than 1077, and it can be shown from the expansion of the radicals in the expressions for t* and T5* that T* depends essentially only on the square of this term as a slight correction factor. Therefore T* can be given by equation (4.23) to four significant figures in the neighbor- nood, of t' = 0% The apparent local periods which correspond to the times and distances given in Tables 1 through 4 are given in Tables 5 through 13. The apparent local periods are given to three sig- nificant places. Table 5 can be interpreted as follows with S155 the use of Table 1. In Table 1, after the envelope has traveled 29.6 kilometers in 1.87 hours, the envelope is 0.5 units high, 107 seconds before its maximum value of 0.8 and 107 seconds after its maximum value. Then in Table 5 at the time t = Be - 107 seconds, the apparent local period is 5.04 seconds and at the time t= t. + 107 seconds, the apparent local period is 4.96 seconds. Thus the first waves to arrive at the point of observation have the longest apparent local period. Tables 5 through 13 combined with equation (4.20) show that the larger the values of o and T, the more rapidly the value of T* departs from T as the wave group travels away from x = 0. In Table 8, for example, after the group has traveled only 10.9 kilometers, the apparent local period for the waves which arrive first is 5.42 seconds and for those which arrive last, 4.64 seconds. In Table 5, after the group has traveled 612 kilometers, the apparent local period is 5.08 for the waves which arrive first and 4.92 seconds for those which arrive last. The dispersive effects of the various spectra graphed in figure 1 are evident. The overall variation of equation (4.9) at a fixed point as a function of time can now be described. A wave height re- corder at some distance xy from the origin would not detect any Significant variations in height until a time corresponding to t. had elapsed. At a time in seconds before and after t. cor- responding to t' waves of the amplitude given in the tables with an apparent local veriod given in the tables would be recorded. For times much greater than too the sea surface would be essentially - 57 - *sATZeSOU ST ,9 USYUM SaeTTdde yj] Jo on~Tea ysesreT ouy *aaoge peqeTngey 1 *T eTaqey ees pefoTdwe stoqmuAs aud JO UOTYTUTJep oy pue *,9 Jo sanz~tea eyetadordde eyy sz0g 00°S 26° + g0°s 26 °v g0°s 26° go°s 06° TEES 8°Y T°S 06°¢ OT°S T6°v 60°S €6°r 40°S 00°S EXO Gow, 082 G6°v 96° 90°S +0°S Z6°b v6°r g0°S 90°S TO? 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TELE 70°96 PAA EL° 94 6L°tT cee Ly Gt°rvt 00° €e€ 02 » 40? 02 26 °ST 06°92 02°ST co°’6c Le°st 46°92 CL°vT Serve 60°ET Le°er €v° vt 6S°2e 94° +vT 00°TE 89°ST 65°42 02 T°O "aaoge pagetnge) * J *y eTGe] ses sAoqe poefoTdue stToqudAs eyud jo UOTATUTIJep ey pue ,4 Jo soenTea oxZetadoadde 9yq azo¥g 02 69°TT 66°v2 vv°9L 95°Sz2 veo°ST 60°62 Sik vil TO* ve [EBL 86°82 0S°ST Liege 8c°9T c6°Se Oz esl) go°oe €L°€z Oe LOnSr= 6662S GLeGr GGerve O¢-9e eOSre C62 Gir CGO Giedair 48°9e O€°Se O06°E? 8O0°9T O9°9T e2e°dT 9v°9e2 ST°Se 90°r2 Co°9r vEecie 19°C Eom vee OO°re= TE 02 Oc 02 v°O S°0 (Ye 70°QT ~y°ce E9SLiL €o°te 68°27 89°ce 02 9°0O ante, sedoTeauyq 02 Oe°sTt Oc*ce 6e°8T 90° ed O¢ L°0 8°O —_—_— — — O¢ Oc Oc JO (O10 GL°T Stv° v6T° wie? 890* 640" 1he(0)2 610° say O02 ‘spuooes Og = 1 pue OC/T = 9 JOS y ATAPI Aq Usat3 squtod pexTy 1e eoeyuns eezy 94g Jo potaed TeooT yuereddy €T STdeL 4 = "66 = undisturbed again. Figures 3 and 4 are a group of graphs which show the wave records which would be observed at various fixed Xj» as a function of time. The graphs have been obtained by con- sidering Tables 4, 12, and 13 and hence values of 9 equal to 1/20 sect and of T equal to 10 and 20 seconds. The phase of the wave crests has been chosen to go through zero at t, =0O. A slight variation in x' and t, within the accuracy of the last significant figure given for them would make this possible. The wave crests have been sketched in from the data given in these tables. It is simply too long and difficult a procedure to evalu- ate equation (4.9) by letting t vary through 2 second increments throughout a range of several thousand seconds in order to graph the free surface. Figures 3 and 4 are sufficiently accurate, however, to show the major features in the transformation of the wave group. From equation (4.12), it would also be possible to discuss the shape of the envelope as a function of x for a fixed time. The envelope is not a normal probability curve as a function of x for a fixed t since D varies with x. This aspect of the problem of evaluating the solution has not been investigated in as much detail as the problem of the variation in time at a fixed x. For T= tol sacwands on l= M/dloonsecas » the graphs shown in figure 5 have been obtained. The slight skewness shown by equation (4.12) (which is not so great as one might expect because the D's in the two places where they occur counteract each other) is not evident in the graphs. It might show up for other values of the parameters. - 67 =- He T=10 T=10 10 1.0) o =! é v= Yeo Oo : 20 4 x = O (n=1.0) x=.7lkm (n=.9) 2 6 t=O an t= 025 hr. 2 TialE K-S0SEC>}T-10 o-l4 12 T=10 20 10] 1.0 * x = 174km.(n=.7) x= 20 T 8] 8 ty= -O62hr. 6] .6 x = 1.18km.(n=.8) 4) .4 tg= .O37hr. 2|.2 [=S0SEC=| T= 10 oho Tae x = 3.12km.(n=.6) ty= .O97hr. FIG.3 FORM OF THE FINITE WAVE GROUP AS A FUNCTION OF TIME WHEN IT PASSES VARIOUS POINTS. = !469 T*#10 T=10 ve v=o 50SEC =| I x = 3.99kmin=5) fy tos 135 hr 8/ .8 .6 T=10 Ps T* oo 6 x = 4.96km.(n=.4) A t = 22hr. 2 —~x Qa > oO = 10.9km.(n=.3) [*s° aoa = .388hr. + py 25 7 an T=20 T1469 x=1.18km t= O18 hr. FIG.4 FORM OF THE FINITE WAVE GROUP AS A FUNCTION OF TIME WHEN IT PASSES VARIOUS POINTS. o= Yoo T=20 45t 25) Ve2O o=Voo x =4.96km. thee | tg = Ihr. 15 9 20 Ov O€ ‘sayLLLLe “wy 98622 ‘asOl =1 0080017 = 5 “SaWil GaxI3 YO4 X JO NOILONNS WV SV dNOND 3AVM JLINIS JHL 40 4d013AN3 g9l4 ok Ol 0 Ol- (0) of— Ov- n bad ‘sau lll? = 4 wyH9B6 22 = * em n27tr ont na Apparent local wave length An apparent local wave length can be obtained by an analysis similar to the one carried out for the apparent local period. In the derivation of the apparent local period, no approximations were made until equation (4.23) was obtained. The derivation showed that higher order effects could be neglected, and so it is possible to simplify the derivation of the apparent local wave length on this basis. Equation (4.24) defines the argument of the sinusoidal part of the solution as a function of x and t. If equation (4.24) is partially differentiated with respect to time, and then if finite increments are taken as in equation (4.25), T* can be found im- mediately in the form of equation (4.26). Equation (4.13) would then give equation (4.23) from equation (4.26). Now the derivation of the apparent local wave length can be carried out easily. Equation (4.27) leads to the apparent local wave length as given in equation (4.28). Note that L* is not equal to g(T*)°/2r. Equation (4.29) shows that t can be considered a fixed value and redefined in terms of a fixed x by means of the group velo- city relationship. Also x can be treated as the sum of two terms. The Xy is the large term which determines the location of the maximum amplitude of the wave group, and the x' determines the distance from this maximum. With these equations, an alternate relationship for L* can be given by equation (4.31). Equation (4.31) shows that L* is almost equal to the wave length of a sinusoidal wave of period T in infinitely deep water Big = Properties of the Solution (Continued) 4n*x 2mt_ 4a%t2x Aa Arg S(n) = Jo Stebel gam iil Mat: Mize gt 1\6a04%x2 1+ g eat) a AS 2 of At res * AED) u : 4o*T tx (1 ee 8 f (x,t) Af a +27 @ x i OIE KED © L* Eee oT iS Ac SAAD eR EG Sie Se pataeNes |: T? o2D | gt la g 7 O12 t, = ar 2. te xX = x, + x! 2 gT? (fp ee | L* 27 g*D g2D ee IGoFxxX'\2 gg 4/ 4x’ \2 T2602 ( g*D Be | g 272D of 9 f -2 7 27 df =. —dt —dx = —drt — dx =-0O ; tr ax x = + . Chae c* = i dt 1) Another form of the Cauchy Poisson wave Agee TAS, bs git? . gt t = — =: {cos —— Sin) 7 (x,t) 5 ag oF a is -1 402x | + a tan ogi =ft(xt) (4.24) (4.25) (4.26) (4.27) (4.28) (4.29) (4.30) (4.31) (4.32) (4.33) problem (4.34) (4.35) when x' is zero. The term Q? g °/2r-D is small compared to one for the values of T and o0 employed in the evaluation. For a fixed X,, as x' increases positively, the apparent local wave length increases. As x' decreases the apparent local wave length decreases. Thus for a fixed t as a function of x, the longer waves are in the front of the group. One final point needs to be made. It was shown that a posi- tive increase in t by the amount T* caused a decrease in f(x,t) by the amount 27. In equation (4.27), it was assumed that a positive increase in xX by the amount L* caused a positive in- crease in f(x,t) by the amount 27. Equation (4.31) then gave a positive value for L* over the range of x and t where the maxi- mum value of the envelope occurs. The derivation therefore shows that those wave crests which are under the maximum value of the envelope as it travels along are moving forward in the positive x direction. Apparent local speed The total change in f(x,t) at a wave crest should be zero if the observer moves with the speed of the crest. Equation (4.32) imposes this condition and yields the result that the wave crests advance with an apparent local speed given by C* in equation (4.33). In equation (4.33) it should be understood that L* and T* are given by equation (4.26) and (4.29). For values of x and t which give a maximum value for the envelope, it then follows that the wave crests are moving forward with a speed twice that of the envelope. a) GR} a Summary of wave group behavior In summary, the wave group studied travels forward with the group velocity appropriate to the value of T employed. It dies down in amplitude as it travels along and spreads out over the ocean surface. The individual waves under the envelope form in the rear of the envelope es waves with a short apparent local period, travel through the envelope with a gradually in- creasing apparent local period to a maximum amplitude where they have an apparent local period nearly equal to T, and finally race ahead with a longer and longer apparent local period to disappear at the front of the group. At any instant of time, the longest apparent waves are at the front of the group, if x and t are greater than zero. The study in this section of the behavior of the solution for values of the parameters employed in the tables is now com- pleted. A study of the pressure caused by the surface disturbance will be made in a later chapter. New form of Cauchy-Poisson problem One very special modification of the solution can be found which yields another fascinating form of the Cauchy-Poisson wave problem. If in equation (4.1), sin 27t/T is replaced by cos 2rt/Z, then in equation (4.9), the negative sine term can be replaced by a positive cosine term. Then in these new equations replace A by a modified form given in equation (4.34), where A* is constant. As o approaches infinity, the modified form of equation (4.1) approaches an infinitely hign spike which lasts only for an instant in time. The spectrum given by a(y) is equal to Sane a constant value everywhere and thus degenerates into a white noise spectrum. Then for x not equal to zero, the free surface assumes the form of equation (4.35). In this modified form of the Cauchy-Poisson wave problem, the amplitude of the waves at a fixed point, x does not increase with time and is finite at all x not equal to zero. The disturb- ance, as in one of the previous cases, is caused by an infinitely high, infinitely long, infinitesimally wide column of water at the origin, but in this case it lasts only for an instant of time. Thus there is not enough energy to produce an infinitely high disturbance at points other than x = 0. Physical reality of problem The physical reality of the whole problem discussed in this section should be considered. If such a wave group were gene- rated on the surface of the ocean, would it travel as predicted? It might not because no ocean is infinitely deep, because the low periods associated with high values of » are really capil- lary waves, and because such effects as internal viscosity, and the friction of the atmosphere against the moving waves have been neglected. The wave length of a sinusoidal wave in water of finite depth is less than the values employed here. Since the spectrum of the waves is defined near p = 0 where the period of the waves is infinite, the group will not travel exactly as predicted in water of finite depth. In figure 1, for the values of the parameters employed, it can be seen that that portion of the spectrum which is affected * In the y direction. *tor the same period. =- 75 - by a depth equal to that of the average depth of the ocean is very small. Note that this is not the case in the Cauchy-Poisson problem where the spectrum is a white noise spectrun. The wave length of a sinusoidal wave when surface tension is considered, is greater than the values employed here if the per- iod is very small. For the range of the parameters considered this effect is very small. Again this is not the case in the Cauchy-Poisson problem. The effect of internal viscosity has been shown to be negli- gible by Sverdrup and Munk [1947] for the dominant spectral com- ponents employed although Johnson [1949] has shown that it is important for very short period waves. Internal viscosity would not be important until the group had traveled a distance equi- valent to several times around the earth. * The action of the air against the traveling wave group or of some type of internal eddy viscosity in the motion is possibly an important effect which could modify its actual travel. For the present, there will be no speculation about the modification of the solution obtained in this section by these mechanisms. *See Lamb [1932], sec. 348 equation (9). For the ten second spectral component, traveling with the grouv velocity of a ten second wave, the distance the zroup would travel before dying down to the 1fe of its former height due to molecular viscosity would be of the order of 10° kilometers. - 76 - Chapter 5. THE PROPAGATION OF A FINITE WAVE TRAIN IN INFINITELY DEEP WATER Introduction There is considerable interest in the problem of what hap- pens to a train of waves of constant height, finite duration, and constant apparent local period as it travels along. Sver- drup and Munk [1947] have given a physical argument based upon the fact that the energy of the wave train advances with the group velocity which shows that the major rise of the amplitude advances with the group velocity of the apparent local period and that only very low waves travel out in front of the main group. Such a finite wave train has a continuous Fourier spectrum. In order to determine the effects of dispersion, it is necessary to investigate the problem mathematically with the techniques of the previous problem. Despite Munk's [1947] assertion to the contrary, dealing with the recorded period, "without recourse to the nature of the underlying continuous Fourier spectrum" always tacitly assumes something about the underlying spectrum which may not be theo- Tetically justified. There is considerable confusion in the technical literature about the difference between the apparent local period of the previous section and the period of a periodic function. In addition the use of the formula c = gT/2r in the above reference is not valid because the formula applies only to a purely periodic wave train of one constant period. - 77 = Formulation Suppose that a wave record defined by equation (5.1) is ob- served at the point x = O and y = 0, as a function of time. The sea surface would be perfectly flat for all times before t = -ntT. After that time waves all of the same height with an apparent local period equal numerically to T would be observed until t = nf. There would be 2n complete wave crests. After t = nT the sea sur- face would become and remain flat again. Thus the wave train lasts only for a finite length of time, and it is therefore referred to as a finite wave train. It would be nice to know what the sea sur- face looks like at other places and other times. It would also be nice to know how the pressure varies as a function of depth as the wave train passes overhead. Method of solution The continuous spectrum, b(y), can be found as usual by in- tegrating equation (5.2). The last result in equation (5.2) has the same property that was found in the previous chapter in that for > O the second term dominates the first term. By arguments exactly parallel to the ones in the previous chapter, the equation for the free surface can be written in the form of equation (5.3). For x = 0, equation (5.3) reduces to equa- ealoja\ (Cojpabe It was possible to obtain a representation for the pressure caused by the disturbance in an integral form similar to equation (4.7) of the previous section. However the indicated integration could not be carried out, so, although it would be nice to know something about the pressure caused by the disturbance, that aspect - 78 - The Propagation of a Finite Wave Train in Infinitely Deep Water Sines ea MIS a n(0,0,t)= (n an integer) O , otherwise 6.1) In O(a) = ae n(0,0,t) sin ut dt - AI sin 22 sin ut dt a, —In sin -f aie cos (= fof) cos (22 —#) t\dt ps sin eta sin (= —-)Tn | | (SF +H) a) A sin Tn singe Tn "aro A ec /am0, [ang te 19 edocs Wl (5.2) ¥ | (SF +H) Gao) | +o. 2 3 Co oes pee (5.2) Lass =) ee (5. 4) A Xena X 4mX_4)\| SIN.an (xt) | sin (228s GE + og my] SET da 2 ne A sin (22 ERP cos [8G X40 (am— 1] Sinan gy ; Gil g g a a cos (SS 2xty [sin a + ale — | ee (S75) PHS ANTE SEX of the problem will have to be unknown for the present. The next step is to integrate equation (5.3). A few manipu- lations in the form of trigonometric identities and a transfor- mation of variable make it possible to put the equation into a form where the integral can be evaluated. The transformation of variable given by equation (5.4) and the formula for the sine of the sum of two angles yields equation (5.5). The trigonometric identity for the product of two sinusoidal terms can then be used to obtain equation (5.6). The assumption that n is an integer is used. Consider the first term in equation (5.6) (Plate X). The term under the integral can be expanded by the trigonometric identity for the sine of the sum of two angles and thus this integral can be written as the sum of two integrals. One of these integrals is given by equation (5.7). The integrand is an odd function, and the integral of an odd function from minus infinity to plus infinity is zero. The other term is even and its integral from minus infinity to plus infinity is equal to twice its integral from zero to infinity. The contribution of the first term is there- fore only the first term in equation (5.8). If this operation is carried out on each term in equation (5.6) the corresponding terms result in equation (5.8). The terms in equation (5.8) can be written as a double Fourier Integral by means of an interesting mathematical detour, and the double Fourier Integral can be evaluated by an interchange of the order of integration. The mathematics will be carried out for the first term in equation (5.8). = 0) < The Propagation of a Finite Wave Train in Infinitely Deep Water (Continued) +0 c puAs 42x 2mt a2x 41x a eee qT? a | feos| + (ist mr) +0 . a2x (42% ant da 5 [sin 22% cos| a gar +nT a O oO 2 7 (x,t) - Asin 42x _ 221) cos a58 sin(a(42% +n 1)) 22 g T2 i mn gT 2 yg om 41x A 4 Boone l| Ree 2zt} [coset sin a oan 5 © A nlie—&p| in sten oft oon @ 2 2" 2fcos ¥ a?cos Bada = [cos 8° 4 sin £2) mae ie} g 4x x (Er B’q B*q COs ae =) [t/a eose—— --.Sin—= Cos Biad a a [A 2x | ax =] BeGe @o 2 2 nek wen eale lo) Eg Ce ee sin 7? - {4 = [eos 5 sin 37> cosBadB Plate xX (556) (32) (5.8) (529) (5.10) (Set) The first step is to find the Fourier Spectrum of cos a°x/¢ as a function of a. Equation (5.9) gives this spectrum as a func- tion of the new variable, 8. It then follows that cos a°x/g is given by equation (5.10). Equation (5.11) would be the correspond- ing equation for the sin xa°/g. Note that this step involves the assumption that x is greater than zero. Slight modifications of the analysis from this point on would also yield valid results for x less than zero. In the second expression in equation (5.12), (Plate XI), equa- tion (5.10) has been substituted for cos a-x/g in the integral which occurs in the first term of equation (5.8). In the third expression, the order of integration has been interchanged as in- dicated by the rearrangement of the brackets. The change in the order of integration can be justified théoretically. The term in brackets in the third expression leads to the conclusion that integration of the term, cos 8° e/4x + sin 8° e/Ax, from zero to the indicated variable upper limit is equal to the original integral as in the fourth expression. Finally a change in variable gives the last expression in equation (5.12). The term in brackets in the third expression in equation (5.12) is considered in equation (5.12) and designated with the letter re It can be shown easily that this integral as a function of 8, after integration over a, is either constant or zero, and its values are given below the integral (see Pierce, [1929]). Now that the integral over a has been evaluated, consider the integration over Bp when 47x/gT - t + nT>0. For 8 <4rx/gT - t + nT, the integrand as a function of 8 is equal to cos B°2/4x ne Salil Be /4x = B82 Oo The Propagation of a Finite Wave Train in Infinitely Deep Water (Continued) fo oa (a ( 222 lees +3/| [ [feos 828 +sin B8)cospaae|sin(a(% am -t+7}| da + FB [roe 2 Boia ee A ipselanials (414 ot} a *—tenT}laa clae se |in By + sin p E9)a8 = x: (2x2 ma nt) Teor (cos £08 + sino!) ae (5.12) Cd 41x cosBasin o(428-1407)) = fea rs ali (5.13) fo) Qa 3/0 UIC ines Sy ay © Gre ABS PS Are © gT gt ‘itr, _ 4mx_ 2B if p+ —Si-t+nT) 0 ‘ cule im 4X --Litp= 4k-t+nt co an OK Bk a tein gt 2 eae 47%Xx 2 gv TEEN ON Orage 2 j2mx a4) Plate XT = 932 times a positive constant. For p>4rx/gT - t + nT, the integrand is equal to cos 8-2 /4x EMS 8° ¢/4x times zero, which is zero. But since the integrand is zero beyond this value of Bp, the integral ean be evaluated by integrating cos 8-2 /4x Se Sien 8° 2/4x from zero to 4rx/gT - t + nT as in the fourth expression in equation (5.12). Then the transformation given by equation (5.14) yields the final expression in equation (5.12). The theory of integration also shows that if 4rx/gT - t + nT had been negative, the integrals would still be correct as written. The integral given in the last expression for equation (5.12) is the sum of two known integrals. They are the Fresnel Integrals which are tabulated, for example, by JahnkeHmde [1945]. For any particular value of x, t, n, and T, the upper limit of integration is some number, and the table gives the value of the integral. Solution Each term in equation (5.8) can be treated in the same manner as the first term was treated. The final form of the solution is then given by equation (5.15) where G(x,t,T,n) and H(x,t,T,n) are defined by equations (5.16) and (5.17). These three equations then are the solution, because they can be evaluated for all values of ty n, and T, and for all x greater than zero. In order to show that it is the solution of the problem, equa=- tion (5.15) must reduce to equation (5.1) as x approaches zero through positive values of x. If the upper limits of integration are plus or minus infinity the values of the integrals are Salta rate by equations (5.18) and (5.19). Consider the expression for G in equation (5.16). Pick any =, aes Finite Wave Train (Solution) Flat) = G (x,1,T,n)sin (AEE — az) gt? + H(x,t,Tyn)cos (472 271) (5.15) S al. G = a (5.16) 2 = (am -t+nr] Fa (224 ~+ at Jao] cosy o SF cite calden cos— o2+sin $0? Pan qT SUS nT] FS, (42% -r4nr] H = = ok TT 52 u ST eS ih tone LI cos of£—sin > v2\de (cos 3 o sin Zo?do| (Saliva) () @ ue, Ae initia 2 erties a ad do fois oe do 5 (5.18) — 0 -@ GED, Sats inno Del Joos Do do fon of“dao > (5.19) (e) O limG =-A if -nT¢ t¢nT (5.20) Xx ~ot limG = O ih estenT, equation (5.21) holds. The sine term approaches - sin 2rt/T. The expression for H can be analyzed similarly and H is zero if t is not equal to either nT or -ntT. Therefore equation (5.15) reduces to equation (5.1) when x approaches zero except possibly at two points, namely t = nT and t = - ni. At these two points, the actual behavior of the solution is clarified if, now, after passing to the limiting value of x equal to zero, t is allowed to approach t = nT and t = - nT from both positive and negative values. The free surface approaches the value zero as t approaches nT or -nT from either direction, and therefore (5.15) can be defined to be equal to zero at t = -nT and t = +nT. Therefore (5.15) equals (5.1) everywhere as x ap= proaches zero through positive values of x. There is a reason for the particular care which must be em=- ployed in the study of the solution near these two special points. It is that the slope of the original expression, equation (5.1), is discontinuous at these points. The effect of this discontinuity in slope causes the solution to have a very peculiar appearance as a function of time for values of x near x = 0. Evaluation Now that a solution to the problem has been obtained, the properties of the solution will be discussed and graphed as in the eG. = previous chapter. The nature of the continuous spectrum, the be- havior of the wave train as it travels along, and its variation with n will be described. Spectrum The part of the continuous spectrum which was used in the integral form of the solution is given as b*(a) in equation (5.23). The second form, in terms of a, can be obtained with the use of equation (5.4), if n is an integer. As #! approaches 2r/T or as a@ approaches zero, the spectrum has an indeterminate form, but the application of standard methods shows that the value of b*(a) ap-= proaches ATn/m at this point. Thus for larger values of n, the contribution to the spectrum near values which are equal to values associated with the apparent local period becomes large compared to other values of the spectrum. 4s n approaches infinity, how- ever, the spectrum does not reduce to one infinitely high spike as in the problem in Chapter 4. The actual behavior of the spect- rum is shown in the graphs of b*(a) which are shown in figure 6. If n equals a small value as in the top graph of figure 6, the spectrum is a smooth curve with important contributions for all Spectral values. Such a disturbance of the sea surface would tra- vel only a short distance and rapidly die out. In the bottom graph of figure 6, there are three different scales on the ordinate, (b(a')), and abscissa (a'), axes. The inner scales apply for n = 10. If n is increased by a factor of ten, the ordinate scale is increased by a factor of 10 and the abscissa scale is decreased by a factor of 10 as in the middle set of scales. Thus, the important part of the spectrum is increasingly concentrated near wp = anr/T, - 87 = n= 1000 D(a) 4000 3200 2400 1600 800 -800 -1600 -2400 -3200 -4000 100 b2(a) 400 320 240 160 80 -80 -160 -240 -320 -400 Fig 6 20 ae] ae 32 24 aWaWaw ae .2 ee ote A Ae -24 “32 -40 “1.0 =d} AS ~.4 =e. fe) -2 4 6 -0.10 -.08 -.06 -04 -.02 fe) 02 .04 .06 -0.01 -.008 -.006 -.004 -.02 ie) 002 004 .006 a=0 Graphs of b(a) for various values of n. Co-ordinate different n_ for differ for bottom graph as shown 8 1.0 -08 0.10 008 0.01 scales (a' = 0), and the side spectral components oscillate more and more rapidly and tend to cancel. For large values of n, the same re- marks about the spectral components which travel in the negative x direction are applicable which were made in the comments about the spectrum of this finite wave group. As n approaches infinity, for any finite values of t and x, equation (5.15) reduces ton (x,t) =A sin(4r°x/gT2= 29t/T) which is a simple sine wave traveling toward the right. Only when the wave train is infinitely long and lasts for an infinite length of time is it possible to apply the usual formula for wave speed and wave length without qualification. Also equation (4.9) reduces to the above form aS o approaches zero. An interesting question is, "Why are the two continuous spectra so different in their limiting forms?" The explanation lies in the way that the free surface at x equal to zero approaches its limiting form. The free surface studied in Chapter 4 deforms con- tinuously into its limiting form, namely 7 (0,t) = A sin 2rt/T, as 0 approaches zero. The free surface studied in this chapter does not deform continuously into its limiting form. The sharp discon- tinuity from full amplitude to zero amplitude is always present, and an increase in the value of n just displaces the discontinuity in time. This difference in behavior thus explains the differences in the type of continuous spectra obtained. The solution to the problem studied in this chapter is valid for all integer values of n. For n egual to one, there would be two complete wave crests under the envelope. The evaluation of the solution for other values of x and t would then be somewhat difficult Laos because all terms in equations (5.15), (5.16) and (5.17) would have to be evaluated carefully over small increments in the variable quantities. The disturbance would die down in amplitude and spread out over the surface quite rapidly. In addition, the variation in amplitude and sign of the terms G and H would be as rapid as the variation of the sine and cosine terms so that the simplifying con- cept of a slowly varying envelope and a relatively rapidly varying Sinusoidal term under the envelope would not be applicable. Simplification of results For large values of n the analysis of the solution and the physical interpretation of the results are simpler. One could con- sider the problem to represent a train of waves of constant apparent local period which takes, say, ten hours to pass a given point, x = 0. Then if T equals ten seconds, n equals 1800, and nT equals 18,000. Now consider the integrals in equation (5.18) and (5.19). If the upper limit were replaced by plus or minus ten in these inte- grals, the values of the integrals would still be very close to plus or minus one half. Thus, interest should be concentrated on times and places where the upper limits of integration in equations (5.16) and (5.17) lie between minus ten and plus ten. One way to do this is to study the variation of the solution at a fixed value of x as a function of time. Pick a fixed value of x and call it X, as in Chapter 4. Then the forward edge of the wave train arrives at xy at a time, t, determined by the group velocity of waves with a period equal to the apparent local period of the wave train. When =OOm= m= 4nx,/gl - nT, the upper limit of integration in the second term of (5.16) and the first term of (5.17) is zero. The upper limits of integration of the other two terms is given by (g/anx, )V? 2nT and its value is greater than 10 for all values of Xy less than ten thousand kilometers. Note also that a wave train with 3600 waves of 10 second period in it would be about 600 km long. For a fixed point, Xy> then, if xy is less than approximately ten thousand kilometers, the transformation of variable given by equation (5.24) can be applied where the time of passage of the forward part of the wave train occurs near t' equal to zero. Equation (5.15) then simplifies to equation (5.25). The first integral in equation (5.16) is practically a constant; the second integral in equation (5.17) is practically zero; and some alge- braic manipulations then yield equation (5.25) where G* and H* are defined in (5.26) and (5.27). The value of x, is fixed; equation (525) does not imply that the waves are traveling in the negative x direction. G* and H* are functions of the variable upper limit of inte- gration. For Xy small, the upper limit of integration is greater than ten or less than minus ten after t' has varied through a small range of values. For Xy large, t' must vary through a much larger range of values. G* and H* are graphed in figure 7 for A =1 as a function of (g/arx,)/7t", The appearance of the solution as a function of t' fora fixed Xj depends upon the choice of Xy° For small values of xy» the effect of G* and H* is to put ripples on the first two or three waves in the wave train and to leave the remaining waves o> onl & 7 Finite Wave Train (Properties of the Solution) py(uy = —Asiputn , Asinatn (5.23) a(S —u wie ie Sina = uty tt” argu + = ty (5.24) an % ] ° 42x, Qrt m(xit) = G (n,thn,T) sin (40% + ant) + H*(xitjn,T)eos (427% + 27) (5.25) @ iG jam +’ 4nX, : Ali 4 pee ay ba _- a (costo 2 sin Lotlde (5.27) if x, is large enough. ax) s=na(Gele + oCHie )= sin(2zt 4r? x, Hi) ) = + oT + tan Ge (5.28) Ti 2 E, = 3° (mix,t ) (5.29) t+ 7 Ey +(92 (nonth) arte S2(ort +t Bea t’ Plate XII -92- eecary v ¢ Z l fe) ie ae Qe v- c- | | | | | | | TEX 7 \ [IS L\_L\ fy LAA DL EIP ZS PO LEN [NN VAN ia <> C\ CY OO 5 | yO uolyouny 0 so paydoib yH pud ,95 2 64 oo) -93- unaltered. For slightly larger values. of X49 the effect of G* and H* is to produce strangely distorted wave crests shortly after t' = 0 and crests with many different apparent local periods short- ly before t' = 0. For moderate values of x,, the effect of G* and H* is to cause the individual wave crests to be modulated in ampli- tude and phase and to have an apparent local period equal to T for all values of t'. Finally for values of xX, near ten thousand kilo- meters, G* and H* are no longer appropriate and the original form of the solution must be studied. Figure 8 shows the forward edge of the finite wave train as a function of t' for various values of x, in order to illustrate the above remarks. T is ten seconds, n is 1800, and xX, has been chosen to be 1/2 centimeter, one twelfth of a wave length, five wave lengths and one hundred wave lengths. The time scales for the graphs on the left are different in order to show the fine details, and the times scales for the graphs on the right are the Same in order to show the overall behavior. For Xy equal to 1/2 centimeter, the effect of G* and H* is to put a few high frequency ripples on the very first wave crest as in the graph at the upper left. The graph on the upper right shows that the major portion of the train is essentially unaffected. For x = L/12 and x = 5L a few crests at the forward edge are affected but the major portion of the train is still unaffected. For X = 100L, about ten crests are found of substantial amplitude in advance of t' = 0, and about twenty crests are modulated in amplitude behind t' = 0. The transi- tion from constant amplitude to small amplitude is quite gradual. For values of x, which are large enough, G* and H* are slowly ~- %- _ oos!i =u 4S Ol = Lt ™% JO SANIVA SNOIUVA YOS 450 NOILONNS V SV NIVYL SZAVM JALINIS SHL 30 SHdVYHD ‘89l4 0453S O0¢ 002 ool O T T la [a al WY9°S| = 100] =X WO82=1G=X woooe! - % =x wo = O0ZIE/ uty O=+ O=4, J1VOS SWIL LNVLSNOO J1VOS AWIL GSQNVdxX3I —95- varying functions of time such that they change slowly over several hundred seconds. Under these conditions, 7) (x,,t') can be approxi- mated by equation (2.8). The wave train now has an envelope given by (G*)° + (Ht)? l/2, Also since the most rapidly varying term is the sinusoidal term, the apparent local period is everywhere equal to T. However H* and G* still vary slowly with time so that the wave crests under the envelope will not all be in phase. The eradual phase shift with time is given by tan™~+ H*/G*. It is also possible to compute the potential energy at each point by the use of equation (5.29). The potential energy can be averaged over one cycle; and by suitable approximations the aver- aged potential energy can be given as a slowly varying function of time by (5.30). Figure 9 is a nomogram which permits the determination of the amplitude of the envelope of the wave train as a function of t' for fixed x, (and consequently tee The straight lines of various Slopes in the bottom part of the figure are graphs of t' = (4nx,/e)/?x for various xy where K is the numerical value of the upper limit of integration in equations (5.26) and (5.27). When t' equals 2,000 seconds as in the example, and Xy equals 439 kilometers, K equals 2e7- The envelope is graphed as a function of K in the upper graph. Thus, 2,000 seconds after t. equals 10.63 hours which corresponds to xy equal to 439 km, the envelope is .92 times the amplitude of the waves in the original train. Note that the forward edge of the wave train passes the point x = 0 five hours before t = 0, and the forward edge actually takes 15.63 hours to travel the 439 kilometers. For this particular value of Xy the waves build up from an amplitude - 9 - 40 20 20 40 l(c x)? +(H*)? ] 1/2 Sen carta [(o*)* +(H*)"] 8000 6000 4000 2000 - 2000 - 4000 - 6000 - 8000 Fig,9. Variation of the envelope, the average potential energy, and the phase of the crests for relatively large values of x: as a function of ft -97— of less than one tenth to maximum amplitude in 2300 seconds (38 minutes). This means that after the passage of 230 waves, the train is essentially constant in amplitude. An analysis of equations (5.15), (5.16) and (5.17) would show that the trailing end of the wave train would pass near the time t' = 2nT after the forward end. The procedure employed in the study of the forward end of the wave train could be employed to study the trailing end of the wave train, and similar equations and results could be obtained. Between the time t' equals zero and t' = 2nT, for the example considered above there would always be essentially 3600 wave crests in the wave train. A few extra would be found before t' = O and after t' = 2nT. From K = O to K = 4 in figure 10, for the example considered above when xy equals 439 km there would be only 300 waves which are not quite of constant amplitude. At the trailing end, there would be another 300 waves. Thus only a total of 600 waves out of the 3600 waves, or 16.7%, would be modulated at the ends of the train. Other quantities of interest are also graphed in figure 9. The relative amplitudes of the potential energy at various times is shown by the dashed graph. The gradual phase shift as a position of t' is also shown above the graphs of the envelope and the po- tential energy. In the forward part of the wave train before t' = 0, the phase shifts might cause waves which are not approxi- mately sinusoidal in form. For the relatively large values of xy employed in figure 9, the wave record as a function of t! which would be observed at Xy - 98 - can be graphed by first constructing the envelope and then by draw- ing in the wave crests with appropriate regard to phase. There will be some point X49 at which the approximations em- ployed in equations (5.25), (5.26) and (5.27) begin to fail because the modulation on the rear edge of the wave train will lap over and combine with the modulation of the forward edge of the wave train. Figure 9 shows that the modulation is only important for 6000 sec- onds if xX, = 439 km. Modulation which is effective for more than 18,000 seconds would affect the rear half of the train. Therefore an estimate of nine times 439 km or approximately 4,000 km is better than the previous crude estimate of 10,000 km for the point at which the use of equation (5.15) directly would be required in evaluating the solution. Some objection might be raised on the: physical reality of the problem because of the behavior of the solution near xy equal to zero. The unrealistic behavior is due to the discontinuous char- acter of the functions employed. Wave trains in nature would not be so sharply delineated. To eliminate this objection, just con- sider the wave train as a function of time as it passes the point x, = 5 km. Let this value of x, be the new point of origin of the wave train. Then the new wave train at its starting point would have smoothed ends, and beyond the new reference point for all times the solution would be a well behaved function. Summary The behavior of the finite wave train can now be Summarized. If the wave train takes a given number of hours to pass a given point, it will take essentially.the same number of hours to pass - 99 = each subsequent point reached by the train in its forward travel. There will be a few extra low waves in advance of the train and a few more lagging behind the train, but there will be only a small percentage of extra crests produced which are of any appreciable amplitude. The forward end of the train will advance with the group velocity of the apparent local period of the waves in the train, and the trailing end will follow with that same group velo- city. After a given distance of travel, the ends of the train will be modulated by a Fresnel interference pattern. Eventually when x becomes very, very large, the wave train will have a much lower amplitude. As x approaches infinity, for n finite, the amplitude of the wave train approaches zero everywhere. All disturbances of initially finite duration and amplitude must eventually approach zero amplitudes because of dispersion. The individual waves will be essentially constant in ampli- tude and period over the central part of the train. The crests will travel forward with a speed appropriate to the apparent local period of the waves in the train. Thus the individual crests are traveling with twice the speed of ane train. Therefore they must form in the rear of the train, grow in amplitude, travel through the train, and die out again at the front of the train. At the ends of the train, a particular crest will not have a speed exactly equal to the speed in the center of the train, because of the effect of the phase shifts shown in equation (5.28). Wave crests are created and destroyed. Agreement with classical theory Finally, it should be pointed out that many of the abstract - 100 - points made by Lamb [1932] concerning the propagation of gravity waves in infinitely deep water, are illustrated in these two con- crete exact solutions which have been presented. Two quotations from Lamb follow which illustrate this point. "It has often been noticed that when an isolated group of waves, of sensibly the same length, is advancing over relatively deep water, the velocity of the group as a whole is less than that of the individual waves com- posing it. If attention be fixed on a particular wave, it is seen to advance through the group, gradually dying out as it approaches the front, whilst its former place in the group is occupied in succession by other waves which have come forward from the rear." “Hence in the case of an isolated group the supply of energy is sufficient only if the group advance with half the velocity of the individual waves." Note also that the solutions obtained in this paper give some information which is not described by Lamb [1932]. For ex- ample the solution for the finite wave group gives information about how the amplitude dies down and how the apparent wave periods change with time. The solution for the finite wave train gives information about how the ends of the train are modulated. Comments The finite wave train is an interesting study. It still has infinitely long crests, and it is still unrealistic in that respect. However, properly interpreted and modified it will be a building block in the formulation of the more realistic models. - 101 - Chapter 6. SOME MODEL WAVE SYSTEMS IN WHICH THE CRESTS ARE INFINITELY LONG Introduction The usual wave record today in deep water is observed as a function of time at a fixed point. A question arises as to whether this one record is enough to characterize the sea surface and as to whether it can be used as a forecasting tool. The actual short erested appearance of the sea surface and the finite width of the storm cannot be determined from this one observation. Some elementary results (without the use of time series theory) can be obtained which demonstrate some of the effects of dispersion, but they will be shown to be inadequate even for the case of infinitely long crests. Equations (6.1) through (6.6) describe free surfaces as a function of time which might be observed at the edge of a storm at sea at the point x = 0 and y = 0. They increase in complexity, and they are described less and less precisely as functions. In equation (6.6), for example, a,(m) and b,(y) are unikmown functions which would have to be described before anything could be said about the behavior of the sea surface either at the origin as a function of time or at other points. Figure 10 shows portions of the graphs of 7 Ty (02t), 1 rq76020,t), and 7 ty(0,0,t). The first graph is regular and repeats itself exactly. The second graph was constructed by pick- ing different values of An» on? and en at random for a few values and graphing the resulting function. = lige Model Wave Systems with Infinitely Long Crests 7 eal 7(0,0,f) = Asin = a (6.1) Asin at al) WS cS (ah AE 7(0,0,f) = (6.2) O otherwise n= —o2(t-nt)* 7(0,0,t) ns eum "sin 2uttene (6.3) n=-o —g2 t—n 2 n(O,0,t) = sh NC a “Vein 2z(t=n7) (6. 4) Ir n=—p T Weve -o2(t—nt+8,)* 7.(0,0,t) =>) Apne "sin(2tli=ae*8n) _ py (6.5) Ir n=-p @ n=+p T,(0,0,t)= D5 fos u(t-nt+8n) +b,(u)sinu(t-ne+8n)]du (6.6) NE =f; Examples. 2 P 14+300F5° 00 oe ee 710,0,t)= Re lOO sin(S2(t+300)+ e)+3e 202 sin( 2204200) 20 .,_2m(t+100) .,/2m(t+!00)), (20)? ;.. 2mt Ont, 7 iaieioo) "|" neo IO )* omerz ei sing +a) 3 stIOO 2mlt-100)) , /,_/t-2002) 6759 20: om 200)_ Pog CO singer (55 oy Jeu to? sin( 2 eee) Ss 25 2n(t-300)..,(27(t= 500) +4 Bat-300) °'" ie 2160 zs) ge oe (t+300)% uieeee) =, (ee 20 2x(t+100) rT) ~ApA AND ~ us 7'(0,0,t) = Ee Re ~ 100 + Se 400 + da1t+100) _O—— -(t-100)? 2 (t-200)? (202 |... 2nt Se Toe _(t=200)) , 760 Teams erty? + e 10 +(I 202 je es. 25 2r(t-300)] ... 2rt "4 2m(t- 300) me sin To (6.6b) Plate ADL (€9'9) puDd (G9) (b°9) Suolyonb3 40 spuawbas Me) sydoi9 ‘obi -104- Equation (6.6a) is a specific example of what equation (6.6) might look like after different functions had been picked for an(#) and b,(y) and after integration overp . It could be graphed as a function of t but the phase shifts indicated from group to group would require more precision than is warranted for the purposes of illustration. An easier function for pur- poses of illustration can be found by setting ante ) equal to zero and the on equal to zero. The "waves" under the envelope then factor out and the term in the bracket represents the over=- all envelope in equation (6.6b). Equation (6.6b) is the last graph in figure 10. The number of different functions which could be constructed according to equation (6.6) is limited only by the imagination. It will be left as a problem for the reader to solve to find out what the functions a,(y) and b,(y) are which yield equations (6.6a) and (6.6b). They are all smooth piecewise con- tinuous and piecewise differentiable functions. In this chapter, it will be assumed that the wave crests are infinitely long in the y direction. The results will then be in- dependent of y, and a y could be substituted for the second zero in all of the equations of Plate XIV. It will therefore be omitted and the free surface will be treated as a function of x and t. Equation (6.1) is trivial. If the wave is traveling in the positive x direction, the only possible motion is given by equa- tion (2.19) where @ = 0, 5 = 3r/2, and for infinitely deep water, hs et /2r. The comments made on equation (2.19) still apply. If the observation were to represent a storm at sea, the storm would have started before the start of time, and it would never - 105 - end. Conditions would be the same at all points. Equation (6.2) has been solved in Chapter 5. It is far too regular to represent a storm at sea. However it does start and stop at the origin, and the wave train actually travels and dis- perses so that it is observed at different times at different values of x. If 2nT is of the order of several hours, at each point, x, there is a time interval of several hours where the dis- turbance can be thought of as having the properties of equation (6.1). Outside of this time interval the waves have either not arrived at a point x, or they have passed the point x, and the sea surface is essentially undisturbed. An infinite periodic train of wave groups Equation (6.3) has not been treated before. It has the faults of equation (6.1), but it also has some other interesting features which make it worth studying. The function represented by equa- tion (6.3) is periodic with a period,t . Therefore it can be ex- panded into a Fourier Series of simple sine waves with discrete spectral components. Thus equation (6.7) shows that it is possible to represent the infinitely long periodic train by a sum such as the one given by the last expression in equation (6.8). Each side of equation (6.8) is multiplied by sin 2rpt/r in equation (6.9). The function is odd and there will be no cosine terms. Integration of both sides of the equation from - t™/2 to t/2, as shown by equations (6.10) and (6.11) yields the values of a. Equation (6.12) is then another representation for n port). Equation (6.12) is much more informative than equation (6.3) because it is a sum of simple sine terms, and the classical theory - 106 - An Infinite Periodic Train of Wave Groups. For Equation(6.3) 7,(0,t) = 9,(0,t+7) (6.7) Net O!) = o2(t-nz 70,1) = yy Ae "sin 2 (t- nt)) = “Dy onsin2% amet (6.8) n=-@ n=+o0 —o2(t—n Ae oe in 27 (t- —nt)sin oe Stn sin 2am sj —— (6.9) nN=-o =| z O m#p 2 [insintetian tater. { (6.10) Tt aE) = 5 am Q m=p N= +o -g2(t-—nt) = o2t [s Ae sn@zir-nepin gmt ar [ae sin sin ema N=—o oe oo a Phe SO her, sf ng(0,1) ny le sec? vos cee Tact oamt io nfo FAL grt Tact ox , 2nmt y Oh, fects _2emty ie Plate XV -107= gives information on the correct spectral wave length to assign to each discrete spectral period. In equation (6.13), these wave lengths are assigned to the periods, and complete knowledge of the behavior of equation (6.3) at other x has now been obtained. Some of the waves are assigned to travel in the negative x di-, rection in order to obtain complete agreement with the results of Chapter 4. For a fixed t, the sea surface as described by equation (6.13) is periodic as a function of x. The wave lengths involved are given by L = et °/2nm=, and they decrease in length by one over the square of the integers. Alternate solution An alternate solution to equation (6.3) is also possible with the use of the results of Chapter 4. If, in equation (4.9), t - nt is substituted for t, and then if the equation is summed from n equals minus infinity to plus infinity, an alternate so- lution is the result. The alternate solution is horribly diffi- cult to evaluate and interpret. It represents a sort of blind alley with little practical application. The difficult terms and derivations of Chapter 4 would have to be analyzed and summed over many values of n before a result would be obtained. For comparison, in equation (6,13) for typical values of o andT, only about ten terms are important in the sum, and the evaluation and interpretation is quite simple. A finite train of regular wave groups In equation (6.12), for typical values of o , t , and T, the first term in parenthesis is negligible and it is therefore neglected = 32906:2 in equation (6.14). Also for typical values of o , Tt , andT, the finite wave groups in equation (6.4), are essentially zero outside of the interval nr - Y/2@ 5 — A Pe.- (6.21 ) tok mat se 2 4r2m2x eum 4r2mx 2rmt\2 1E.= T p> a any sin (Ze 2rmt\vat= Ups $a (Si aa = emmy)? dt athe Gr2m2x 2mrmty... ,47°UX 2nrqt += Px > 2) &maqsin(=" TS Sea) S10 ema acta =—)\dlt (6.22) {* m=| q=l | Ee gra? 2-28" - 2h oe lim == Spo 2 pom TEpeT Wee poPE. “24g am Se ee (6.23) for fixed x= x, and t=t* gh? if -nT+E(x,,n Uses t*-T 0; the disturbance is present for 2nT seconds, and the 2nT seconds could stand for ten or twenty hours. If, in equation (6.19), t* were a time after the train had arrived and if t* + T were a time before the train had passed, then PoE. would very nearly equal oga/4. Note that for the elementary cases discussed above, the average over ten cycles is only two per cent in error. The modulation of the edges has to be considered, and the value of P.E. is not given by oga-/4 if equation (6.19) is evalu- ated in the modulation zone. Equation (6.24) formulates the above discussion in terms of inequalities. Apart from the modulation effects of the edges expressed schematically by E(% 5 n, T) (a positive number), the value of P.E. will be approximately pea-/4 if the first inequa- lities indicated are satisfied and it will be approximately zero if the second two inequalities are satisfied. If none of the 2 dag four inequalities are satisfied, P.E. is of some value between ogA@/4 and zero. The average potential energy at the modulated edge of the train was discussed in Chapter 5. Figure 9 shows that at least to the eye the area under the dashed curve is equal to the area under a jump function given by f(t) = 0 for tbe andi byt) for t>t,- To a good degree of approximation, then, the potential energy averaged over a time short compared to the total duration of the train but long compared to a cycle is constant when the train is present at the point of observation. Since the train, if it has not traveled too far, takes 2nT seconds to pass, the total amount of energy present is the same as at the origin at each point of observation. For great distances of travel, dispersion modifies the re- sults, and P.E. decreases. No energy is lost; it is just spread out over a greater time interval. Energy balance for the finite regular train of wave groups The finite regular train of wave groups was broken up into finite wave trains of different periods. At a given point of ob- servation, X;, Some trains will be present, some will have passed, and some will not have arrived as shown by figure 11. If the train for m equal to K is present, and if the train for m equal to K + li is present, then the trains for m for values in between will be present. Equation (6.25) expresses this formally. Over a long enough time, all of the energy in the original record is accounted for at each point of observation. - 120 - ae ee Sere mir ne ee ee Even if the waves on the sea surface had infinitely long crests, the models studied in this chapter would be inadequate for very subtle reasons which will be discussed in the next chapter. For these reasons, equations (6.5) and equations (6.6) were only indicated in Plate XIV. Various approximate results based upon the assumption that the individual groups in the sum of wave groups do not overlap to the extent that the potential energy associated with one group is affected by the presence of the neighboring groups can be obtained. The difficulty in the analy- sis of equations (6.5) and (6.6) lies in the fact that it appears that the wave record cannot be expressed as the sum of a number of sine waves in a form which applies to the whole sea surface. Even the most general model described by 7) Ty 60209) could not be made to fit an actual observation of waves on the sea surfacee The methods of Fourier Integral theory have been pushed as far as practicable, and it becomes necessary to introduce new con- cepts in order to obtain more realistic models. 4G - 121 - Chapter 7. THE MOST REALISTIC WAVE SYSTEMS WITH INFINITELY LONG CRESTS Introduction In Chapter 6, model wave systems were derived which be- came increasingly more complex throughout the chapter. The most general wave system mentioned only briefly in the last chapter depended on the assumption that the waves came in groups separated by an average time interval, T , throughout the storm and that the waves were low near the times t =nt + ‘2. If sample wave records are examined, it will be found that portions of the record do exhibit groups which appear to be of a length t . But also it will be found that there are long stretches of the record which do not show groups, and which appear to be just irregular bumps of assorted heizhts and various time separations of the crests. Such records are not adequately described by any of the models which were dis- cussed in the previous chapter. The author has spent many hours in conversations with those whose activities are connected with waves. Along the New Jersey coast, for example, a fisherman once solemly in- formed him that “every seventh wave was the highest." Another fisherman was equally positive that every fifth wave was the highest. In fact, opinion was well scattered over all values from three to seven. A fisherman (or for that matter, any one) with a profound faith in any one of these particular integers will some day be bowled over by a wave higher than either its - 122 - predecessors or successors with a label “favorite integer plus two." The sea surface is irregular, it does appear that the waves sometimes come in groups, but the groups do not persist, they do not have a mean time of separation, and they do not contain the same number of waves. Figure 12 shows some wave records. They are on a greatly condensed time scale such that the crests are all crowded to- gether. Note the basic features of these wave records. Iso- lated high waves frequently occur as at the points marked A. Sometimes groups appear as in the intervals marked B. At times the trend in the amplitudes is quite high as in the intervals marked C. And at other times the trend in the amplitudes is quite low as in the intervals marked D. The basic feature of the records is their irregularity, which is a type of irregu- larity which would almost appear to defy an adequate mathe- matical representation. Obviously any of the mathematical models employed in the past chapters do not represent such a wave record. Consequently, better models must be found. The next step then in increasing complexity is to find some way to represent the sea surface which is general enough to include this very irregular pattern. It will be found that Fourier Integral Theory is not enough and that an extension to a type of Lebesgue Stieltjes Integral is needed.* The ex- tension to this type of integral and the inclusion of some very interesting statistical methods simplify the problem once the basic concepts are understood and permits a tremendous stride *Phe Lebesgue-Stieltjes Integral is defined in James and James [1949] for example. = 323) 3 “SQ¥YO0934Y S3AVM 3J1IdWVS G3SN30NO9 AY3A 3WwOSs él Sid —P = ne eR fi , = Airis ve views weal em == Boll Ba i a a Mala vs dds bed v fl = r =I ofiz miata. 1661 ‘81 ‘LOO 8Szz : te OUR Sl ae ee oes == SS Ss = ae a Vee ee e ert | = y a at H rete SS — = BS r = iS6i ‘Il 3nar PACA —124— forward in the problem of understanding and forecasting ocean waves. The generalization which will be developed in this chapter will make it apparent that the finite irregular train of wave groups mentioned in the last chapter as given in eouation (6.5) involves too many special assumptions to permit its development to a completely realistic case. The Lebesgue Stieltjes Power Integral In order to extend the techniques of wave analysis, it is necessary to discuss a new type of integral which is well estab- lished in theoretical mathematics, but which is unfamiliar to Many people. The ordinary Riemann Integral is the one which is well known. “The concepts of the Lebesgue Integral and the Stieltjes Integral are employed in theoretical statistics, and Cramer [1946] is a reference for such a study. No attempt will be made for complete mathematical detail and for complete generality, but the derivation will be general enough to in- clude those properties which are needed for wave record analy- sis. The reader who is interested in greater detail is referred to Tukey and Hamming [1949], Tukey [1949], Levy [1948], Cramer [1946], and Wiener [1949]. The methods by which these con- cepts can be applied to wave analysis most directly are given by Tukey and Hamming [1949], and many of the arguments herein will be based upon quotations from and explanations by Tukey and Hamming [1949] and Tukey [1949]. Consider the Lebesgue Stieltjes Power Integral given by equation (7.1). 7(t) is the free surface as a function of OSTA me The Lebesgue Stieltjes Power Integral n(t) - [eos + ¥(W)VGE(H) (7.1) fo) where E(H) = O vole (s S10 (7.2) and E(w) = E42) if Of, * <00 (7.6) then m(t) = lim > (Hone 2) ~E(Hon) “COS (Hontit +¥ Hon.) (7.7) max(Uy, 4.) —U,)->O p= H-2R--@ where O£=V(y,.,,) = 2m (7.8) Partial Sum n(t) = > ional E(Hon) © COS(Mo ng) t+ V(Hone))) (7.9) n=O MIN (Hg Pu) = Ap MOX(Hy+) —Hk)= Aa# fie ci a5 Pg «ig PE. = Lim Satoh ‘ tim 3 [EtHonea)-EHan)| [SP] cia A,p->0 Ptul PA 2 > [-Etwen)+ E(uan+a)] A2n+0 eres rag (-O+ E(u2))+(- E(H#2)+E(H4))+---- (-E (Hp) +E (Hor 2))| Zz PG ; pre E(Hor42) = ae Emax (7.10) Agno Plate XW -126- time at the point of observation. The function, W(p), is a point set function which will be defined later. The notation Van Cp ) at first does not make sense, until the process by which the integration is to be carried out is defined. The properties of E(/t) are given by equations (7.2) through (7.5). It is zero for p less than or equal to zero. It is GING OHNE UI 3) non-decreasing for uw greater than zeros; that is, if pe 2 is greater than pu ? then E(p 5) is greater than or equal to E(u 4) as stated by equation (7.3). Finally, for allp , E(u) is less than some positive constant, M, as required by equation (7.4). If E(y) is monotonically non-decreasing and if it is bounded from above, then it follows that E(p) has a definite maximum value E,., (equation (7.5)) which is either actually reached at- a finite value of w or which is approached asymptotically as w approaches infinity. In statistics, a function with similar properties is referred to as the cumula- tive frequency function, or ogive, as defined, for example by James and James [1949]. E() will be referred to as the cumulative power density. It measures that part of the averaged square value of 7) (t) which is contributed by those spectral frequencies less than or equal top . The word "power" in the definition is unfortu- nate for wave theory because the averaged squardl value of ee is most nearly connected with the potential energy of the record averaged over time. In electronic theory where these concepts were originally developed, equation (7.1) usually described i voltage produced by an alternating current, and the voltage <«. ws le ae squared working into a known load was a measure of the power involved. By extension anything involving the square of the sample studied has been described in terms of power which ex- plains the origin of the term cumulative power density. Later on when actual wave power is studied, it will always be referred to as wave power in order to eliminate confusion. . To proceed with the definition of the integral given by equation (7.1),* in equation (7.6) the uw axis has been marked by a series of points, Op ysH arp zeeeee* Wore Such a division of the range of integration into a number of small intervals is called a net. The Bh are not necessarily equally spaced, and they are not necessarily rational points. Now form the sum of terms represented by equation (7.7) before the limiting process is applied... The first term is given, for example, by the square root of the difference (which is greater than or equal to zero) between E( p>) and E( py.) times the cosine of #,t plus W(t) where, as yet, w(y4 4) is not defined. The function, wy (p4)> can be defined in many ways. One definition would be to give a set of points between O and 27 from which a value could be picked by some rule once V on+1 was given, and the fact that the integral involved such a set of points would then make it a Lebesgue integral. Suppose then that such a rule is given for picking the value of W(p a43). Then the integral of equation (7.1) is the limit of the sum given by equation (7.7) as the mesh of the net approaches *For additional information, see Levy [1948]. - 128 - zero. That is, the integral is the limit of the process defined by equation (7.7) and the law for picking YC bony) as the greatest distance between two successive / 's in the net, say H+] and ,, is shrunk to zero. Note that the partial sums in equation (7.7) are almost periodic functions as defined by Bohr [1947] if the 4's are irrational. This integral has one very valuable property. The square of the function given by the integral averaged over time, is equal to (1/2)E. ax: This can most easily be shown by consider- ing the partial sum given by equation (7.9) in which the small- est distance between two successive #'s is Aj 9 a Small but finite value and in which the largest distance between two suc- cessive # 's is A,p#.* The potential energy of 7 (t) averaged over time is given by the integral expression in equation (7.10) (see equation (3.10)). Since the 4 5,,, are different, the cross product terms in the Square average to zero when (7.9) is substituted for 7(t), and the second expression in (7.10) results (see, for example, equation (6.22)). Upon rearrange- ment and evaluation of the sum, the plus E( #5) in the first parenthesis is cancelled by the minus E(w.) in the second parenthesis, and the r'th partial sum is EC Hongo) As f ap=- proaches infinity, P.E. equals (pg/4)E,.,. Equation (7.10) holds for arbitrarily small values of Ask and hence it holds in the limit. *A, b is the smallest segment in the net; Ap is the largest. The lengths of all others lie in between. = 129 = Some examples The Lebesgue Stieltjes Power Integral just defined in- cludes as special cases all of the representations of the sea surface in the previous chapters which were infinitely long in duration. From the definition of the integral, it is evident that the function 7(t) never attains a constant value of zero. Example one is another way to express equation (2.19) when x and y are zero and 6 equals 37/2. E(p) is given by equation (7.11) which shows that it is piecewise constant with a value of zero below 21/T and of A above 21/T. The function, (ph), could be given, for this example, by equation (7.12), but act- ually W(p) could be anything outside of a small interval about 27/T. If the limiting process defined by equation (7.7) is car- ried out, the value of [E(H5,45) - Bp ig JIE is zero for all n except for that particular n, say n = p, for which m ops 2n/T WS we . he Whe >1>0 ia {nb ol Le pajdwoxg (m)3 w@>vsth2 'Svtively "LAgans hs: ev +sy - Wag Ws Yaa! sup (3 Az 5u50!0 $2 00) Zz izo a5 “ieee Fil Agenta v s etd Roh All 2V +20 +2 2 ajdwoxg (m)3 yO sajdwox9 Veg 39 Tfaguisy =) SNOINDA ayy ul PaA}OAU) SUOI;IUNY AY} yO SYdoDIdn Ey'BIy m= tgup Kg 2 Yazlit Wms (A240 Ch a(n) Vip 25 240! 4p Yiz>n'o 01 $2 00! rr iz Leo Che = (MA € 8}dwox oatslyoty = (m3 Lz > 1s 0'0 = (13 2+ Liz =Wtie lejdwoxg ()3 == infinite. However, P.E. as defined by equation C7226) nies still finite. The appropriate graphs for example four are given in figure 13. In this case, there is no jump in E(u), and dE(y)/dp equals a if p is less than 27/T). The Gaussian case, or the principle of independent phases The examples of the integration of equation (7.1) which have been given so far have not introduced anything basically new in the nature of the sea surface. The integral is so general, how- ever, that it includes many cases which can only be represented by such an integral. One special case is the Gaussian case which is of extremely great importance in the theory of noise and which will prove to be of equal importance in wave theory. The integral considered is still equation (7.1), and the conditions given by equations (7.2), (7.3), (7.4), and (7.5) are still imposed. In addition, the condition that E(p ) be a continuous function is added, and the point set which defines W(#) is very specially defined. Continuity in E(p#) yields all necessary qualities. It permits a very peculiar mathematical form for the derivative of E(#) which will be discussed later in Chapter 10. Continuity of E(u) is imposed by equation (7.27), which states that the difference between E(y 5,5) and E(y 5,) can be made smaller than some delta if 5,5 - pop is made smaller than some epsilon (which may depend on delta). * In examples one, two, and three, this condition is not fulfilled at the jumps. In example four, E(w) is continuous. *See Chasant [1937]. -134 - The Gaussian Case Equations(7.2), (7.3), (7.4) and (7.5) hold in addition OT" E (Long oa eee if Pons2—Hean < 8(e) and O S W(uone1) Ss 27 such that p(W(Hons))< 227) = a when O chosen at random, that the ampli- tude of the sea surface will be less than the value K is given by the normal probability distribution. K is the departure from the mean, assumed to be zero, of the record. Equation (7.34) is another way to express this condition. It gives the probabi- lity that a point chosen at random in the record will lie between - 139 - the value of K and K + dK. The representation of wave records by the Gaussian case of the Lebesgue Power Integral If a portion of an actual wave record is to be represented by equations (7.1) through (7.10) and (7.27) through (7.32), then it must, at least, approximate the properties of the inte- gral, and satisfy equations (7.33) and (7.34). One property of the integral is that 7 (t) never repeats itself. Another is that if a time interval t is chosen, which is large enough to eliminate autocorrelation effects, then the values of the heights of the sea surface measured at tis tit Fy ay 2T .acee and so on, will be distributed according to equation (7.34). Herein lies the fault of the models in Chapter 6. For different ty in this model, (since it was assumed that the groups were spaced T units apart plus or minus a small deviation) the values of 7 (t) will not all have the same probability distribution and therefore the model is not Gaussian. It is very easy to test a wave record to see if the dis- tribution of points chosen from 7(t) at time intervals suffi- ciently great is Gaussian. The test has been made on some actual amplitude wave records and on some pressure records. Some re- sults of the tests are given in figure 14. The first histogram in figure 14 is from a wave height record obtained with the Beach Erosion Board Tmetrenene described by Caldwell [1948], which was located on the pier at Long Branth, New Jersey. It shows that the distribution is not quite Gaussian because the median value of the histogram is below zero and the - 140 - Frequency 24 22 20 18 16 Emax = 45 000cm2 14 — ay RE. = 11,250,000°'9% mp2 ! (equivalent to a sinusoidal wave 10 2.12 meters high) 8 6 4 S (| unit equal to 6.6 ft.) Math -1,795-1395-0995-0595-0195 0020506051005 1405 1805 2,205 2605 Limits Histogram of samples from a wave record for 5-l4-48 0000 to 0007 EST, Long Branch, New Jersey. x? = 23. with 8 degrees of freedom — Eliminate last group.x?=78 with 7 degrees of freedom. 35 out of 100 times sample could come from normal distribution. Frequency 20 E pmax= 601.5 cm? (| unit equal to 6in.) Math -27495-22495-17495-12495-749524.05 0 2505 750512505 1750522505 2750532505 Limits Histogram of samples from pressure record for !0-18-5l, 2258 to 2323 ES.1., Long Branch, New Jersey. x*=9.128 with 9 degrees of freedom. 43 out of 100 times sample could come from normal distribution. Fig.14. Histograms of wave height and pressure amplitude distributions from sample records. —141- frequency of large negative departures from the mean is not as great as the frequency of large positive departures from the mean. However, the departure from the Gaussian distribution is not so great that the resemblance to the Gaussian distribu- tion is lost. It is interesting to consider the probability that such a histogram composed of one hundred values chosen at random from a seven minute wave record could have come from a Gaussian distribution. The Chi-Square Test can be employed to determine this probability by standard statistical methods. The needed values computed by the methods described, for example, in Hoel [1947] are given in Table 15. Table 15. Chi Square Test of the upper histogram in figure 14. X, 9(X,) ee ta eS Shs De (f, - BoA 2.30 .029 167 - .669 0.448 0.268 eye 2091 5.24 -. 2237 0.056 0.011 ibeald! R20 Ze = mate, 9 4.09 16.70 1.40 ORbeCaes 39) 849.5 -1.51 2.28 On2a7 0.00 -399 23.0 -3.96 W567 0.684 0.583 .337 19.4 Bical 13.00 0.670 Thy alte) S20 So aay) 2.26 Big dla 0.435 au 7/3) .088 5.06 -3.06 9.39 1.854 Peso) .028 TA(ERE -1.61 2.60 ee oualt 2.89 .006 0.345 2655 0.26 0.754 3.46 -OO1 0.058 -942 0.89 15.29 (Chi)° = 23.1 Less last term Fats zpq4ole The value of Chi Square is 23.1 with eight degrees of free- dom. It is highly improbable that such a distribution could come from a normal distribution. But note that it is the very last value in the sum which makes the sum so high. If this last term is omitted Chi Square is 7.8, and the sample could have come from a normal distribution 34 out of 100 times if chosen at random. The departure from the Gaussian case can be explained on the basis of the actual non-linearity of the sea surface. Be- cause of the non-linearity, crests are higher and troughs are lower than in a surface described by the Gaussian case. The peak values of the crests are what have produced the high value of Chi Square. It was pointed out in the second chapter that little could be done about the essential non-linearity of the sea surface, and these histograms show remarkable agreement with the hypothetical Gaussian case within the limits of the lineari- zation assumption. The potential energy of an actual wave record averaged over time can be computed by squaring the wave record and averaging over time. Such a computation would require rather leneeie computations. The histograms show that E ax can be estimated easily be taking the second moment about the mean (square of the variance) of a sample of one hundred or so points from the record. The computations involved would be considerably less than by the other method, and the reliability of the estimate would depend on the size of the sample, on the magnitude of Enax? 2nd on the function E( jy). The value of Enax and of P.E. =o as computed from the histogram is given beside the histogram. Bax and P.E. are reliable and theoretically definite statis- tical quantities obtained from wave records. These points will be considered in more detail later. The remaining histogram in figure 14 was obtained from a pressure record which was taken by a pressure recorder in 30.5 feet of water (mean low water) offshore from Long Branch, New Jersey. It was obtained in October 1951, while an east coast storm passed Long Branch just a short distance out over the ocean. This histogram is more nearly distributed according to the Gaussian law than the height record was. In fact, 43 out of 100 times at random, this histogram could have come from a normal distribution. The better agreement can be explained on the basis of the fact that the second order non-linear terms die out more rapidly with depth. Consequently in the pressure re- cord the higher ridges and shallower troughs which were at the surface are less accentuated. From these histograms, EY max has been computed which is analogous to the quantity Ea in the height record. However, P.K. cannot be computed from E ° sities max P.E. can be computed if either BC pe) or (AH) is known, and if a certain linear operator is applied to EC) in order to get the value of E,., which applies to the free surface (see x the discussion of pressure records in a later chapter). A number of other samples of points equally spaced in time were picked from this same pressure wave record. Of these, several values of Chi Square were so high that they were not in the tables. Other values were quite reasonable, and the S44 probabilities that the samples could have been picked from a normal distribution were 30 in 100, and 50 in 100. There were two cases in which Chi Square was very large. The skewness of the histogram of the surface records is one way in which 7)(t) does not follow a Gaussian distribution. A second way in which the distribution will not be Gaussian comes from the fact that equation (7.34) yields a finite probabi- lity for very high crests and very low troughs. For low waves, this finite but very small probability is not important. The Gaussian distribution is only a statistical ideal; for example, it predicts men twenty feet high from a population with a five foot mean and a variance of one foot. In short, all statistical theory must be used with judgment. Actual wave heights cannot exceed a certain value since the crests will break. It is to be expected that for high seas the histograms will be both skewed and chopped off at the extremes. The effect of breaking ina complex irregular sea surface is again a non-linear problem and cannot be treated by the methods under study. Figure 14 shows consequently that actual wave records very closely approximate the requirement that 7) (t) have a Gaussian distribution of the amplitudes. Berkoff and Kotig [1951] have commented on the fact that certain symmetry requirements for (tt) are not met in actual wave records. This failure is a consequence of the actual non-linearity of the problem, but again the departure from the Gaussian case is small. This fact is indeed fortunate. The theory of the statis- tical analysis of functions of the form of equation (7.1) in = IAG = the Gaussian case has been presented by Tukey and Hamming [1949]. For the non-Gaussian case very little is known. Paraphrasing Tukey, it can be said that "This restriction to Gaussian (wave records) will presumably not be a serious hindrance to our analysis of actual (wave records) which will be non- Gaussian to a greater or less extent, if we use the quantitative expressions for the fluctuations as warning signs, and realize that fluctuations larger than those predicted by Gaussian theory are likely. The recommended procedures (in the paper) are known to be good for Gaussian (wave records). For moderately non- Gaussian cases, the analogy with simple problems suggests that the procedures will be quite good." For wave records the modifying effects of non-linearity must be kept in mind, at least in a qualitative sense. Tukey (personal communication) Says that the values of Chi Square given before are just what one might expect from random noise and that the better results for the pressure records show that the system is non-linear in the high frequency components. If the qualifications and explanations in the above section are taken into consideration, it can be concluded that the best possible known way to represent a wave record and consequently the sea surface as a function of time at a fixed point is given by the Gaussian case of the Lebesgue Power Integral. Any portion of a record of the sea surface as observed as a function of time, if the sea surface is in a stationary state, can therefore be thought of as a segment of one of the statistical ensemble of functions which would result from the indicated limiting process = 146% which defines the integral. Stationary processes and stationary time series The first three examples given in Plate XIX and the Gaussian case of the Lebesgue Power Integral are all specific examples of stationary processes. A stationary process is simply a function of time, say (t), such that the essential proper- ties of the function are not altered by the substitution of _t +h for t in the functional representation. Substitution of t + h for t in the first three examples simply changes the phase of the various sinusoidal waves in the function. The power spectrum is still the same, and the function is still composed of the same sine waves. Similarly, in the Gaussian case, sub- stitution of t + h for t simply changes the values of W(p), and the function is still an element in the class of all possible functions which can be found by the procedure of integration de- fined above for a particular E(p). A stationary time series can be made from any of these functions by giving their values only at separate points; say, at ti to» taeeceeths preferably separated by the same length of time. The height of the water against a wave pole in suc- cessive frames of a motion picture film strip would be a practical example. Special note The Gaussian case of the Lebesgue Power Integral does not have to have any special form for E(w). E(p) can be any function as long as it is continuous. In previous chapters, the normal probability curve has been used as a special example = NA of an ordinary Fourier spectrum to study the propagation of a finite wave group and to study the propagation of various finite wave trains. The accidental fact that the spectrum was connected with a normal probability (or Gaussian) curve should not be con- fused with the very important fact that values of 7 (t) at greatly separated values of t, chosen at random, are distributed according to the Gaussian probability law. Wave record analysis Any given wave record as a function of time can be considered to be a short piece of an infinitely long record which is one of the infinite number of records possible from the integration of the Gaussian case of the Lebesgue Power Integral. The problem is to find (An ))*, given the short piece of the record. This problem is the basic problem of wave analysis if it is general- ized to permit representation of short crested waves. The function, CG a. and the extension to what corresponds to it for a short crested sea surface can only be estimated because of the finite length of the record. The longer the record, the more reliable the estimate of (a(m))°. The problem of wave analysis will be considered for the short crested case in a later chapter. Wave forecast models for wave systems with infinitely long crests in the Gaussian case Consider again equation (7.1), for the Gaussian case. Instead of %(t), equation (7.35) employs 7 (0,t) to point out the fact that the function is presumably known only at the origin of the x coordinate system. Assume that (AC pw ))® is known. This wave record as a function of time at the origin never started SWS The Forcasting Problem for a Sea Surface Represented by the Gaussian Case of the Lebesgue Power Integral with Infinetely Long Crests in the Y Direction 0 7(0,t) -/oosts +h (p))¥d E(p) : to wex 1X, 1) feast —pt—w( ps) GEC a) oO oi 2 (x,t) -feosture(vu- E>) aE) {o) 2 Uy OS Vibanei)— HPN Dan = Wwansi) S20 F(t)-7(0,t) = F(t) [oostut WusETH (7.35) (7.36) (7.37) (7.38) (@) ls Z(t) >. VE(Han aa) —ElP2n) COS(Hanay t+ ¥(H2n4i)) (739) n=0 MiN( HK gH HK) = Ale MOX(Hp+T Hp) =Apy O5,41); which satisfies equation (7.38). The W'(u5,,,) will be distri- buted according to the same probability laws that govern the distri- bution of the original W CH ony)? and consequently W'(#) is another point set function like V(H). Consequently 7 (x,t) at any x is Gaussian and has the same cumulative power distribution that the record at the origin had. 7 (x,t) also has the same power spectrum. Ina statistical sense, then, the sea surface has the same properties at all points. Wave record of finite duration In order to generalize the model to a record at the source of finite duration, consider the multiplication of 7 (0,t) by F(t). F(t) is any function of time which varies very slowly compared to the individual waves in 7 (t).. F(t) should also be essentially zero outside of a certain range of t. F(t) operating on the inte- gral as in equation (7.35) is equal to the effect in the limit of F(t) operating on one of the partial sums which represent 7 (t). SiG. 6 But it is relatively easy to determine the effect of F(t) on a simple trigonometric term of angular frequency, w Ont1* The result of operating on any partial sum with F(t) can consequently be found very easily. In the limit, then, the complete effect on the inte- gral can be determined by considering yw to be a variable. One of the many possible F(t) is given by equation (7.40) where Do is the duration of the waves. The wave record builds up to full amplitude instantaneously at t = O and dies out instant- aneously at t = Die When this particular F(t) is applied to one of the terms in the partial sum indicated in equation (7.39), it can be seen that the problem is essentially the same problem that was solved in Chapter 5 except for a shift in the time axis. If F(t) is applied to 7 (t), the result is no longer a stationary pro- cess, but a sample taken during a time interval in which F(t) is essentially one would yield a power spectrum upon analysis indis- tinguishable from the one obtainable from the unmodified function, 1(t). In Chapter 5 it was found that the forward edge and the rear edge of the wave train advanced with the group velocity, and that the edges were modulated by Fresnel Integrals. For the moment, although it is physically impossible, assume that the amplitude of the train is either zero or one at any x and that the Fresnel modulation effects are not present. The square cornered or sharp cutoff filter The time, te» required for the forward edge of the wave train to reach the point x for a fixed # is given by equation (7.41). The rear edge, t,, passes De seconds later. Consequently, for a - ALSyiL as ] particular » and for a fixed value of x, if the time of the observation, top» lies between t,, and t, then the component sinusoidal wave for that particular # is present as shown by equation (7.42). Substitution of equations (7.41) and (7.42) into (7.43) yields equation (7.44). Rearrangement of equation (7.44) then yields equation (7.45). For a fixed time and place of observation, and for a fixed duration of the waves, those spectral values of # are present which lie between the values g(t.) - D)/2x and gt op / ax. The other values in this simplified case are not present. The upper value of # which is present is given by equation (7.46), and the lower value, by equation (7.47). The band width present, Ap , depends directly on D, and inversely on x as shown by equation (7.48). Figure 15 shows how these considerations can be used to con- struct a forecasting diagram. The top part of the figure is a graph of the straight lines given by equation (7.46) and (7.47) as functions of t and » for various fixed values of x and for Do equal to 10 hours. Pick a time, t = top? say, twenty hours and a fixed x, say, 200 kilometers. The line for t = 20 hours intersects the two parallel lines which apply to x = 200 km, and a segment of the # axis is cut off between the two parallel lines. The orojection of this segment onto the # axis then gives the band of frequencies present at x = 200 km, 20 hours after the start of the storm. Practically nothing is known about the power spectrum of waves at the edge of an area of generation in a storm at sea. =e in hours t +700 hm “Ay oe ee X*0008m . Our ee = K+3008m 4'020° as SS %-28018 - oS 2150 8° = X-200te te20 hours t= 40 hours t= 60 hours 20 z 36 ee = = = a t=80 hours t=100 hours Aa Gd mat Boot & & = = reas | 3 5. FORCASTING DIAGRAM FOR THE SQUARE CORNERED FILTER AND THE FRESNEL FILTER FOR THE CASE OF A VERY SHORT FETCH Seas Consequently, for purposes of illustration, a form for the power spectrum has been assumed. The assumed form of the power spectrum has been plotted below the part of the figure just discussed. It represents what mead to be known about the power spectrum at the source before the power spectrum at any other point of observation can be forecasted. The power spectrum at the source is given by the dash dot curve. The band width determined above has been used to multiply the power spectrum at the source by the square cornered filter in order to find the power spectrum at the point and time of obser- vation. The square cornered filter is given by equation (7.49), and to apply it to the given power spectrum set the forecasted power spectrum equal to zero outside of the segment described above and set it equal to the power spectrum at the source inside of the segment described above. The heavy solid lines show the effect of applying the square cornered filter to the power spectrum at the source. The forecasted power spectrum is an instantaneous power spect- rum, and in terms of our original definition, it has no meaning. However, if a wave record taken at x = 200 km from twenty hours minus ten minutes to twenty hours plus ten minutes is analyzed for its power spectrum, it might be expected that something like the above pattern, except for slight smoothing at the edges, would be obtained because the filter function is a slowly moving function of time. The remaining power spectra show the forecasted power spectra for various x's and various times. For a fixed x, as time increases, Erie ee the filter tunes through the power spectrum at the source. High period waves are received first followed by low period waves. The band width is constant for a constant Die For a larger fixed X, aS time increases the filter tunes through the power spectrum at the source more slowly and its band width is narrower. A square cornered sharp cutoff filter is physically impossible. It is, however, a relatively simple step to extend the forecast diagram to a Fresnel Filter. The procedure is to return to the methods of Chapter 5 and solve the problem given by 7(O,t) = cos(Sze + W( Hans) if O WsIS w9¢ol =_> be applied in this paper. The conditions which will be imposed are the following. First, the functions, f(x) and its first derivative, f'(x), are continuous as stated in (8.6). Secondly, the absolute value of f(x) is less than some positive constant M as stated by equa- tion (8.7). And thirdly, the absolute value of the derivative of f(x) is less than M. If these conditions are satisfied, then it can be proved that equation (8.9) holds. The proof follows. The integral from -A to A can be broken up into three parts as in equation (8.10) where «€ is some small but fixed number. The integral from -A to -e and the integral from e to A can be shown to vanish. The integral from - « to € contributes the whole value to the entire integral. Consider, first, the integral from ¢«€ to A. It can be in- tegrated by parts, and if absolute values are taken, the first inequality in equation (8.11) is the result. Estimates based upon M, the value of « , and the length of the path of integra- tion, then yield the second inequality. The second inequality as N approaches infinity tends toward zero. Therefore the inte- grals from € to A and from -A to - e tend to zero as N approaches infinity. Consider next, the integral from -¢« toe . In equation (8.12), the transformation of variables given by equation (8.13) yields the first expression. As n approaches infinity f(x'/N) approaches f(o) over a large range of x', and f(o) can be factored out of the integral as a constant. The integral from - N to N of (sin x')/x' approaches the integral from minus infinity - 178 - A Useful Lemma Given f(x) and f'(x) continuous (8.6) | f (x)| co aN N> Le stim fa f (x)dx (8.10) € A [eG )ax € A A lim {| ae f(x) dx | < lim | - 49 <9 | [+ tim € aon € N-~ oo N->co < lim (24 +204) <0 (8. 11) m™ EN lim] S85% £(x)dx= lim] 5% Bin. x. filer “)dx' = Jim 9 [88x m f(o) (8.12) a Nooo) Neo J Where x=Nx (8.13) — Plate XXIV SoS to infinity of (sin x')/x' which has the value 7. The final expression in equation (8.12) is thus the desired limiting value. If A is permitted to approach infinity in equations (8.9), the same result will hold. The results hold for any preassigned A no matter how large, and therefore they hold for A infinite in the limit. The initial value problem in the y,t plane for a disturbance of finite duration and width In a storm.at sea, the waves are quite irregular. There are high waves, and low waves. The high waves sometimes appear to come in groups followed by times when the waves are relatively low. In addition, even when the waves are high the crests are not very long, possibly only ten times the distance between suc- cessive crests. .Consider then a wave record in deep water ob- tained by a whole line of wave recorders along a segment of the line x = O parallel to the dominant orientation of the crests. If all of the wave records were properly synchronized, it would be possible to obtain a plot of wave height as a function of time and the position of each of the recorders on the line x = 0. The free surface would then be expressible as a function of y and t inside of a closed curve given by some function of y and t. Inside of this closed curve let the sea surface be given by the observations. Outside of the closed curve, let the sea surface be identially zero in amplitude, and if desired smooth the sharp edges off the boundary. The result is a finite short crested wave system observed as a function of y and t at an arbitrary origin in deep water. It could represent a whole storm at sea - 180 - ~AXX F4Did ( 21°8) 7 ee 1p [4(,7/+71) soo ee, ) soo4]- ode 018Z jonba YdIYM SW4a) + rc) N ae 1p Ap gp 7/p fu S09 (A0 Us a) soo |7/s09 (46 Uae eo2 (Q‘7!)o WT anit Clas ip Ap 47/s09 (46 uIS Ga) s09 (j‘A‘o) & an ull| (918) gp 7p firfuis (g us K -f) soo (g‘7!) q- Wr \‘K‘o) & in Aceh \7!s09 (Quis K S)uis(g‘7/)q +1T7u1s (Quis 6) uis (971) 0 +4709 (Quis K -6-) soo ( g‘7!)o es (S18) gp Mp7! Quis kK) uis(Q‘7!) q + (i7—Quish ei) $09 ( 9'7/) =| (v1 °8) gp mip [(mi-(@ ulS + @SO9 x) +) us (97) q +(47/-(Quls A + Qs09 x)£-)so9(@71) 0| iff (K‘x) oz: UJPIM PUD UOYDING ajlul4 JO eoUDGuNISIG D 40} aUD]q 4'A ayy UI Wia|qosg AN|DA OIsU] SYL -I8l- as it passes the line x = 0. Before a certain time its ampli- tude would be zero and after, say, ten hours had elapsed it would again be zero. In addition, outside of a certain range of y at x = 0 it would never be observed. The immediate problem is to find out how this disturbance behaves at other values of xe In anticipation of what is to follow, though, think of a storm at sea as a sum of many elemental sine waves traveling in various directions but bounded by the Closed curve described above. The disturbance is, by the principle of superposition, and due to the linearity of the system, equal to the sum of the individual disturbances, no matter how they differ in direction, amplitude, phase, and period. The function, 7(0,y,t), has now been obtained. What is the function 1(x,y,t)? Strictly speaking, 7 (0,y,t) does not determine 7 (x,y,t) because there is an ambiguity in the possible directions of the individual spectral components. For a com- pletely general problen, 1) (0095) would also have to be measured. However waves in a storm at sea travel in the direction of the wind and if the reasonable assumption that each spectral component has a component of direction of travel in the positive x direction is made, then a solution can be obtained. Equation (8.14) postulates that the free surface is composed of spectral sine waves of special frequency pm which travel in the spectral direction x cos 6 + y sin 9 (see equation (2.29)). Note that the limits of integration are over only half a circle in the p , © polar coordinate system, and that mw is always posi- tive since the integration is from zero to infinity. With this - 182 - assumption about the limits of integration, it is not necessary to know N x(¥ot) because all spectral components have a direction component in the positive x direction. If equation (8.14) represents the free surface everywhere, then equation (8.15) represents the sea surface at x = O and it can be expanded into the form of equation (8.16). Now the left hand side of equation (8.16) is a known function, and if the values of a(/# ,©) and b(#,0) were known then the step back to equation (8.14) would be simple and the problem would be solved. Take the known function 7 (y,t), multiply it by cos((p *°/g) sin 0*-y)-cos u*t, and integrate it over y and t from minus N to plus N and from minus M to plus M. Consider the limit as Mand N approach infinity. The first expression in equation (8.17) formulates this operation, and in the second expression (8.16) has been substituted for 7(0,y,t). The sec- ond, third, and fourth term in the bracket from equation (8.16) are not needed because the integration is even, and since, for example, cos w*t sinwt is odd the integration is zero. The third expression in equation (8.17) can be obtained from a trigo- nometric identity. The integration under analysis is continued in equation (8.18). Two transformations of variable are employed in order to get from the second expression to the third expression. The second expression can be expanded into one integral which involves (sin( - p *)M)/( » - » *) and another which involves (sin(# + » *)M)/(p+ p *). In the first integral the transfor- mation of variable given in equation (8.19) is used and in the - 183 - TAXX = 940d (02'8) p+ 71-7 42) wa} puodas (61'8) D+ 1=7 a) Wa} 4suij : [o)(0) te (81°8) @P Pp WD? uls N( gus— - 3* apr 5) Jus ee 6 . pus —® oles u sTintr on (pu 377 OM pati) (0 War 8 * Leng) ('0 ano] | GP Pp aR + ; N uls N ad Gus) ,(?+7/) 5 ( oe g uls a | wld 7- = gp tp Pulido uot sous, +auise puis 2 —guis ay gen Wied) us Wl (i 7)us | 7 aD) + N[@us E gus vue (Q ) Of UI] = 2¥ CAT, UJPIM PUD UOIJOANG aplUly JO aOUDGUNJsIG D 40} aUDdid ‘A SY} Ul Wa|qosg AN|DA |OI}U] a4 —184- second, equation (8.20) is used. As M approaches infinity, the results of the lemma given in equation (8.9) can be applied. In the first integral, the range of integration includes a = 0, and in the second integral it does not. The limit in the first case is consequently a definite value. In the second case, it is zero. The second expression in equation (8.21) is the limiting value as M approaches infinity. The limit as N approaches infinity can now be studied. There are two terms in the bracket in the second expression of equation (8.21) and the integration can be written as the sum of two terms. The transformation of variables given by the upper sign (where applicable) in equation (8.22), (8.23), and (8.24) can be used in the first term, and the corresponding relations with the lower sign can be used in the second term. The result is the third expression in equation (8.21). The range of integration in both integrals includes the origin, and as N approaches infinity both integrals have a limit. . The limiting value is given by the last expression in equation (8.21). It is an even function in 9* as should be expected from the form of the original integral over 7 (o,y,t) times even cosine functions. In equation (8.25), a similar integral is evaluated where the cosines have been replaced by sines. The result is an odd funetion in O*. When the two equations (8.25) and (8.21), are taken together, it is possible to solve for a(# *,6*), and the result is given by equation (8.26). Similar operations with cos (m *)*/g sin@]sin p*t Sab = MAXX 401d (p28) (gf 242+ Quis z) uls =@ (247) 4 gus =) —1 (e299) +H, zl). 9p (228) g- (,9 us sus} : @ 809.(,7/) Z ( ld 8) [a-7) 0+( 9571) | 6,4 (@ Ms-1~) 7 g (4) F fmen gp co Gi uls— Sean wi] + 9P N@ us +9 uls), us‘) OB x Wi} = vamening g Lgms-n | os eae a g uls- OM (a ws - gus) cop (9471) a ie (N ( Soe ag (NC g uls—@ U8) cay Jue * Z 3 NSAWa es 1p Ap 471 $09 (Ag uIS £.)s09 (4'K‘0) & | [ai oF UIPIM PUD UOI}OING ajiuly JO BOUDGUNySIG D 4O} auU_]g 4°A ay, UI Wal|qo’g AN|DA JOIJlU; aYyL -186—- and sin[( uy *)*/g) sine ]eos p *t, make it possible to find b(#*,e*). The spectrum of the disturbance given by a(mu *,0*) and b(y *,6*) has now been found from the function 7 (0,y,t) which was known. These known values can now be substituted into the original formula given by equation (8.28) which is known once N(o,y,t) is given. The integration in the square brackets must be carried out before the integration over # * and 6*, and in order to emphasize this, the y and t which disappear due to the process of integration are not starred, and the ones which will remain in the final solution are starred. a i i ee ee es ee ee a ee SS Given (x,y) at t equals zero, it is possible to find 7 (x*,y*,t*) by the methods used above. Formulated in terms of Y, where Y is the spectral wave number, equation (8.29) pre- scribes a motion such that each elemental wave in the motion has a component of travel in the positive x direction. A derivation which follows the procedures used above very closely then yields the final result as given by equation (8.30). The values of a(vy *,0*) and b( VY *,e*) are given in the brackets and can be found given 7(x,y). The use of vy instead of # is more convenient in the deri- vation but not essential. The variable could just as easily have been #. The transformation of variables given by equations (8.31) and (8.32) then yields equation (8.33). The wave system conse- quently has the same form as the previous system. - 187 - WAXX. 940id (82 °8) OP AP (a Quis, K+ 9 Soy) ae us yp Ap(4y7/— A UE =p Bus (4'K'0) Ul] | Seer || [+ 2 cot “oot cee ey 6327 Pp (4 7- FUE BEeL Bee ip Ap (471K 6 ey) SD ph fitate IT: (aif (x) & (12°8) lip Ap (ag OE: ) us’ viet | f] laa = (O17) q (92'8) 4p AP (4 Kg uss Sy iso2 witoy | Fh tee (9‘7!)o (SZ 8) #9 $09,(47) 2 N-,W- = =P Ap ius (A @ ws) us (460) | a YIPIM PUD UO!}OING ajiul4 JO BOUDGUNISIG D 40} aU_|q 4A BUY Ul WalgGOsd SNjDA |OWIU; ay] —|86- XIXX 401d (€¢'8) ry O pte ( Quis A +6 S000 a uls |xp Ap (() Quis A +9809 a) ) uis.(A‘x) al Hel ag (Quis k+@s09 *) 4,]s00|xp Ap (( Quis A + S09 per iney a vo} | ae Nic (ze's) ,ap b= 7p rz 2" (O¢'8) con «9 P,AP (4 bah -(9 uIs,6+49 S09, x)juis. xp Ap ((@ uls f+ 9 soo x) A) uis . (A‘x) ef Jef ony on de (Quis, K+9s09 x) 1)s09- xp Ap((@ us K+9 S09 x). f)SO9. a 22 EINE (ao hy x) & opp (3 afl -( (62'8) Op aphy 6al\-( 9 us +9 soo x)a) us (Q‘2)q + (46a)\-(g uls A+Q S09 x) 2) soo (Q°a)o = (1A) GY fies D8JY AflUl4 D JAAD BDUDGINSIG D 40} auUDig A‘K aU} Ul Wa]qo‘g anjDA jONIU] ay) == The initial value problem in the y,t plane for a wave train of finite width and finite duration The problem to be solved now is the logical extension of the problem in Chapter 5 to the case of a storm of finite width. Equation (8.28) is the starting point and when 7)(0,y,t) is given, the problem consists of evaluating the indicated integrations. The initial values are given by equation (8.34). Outside of a certain range of y at x = O given by plus and minus one half the width of the storm, W.,no disturbance is observed at the source. Outside of a certain range of time at x = O given by plus and minus one half the duration of the waves, Do no disturbance is observed at the source. Inside of the indicated rectangle in the y,t plane, a disturbance given by A sin((y4°/e)y sin @, - #4t) is observed. For y fixed, the disturbance inside the rectangle is a disturbance whose record, as a function of time, would look very much like the disturbance produced in Chapter 5. Since # 1 is a fixed number, the apparent period of the disturbance would be given by T, = 2 / 1 Within the time interval given in equation (8.34). For a fixed 6,, },, and t, as y varies, the disturbance is a slowly varying sinusoidal function, if 8, is small. The small- er the value of Q1> the more rapidly the crests of the disturbance move in the y direction. The crests in the y,t plane do not move in the y direction with the speed of gravity waves because they are really only a component of the wave as observed at x = 0. Given the graph of 1(0,y,t) the use of equation (8.34) determines By and 0, uniquely. Several integrals must be evaluated in order to obtain the - 190 = XXX Did 6 n'a Se qd il uls 7 — gus (9¢°9) (i ) — g uls fg) ooh Se lew se grag! Sell pe ee ig 2 ( dst) urs 2 (gu x +@Quls wus ra we sql Out ls uls mas Mg IPP EU Hy s00 (ACO US 5, +8 uis © )) soo - Ue 7 soa OS salt, £)) soo <: 37 3 iz Sm "a . ang ip Ap (4,779 us KS) uis (¢7/-'g us KS) us y u i gf : 2 Je iT SM AG 62 3 5m Fa, (s¢') O = 4p Ap (471K Q uis-£)soo (1'7/-' uls Nag 6) uIS ¥ o* Zu “ ¥q BSIMJBU LO O (pes) = (}A%0) & 2/*M>k>Z/*M- PUD Z/"C>4>z/"C- H! ('7/—'9 us K6) us y : é SANTVA TWVILINI UOILDING AlUl4 PUD UIPIMA O4flUl4 JO UID4] BADM OD 404 aUu_|G 4°A ayy Ul Wa|qosg AN|DA jOIju] SYL spectrum of the disturbance. The first integral in equation (8.28) involves the evaluation of equation (8.35). All of the terms in the integrand are odd, the integration is even, and the result is that equation (8.35) is zero. The second integral in equation (8.28) involves the evaluation of equation (8.36). The integration is straight-forward and the result is given in equation (8.36). The results of equation (8.36) can be substituted into equation (8.28). An integral would then result over the sum of two terms from zero to infinity. The integral is approximated in equation (8.37), for ease of evaluation, by an integration from minus infinity to plus infinity of the term which gives the important contribution for p * positive. For all equations subsequent to equation (8.37), all of RS es re en SS ES stars for simplicity of notation. By the transformations indi- cated in equations (8.38), (8.39), (8.40), and (8.41), equation (8.37) can be put in the form of equation (8.42). The pair of equations given by equations (8.43) and (8.44) define a transformation of the spate over which the integration is to be carried out. The inverse of the transformation is given by equations (8.45) and (8.46). The Jacobian of the transforma- tion is given by equation (8.47). The application of this trans- formation to equation (8.42) yields equation (8.48) in which the original strip over which the integral was to have been evaluated in the # ,p plane now maps into the whole a,8 plane. | Were it not for the very complicated coefficient of x in - 192 - . D D 6 (pn +!7/) (8p'8) OP P| did ( (Sander sa alll cae a (2pg) we. = Nvigoove =: (98) ve (Gp8) ‘40-7 (ppg) o=1-'7 (ope) Berane yang W-p I- . a ge \ (2v'8) 9p Mp (a7! (0K + =I ¥) 47) us: oF vy =(4'ACx) G@ ° wy | \ (ive) %=@uis (Ove) %P 2-\/1-OP (6e'8) =9s00 (g¢°9) %=@uls ALIONTAWIS HOA S,, T1V LINO (2¢'8) a a Tp (\ 71 = £\uis. ! 14ers -OP, P(t, (0 uisiA £ 500) us ~ (2 -—2 us pwns ists] fav _ SA 9 uIs7 \f 19 us if UOI}DANG PUD UIPIM alUl4 JO UIDJ] ADAM D 40J BUD[q 4A YY Ul Wa|qOsd AN|DA jOI}lu] B4L the last sinusoidal term of the integrand, the integration of equation (8.48) would then yield the final result. Note that as De and We approach infinity, the application of the lemma given in equation (8.9) yields a simple sine wave of the form A sin((u 4°/g) (x cose, + y sin®,) - Ht) without edges. The integration as it stands, for D, and W, finite, is too difficult to carry out and it must be approximated. The term involving x in the last sinusoidal term of the integrand is approximated in equation (8.49). The second expression in equation (8.49) is simply a way to rewrite the original expression. Since the major contribution of the integral occurs near a and 8B equal to zero from the behavior of the other terms in the integrand, higher order terms such as those involving a> and at can be neglected. The third expression in equation (8.49) employs this approxi- mation. Also since the major contribution is given near a and B equal to zero the square root can be approximated by the first term in its binomial expansion, and the fourth expression is ob- tained. The final expression in equation (8.49) is the result of clearing fractions. Equation (8.50) is the aippioeinate result which is obtained when the approximation given in equation (8.49) is substituted into equation (8.48). The first term in the argument of the last sinusoidal term of the integrand is simply a constant as far as the parameters of integration are concerned. The remaining terms are functions of ae. a, Bo and B alone without cross product terms of the form of, say, a8. For simplicity let /1 - pie = K as defined by equation (8.51). = 194 = UXXX ®10id ATX -a/0N (¢¢°8) JF =Ay4x!'0- (2S°8) XK='Ih+'yx (1¢°g) eadeif\ 27 (OS ‘8) gp ooh x e10 Sepeatrare (iC ips) us} P g yy = 7 d3 ua & Dus zal feu xX) (6b 8) 27=1N J-\6 td_ x! “xgid xpogt UO!HO4ANG PUD YIPIM AylUly JO UIDAL AADM D 40} aUD|q 4A ayy UI Wa|qosg ANjDA |DI}IU) au, —195- The terms, xK, * Yer and =p x+ Ky, occur as units in the inte- grand. The term, xK, + YPyz> is the direction in which the wave crests travel, and the term, -p,;x + Ky is the direction of orientation of the wave crests. They are the coordinates of a rectangular coordinate system which has been rotated through the angle 8,, and they are designated by X and ¥ in equations ik (8.52) and (8.53). Note that X and ¥ are perpendicular. The constant term with respect to the variables of inte- gration can be factored out by a trigonometric identity, and the use of the notations given above them yields equation (8.54). The notation for the various constant terms can be shortened by the use of the symbols, C, D, E, and F, as defined by equations (8.55), (8.56), (8.57) and (8.58). Then the trigonometric terms under the double integral can be split into a product of two integrals by expanding them by a trigonometric identity and the result is equation (8.59). Each of the integrals in equation (8.59) is an integral over only one variable and if one of them can be evaluated, all can be evaluated by similar techniques. The integral of one of integrals is given by equation (8.60). It is integrated by the very same techniques that were employed in the integration of a function of similar form in Chapter 5. The steps from equations (5.9) to equation (5.12) in Chapter 5 could be carried out (with a different variable for the notation) in order to obtain equation (8.60). The integration of equation (8.59) would then result from the substitution of equations like (8.60) into equation (8.59). Delta is the dummy variable of integration for thos® expressions - 196 - (09'8) (eta) 8S/ +89/ =gP( Seuis+ ¢-Zs09) SMe oo (65°) [Pp(eq+P9)uis—2 _+#/ gp(gl4+g/3) us—z2_ 4 | -pp(nq+,09)s00.,2 +, mane Mg. qb." op (S4+.9/2)s00. 2 _# ark: pp(2g+)soo—> _# | Pp(gy+g/a)uis 29 + | Sug 8. Rqoe's é aT 5 D g ee Pee EY ag JP f4+ 3)s09- Ge 4 (ih 7) S99 v=(VA'x)& d t ' : hy lof ' (es) S+4 (498) 482-3 (9¢°8) 1-R-a (G98) #P-9 (pS'8) AP P (f(A) 9(MEZ) +0()-A8)+,0 48) s09- ars ct/ (Wirt -sroq) UIs V+ gp p(X), (22) +4- Me) +08) us. 2. g +] [regrow (y'A'x) & x7 UOHDING slUI4 PUD UJPIM SJlul4 JO UIDAL BADAA D JO} auDd|g 4‘A aU} Ul Waqosy aNjDA JOIjlu; aUL -9P(g/4+9/3)s09-—2 _ 4 / -I97—- which originally involved beta, and gamma is the dummy variable of integration for those expressions which involved alpha origin- ally. Let the last expression in equation (8.60) be a short hand notation for the, expression which precedes it. Equation (8.61) is the result when this short hand notation is substituted into equation (8.59). Each of the indicated inte- grals is a Fresnel Integral and its value is consequently known. The terms preceding the two trigonometric terms in (8.61) deter- mine the envelope of the traveling wave, and since an expression of the form G cos6 + H siné can be written in the form (a? + q2)V2 sin(e + tan7/c/d), the last expression in equation (8.61) shows the results of this transformation. The expression, FF(x,y,t),is equal to the sum of the squares of the two coefficients in equation (8.61). It will turn out to be the two dimensional equivalent of the one dimensional Fres- nel filter described in Chapter 7. When the process of Squaring and clearing terms is carried out, the final result is the last expression in equation (8.62). FF(x,y,t) will be referred to as the Fresnel filter for a storm of finite width. Interpretation of results The expression for FF(x,y,t), given in equation (8.62), is a product of two terms which involve Fresnel Integrals. The function will first be treated for a fixed value of x as a function of y and t. Each of the terms in the large bracket is essentially two inside of a certain range of t and y for a fixed value of xe Outside of this range at least one of the terms is nearly zero and the product is therefore nearly zero. Consider the first - 198 - NXXX 94DIid (ge-41- 48) aah, (or he) 9h (29'8) \- (§- Dats (2 - pats Ap AF urs jee (doceetlen gt )+ *£9P,8 g £so9 ) |+= (B-s- — ee XG (we sPE) BN Se 2k) mye i) ng 2 a +4, 9442 (1 )+(A9/)] (gs) + (go)] # = [4s | = a ef). (As/ Aa \(gs/ go/)]+[(4s/+40/ )(gs/ — gof}+ (Asf — Aof (gs f+ eft - AX) (19 8) [ WR) 4 Ulm ste Juss» [445 34 | = (orga) us [CAsf— Aa Negsf — 92f) (As) + 4of 1(ge/+g0f)] $+ (i! 945) soo [4s] +40 )1¢s/ ~ go )«(As| 0 igs] +99/ <= ('A'x) UO!}OANG A}lUl4 PUD YIPIAA AflUl4 JO UDA, BADM D 40} BU_|G 4‘A BY} Ul Wajqosy ANjOA joI}iU; au] -—199- bracketed term. If Y is zero and W, is fairly large, for an x which is not too large, the integrals will have to be evalu- ated from a large negative value to a large positive value and by the same arguments employed in Chapter 5, the value of the bracketed term will be essentially two. If, say, ¥ = K,W./2 then the integrals will have to be evaluated from zero to a large positive value, and the value of the bracketed term will be one half. Thus when Y = KW, /2 or Y¥ = -K,W./2, the potential energy, averaged over a relatively short interval of time, of the waves at that point under the envelope will be one fourth of its value near the center of the disturbance. Similarly in the second bracketed expression if (2 p4x/eK,) -t- (D/2) =—0), or if (2m 4x/eK,) -t +(D,/2) = 0, the average potential energy will be one fourth the value at the center of the disturbance. If the four equations treated above are put back into their original form as a function of x,y and t and 85 by the use of (8.51), (8.52), (8.53), and (8.41), then y and t can be found as a function of x and the other parameters of the solution. The result is equations (8.63), (8.64), (8.65) and (8.66). Fora fixed value of x, O55 WW, Dis and p 1? these equations are equa- tions of four straight lines in the y,t plane. Segments of these straight lines are graphed for x = xy in the upper right of the y,t plane shown in figure 18. Their intersection determines a rectangle in the y,t plane. Inside the rectangle, the disturb- ance is at essentially full amplitude, and at the heavy bound- aries as indicated on the figure, the average potential energy if one fourth of what it is in the center. - 200 - Interpretation of Results Quarter Power Points of the Function FF(x, y, ft) For fixed x in the yt plane: _ Ow 4 2px 1= 2 +o cos 0, __ Dw , 2px 2 *gcosG, For fixed t in the xy plane: W. Wiy= cos G5 W na =— cos As _ gcos 8, (3+ t) X= Diy gcos 6,(St +t) an PLATE XXxV — Olle (8.63) (8.64) (8.65) (8.66) (8.67) (8.68) (8.69) (8.70) y = SS+tan8x—— —— =-—=*+tan§ x -— y 2 il | K= x, 241%, Ow 2uiX, , Dw -D 2S SS peel =o joo =u ! gcos@, 2 gcos0, 2 gcosO(t+2") > ¢ a yay 2 ye eyo Fig!8. The Quarter Power Boundaries of the Envelope of the Solution. The heavy lines are the boundaries.. The dashed lines are portions of the equations of the boundaries. The wave crests are shown in the xy plane. ! eG o0s8Dw _gcos6;Dw —-202- When x is not zero, as a function of y, the profile of -FF(x,y,t) where it exists looks at an edge like the dashed curve in figure 9. Similarly when x is not zero, as a function of t the same profile would be found. The product of the two profiles is rather complex at the corners of the rectangle. As X approaches zero in equation (8.62) the radicals in the integrand become infinite. The quarter power lines move to the position indicated at the origin of the y,t plane in figure 20. This shows that the solution reduces to the initial values given in the formulation of the problem despite the approximations employed in evaluating the integral. N ow the function will be studied for fixed values of t as a function of.x and ye A second set of coordinate axes defined by X and ¥ as given in equations (8.52) and (8.53) are also use- ful. The bottom graph in figure 18 shows the two coordinate sys- tems. The quarter power points in the Y direction are given simply by equations (8.67) and (8.68). In the x direction, they are given by equations (8.69) and (8.70). The area in the x,y plane occupied by the waves is consequently a parallelogram with sides parallel to the Y axis and the x axis. The individual wave crest segments are parallel to the Y axis and travel in the positive X direction. On the X axis, the value of X for the forward edge of the disturbance is given by X = x/cos0, = g(t + D,/2)/2y, which shows that the forward edge of the disturbance travels in the positive X direction with the group velocity of waves with a period }# j- = 20Oe) = Two approximations were employed in the derivation of the solution. The first approximation was in the formulation of the integral representation given by equation (8.37). This in- tegral representation permits some of the spectral components to have a component of travel in the negative x direction. By the arguments given in Chapter 5 for the simpler case, this ap- proximation is probably not too bad. The effect of the other approximation, namely that given in equation (8.49), is probably more important. The approximation is more accurate for small values of ©,. With ©, greater than 7/4 or less than -7/4, the approximation becomes poorer. A more accurate evaluation of the integral might show that the parallelogram form in the x,y plane shown in figure 18 would tend to lose the sharper corner and assume the shape of a rectangle with sides parallel to the ‘ X and Y axes as it travels along. The approximation is adequate in the sense that it locates the disturbance fairly precisely and shows where it is not located to a great degree of accuracy. Additional comments It would be possible to take the two initial value problems given in this chapter and manufacture some model wave systems from storms at sea which have properties which would be analogous to those models studied in Chapter 6. Models with discrete spectral components which would travel in all directions within a 180° sector could be manufactured. They could be made to be infinite in duration and width, infinite in duration and finite in width, finite in duration and infinite in width, and finite = 2040 in duration and width. The elemental unit of construction would be a finite wave group which would last for only a few minutes and be only a few kilometers wide as it passed the line x = 0 near a point’ y = Yor The most realistic model possible by these methods would be a model something like the last model described briefly in Chapter 6. The analysis of the properties of the model would turn out to be very complex, and the results would be indecisive because of the difficulties involved in evaluating the potential energy. None of the above possible models would be Gaussian in character, and since there is evidence that a wave record is very nearly Gaussian in character as shown in Chapter 7, they would all be unrealistic. For this reason, none of these models will be treated. Instead, in Chapter 9, the Gaussian case for a short crested sea surface will be treated, and FF(x,y,t) as derived in this chapter will be applied as a filter to the Gaussian case in order to forecast the spectrum of the waves in the area of decaye 5 208. Chapter 9. THE MATHEMATICAL REPRESENTATION OF A SHORT CRESTED SEA SURFACE BY A LEBESGUE STIELTJES POWER INTEGRAL AND THE PROBLEM OF FORECASTS FOR A STORM OF FINITE WIDTH AND FINITE DURATION OVER A FETCH OF LENGTH, F. Introduction Now that wave systems have been represented for the short crested case by Fourier Integral Theory, the next step in general- ity is to represent an area of a short-crested sea surface by a Lebesgue-Stieltjes Power Integral. It will then be possible by extending the methods of Chapter 7 to devise a forecasting proced- ure for an actual short crested irregular sea surface in a storm at sea of finite duration and finite width. The Lebesgue Stieltjes Power Integral for short crested wave systems The logical extension of equation (7.1) to a short crested sea surface is given by equation (9.1). The integral could also be considered as if x were zero and the term x cos 9 were absent. It would then follow from the properties of gravity waves in deep water that the term x cos © could be added immediately under the assumption that the disturbance occupies all of the x,y,t space. Note that the integral is not completely general in that it does not represent waves traveling in all possible directions. All component waves have a component of motion in the positive x di- rection. A more general representation would later on necessitate evaluation of derivatives of the sea surface. For most systems, equation (9.1) is general enough. - 206 - The Lebesgue Stieltjes Power Integral for Short Crested Wave Systems ates | [cosltncosd-ysind)-etey(y,0) PE(u,@) (9.1) E2(0,6)-O (9.2) Ex (wjZ)=O (9.3) OEAp x » Qe) Ealn) (9.12) and also En eiees the) = Catena Gs)) Celie Ge) Hea itxs 0s On COeNS) let O < pri< plo(M 9,0) and Ej(f g,6,5)- The only one not zero of those that are not can- celled out is E,(H2,05)- The next step is to integrate over y and t and pass to the limit as L and T approach infinity. The only term remaining is 48, ( fog )8oR) (See for example equation (6.64).) Then as Q5p approaches 1/2, the next expression is obtained. And as p# 28 approaches infinity, /2 is obtained. Finally, the same re- Bo max sults hold as A,b and A,° approach zero. Therefore equation (9.24) is proved. The results hold for any value of xe Consequently, they hold for an average over x also. In other words, equation (9.24) and (9,25) could be modified by another integration from x* to x* + L* and a division by L*. Then the limits as i L, and L* approach | infinite would be the same as the limits as they are given. Some examples Various examples of the integration of equation (9.1) will now be described. These examples will all be examples of the non- Gaussian case. They are of interest because they show that all of the systems which were infinite in duration and width in the = Bales Figi9 The Properties of the Net in the (u%,@) Polar Coordinate System. —214- past chapters come under the properties of this particular integral. The form of Eo( 39), for example one is given by equation (9.29). W(p,8) is zero. E,(p,0) is indicated schematically by the little polar coordinate sketch on the left. Graphed as a surface in the three dimensional ,0,E,(,0) space, E5(p ,e) would look like a vertical cliff along the curve pm = Ho be- tween © = O and @ = 7/2, and the curve 9 = 0, between M= Hy and = 00. There will be a sharp corner at the point (#,,0). ; would exist to the upper right behind these A plateau of height A two curves, and E,(#,®) would be zero everywhere else. Now consider a partial sum such as equation (9.22). For any net, a portion of the net will look like the magnified part shown in the plate. There will always be an area element in the p ,0 plane which encloses the point (70). For this particular net, the appropriate corner points are given by C5528) 9 CH 5440982) (4554028 omeo) and (5518 om40) in a counterclockwise order. The Square root of the appropriate term in the partial sum then has the value, A, for this set of four points as shown by equation (9.31). All other elements in the net contribute nothing to nee partial sum. For example, the contribution of the element to the right of the one just considered, yields a value of zero as shown by equation (9.32). Consequently, for this particular partial sun, the value of the partial sum is given by equation (9.33). There is always some set of four net points such that equations (9.34) and (9.35) hold, and as A, and A,° go to zero, the final re- sult is equation (9.36). Consequently, example one is simply a single sinusoidal wave traveling in the positive x direction of - 215 - Some Examples Example L as Px 0:8<0 3 E.(u,@) O;

ei ( re and Grue0>Oy (9.34) in the limit (x,y,t)=Acos (4g-x-pst) (9.35) Plate XXXIx = 35> constant period and amplitude in deep water. The form of E,(#,0) for example two is given by equation (9.36a) and the appearance of the function is given on the little figure to the left. wW(,0) is defined in small strips which cover the jumps in Eo(H 50), and it can be anything otherwise as shown by equation (9.37a). A partial sum, if the net points are close together, for the element which encloses the point (#7729), then yields for the square root term the value 2 2 2 W/y The final result, in the limit is given by equation (9.38a). From the above two examples, it is evident that all of the examples given in Chapter 7 are special cases of the integral given by equation (9.1) in which the sea surface does not vary in the y direction. In example three, Eo( # ,e) is given by equation (9.36b), and W(#H,°8) is zero. For the area element which encloses the point (17/6), the radical in the partial sum for that term is given 2 aO)l/] =iae The dnteprel We consequentayeieen ip Ge Seen by equation (9.38b). Equation (9.38b) is just a specific example of equation (8.4) as far as the direction of the two waves is concerned. In example four, the form of E,(#,0) is given by equation (9.39), and W(#,@0) equals 37/2. The integral is then a special case of equation (8.5). Infact, the integral is equal to the equation given in figure £3 when t is zero. E> (# 50) is an inter- esting function in this particular case, and a three dimensional sketch of the surface involved is given in figure 20. Note the = BG < Some Examples Tr 2 Example IL O; @ poe 0; ws" m0 ' 43 FP cpioo sin'¢z)<@(# 4) at the point and at any other point in the circle be smaller than an epsilon (which may depend on delta). Stated another way, E,(y ,®) is a continuous function in both variables, and it is monotonic non-decreasing in both variables (see Courant, Vol. I). Equations (9.41) and (9.42) are another way to impose these conditions. Finally, ¥(#,e) must have a value between zero and 27, and its value is picked by the proba- bility law given in equations (9.44) and (9.45). If equations (9.2) through (9.13) hold, and if the conditions - 220 - The Gaussian Case of the Lebesgue Power Integral for Short Crested Wave Systems. Equations(9.2) through(9.13) hold in addition | 2 2 2 if (Ls HK) +(8,j—-8)<8 then Eo (sK41,9)+1) —E2(wx,9j) < €(8) and O< ¥(u,8)< 27 P(O< V(HansG2p+i)< 4 277) = @ where OSa=) then equations (9.22) and (9.24) still hold. also if E,(n,8) has continuous first derivatives 67Ea(u, 4) on 08 =[Ao(#,8)]°aude alternate formulation of integral Tr 2 2 (X,Y, 4) [foe cos8+ysin8)—pt +¥(1,8)|VAa(# Olean de iyi wT 2 d°E.( 4,0) = dud@ (9.41) (942) (9.43) (9.44) (9.45) (9.46) (9.4 7) S-! RI 2 MX, tf) ee 2 -cos(B20tl(xcos@,,,,+ysin8,,4,)-Hansit+¥ Hens! Gopal —— + . ano an z O50 V (Aol Hons Pope (H2n+2 —H2n(Or945- 929) A\po Abo (Monet) =lim Re>d-> @ s-1 RoI i eacaalh cos 855 atysing, o+t) Hansit +¥(Hansl,O2p4))) H2s—>@ Tia picks, aii ea a (Az(HanstO2pei)) (H2n+ez2 2a) (O55407 65) (9.48) Plate XLL -22I- given by equations (9.41) through (9.45) are added, then the limit of the partial sum defined by equation (9.22) still exists, and the integral given by equation (9.1) still has the property given by equation (9.24). If E,(,) has a continuous second mixed partial derivative, it must be everywhere greater than or equal to zero. Consequently, it can always be written as the square of some function An(H 58), and under some conditions equation (9.46) is another way to write denn GH 40). Substitution of equation (9.46) in equation (9.1) yields equation (9.47) which has a meaning only in the Gaussian case. The expressions for the partial sums can then be written in the forms given in equation (9.48). In the last expression in equation (9.48), the partial sum has again been expressed as a vector in the complex plane. It will be shown in Chapter 10 that for a fixed x and y as t varies, the short crested sea sur- face as observed at a point has all the properties studied in Chapter 7 for 7(t). The exact relations between E,(H ,6), E(u), (Aj(p 4e))? and [a(p )]° will also be discussed at that time. Some examples of cumulative power density functions and their power spectra Values of [Ay (m ,e)]? have never been obtained for an actual sea surface because the observations needed on which the compu- tation of the function depend have never been obtained. Some examples of what the function might look like can be given, and then the consequences of the form of [Ap (nu ,e)]° in the results of a hypothetical forecast can be described. It will be seen that the nature of the forecasted values depends critically on the - 222 - Some Examples of Cumulative Power Density Functions and Their Power Spectra. Example I A; E (1,9) =( B; Eo(u,9) = C; Eo(4,8)= (u | ae —— a za 2 D; Eo(1,8) = (u—- £710 a rail E, Eo(u,8)=0 pee HE NM 27 2 oa A; [Az(u,0)) = 0 A ie) B, [ao(2, 0) uA K* ZY : Example IL Ex(we)= 0 if -%<@< -3% HT, E,(#,9) = [xe Yen \+ cosSe\dude - 3% 0 K(2 Hor pt, atl ce sHiapE ee “read ley Ue Ver organ te if - Sh <9< 50 ‘g *T, T wT PA eee SU. ph ies. canor-[aznste 27™du" = 72 ale em-Ze 277) fo) if SPe8< FZ —pT [ap(2,8)] =Kue "72714005 88) if -S7<0<3% [a.(u,8))° = O otherwise Plate XLII =223—— (9.49) (9.50) (9.51) (9.52) nature of the sea surface in the storm. Example one is a possible form for E,(#,6). The ,© polar coordinate system is broken up into five areas as shown in the little sketch on the side of the plate. In area A, the value of E,( 48) is constant and equal to the values given in equation (9.49). The figure is cut off at finite values of ” , but the same value holds for © between 7/4 and 1/2 and # greater than 27/5. In the other areas, the values of E,(u,@) are given by the appropriate functions in equation (9.49). Note that E,(p ,0) is continuous. Area D is the only area in which E5(p 40) is a function of both 6 andw. The second mixed partial derivative is different from zero only in this area. [a,(p ,0)]1° is equal to K* in this area, and it is zero everywhere else. For any partial sum, with a small enough net, there would be no component wave crests travel- ing in directions between -m/2 and -17/4, and in directions between t/4 and 1r/2, and there would be no component wave crests with periods greater than 10 seconds or less than 5 seconds. In the limit, if the phases were random, the wave system would still be Gaussian. In equations (9.49) and (9.50), if K* equals 1.69 10? em sec, then E equals 2.5 10? om’. The potential energy in the 2max system averaged over y and t is then equal to 6.25 10” ergs/em- (if the product pg equals 103 em/cm*sec“). Example one is physically not a very realistic example. It would not be expected that a turbulent process such as the one which produces waves in a storm at sea could produce such a sharp = 224 - cornered power spectrum. Example one is given so that it can be compared with the next example in order to show how remarkably different the forecast results can be. Example two is somewhat more realistic, although it must be emphasized again that very little is known about the actual values of Ej(#,©) in nature. In example two, E,(#,0) is zero for © between -1r/2 and -37r/8. It is given by equation (9.51) between -37/8 and 37/8. For © between 3r/8 and 7/2, it is a function of / alone. The second mixed partial derivative is different from zero only for © between -37/8 and 37/8. [ay (pw ,0)]° is then given by equation (9.52). E,(# ,©) and [a,( #,0)]° are shown in figures 21 and 22, respectively. The isopleths of constant Eo(# ©) for 8 greater than 37/8 follow the circles of the coordinate system since there is no variation in ©. The power spectrum has a peak at 6 = O and B= 2n/T,° The values of the parameters in the equations for the evaluation are given by K = 2.68 10? em*sec*, and Ty = 10 seconds. is then given by 2.5 10? em>. The Eo max average potential energy in the system is then equal to 625 107 erg/cn@ (if the product pg equals 103 g,/em-sec°). This is the Same amount of potential energy averaged over y and t which would be found in a simple sinusoidal wave with an amplitude (crest to mean water level) of five meters. The same amount of energy as in the power spectrum given in example one has been used for purposes of comparison later. =) 2255= 208 e- ! I= E se Sem ce Onsen Kim py ee [o) = xa & 19 /] if WY on oa nO oes ee Be A= T=10 p= (2,8) FOR EQUATION (9.51). THE FORM OF Ea OPK DEK Gi —227-— The forecasting problem for a sea level surface represented by short crested waves, a Gaussian Lebesgue Power Integral, and ee ES SS SS ES ST the disturbance exists everywhere and has the same character every- where, once E,(# ,®) is fixed. If the disturbance were observed as a function of y and t at x = 0, it would be represented by equation (9.53). The limit of the partial sum given in equation (9.54) is again a representation for equation (9.53). In order to produce a localized storm, instead of a disturb- ance everywhere, the representation given by equation (9.54) can be multiplied by a slowly varying function of y and t as given by F(y,t). For a particular example, F(y,t) can be represented by equation (9.55). Other functional forms for F(y,t) with smooth- er sides might be employed for somewhat more realistic results, but the form given in equation (9.55) at least has the property that the area covered by the waves has a finite width and that the waves are of finite duration at the source. Waves produced by storms in nature are not so sharply defined as this model. The argument now follows the same line as the one which was used in equation (7.39). If F(y,t) as given in equation (9.55) is applied to 7(0,y,t) as given in equation (9.54), the disturb- ance which results is observed at the source only over a distance of length Wy and only for a time Doe If the disturbance is ob- served within the time interval indicated, it will be indistinguish- able from a similar short observation at any time and place in - 228 - 1x 41W1d asimMsauLO ‘O (¢9'6) = WO4! Yo =, 5 _Ud ey s As _X ‘ 94'S qi / > (Mq 991) 6 }I puD Sak ,-UD} > GQ > aah UD }! | aoe aSIMJQULO * O XA —fS09— xe i is Sek Beg LX oe EMO Gog > (Re SOG }| puD ele Se UO} ft | Ney aah CS) oth seh (196) * =epud} (09%) Fae (6G'6) 1g s09-6 = Kigsoo+ x iguis --1 (896) oe = "gud} (ZS) "9809-& -= k"gsoo + x"puis- = "x ee XZ (sey hue XZ 68>) _ bug (eo) bxuZg | 6x2 gsooxal{ (“ES)6 7) QS0OXZ soar SM oa gsoare (G25 7, \|SP2 Si é uls 4 ae Or € soo ite Y_(9‘r1' ‘KX) 4 4 (9¢°6) oP 8a 2 uls a he soo | aSIMJ9YLO * O ($6) Lok> Z Mqsisg PU AS ommy oto. ee i oS (y‘K‘o) L : dz 2402 \ uz 2+u 402 14UZ g +42 |4u Wu i tose) (Bene) [BH] [OB Mpa He Bus K rete 509 . gies . (€S°6) (gi7/ Ne a2e)(g'71) ) + yr1- guish—7 6) )s00,] | = (4‘A‘o) & “SAN ULDIM 40 W401S D jo abpy ay) $0 spuodas “q jsp} yOu} SaADAA PUD “aiaitn Jamod anisaqeay UDISSNDO D “SAADM Payse4D J40YS hq pajyuasaidey adDJ4nS Das D 40} W|qOig Huljspd9404 ay] =O) = the disturbance covering the whole y,t plane. F(y,t) can be applied to each term in any finite partial sum. As the indicated limit is approached, the result of the operation by F(y,t) can be treated as a filter operation on [ay (ye) ]° in order to find the power spectrum at other times and places as the concentra- ted disturbance at the source disperses and spreads over the x,y plane. The problem of the result of the application of F(y,t) toa particular term in the partial sum was solved in Chapter 8, apart from minor modifications necessitated by the arbitrary phase, VC Home P2pe1)* These modifications only serve to complicate the algebra in the analysis and the same filter function is ob- tained. The end result is FF(x,y,t) as given in equation (8.62). This filter function is repeated with modifications in equation (9.56). The time variable has been referred to top by a change of variables, and the filter is also given as a function of w and 0. IGP sey NACE D, and W, are fixed, FF(x,y,t,#,®) can be varied as ob? “w a function of “ and 6, and the filter properties can then be deter- mined. The 6-band cutoff points The Fresnel filter, FF(x,y,t,#,9), has the disadvantages of the corresponding filter given in eauation (7.50) in that it oscil- lates rapidly near the quarter power points as a function of # and ©. It can be approximated as before by the square cutoff filter. In the first term given by equation (9.56) the quarter power points occur where equations (8.67) and (8.68) are satisfied. With the use of equations (8.53), (8.51) and (8.31), equation (8.68) can be - 230 - put.in the form of equation (9.57). The points x and y are treated as constants. © is treated as a variable, and when 9 equals 0,,, as © is increased, the term in the filter passes through the value one half. For © greater than ey the term rapidly becomes zero. The result is that tan ©, is given by [y + we/2) 1/x. Similarly equation (8.67) yields equation (9.59). For ® less than ®,, the term in the filter is nearly zero, when © equals 6, it is one half, and for 9 greater than oy. but less than Pee) the term is essentially two. The properties of the filter which cause it to cut off all ‘ but a certain angular band width of the power spectrum at the source can be explained by reference to figure 18 and to figure 23. Figure 18 shows that for a fixed value of 6, the disturbance in the x,y plane remains between the two lines, ve = cos 6,W/2 and Y, = - cos O,W/2- Consider then in the x,y plane, the area which can be occupied by a disturbance which travels along the line da = 0, for a fixed point x = x, and y = y,. The lowermost part of that disturbance as shown by the dashed line a = - cos @,W/2 will just miss the point XiYy> and any disturbance which leaves the source at directions greater than 8. will never be observed at the point X1Yz°- Similarly the disturbance which travels along the line X = 0, will pass just below the point x,,y,, as shown by the dashed line, Y, source with a direction less than 8, will never be observed at the = cos 0, W,/2- Any disturbance which leaves the point X,,y,- Equations (9.58) and (9.60) have a simple interpre- tation in terms of these considerations when interpreted with the = 2 = aid of the upper part of figure 23. Another important direction namely the direction to the point x,,y,, is given by equation (9.61). The # -band cutoff points The second term in equation (9.56) can be studied as a function of # and © for a fixed value of x and tobe The upper and lower ranges of integration when set equal to zero, yield the information that when # /cos © is less than g(top - D)/2x, the disturbing ele- ment in the partial sum will already have passed, and that when P/eos © is greater than gt ,/2x the disturbance will not yet have arrived. When the © band width is small, the variation of cos ® is small and the range of the values determines the range of p essentially. The square filter for the Gaussian case of a short crested sea surface in a disturbance which lasts Do seconds at the edge of a storm of width, Ws Under the assumption that the Fresnel fringes will cancel out because of the finite time of observation, the square cutoff filter for this model wave system can be given by equation (9.62). Since 0,< 8p <0,, and if ©, - 9; is small, the value of H/cos © can be approximated by M/eos 8p- 8p is the angular direction of the point X59¥,, from the point x = 0, y = 0. The second inequality in equation (9.62) can then be multiplied through by cos 6p, and the result is a factor of the form (cos Op )/x- The factor, (cos Op )/x is simply equal to the reciprocal of the distance a given elemental disturbance must travel to reach the point (499) and consequently it is equal to 1/R. The value of R is measured from the center of the forward edge of the storm to the point of forecast. With this = Dee Slight approximation the square cutoff filter is given by equa- tion (9.63). The result is then an equation analogous to equa- tion (7.45), in which R has been substituted for x. Equations similar to equations (7.46), (7.47) and (7.48) would also result and remarks similar to those in Chapter 7 could be made about them. The @-band width From equation (9.58) and (9.60), it is possible to form the difference given by 0 TeeE and determine the 6-band width. The result is equation (9.64) where A © is the angular width of the Square cutoff filter. The 6-band width is not of equal width above and below the value ©)- This is shown in equations (9.65) and (9.66) which show that Ae.) the variation in radians from 8p to 8 (the upper cut- off angle), is smaller than Ae, the variation in radians from the lower cutoff angle to ®p- The square filter for the Gaussian case of a short crested sea surface generated by a storm of finite duration D., finite width, We» over a fetch of length, F. With the realization that the wave systems under study are only 3 first approximation to actual wave systems from a storm at sea (mainly because of the nature of the functional form of F which has been assumed; and not because of the inadequacy of the Gaussian Lebesgue Power Integral), it is possible to account for the effect of a fetch of finite length. If a wave record is observed at a point x = 0, y = 0, (or any value between + W,/2), there is of course an ambiguity as to where the waves come from = 233n— ATIX ®1091d : , fee ten . Ja;0M de2eq (12°6) ep7p-[(9‘7/)*v] [4M'9'3'S] Siege eas S00 ic JO} DiNws04 ys090s04 |OuUl4 v (Fa) (696) UNECE He ry — asaym aSIMJa4UjO O (896) x oe eX =i ae u)2 = JM9AS aSiIMJayjO O | 6 s(AVKY om (19'6) gok> E41 pun {yp ttGet>0 s! (7 AYS : 2Xo 2x (99°6) [eg (S9'6) [ XC _X Ast oh v ( k) +3 (796) Sid Tet aa Ss UDy= X_) up -(s— u}) D4) uD Eee =1p _"p = = 5 (za) ; (Zan) ) 4D4)uD4), uo) = (Zn, zu (Zan ), us "9" = @V $O W401S DO JO} 1; 04ba4uU; d*yybuay 30 yoyay 0 araectoneone yo°*M*ULPIM Jam0g anbsaqe7] udIssnDd Dd ‘sanDmM paises 4404S Aq Pejuaeseiday @9DjJ4NGS DAS D JO} Waqosg Buljspdes04 ay} PUD YIPIM PUDgG OE PY4L behind the y axis. The lower part of figure 23 illustrates two possibilities which could occur. Suppose that an area ofr relatively strong winds covers the region between x = O and ena y= W./2 and y = - wo/2. Also suppose that the winds last Ds units of time. Then at the time t = D, an ele- mental wave which has left the storm area would occupy the area ahead of the y axis which is bounded by the solid lines. When the winds stop at t = De» the area behind the y axis bounded by the solid lines might be occupied by an elemental wave component which could travel off in the 6. direction after t = D,- This al would imply a rather peculiar behavior of the wave component, and the two obliquely oriented areas would sweep out a rather peculiar area as they travel along. In an actual storm the area covered by the winds merges gradually into the area of relative calm. In addition the wind direction varies turbulently over the storm area. It would there- fore be equally consistent to assume that the area occupied by the wave element traveling in the 8, direction could be given by the dashed area at the time, t., = Doe At any time after t,, = 0, one observation, sufficiently detailed to determine E,(p,©) at x = 0, y = 0, would yield only enough information to show that either of the two assumed areas could be used. A system of weather maps which could yield such a pattern and which could occur in an actual meteorological situation, will be described in a later chapter. The filter under discussion will be derived under the assumption that the elemental wave system occupies the area bounded by the dashed lines. = 235 - The determination of the angular bond width of the filter for a storm of finite width. ———e=== Possible area covered by waves at ¢ =D, for storm of finite width. =<. === Assumed area for the computations of the square cut-off filter for the storm of finite width. Fig.23 Filter Considerations for a Storm of Finite Width. —-2356- Under the above assumptions, the modulating function Fp(y,t, +) is given by equation (9.67). Equation (7.51) of Chapter 7 modified by the requirement that the storm be of finite width has been employed, and the argument that shorter period waves require a longer time to travel from the rear edge of the fetch to the point, x = 0, y = 0, is employed. The square cutoff filter for the Gaussian case of a storm of finite width and duration over a fetch of finite length under the above assumptions, is then given by equation (9.68). The HM cutoff values are given by an equation similar to equation (7.57) except that they are determined by the distance R and not by xe The © cutoff values are given by equation (9.64). The # band width is given by equation (9.69), and the © band width is given by equation (9.70) in which R° equals x° + ye It should be noted that equations (9.64) and (9.70) hold only if R°>wW,“/4, that is, outside of a semi-circle with a center at x = 0, y =0, of radius W./2. If the point of observation is inside of this circle, the expressions for Ae, and Ae, given by equations (9.65) and (9.66) can be employed. The filter might be smoothed by arguments similar to those employed in deriving equation (7.61). The final forecast formulas for waves in deep water If the entire forecast is to be carried out in deep water, the final forecast formula is then given by equation (9.71). If the short crested disturbance at the source is given by equation (9.47), and if the disturbance is produced by a storm of finite width and finite duration over a fetch of length, F, then the forecast formula states that the short crested disturbance in S207 3 the vicinity of the point X19, ot op is given approximately by multiplying [a5 (# 50) ]° by the cutoff filter given by equation (9.68) and integrating the resulting Lebesgue Power Integral for the Gaussian case. More precisely, the wave system at the source is one system from a whole statistical class of systems given by all the poss- ible forms the free surface can assume upon forming all of the possible limiting partial sums which can be obtained from equation (9.48) with all possible combinations of the random phases. Thus the disturbance at the source is one of an infinite number of possible disturbances for a fixed functional form for [ks (ese Nlan and the disturbance in the area of decay is one of an infinite number of possible disturbances for a fixed functional form of S.F.G.W.F. times [A510 le Also more precisely, when, as in the last paragraph, integration of equation (9.71) is referred to so glibly, one should think only of some finite partial sum evaluated with a sufficiently small net to yield a result ade- quate for the problem under study. In addition, the indicated forecasted sea surface should be considered to be valid only for a relatively short time, and only over a relatively small area of the sea surface. CS SS Three sharp cutoff filters have been described above in equations (9.62), (9.63) and (9.68). For a fixed value of x, y, and top? they determine an area in the # ,@ plane. Inside of this area in the # ,© plane the power spectrum is the same - 238 - as at the source, and outside of this area the spectrum is zero. In all three filters, the © cuteff values are given by the same expression. In each of the three filters the # cutoff values are slightly different. The determination of the 86 cutoff values and the ©-band width will be described first; then the various ft -band width determinations will be describeds and finally some sample filters will be graphed. Determination of the 0-band width and the 6 cutoff points The angles ®p, A®,, and Ae, are functions of x, y, and Wee For a storm of given width, the values of x and y then deter- mine these angles. If the width of the storm is doubled, and if the values of x and y are doubled, the same values for the angles result. Consequently, if x and y are measured in units of We» the angular cutoff values are in a sense independent of the actual width of the storm. In figure 24, lines of constant 6), Ae, and A 8, are shown on an x,y coordinate system with units marked off in terms of the width of the storm. Note that A 8; is slightly greater at each point than A@,. Given [a,(# ,e)]° and the point in the x,y plane at which the forecast is to be made, measure x and y in terms of Wy and enter the point on figure 24 to read off @), Ao, and Ae,. Then in the ,© polar co- ordinate plane draw a line through # equals zero along a radius at the value 8p- Mark off an angular increment in the positive 6 direction equal to Ae, units from the values 9p, and a decre- ment in the negative © direction equal to Ae,.- Draw the two radii obtained. The two outside lines then cut off a sector of =239- “SM ‘HLGIM WHOLS JO SLINQ NI JNVId A‘X JHL YOS ev gny ‘"av'"@ 4O SHLIId0SI +2914 19@V ONY "gy 4YO4s SYINWHOS 3SN a Sh +x 4O4 z Io) = So isnoo= '@ VY -———— NSNOO = 26 —— eee, ,0b="9V M 58 SN SMZ SME of 7O=X* zi) SS “4 oN NOYHL SSvd SMb LS ~ y, Ye ee y *y uD; OE SMpb YY —_— . ‘gh =A $Y3LNIO a 909 Emo fe) guy wea *SsaAuno Z Sa . — ole a5 F 4. 50S ¢ ANIOd 3HL HONOMHL SSVd ABHL 4 ‘wuney/Sm =x ¢-Fo-=, ly YaLN3O SMG LL etl HLIM S370NI0 ayv 'H="9 Y‘SIAUNIT 5. < Ss SMB L SM9 SMG “Mb SME SM2Z SM oSb > YOS BLVYNDOV AYBA LON AYOSHL the # ,® plane which determines the 9 cutoff values. Note that the @ cutoff values do not depend upon time, and that for a given storm and a given point of forecast, they are fixed once and for all for all forecasts. Note the complete symmetry upon reflection in the © equals zero axis of the coordinate system as given by the equation. Only the first quadrant is shown in figure 24 for this reason. Determination of the F cutoff values In equation (9.62), the ratio, H/cos 6, mast lie between two fixed numbers, once top? Do and x are fixed. An equation of the form, -H/cos @ = const, is an equation of a circle in the #,@ polar coordinate system which passes through the two points (PH = 0,0 = 7/2, and # = const/cos @),9 = 6p). The circle has a center on the line © = 0. The intersection of the two curves H/cos 0 = const, and M/cos © = const, and the two lines, © = Op + Ae, and © = ©) - Aé,, then determines an area in the /@,® plane bounded by segments of two straight lines and two ecimedesye In equation (9.63) in which cos 6) has replaced cos 0, H., is given by t.)/2R and #, by g(t,, - D,)/2R. Figure 15 can then be employed, upon reading R for x, to find the band width and the upper and lower cutoff values for the point and the time of the forecast. In the M,® plane the area bounded by pm = Ha b= Hy, © = Op + A®, and © = 6) - Ae, then determines the edges of the filter given by equation (9.63) once x, y, Wso tops and D, are given. (D, is ten hours for figure 15, but the extension to any value of D, is simple. ) Sah In equation (9.68), figure 16 can be employed to find the # band width and the # cutoff values for a storm with a duration of ten hours and with any length fetch. The # band width depends upon Ds; R, F, and tobe Beebe cas vadtaenals Figure 24 and figure 25 are all the equipment needed to deter- mine the filter characteristics for the filter given by equation (9.68). Figure 25 is simply a graph of the straight lines given by top = 2p R/g (see Chapter 7, page 162) ona top» # coordinate system for various values of the parameter, R. Given De Fy top? and R, the appropriate graph of the straight line oa - D, = 2H (R + F)/g can also be found, and the intersection of various lines on the diagram then determines ,, and w , for the filter. The entire procedure for the evaluation of equation (9.68) for a fixed set of parameters with the use of figures 24 and 25 will now be described. The given parameters, which could theo- retically be evaluated from weather maps are: storm width, 200 km (W.)3 fetch, 200 km (F); duration of storm, 15 hours (D,); x, 600 km; y, 600 km; and time of observation, 40 hours (top The evaluation of the @-band width proceeds as follows. Tan ©» = y/x = 1, and Op = 45°. x = y = 3W,, and from figure 24, Ae, = 4.4° and A®, = 5.2°. The ©-band width is therefore 9.6° and the © cutoff points are at 49.4° and 39.8°. (From equations (9.65) and (9.66), Ae, = tan7+(1/13) and Ae, = tan7/(1/11). The evaluation of the # band width proceeds as follows. R equals “2 600 which equals 848 km. In figure 25, locate the line t., = 40 hours, and the line labeled 848(= 850) km. The - 242 - N \ a z . point of intersection in the pw stop plane then determines the value of 7 which is equal to .81 radians per second. Next lo- cate the line t., - 15 hrs = 25 hrs, and the line labeled 1050 km(= 848 + 200). The intersection of these two lines in the M yt Q4 plane then determines the value of / 7 which equals e402 radians per second. The value, .402, corresponds to a per- iod of 15.63 sec, and .81 corresponds to a period of 9675 seconds. All spectral periods greater than 15.63 seconds or leS$ than 7.75 seconds will not be present at the point and time of *bservation. The filter for the given set of parameters thenmequals one inside of an area element in the p» ,© plane bounded by oF = 49.4 degrees, ©, = 39.8 degrees, # , = .81, (a segment of a circle), and p, = 402. Inside this area the forecasted power spectrum equals the power spectrum at the source, and outside of this area, it is equal to zero. The wave system at the point and time of ob- servation is then the Lebesgue Power Integral over this forecasted power spectrum. Some examples The filter given by equation (9.62) can best be evaluated by brute force. The filter given by equation (9.63) is simply a special case of the filter given by equation (9.68) when F equals zero. Figure 26 shows the cutoff boundaries of the three filters described above for various values of the parameters. The values appropriate to equation (9.62) are shown by the dotted lines when needed. The values appropriate to equation (9.63) are shown by dashed lines when needed and the values appropriate to equation (9.68) are given by the solid lines. The © band width is the same for all three = 244) = ey ae fz | FILTERS I, 10, IL, AND Iv Cogs R=850 KM. W,7 200 KM. F =200 KM. Top? 40 HRS. D,= 15 HRS. FILTER I 9,=67.5°, y=790, x=326 FILTER IZ 0)-45°, y=600, x=600 FILTERTE 6,- 22.5, y-326, x=790 Pili Cho 5 MoO 4 necsO FILTER ¥ FIG. 26 SOME EXAMPLES R=I700 KM. bed We-200 KM. Ob TE SSHARP = CUO BE F =200 KM. FILTERS FOR EQUATIONS caer (962), (9.63) AND (9.68). 20.50 (NOTE THE SMALL EFFECT OF x-1580 THE 200KM. FETCH EVEN FOR =-652 SUCH A SHORT DURATION STORM). —— SOLID LINE FILTER FOR SFGWF (9.68) ——— DASHED LINE FILTER FOR SFGW (9.63) (NO FETCH, 92 8,) =S=S DOTTED LINE FILTER FOR SFGW (9.62) (NO FETCH) —245— filters and the lines are given by solid lines for all three filters. See the legend at the bottom of the figure. The greater the departure of QD from zero the more the filter for equation (9.62) departs from the other filters. The approxi- mations employed in obtaining the other filters is therefore most accurate for small 6p. In addition the original Fresnel filter was more accurate for small values of 0). Consequently, these re- sults should not be applied too strictly at large angles. If the power spectrum given in figure 22 for equation (9.52) were to represent the disturbance at the source, then the appli- cation of the filters given in figure 26 would result in various quite different sea surfaces at the various points and times of forecast. There would be a very small disturbance at the point and time used to determine the particular filter labeled number I, because the power spectrum is identically zero for 9 greater than 67.5° and very low for © near 67.5°. In contrast for the power spectrum given by equation (9.50), the disturbance would be identi- cally zero. For filter number III, the power spectrum given in figure 22 would result in considerably higher waves at the point R = 850, and @p = 22.5° (corresponding to y = 326, x = 790 km) than at the point determined by filter number I. For equation (9.50) and the value of K given for equation (9.50) the waves determined by the filter at the above point would be considerably lower compared to those determined by figure 22 because only the components from 10 seconds to 7.75 seconds would be present (due to the original nature of the power spectrum). atoaG) 2 A study of the effects of given filters upon the two power spectra under consideration thus shows that the forecasted values would be completely different in many cases for the same storm parameters and the same point and time of observation. For many forecasts based upon equation (9.50) there would simply be no waves present, whereas for the same forecasts based upon figure 22 (equation 9.52) an appreciable disturbance would be present. These two examples therefore make it evident that there is no hope for consistently accurate wave forecasts until E5(p 8) has been measured for wave systems at the edge of an actual storm at sea. Dealing with the significant waves at the edge of the storm without regard to the underlying power spectrum can never yield consistent results. At this time, the hope that E,( ,©) will in some way be a function which depends consistently upon the wind velocity, and the air mass in which the winds are blowing so that it can be predicted is expressed. Methods for measuring E5( 49) will be given in the next chapter. Decrease in wave height with travel For the same power spectrum at the source (say figure 22), the effect of doubling R and top is interesting to study. Filter Vetior Cha = 22.5° for example could be reflected in the 6 = 0 axis. Then it would correspond to the filter for the same para- meters given on the figure except 85 would then equal + 22.5°. Thus doubling R (or x and y) and t.,, results in a power spectrum at the new point of observation with the same value ofp .,, but u A p and A® are approximately halved. Consequently the potential Saas energy averaged over y and t are the new point of observation is only one fourth of what it was at the closer point. The wave height which is (crudely) proportional to the square root of the average potential energy therefore decreases like 1/Rk. In particular, for waves observed on the x axis, at large values of x, Ap= gD,/2x and Ao = W./x. The effect of the short crestedness of the sea surface at the source is then of the same order of magnitude as the effect of dispersion, and the average potential energy decreases like 1/x°. Consequently, the actual short crestedness of waves from a storm at sea cannot be neglected in an adequate wave forecasting theory. At this point, reference is made to H.O. Publication No. 604, Techniques for Forecasting Wind Waves and Swell. This book contains the lates theory for forecasting significant waves as developed by Sverdrup and Munk. Consider, in particular, Plate VI of the above publication. It can also be found as figure 3 in Forecasting Ocean Waves by Munk and Arthur [1951]. It gives values of Hp/Hp as functions of T,. For T, in the plate, equal to 10 seconds, Hp/Hp is 0.8 at 200 nautical miles, 0.63 at 400 nautical miles, 0.43 at 800 nautical miles, and 0.26 at 1600 nautical miles. The numbers squared are given by 0.64 at 200 nautical miles, 0.40 at 400 nautical miles, 0.17 at 800 nautical miles and 0.07 at 1600 nauti- cal miles. Roughly these values decrease by a little more than one half as the decay distance is doubled. The theory discussed above in this paper says that at great distances the values should decrease by one fourth as the decay distance is doubled. The methods employed in the derivation of the theory on which the figure in H.0O. Pub. No. 604 is based, - 248 - depend upon, among other things, the assumption that the decrease in wave height is caused by friction against the air. The width and the duration of the storm are not considered. Groen and Dorrestein [1950] attribute the decay of waves to eddy viscosity in the water, but again their theory does not account for wave dispersion and lateral spreading. The theory discussed in this paper predicts greater decreases in wave height simply due to dispersion and angular spreading from a storm of finite width and duration than are predicted by the Sverdrup-Munk theory without these considerations. Storms are of finite width and duration. A storm which is wide compared to the decay distance but which lasts a relatively short time would cause waves at distant points which decrease in height like 1//R simply due to dispersion. A storm which is narrow compared to the decay distance but which lasts a long time would cause waves which decrease in height like 1/VR simply due to ) angular spreadinz. Other small, short duration storms would be- have differently. Storms which cover a large area and which last a long time would behave still differently. The curves in Here Pub. No. 604 are based on wave observations from many storms of many different widths, durations, and fetch lengths. Consequently, the curves average in many errors even if there is some slight loss due to friction. From these considerations, and since the significant height and period have been shown to be inadequate in many other respects, it must be concluded that the decrease of wave height with distance traveled can best be explained by the methods derived herein and =249N— that friction effects are negligible, or of second order in . importance, in the problem of wave forecasting. Transformation of sea into swell The results of Chapter 8 and of this chapter also explain all of the known effects which accompany the transformation of sea into swell. Short crested waves are simply sums of waves with infinitely long crests such as equation (8.5). A short crested Gaussian sea surface is given by an integral of the form of equa- tion (9.47). The greater the variation of [An(p ,0)]° over / and @ the more irregular, choppy, and short crested the sea surface will be at the source. The apparent crests will actually vary in direction depending upon what particular terms happen by chance to reinforce at a particular time and place. For example, if [en 0) 1 were given by, say, figure 22, and if a partial sum such as equation (9.48) were formed over a net containing about fifty elemental net areas, the resulting equation for 7 (x,y,t) would represent a very complex irregular short crested sea surface which would approximate (even for such a coarse net) many of the features of waves at the edge of a storm at sea. Now consider the power spectrum given by applying filter V to figure 22. For any partial sum, all the elemental waves would be traveling in directions only a few degrees from -22.5° and all the elemental waves would have nearly the same spectral periods. The sea surface would therefore have to consist of large areas of waves of nearly uniform height with quite long crests all traveling in the same apparent direction. Arguments similar to - 250 - those in Chapter 7 show that a wave group would have to last a considerable length of time before the elemental vectors in the partial sum become sufficiently out of phase to cancel out the wave amplitude. Note the change in the direction of the apparent crests. The crests appear to be coming from a point source at these distances. Period increase of swell If figure 22 were actually to approximate the power spectrum at the source, the period increase of ocean swell can also be explained by this model. The "significant" period for the highest waves passing a point of observation would increase from a value of approximately seven seconds in the storm to a value of ten seconds at distant points of observation, but it would not in- crease indefinitely. Complete reality of the final model The decrease in wave height with travel, the transformation from an irregular choppy short crested "sea" to a regular "long crested" smooth "swell!' the arrival of waves at points at an angle to the main direction of the winds in the storm, the period increase of the swell and the so-called forerunners of swell are all explained by this model. Note that the "swell" is still Gaussian. The author has yet to see a natural wave record even approximately equal to A sin 2rt/T over a time interval of 20 minutes. i ee a SE. ES SS The final forecast formula, given by equation (9.71) and the auxiliary formulas given by equations (9.24), (9.25), (9.46), 4 etal = (9.47), (9.48), (9.65) through (9.70), and equations (7.55) and (7.56) (with R replacing x), is the most realistic forecast for- mula of all those that have been presented. The above formulas are the only ones out of over three hundred in this paper (so far) which are needed to carry out a wave forecast. Actually only two diagrams given by figures 24 and 25 are needed along with the concept of the Gaussian case of the Lebesgue Power Integral for short crested sea surface. All of the other attempts to represent the sea surface and to forecast ocean waves serve only to illustrate forcefully the inadequacy of the models employed. A system which depends on the gross characteristics of a storm at sea, namely its duration, width, and fetch, and on the properties of a very special integral has yielded results which explain all known properties of waves from a storm at sea by the use of the classical concepts of gravity wave theory. In actual practice, the square cutoff filter will be only a first approximation to the actual wave systems because the winds which produce the waves require time to build up to full amplitude and die down from full amplitude, and because of smoothing effects due to the finite time required for observation and the finite area which must be observed. The waves build up with the wind, and they have different characteristics at the edges of the storm and at the rear of the storm than they do at the center of the forward edge. The actual filterswill then be smoothed in some way with respect to the theoretical filters. Their actual nature awaits detailed analysis and study of the sea surface. me fay BORE, Bice) ae issues es Storms with rapidly moving edges (from which the waves leave the storm) and hurricanes produce wave systems which are not covered by the above theoretical considerations. The theory can probably be applied to slowly movinz storms without too great an error. Also various successive temporarily stationary positions of a moving storm might yield fairly good results upon application of the theory. Hurricanes just do not come under the scope of the theory for reasons mentioned in Chapter 2. The Gaussian Lebesgue Rower Integral has a different form and a hurricane has no width because it is circular. Possibly in another paper and at another time, the problem will be treated for moving storms and for hurricanes. The above forecast model ought to work for a large number of practical cases. ees a eee There is still a joker in the deck. The functional forms of E5(# 58) for storm waves at sea are still unknown. From the re- sults of Deacon [1949], Donn [1949], arthur [1949], and Barber and Ursell [1948], a particular E,(y,9) must vary considerably over a range of 4 corresponding to a period range exceeding values of from below five seconds to above twenty seconds and over a range of © from forth-five to sixty degrees above and below the dominant direction of the winds in the storm. In the next chapter, adequate methods for the analysis of ocean wave records, and adequate pro- cedures for the determination of E,(y ,®) will be given. Then Sage after a sufficient amount of correctly obtained data has been assembled and analyzed, and after the variations of E5( 42) as a function of the properties of the storm have been obtained, it will then be possible to prepare correct wave forecasts. - 254 Chapter 10. METHODS FOR THE DETERMINATION OF POWER SPECTRA Introduction In Chapter 7, the sea surface as a function of time alone, at a fixed point was first studied. In Chapter 9, some properties of a short crested sea surface were derived. It is still necessary to show that a short crested sea surface observed at a fixed point is a Gaussian case of the Lebesgue Power Integral as a function of time. When this is accomplished it will also be possible to show that the functions, E(H), Ej(#,®), [aC 1° and [a5 (pm 6) 1° are interrelated. The techniques of Tukey [1949] and Tukey and Hamming [1949} will then be applied in order to obtain the relationships between the non-normalized auto-correlation function and the power spect- rum, and procedures for the estimation of the various power spectra will then be described. Other properties of a short crested sea surface will then be obtained. Finally methods for computing [ay (p 6) ]° will be pre- sented. Where and when the methods apply The methods to be presented in this chapter strictly speaking apply only when the sea surface is in a steady state. That is [a(n 1° or cay oh when determined by these methods should have the same value about any point of the sea surface at any time. It has already been pointed out that under these conditions wave fore- casts and methods of wave analysis would not be needed because the = 255 - waves everywhere would be the same. Waves at the forward edge of a storm can be thought of as being in a steady state along that forward edge if [A,( py Seni and [A( ie do not change with time during the duration of the storm. Since the methods of analysis to be presented can be applied to time intervals which are short compared to the duration of the storm and to areas which are small compared to the dimensions of the storm, the methods are valid in the analysis of actual wave records at the edge of a storm or over a wave generation area. The methods of analysis which will be presented are also valid in the area of wave decay. The filters described in Chapters 7 and 9 are slowly varying functions of time. If [ace ae is analyzed from a wave record thirty or forty minutes long, the wave system will be so slowly varying that the methods will be valid. If anu) ]- is determined over an area of the sea surface thirty or forty miles on a side, and for thirty or forty minutes, the wave system will be so slowly varying that the methods will be valid. An analogy to electronic practices might clarify the situation. An engineer designs a radio to operate on 60 cycle AC current. The design of the power supply is based mainly on the formula E=E sin 2rt/l. For nearly all practical purposes, the fact that the radio is turned on or off can be ignored, and the fact that the voltage is actually given by an equation like equation (5.1) is not important. Similarly, the amplification sections of a radio are treated as if they were amplifying constant musical notes. A small enough SOs section of speech, although in reality the frequencies are slowly varying (compared to the duration of one cycle), can be treated this way without any serious consequences. Consequently, the results of this chapter will apply to almost any wave situation. If there is reason to believe that the waves are changing very, very rapidly, the results of the analysis should be questioned, but for slowly varying situations the results can be interpreted in the light of the theoretical considerations given in the previous chapters. For these reasons, [a(n 1° and [a,(# ,e)]° will represent power spectra for any sea surface either at the edge of a storm or in the area of decay. No special notation will be used to desig- nate special conditions. Non-Gaussian short crested sea surfaces Consider, for example, the short crested sea surface given by equation (8.1). For this representation of the sea surface, it is possible to pick some fixed point on the sea surface, say xy and Yu and observe the wave system at that point as a function of time. ividently, there will be places at which very small (or zero) amplitudes will be observed, and at other places the amplituces will be quite large. In equation (10.1), the potential energy averaged over time at the point (x5 9 ¥z) is computed. The result shows that the value t is obtained is still a function of Yy° At some points, P.E. pean /4; at others, P.E." is zero. The potential energy averaged over y and t is peA-/8. (See also equation (8.5).) At first, the point just made above does not appear to be - 257 - AIX 240ld (6 01) (ide MCN) ey 4 (UAtGeplZenZyy2y_(PO2p + Zeler2q 2a —(fereep 2eleen2q 25 : 0=4 0-! w 2h [‘1+d2. ‘1+[Z4uz 1t(Z+UZ [*|ed2 {+02 6 och ACPA AY) ( 9 d)p+4 ni ( guis'k + 9509 ‘irene sok ns wip = (4A) Vi [ + (8°01) ee Dy ete "er uaamyaq SUOIAIDGNS 4aU BYY JO} G }O ONIOA 4S4ij BUY SI BQ aIaUM (201) of C3 Muze, ae nt Rea aa tea > { SHC = as e+e +42r 10 anjDA yoda 10% 4aA0 Jau ||N} D 1apisuod puD (9°Ol) Ae tM V = 27] —N2+%27/ Qiaum (S‘Ol) Come ay = NOt. 2. Ite HAS ays fet Nea LS Cte SI Ne Ney Mr} 4044 YOnS (PI2) UOIJONba UI WY aWOS JO UOISIAIPGNS Ja|jOWS D sapisuod Y-, 0 (v'Ol) (9'71)*aap/((a'7! A +471 - (guis+ gs0o'x)*)s00 = (sx), yoodd A (< Ol) ‘Syulod ||]D 4D xo w 2 eel @SD9 UDISSNDO au) 4104 (2°Ol) UONOAJASGO Jo ,UIOd Jo UO!}OUN} D Si'Qq @SDD UDISSNOH—uoU au JO} }DIaueH U| at : 16 v L 16 15 ike Pee ee (101) [Avie soolaveG = Pall pas -™ 2D—| rye) Or koe sooy] /= = wil = (ad it? ‘"a9DdS Ul! JUIOd paxi4 D 4D OWif JO UOIJIUNY DO SD PeA.~.aSqO SD B2IDJING DAS palsaiO JOYS O JO Saifsadoiqg ay —258 — very important. But wave records at present are observed at a fixed point as a function of time. If this very simple example of a short crested sea surface yields such widely varying re- cords and such widely varying values of the average potential energy, what assurance is there that actual wave records as a function of time represent the sea surface in the neighborhood of the point of observation, and that the average of the squared wave record is actually related to the potential energy of the sea surface? For the non-Gaussian case, that is for sea surfaces of the form of equation (8.5), the potential energy averaged over time varies from place to place. Stated another way, it is a function of the point of observation as shown by equation (10.2). Gaussian short crested sea surfaces For the Gaussian case of a short crested sea surface, it can be proved that the potential energy averaged over time at any point on the sea surface is given by equation (10.3). This pro- perty of the Gaussian case is very important because it shows that the current wave records as obtained as a function of time do contain important information worthy of more detailed and re- fined methods of analysis. The proof of the statement made by equation (10.3) is some- what lengthy, and some other important results are also obtained. Consider first the integral definition of the short crested sea surface as given by equation (10.4) in which xy and yz are given subscripts to point out that the sea surface is being observed as a function of time at a fixed point. In Chapter 9, the inte- = 250n= eral was defined by a net over the # ,© plane as the mesh of the net was shrunk to zero. Consider the two values of / given by Hx and # y,5 in the net defined by equation (9.14), and break up this small increment, Ap , into N much smaller increments as shown by equation (10.5). The relations between the me iods ji is involved are given by equation (10.6). Also consider a full net over © from -1/2 to 7/2, for each of the smaller nets given in equation (10.5). The values of 96 at the net points will also need to depend on the particular net interval, uv 2k+2j to P oto 5429 and they are therefore designated by subscripts like °1; as shown by equation (10.8). One property that these Lebesgue Power Integrals have (and which has not been proved in this paper) is that they are the same as the ordinary Riemann integral in that it is possible to break up the area of integration into small touching but non-overlapping pieces and the total integral is the sum of the integrals over the smaller pieces. Consequently, the contribution to the total disturbance created by the power in the semicircular strip from Py» to My, and from -1/2 to 7/2 is given by the limit of the partial sum given by equation (10.9).* NA p(X st) is thus the contribution of this strip to the total integral as observed at the fixed point X493° The proof of equation (10.3) consists essentially of picking an appropriate sub-net in this semicurcular striv to obtain the desired properties. In equation (10.10), it is pointed out that for any *Note that the R here has nothing to do with the R of Chapter 9. it is just an integer. - 260 - be 2k+25+1 and Oo pt, j with Ky and Yy fixed it is always possible to subtract and integral number of 27's from the sum in order to find a new '(jap5)s (short notation for V'C H oy404412% oper, 5) such that ¥'(3,p;) has the same probability distribution as the original } . In equation (10.11), for a fixed j, the net over 6 for this small subdivision of the original strip has been picked so that each of the terms under the square root sign in the evaluation of the integral has the same numerical value, given by the increment in E(#,7/2) from Hon4053 tO Hoy4oj40 divided by R, the total number of elemental areas in the small strip. This can obvious- ly always be done. Each term in the sum over p is thus given by AE,/R. Equations (10.10) and (10.11) are next substituted into equation (10.9) in order to obtain the first expression in equation (10.12). The cosine term is then expanded by trigo- nometric identity in the second expression, and the summation over p is moved inside. An expression of the form A cos® + B sine / oe aos (6 | W ) and this has can be written in the form [A° +B been done in the last expression in equation (10.12). The conm- plete expression for W '(j) is given by equation (10.13). The next step is to simplify the coefficient of the cosine term in equation (10.13). In equation (10.14) this is done by writing the cosine and sine in complex notation. When the first expression on the right in equation (10.14) is expanded, only the cross product terms remain and the second expression results. The sum from p equals zero to R - 1 of exp(i ¥'(jsp;)) is a sum - 261 - (E101) (2101) (101) (O10!) IATX 8101d o=d o-4 Z {io [) huis —— ak) [('9'0,hs09 ) | I-y eT y O=zd j-UIS = ({}, mi (d‘() Auris Cf) paylt le ener (i Ord 4 028 O=[ een Ry) 4 SOQ: d‘ uligs ——— d‘ [ iB ; )fiut avs) Atl ) h $09 eal yi wi A -u | N I+!Z+Hz , 4 0:4 ]0-[@e8 1!) UIS fd‘( I+ $2427] (4 4 (4 ) alll pause +t soo Fay] (t',# 809< cea o-d of pe Bi al ('d‘() sh +4! 4 !2+%271-)s09 Lu No ytlalxy Me I oN Y | (BMeenyeg— (2 (< ‘2+(2+Her)23 (Leg fet ner jeg 4 (Mg !etw2q/ jeg — (le pi2+letnen 2g — (Mbp'z+letwen)23 - (Peptie+ A2q/)2Q + (ep fe+ A2q1)eq- (fOgez tier 421 23 — (2p'e+!e+n2) 23 12} use ('d'l) pe wag —({l4+4%p ‘14!24H271) pa (142 uis'K+!'#42pso00!x b >0 g 8 ) (ir teaea7) a90dS ul! jUulIOg paxi4 D 4D awi| JO VOIWDUNY D SD PaAsJAaSGO SD BIDJING DAS paysaid 441OUS D yO Saljsadolig ay, Oe. of R vectors of unit length all pointing in randomly picked directions. Let the sum be the complex vector given by B pexp(io,). The other sum is a sum of R vectors and every vector in this sum points in exactly the opposite direction to the corresponding term in the first sum. The sum is therefore of the form B pexp(-ie,) and it points in exactly the opposite direction. The complex pro- duct is therefore always a real positive number (Bap) as given by the last expression in equation (10.14). From equation (10.13), the sine of W (j) can be written in the various forms given in equation (10.15). The results of equa- tion (10.14) permit the use of terms like Bypexp(iese), and in this case the Bar cancel out. The sum therefore represents the Sine of some angle, 5R° But from the nature of the sums dis- cussed in the paragraph above, the value of OR for a large value of R is equally probably any value from zero to 27. Equation (10.16) is therefore the result, and the probability distribution of W'(j) is the same as that required originally in equation (7.28). These results are next substituted into equation (10.17). Since the values of the Bir are not one, one begins to suspect that things are getting complicated. Also the results are be- ginning to look something like the results which were obtained in Chapter 7. So far in the proof only one small strip between pz 2k+25 and pu Det25+2 has been treated. There are N of these strips between # » and # y,5- The values of AE, can, by picking the points p 2k+24? all be made equal to [E5( byyoet/2) - E5(H 4.7/2) ]/N as shown in equation (10.18). = 26a" (91°Ol) (S1'O1) (1°01) Osd ae = Fi TIATX 9 ?!°ld 1\>7 >0 asaym p =(#Z0 > ({),A>0)d as0jyauayy pup ajqoqosd Ajjonba asd suoljsaJ4Ip ||0 abun) yo soy yng ¥1p,0 8! 12 ¥ip,_9 YG oo l wig? | wi} ¥ig-9 “'E | vig? wg | | wi| 2 se TT | O=d (!d' es & (!d° [ al? a, = Piya ta aus el (a iY tee : =u 12 | 5 H(A (1) At (()m! uf Yo oYug >00-)q aduis Z un Clea © og YNZ eon ( I" ) — Ww Sas S = ul aEUELEEEe aE I[= tat (W2+)3V fu'g) S 'Oreewav | 2 aac 2+WaV Sy 3 ; -{ O °@ S09 _ He OW yore= 6 Pus — {0s'01) £9502. of VOY (6°01) oe 2% ~=puD (8% Ol) (,0 -@) soo a 487 z ° (2°01) ap 7p {on *¥] [ome + al N= -g)uis A + (.8- 9) 809 x] S| 502 = ; EJ ig QP 7/P [tg'7) 2] [te 4 + 47—(g us fk + 9809 x) Hee []- (y'A'x) asp) ~upissno9 ay (9%°01) o 3° 36 | (Sv'01) te ene 2°94 2 BS ry) t (vb Ol) x (6-'6) (Toye) cae srr aye ya} pun ‘4 puo A xy (Ev Ol) [a — (6-8) us A + (40 -'9)s09 x) soo y= (HL z (2v Ol) pf 809K + 46 us X= (Iv Ol) 4G us A = 46 S09 =x 1a (O01) (4! —(‘g uls A + '@ S09 ‘gy ne V=(t)u uoHDHIySaAut Aspuiwi|ajd cay °y yO vuolpoulwsajad aul ke ny flying in the true direction of travel of the crests (i.e. 6, = @*), the true wave length of the waves will be observed. If not, some wave length greater than the true value will be observed as shown by equation (10.46). It is not possible to record a wave length shorter than the true value. Conversely, the wave number given by V, = 2r/L., will vary between zero and its true value, and it will not be greater than its true value. The problem is quite simple in this case if the pilot wishes to discover the true wave length of the waves below the aircraft in the fog. Many passes are made over the sea surface at various headings until a heading is found such that the length of the re- corded waves increases when either the aircraft is turned to the right or left. This minimum length is then the true wave length. By then flying very very slowly, or sending out a helicopter, the direction of wave travel could be determined by the Doppler effect. For the other simple cases discussed previously, similar techniques could be used and the resolution of five or six sinu- soidal waves of different periods and directions would not be too difficult a feat by ordinary techniques. However, the true sea surface is best represented by a Lebesgue-Stieltjes power integral over LaXGa euler and, as such, it is composed of an infinite number of infinitesimal sine waves traveling in all directions from (it is hoped) -1/2 to 1/2 with respect to the dominant dir- ection of the crests and with all possible spectral freouencies over a considerable range of the M axis. For these reasons, the determination of [Ay ( m ,e)]° is a complicated problem. 200) 8 LS SE Consider the short crested Gaussian sea surface given by equation (9.47) and apply equations (10.41) and (10.42) so that the sea surface can be studied as a function of x'. The result is equation (10.47). For convenience © equals zero should be picked to be the dominant direction of travel of the apparent crests, and the direction © equals zero is therefore along the x axis. The angle, 0©*, then measures the angle between the x axis and the line of flight. The observed spectral wave number, Vo? then depends upon the spectral frequency and the cosine of the difference between 6 and 6*. Angular directions above and below @©* are determined by an angle, 85° The procedure is now to transform the / ,@ polar coordinate system and the integration over [a5 (um ,0)1° to a v.40, polar coordinate system and an integration over a new function LA. ( 1598590") 1°. The variables, v, and ©, are defined in terms fo) of # and © by equations (10.48) and (10.49). The inverse trans- formation which defines # and © in terms of ve and oF is given by equations (10.50) and (10.51). The Jacobian of the transformation is given by equation (10.52). Substitution of (g y,/cos 5 one for @ and ©, + @*for © in equation (10.47), and the use of the Jacob- ian to preserve the mapping then yields equation (10.53). The Jacobian is needed because the function [A> (m5) ]° has been dis- torted by the mapping, and in order to preserve the total power in the wave system it must be amplified for low values of vie and cos 6, and decreased for high values of v.. Stated another way, an increment, Am , maps into a much smaller Av if it is at the ie) = On low end of the # scale than it does at the high end of the wv scale. From the properties of the Jacobian, it then follows that equation (10.54) holds and Be i is given by both an integration x over the @ ,@ plane and the y,,0, plane (see Courant [1937]). The integration over the 8% coordinate then can be defined to yield the function [A( v 598,71 as given by equation (10.55). The angle, 0*, is, of course, fixed for each single flight. It is now possible to describe the functions which can be recorded by current instrumental techniques. First, the free surface as a function of time at a fixed point in space can be recorded. The function which results is a Gaussian Lebesgue Power integral of the form of equation (10.56) @r (7.1)) as has been proved at the start of this chapter for the case of a short crested sea surface. Secondly, for a fixed 6* the sea surface as a function of x' can be recorded. In equation (10.53), if y' and t' are fixed, then by exactly the same techniques that were employed to study the short crested sea surface as a function of time it is possible to prove that the free surface as a function of x' for a fixed y' and t is given by equation (10.57). Both functions are samples of stationary series and both can be analyzed by the methods presented above in order to determine [a(n )]? and [A( v9") 1° for a finite net. In addition many dif- ferent values of 6* can be chosen and a whole set of functions of the form of [A( v.90") 1° for different ©* can be found. Thus the observed power spectra are given by equation (10.58). From these data, the problem is to find an estimate of [ete (Gu peDiloab Boe THE PEP ee Emmaxt f fi admerl d0 dp® af fe antar IE ie cats) Statement of Problem. The observed functions Proble i 2m Solution, Let Fc" DAT 2 2 can find [a(o)] | [a 425] , which can observe for are (8, +e + fet A Lect de [bs OF cos (6-6 ) 9G Vo axt= feos [ot +¥(u) | _| [AHI dp n (x,y, t) =f COS | %oX ‘+ yl] [Taw #77] &)| dv The observed power spectra are fui and [avo @ m; Find [A (4,8) will be designated by which will be designated by Ome and for a fixed & can. find which will be designated by A(h This total since fate, 2), fae yields m+l numbers of (2q+1)(m+l) A (h,-9) = A(h,q) —28))— [A2 (#6) ay (= 9 y2dBadae Wl 7) d8. dy» 42 (10.60) and Y% Sine/, 2 2 [a #e2)] , [ated] ND) 3 WEOgUocossco.s - z_ /-49+! mee ee Oy ae in a ( q 26000000 2q 2q * * bed a: j) SC) aneecoon +q Fe 7 ) ee ti), [are $4) ine (HE: O)4 Dh, Ssaed ooo from (10.63) and 2q(m+l) values Numbers. PLATE II Hy pelo zee )2 8)| 2 ETE AEs Wee do=[Al», EAB’ +4) from (10.67) ) or a (10.54) (10.55) (10.56) (10.57) (10.58) (10.59) (10.61) (10.62) (10.6 3) (10.6 4) (10.6 5) (10.6 6) (10.67) (10.68) (10.69) The determination of [A. ( “,0)1°3 solution ny For convenience in notation let ae be defined by equation (10.60) and vy, by equation (10.61). These are the cutoff values of the spectral wave frequencies and the spectral wave number. If there is no aliasing for mw greater than Ho then the largest spectral wave number which can be observed is v @ Then for m lags of 7)(t), m+ 1 values of Ke can be found as given by equation (10.62). These values will be desig- nated by A(h) as h runs from 0 to m for simplicity of notation as given by equation (10.63). (ACH, hy 3° is given by equation (10.35) after the use of equations (10.30), (10.31), and (10.32). It is possible to pick 2q + 1 directions for 6* as given by equation (10.64), and 9* can be designated by rj*/2q as j* is sumied from minus q to plus q as shown by equation (10.65). The 6* are equally spaced angular values above and below © equal to zero. For each value of ©*, the stationary series which is observed can be analyzed by numerical methods exactly like eauations (10.30), (10.31), and (10.32). The space separation of points in the series is given by A X, equals (2 At)°2/2r. For each value of ©0*, that is for j* fixed, the power in m+ 1 bands of the ue axis can be estimated. Values of the form [A( Ve a, Ti*/2q) 1°, can be obtained. These values will be designated by A(h',j*) as h' runs from zero to m for simplicity as given by equation (10.67). From equation (10.63), m+ 1 numbers are the result. From equation (10.67),(2q + 1)(m + 1) numbers are the result since j* ranges from minus q to plus q, but for j* equal to plus and minus =E2Q7m= q the results are the same because the same track is retraced in the opposite direction. Equation (10.69) states formally that A(h',-q) equals A(h',q). Thus m+ 1 numbers are duplicates and can be discarded. Finally (2q+ 1)(m + 1) numbers result from the application of the procedures given by Tukey and Hamming [1949] to the recorded data. This is stated in (10.68). As a check, the total area under A(h) in em* should equal the total area under ACh',j*) for each j* to a high degree of accuracy. If not, the value of N is too small and the value of m is too large for re- liable results. It is now necessary to study how the #,9 plane mans into the Yo 5 plane and how area net elements in the first plane map into area net elements in the second plane. For this purpose, pick the values given by equation (10.70) for the mw axis and the values given by equation (10.71) for the 9 axis. The curves de- fined by the double lines are boundary curves of area elements. The unknown number, An(h,j), will designate the value of over the net area element defined by equations (10.73) and (10.74). It will be assumed that Ap (h, j) is constant over the area element. Figure 27 is an example of such a net of the pw ,& plane for the special case of m equal to 10 and q equal to 3. The values of 4,(h,j) are shown in each of the area elements. The A,(h,j) are the unknowns which must be found to determine an approximation of [a5 (p ,@)]*. Near the origin of the coordinate system some of the values of A,(h,j) are not shown because the figure would be too crowded. There are (m + 1)(2q + 1) unknown values of A,(h,j). = ssa AL 10 ld (6:01) (ae) o> O>( G8 la I+ (82°01) (=3 n>n>( 2) % Aq pouljap Daud ay} JAAO juDJSUCO pawnssD Ss! ()/°O!) 424M (22°01) CUR [tree quell 2a) (OE ON ae (C22 Orr =e (Grol) Operon: ae we ae Or jau Huimojjo} 84} 410° Paxil} D 10} auojd °9'% auy ul (201) GPE Z-gri BZ (e201) (aug Prt agg PM Aq pauuep DeuD eu) 29A0 juD}SUOD PawNssD S! (Z/'O|) e4e4M (22°01) ("y)?v= [Fe salu ja] ‘Sjulod 4ajuad sauljyap — ajbuis au} Aq pauljap saasnd ay] “SeAsnd Auppunog 340 == ajqnop eu} Aq peulyap seasnd ay (12°01) < ‘oe7! yau Bulmojjoy ayy yold ‘Q‘7! aus Ul Wo, we Woe w, We 6 WZ? Ww a7] ‘O71 Sage Te Ge He ERT Oa (O20!) l ((9'71)?V] yO UOIJOUIWIEJ9G YL N —-296- In the V 5995 plane for a fixed 0*, pick the net defined by equation (10.75) for vy, and (10.76) for 0,- Over the area element defined by equation (10.78) and (10.79), the value of [A, ( v 59030*) 1° can be designated by A,(h',j',j*) in equation (10.77). The net for the v.,0, plane for @* equal to 7/6 and for m equal to 10 and q equal to 3 is shown in figure 28 by the dashed lines. The circles shown by the solid lines show what happens to the lines # equal to a constant in the m,9 plane as they are mapped into the v,,@, plane. (See equations (10.48) and (10.49).) Consider what happens to the boundary curves which define the area covered by A,(h,j). The curve w equal to mw ,(2h + 1)/2m maps into v, equal to y, cos e(2h + 1)°/(2m)* as stated by equation (10.80). Similarly equation (10.81) is the mapping of # equal to #.(2h - 1)/2m. The straight line © equal to m(2j - 1)/4q maps into ©, equal to m(2(j - j*) - 1)/4q as stated by eauation (10.82). Similarly, equation (10.83) shows the mapping of 6 equal to 7(2j + 1)/4q. Equations (10.80)and (10.81) are equations of circles which pass through the point v 6 equal to zero and 85 equal to plus or minus 1/2. They are shown, for example, in figure 28. From these considerations, the area element A, (8,1) maps into the shaded area shown in figure 28. It therefore covers part of A,(6',0',1*) and An(7',0',1*). Power is conserved by the mapping. Since A, (hy §) is the con- stant value for the power spectrum over an individual area element, the integral over the area element in the # ,© plane is given by equation (10.84). After the mapping the integral over the area de- fined by equations (10.80), (10.81), (19.82), and (10.83) becomes an - 297 - Al Did ; b (88°01) (=, 2)s00 (aa SLI (28°01) ("2 $)s0o( we) a ({y)*v 0} yoadsay usm (Qg°0)) 0:!'O/ euinssy b ( Fisie . voolreua) (eet (4: 8°01) wPz (yyy = opt (8)((uFy f= Ap |S (“8,22) (fu)? obs Ty WZ \9 x bz 9s0A— 5) 4 +(e i (0 S801) =i 9 s00 (Se 2) an (3. = eagle (Agena) gure eqe a4 re ({4)?y = °g pap, (M208) 24 ("yy uv (p8'Ol) (uy 4 22 = ap Mp (fury Wo b 09 809 (U2-)% (aren Geugh (Gres (8°01) rene mn ee eo) ieee) (sng) =°8 ‘°G 19M07 Ue Il (1801) °@ S09 (5% nay) )®a=% ‘Kiuopunog seddy (Q8'0l) ON = ‘Kippunog JamMo7q Bug t20 Paxil} D 403 auDjd °Q‘°A ayy Ul SaAIND BHulmojjo} ay} Oyu! dow ({‘y)*y Jo SaluDpuNnog ayt *: _ [la 2v] 40 uoupuwieyeq out -298- integral over a portion of the Von plane. The integral is evaluated in equation (10.85) which proves that power is conserved. Consider figure 29. The dashed lines bound the area elements of A5(h',j',j*) and they form a net over [A,( ¥5 18590") I For this particular case 9* equals zero. The shaded area shows the area mapped into by Ay (8,1). For this case, j is greater than zero and j* is zero, as assumed for a special consideration in equation (10.86). Then after Aj(h,j) has been mapped into [A, ( ¥598529 1°, the greatest value of y, y, , is then found by substituting the smallest value 85 can have, Aaneiy equation (10.82), into the upper boundary curve, namely equation (10.81), and the result is equation (10.87). Similarly equation (10.88) gives the minimum value of v o° For the net in the v.,0, plane, A, (hy J) therefore occupies part of several area elements given by Ay(h',j',9 4. In fact, there exists some value of h’, say K, such that eon a and “oe. are sandwiched between »y(2K - 1)/2m and y, (2K + P)/2m as stated by equation (10.89). Finally on mapping A,(h,j) into A,(h',j',0*), A,(h,j) contributes part of its power to the Aj(h',j',0*) for h' ranging from K to [K + (p - 1)/2] and for j' equal to j as stated by equation (10.90). Consider equation (10.91). The right hand side of the equation gives the power in the area element, 4,(h,j) in the m,®© plane. Ay (h, j) has the dimensions of om°sec/radian, and the whole term has the dimensions of em*. The number B(h,j,h',j',j*) is a number which partitions the right hand side into contributions to the various elements a,(h',§',J5*) in the v,,0, plane. Equation (10.92) - 299 - IA 1 42d f (66°01) “24 <4 40} (B601) ((u)v) att = (fu) VAR APS (26°01) ws" VO =y pun b of j+b-={ 40y (9601) ( ‘Uwe =((u) v Glia fue me (uz =ua)™ rE (6°01) (La) Ve ween iuyy (peor) (fu) v BAe OPMPLL LY) V (sale aR)e (¢6 01) 2 +y<,Y fl PUD Hay J! O= (o‘fy‘{‘y)g ‘ajdwoxe siy} 104 ({‘{y) y yuawaja you ay ul ({‘y)*y 40 japd jouoljo04 ayy seuiwsayap (_f*fy'l’u)g | Jffuluye 2e% esaym (26°01) (uty) 8 K aot (601) (MU) V > Ho - ({*u)*u(,fifiu' ua % 9uijad (0601) [={ puo (~4)+4" 1e¥H=,4 404 (9 {FUP OF Sainqi4juoo ({‘y)*y esojosey | (68°01) WE Mos MA > Eon NM se <2 yOu) YONs "y Kos ‘y 40 aNjOA aWOS s{sIxe B494) ‘rf le Jv yo vopuluayeq aut — 300- follows from equation (10.91). The dimensionless number, 2rqm B(hyj,h',j',3*)/p OT, determines that fractional part of A5(h4j) which is contributed to the value of A,(h',j',j*). For the examples in the plates, B(h,j,h',j',j*) is zero if h'(2K + P - 1)/2 as stated by equation (10.93). It is also Zoron dias * jue The power in an area element in the v 0926 plane is given by equation (10.94). Aj(h',j',Jj*) has the dimensions of em>/radian, and the right hand side of (10.94) has the dimensions of em. The integral over 85 (see equation (10.55)) then becomes the sum given by equation (10.95). A(h',j*) has the dimensions of om? and the right hand side has the dimensions of com>. All of the terms of the form of B(h,j,j',j', j*) A,(h,j), which have the dimensions cm’, can be treated for a fixed h' and j*. Summed over all possible j', h, and j, they will be all con- tributions to the net elements in the vy ,,0, plane for a fixed h'. In fact, they must again equal the right hand side of equation (10.95) as is stated by equation (10.96). Equation (10.96) thus involves known values of B(h,j,j',j',j*) and a known value for the right hand side given by the values found in equation (10.67). The unknowns are given by the A,(h,j). Sepa- rate equations for each value of h' and j* result as shown by equation (10.97), and equation (10.96) with equation (10.97) there- fore stands for a system of 2q(m +1) linear equations. Also equation (10.98) follows from equation (10.24). The right hand side is known from equation (10.63). There are m+ 1 equations of the form of equation (10.98) as stated by the con- dition (10.99). The unknowns are A,(h,j). = Xen Equations (10.96), (10.97), (10.98) and (10.99) therefore de- fine a system of (2q + 1)(m + 1) inhomogeneous simultaneous linear equations and there are (2q + 1)(m + 1) unknown values of A,(h,j). Such a system has a solution if the determinant of the equations is not zero. It has not been proved that this is the case, but further investigation has shown that sub sets of the equation starting with h and h' equal to m can be solved. It appears that a process similar in the abstract to the concrete process of peeling the outside rings off one half of a slice of a Bermuda onion one by one will yield the values of Ao (hyj). Corrections to the equations Some of the area elements in the “ ,O plane and in the Y 4599 plane contribute only half of the power to the total power that is contributed by area elements in the center of the system. Others at the corners of the system contribute only a quarter of the amount of those at the center. Equation (10.94), for example, must be modified if h' equals m. Also the terms in (10.95) for j' equal to -q and q have a factor of one half in them. At various places in these equations, then, factors of one half and one fourth must be inserted. These factors have been omitted in order to simp- lify the notation since it is not intended actually to solve such a system. (For one reason, the needed data are not available.) Further explanations Figures 27, 28, 29 and 30 can be studied together in order to understand better the procedures described above. In these figures A, (8,1) in the w ,® plane is traced as it is mapped into = 302) = 7. 2 5 Az(10,3) se He eS —— / 19, iE Se Pid Ee | ~ ia SESS Wh 3 ot Ae(9,3) } SSS) SSAIO2) Vue | / Re SS Ge Naas 4 Bes or Mee) OP SS > S ae iia Soa Uae V (cir) eS m3 Oe aon oe ! Sk xe yo Tey UEC ie Gia) SN 13He | — / oS \ Z N N Zo r Rey ee > << ih NK S Tae SE) SS GAIN 4 SHAN iy lle XN 4 S Ls a ek-— J > \ NN SG} > SS 7 an H, i~ A2(6,2) SX os ne YS ‘Ai0,!) Tey ts A2(5, 3)/ ~ Se NO, ~< ‘ \ \ 9c 20 | =o AMB2IS oN Vato ~ To + A,(43)) S Ds Ae \A,(8)!) \ \ x Gl XS \ Hab oa nw Ne (7) . \ 4 bec, . 1 IAG ENS gre ANNE, 1). \ \ \ \ \_ et 5Hc = lol aaa b Aa(32) x \ Ad6,!) § \ Vi Nhezan \ fo te aload — \Be an 4 Leet, vii 3He eae 2 WN A (3). \ Zao \ \ ; \ ine alt Oi. Nan cae’. \ ; ' van epee Ne ue Nee | =, bh Balh)\ cat ! | 1 OK TA,(10) }A A,20) 2 oy £4,0)!4,(50) {Az(60)/A,(7,0) Al6,0)|Aa12,0) 1A,(10,0)—o BA (EIA ta, 1 | } 1 | WES i Saztenly, ha ae! ; i A ane aN Fe i) ! if ~ ae i] z XS ‘y 4-| Y sell ! ety Za ve al we Eh oN [Ping Pama Gass: ny ARG), oe I-I~<26 ASS NK 7 PLE hae a I = 7 Aheod AX ayaa) Ww < oh 7 AdT-ly a / NC 3) A e oe Sas, /Adl851),/ , — A i, SN UL ie Newt = ANN f ALM ‘A,(10-1) 1A,(5; -3) \ i NED / ea ae \ --" A622) ee S 7 7 y ees Le —- tes 7 / 23 _— 4 SY, vA v4 in (G53) AIMED) pre y at eh ae ‘ / Ke Seg Mi SA A Ws SK Yi Figure 27 : \ =a AN(8-2) VAN, 1A, (7,3) 2 7\ 7 3 ‘An example of a net over Bee a wee OW, Se the 2,8 plane WU Seely ae sowacs 1A, (853) = VAG Dey 7 2 < Zi He = | -----——" \ Row ye A(7,!) = ae 2 3] Wien See ~ A,(10;2) | A{9;3) aoe ee \n 2 L gee-—= Wee 23 to A(7/0,1") 2 \ Plies 2 2qm \ Pee ¢ Vi > pe Pet vass=- 8.= 3 gi-==S 55 aliased by rotation into upper part os for as 3 integration is concerned. —304- =- —_— Sone ——— — i re es T \ \ | | / [be / } / | | Ko | 1 / afi). 52 eer 2 “Wy, Yo Ah, 3k pa er oe wa vi / ih Zs ! \ a “ SS Z 7 | \ Ie 7 S27 Yo 4 Figure 29 | a A en Tus YS, F 2 ihe eae Mas Y ‘ie Aer? The Mapping Of [A,(u,8)] ! ‘| ee yf DX {7 Into [A2(%,80,6*)]* es ee ae Dashed Lines; Net For [Aq(v,0.,0*)I = een aaa wee eae oe Solid Lines ; Boundary Lines ; Nephi o> ye For Areas Covered By A,(h,j) lee tex ra @* Equals Zero m=10 E=—— ah N 2 “i 23 t \ yee Ly, 2 ee : 1 es eo 25) eS ae mON pg | ee 1 A* ‘NA (eee B(8,1,4,0*) A,(8, 1) 8(8,1,6,1'0*)A,(8,1) Dsecer y N the contribution of the contribution of The LIV ky TY, The 1 Ak, TY, A{8,1) Fgn'0 AL4; .0")5r= AJB) = toA (6,107) Fae B(8,1,5;1/0") 4 (8,1) y ie the contribution of B(8,1,6,10 )AS8,1) Tie AY) ee th tributi f rhea | \>qm 9 Fas) 2am -305- 8, = Z Foliosed by the rotation into lower part as far as integration Figures sO is ae us nee The Mapping Of a(n, oi Into [A,(%0,400" POV Dashed Lines; Net For [A,(%,60,8™)]° Se, SSN PS Solid Lines; Boundary Lines For Areas Covered by A,(h,j) Ler SK ~ gt=—= : S04. SN = = tu) 8 {X0) Cpe 8) CY NS NN 2 3 Ko) SUC SS / XN , XN Ni B(8,1,2”.2/-1) 4,(8,1), the contribution of pe Ee We SIN ya Tie Fo! ey TY ~/ ‘ aS NGnIN A(8,!)5 qn t0 A,(2,2,- ) qm ooh. Se ~ s Se nad OS N 7 = Sir 2 B(8,1,3,2,-1")A,(8,1); the contribution Me By Ane aN \ Ba > OO” NN SM 5 Resto Ae eS a : \ \ - 2qm (3,2, qm er \ KE NN NN ‘ \ NS &(8,1,4/2,-1*)A,(8,1), the contribution ae BX ‘ x \ * ‘ T fle ba (ey TY, SOx SUZ \ \ \ \ \ZZN of A(8,)) 565, to A(4,2/-I oan RG x \ \ ‘ \ DAN ly! |* Sou ‘Ss Ba SN \\ \ \ \ \ 4 B(8,1,5,2,-I")A,(8,1); the contribution sO% 6 s \ \ \ es Tle B74 ; . an of A4(8,l) 55m to oy \ “ko \ ; / Z Duisieei ae Se NE 2qm ~1 I 1 / / / Hi 7 8,=0 Ke i Ms / i / eats / TS Tk ya SK L ~ [eee a ~ AS OSES ee = = 2 “Sn eae Bese -306- the functions [A5(v 5,85;7/6*) 1°, [Ao( v518510*)]° and [A, ( Vp 1959-0 /6*)]°.* In figure 28, the elemental waves in A,(8,1) are traveling very nearly in the direction 9* = 1/6. Ay(8 1) then goes into a symmetric figure which contributes part of its power to A,(6',0',1*) and the other part to A,(7',0',1"). Note also that the range of integration in equation (10.53) is from -7/2 - @* to 7/2 - 9* and that the figure shows A(v 52959776") as if it varies from -7/2 to 7/2. The true figure can be obtained by slicing figure 28 along the line 8, = 27/6 and rotating the pie-shaped sector obtained counterclockwise until aS 1/2 touches a= -7/2, Then ®, varies from -417/6 to 21/6. For ©* equals zero, the mapping is given by figure 29. The shaded areas again show what happens to Ap (8,1). The power in A,(8,1) is distributed over A,(4',1',0*), Aj(5',1',0"), Ap(6',1',0°) and A,(7',1',0*). The wave elements in the elemental area, A,(8,1), are now at an angle to the direction of observation. For 6* equals -7/6 the power in A5(8,1) is contributed to A,(2',2',-1*), An(3',2',-1*), A5(4',2',-1*) and A,(5',2',-1*) as shown by figure 30. The angle between the wave direction and the direction of x' is now greater. The computation of B(h,j,h',j',J* The value of Gah dgale gat oa“) depends only on the properties of the net and not on any unknown quantities. There are (m + ea + 1)3 possible values but most of them are zero and many of them are numerically the same. There are a possible 77 values for B(8,1,h',j',0*) if m is 10 and q is 3, but for j' not *Note how the circles shown by the solid lines squeeze down to the origin. One of them is so small, in fact, that it is not shown. = 307 = equal to 1 they are zero, and for h' less than 4 or greater than 7 they are zero. Therefore only four values have to be found out of the 77. From figures 29 and 30, B(6,1,7',1°,0*) and B(8,0,7°,aa—a0 have the same numerical value. Thus only a few of the values act- ually have to be determined. The evaluation of B(8,1,5',1',0O*) will be discussed as a par- ticular example. It is the number which results from the double integration of (1/2)(g/cos OyY ye d vdeo, over the shaded area indicated in figure 29. Three sections of the boundary curves are given by constant values of y and 85° Two of the boundary curves fo) are functions of v é and 85° By breaking up the area shown into three sub-areas shown by the heavy lines, and then, by integration over vy, first, the center area works out immediately with respect to the next integration over 86° The other two areas become elliptic integrals over 85 which can be evaluated from tables such as those in Janke-Emde [1945]. For a complicated function, [A5( v ,9)1°, and for m equal to 10 and q equal to 3 as in the figures, seven analyses of the form described by Tukey and Hamming [1949] would have to be carried out. Each might require four or five hours on a computing machine. The evaluation of the B(h,j,j',j',j might require several days. The result would be 77 linear inhomogeneous simultaneous equations with 77 unknownse The matrix of the equations has certain symmetry properties and if its inverse could be found easily, then the numer- ical work would involve another day of work. With time for checks of the computation, it would take about ten days to determine the = 3080- function. For larger values of m and j, the time required in- creases very rapidly, and the use of electronic computers might be advisable. The reliability of the results ie [ay(# ,0)]° were determined by the procedures described above, there would be some doubt as to the reliability of the re- sults, especially for small values of m and qe They would at least give an indication of the values of the function but the degree of confidence in the final numerical results in terms of the number of degrees of freedom cannot be given at this time. The airborne altimeter might introduce additional error by reflecting in part the effect of atmospheric turbulence as sug= gested by Tukey in a recent conference. A stereoptican measure- ment of 17 (x') for different e* from a photograph would eliminate errors due to the effect of turbulence. Ge [a,(#,0)]° is a function which has been filtered by the travel of the disturbance from the source so that it is confined to a small area of the H ,© plane, just a few directions of & would yield, along with the observation of 7(t), a great deal of inform- ation about Ean cu seule Other methods for the determination of [a ( #0) 1° es cS SS ST In a recent conference, Tukey suggested another method for the determination of [a5 (m ,e)]° by the use of the stereoptican measurements. The method depends on many parallel measurements for different y' in the x' direction. Waves not traveling in the x! direction can be partly filtered out by the addition of values on a line y' equals constant. The details of the procedure = 309 - have not been investigated by the author, and possibly they can be worked out in some future paper. The method has definite advantages over the method described above according to Tukey. A final method for the determination of [ay (pw ,0)1° depends upon the acceptance of the results of Chapter 9. The oceans act as a filter on the waves which propagate from the edges of the storms over them. Swell simultaneously recorded on a line of pressure wave recorders (on the California coast, for example) spaced several hundred miles apart can be analyzed by the equa- tions given in this section. The 96 band width and filter char- acteristics could be determined from the dimensions of the storm, and the power spectrum at the edge of the storm could be computed from the observed power spectra after the propagation of the waves over a long distance of decay. [a5 (m ,0)1° at the edge of many storms must also be determined by the methods described in previous paragraphs in order to verify the statement made in this paper that friction effects are negligible. If this statement is veri- fied, and many arguments have been given which make it appear to be true, the method described in this paragraph will then be a very important way to study the variation of [a5 (m ,0)1° with wind velocity and air mass properties. = s71One APPENDIX The foregoing chapters are all of the material available in finished form for publication at the present time. Some detailed wave analyses will be carried out in a later chapter. This ap- pendix has been added in order to show six very interesting figures which illustrate the great range of possible wave records and the interpretation which can be put upon them in the light of the fore- going chapters. Figures A-1, A-2, A-3, A-4, A-5 and A-6 are from the original paper by Klebba (1946). They have been furnished through the co- operation of Admiral E. H. Smith of Woods Hole Oceanographic In- stitution. The figures which show wave records are numbered on the right and the corresponding wave record analyses on figures A-3 and A=-6 have been numbered to correspond to the wave records for comparison purposes. , The sharp jagged tops of the spectra shown (which are not necessarily power spectra) are probably due to design faults of the instrument and to sampling error. The band width of the tuned circuit of the instrument is probably so narrow that the instrument responds very erratically to portions of the record near certain critical frequencies. For example in figure A-3, the sharp jagged tops in the spectra numbered 13 and 14 could be drastically smoothed in order to obtain a spectrum more like the one numbered 15. These data are not quantitative in any way because the instru- ment has a gain control anc the gain of the electronic circuits was readjusted for the various analyses. (See for example records = ai = 17, 18, and 19 and spectra 17, U6,/andyd9. ) Finally note that the spectra yield very little information for periods less than 6 seconds. The depth of location of the pressure recorder was 78 feet in one case (B Station) and 103 feet for the other case (A Station). Periods less than 5.6 seconds and 6.5 seconds are hardly detected by the pressure recording instru- ment. With these qualifications in mind, the records can first be studied gualitatively and then compared qualitatively with the spectra. Note record 22, for example, in figure A-4. By scanning the record, it is seen that the departure from the mean value is most of the time much less than the peak values of the record. It is not too difficult to accept the hypothesis that enough points taken at random would have a Gaussian distribution. Now note the tremendous variability of the record as a function of time. The time intervals between successive apparent crests vary over a wide range. An autocorrelation of the record with itself would rather rapidly die down to zero which would mean that what happens, say, one minute in the future has very little to do with the be- havior of the record at the time of observation. Now note record number 6. It is much lower in amplitude, but again the departure from the mean of the record is much less most of the time than the departure when the few peak values occur. Again, it is not too difficult to accept the hypothesis that the distribution is Gaussian. The variability of this record is much less than that of the former record in that the time interval between successive crests is much less variable. An autocorrelation - 312 - of the record with itself would die down to zero much less rapidly than in the former case (if both were normalized to one at the start). In fact, it is even possible to imagine that one could say something about the behavior of the record one minute into the future given that, say, one of the "oroups" was just starting upe Now compare the spectrum for record 22 with the wave record. The spectrum has amplitudes of importance in the entire band from six seconds to twelve seconds. The wave record is just about what one might expect from such a power spectrum. Finally compare the spectrum for record 6 with the wave re- cord. The spectrum covers a much narrower band from eight to eleven seconds. (It even looks as if it could have been obtained by the forecast procedures.) The character of the record fits the nature of the spectrum qualitatively. Trouble occurs though in trying to apply too precise a reason- ing to the records and the corresponding spectra. Record number 5, for example, differs only a little (to the eye) from record number 6 and yet the two spectra are very different. It is believed that the differences are due to instability of the instrument and not to a marked change in the sea surface during the three hour interval from record 5 to 6. More precise analysis along the lines described herein would eliminate this trouble. In conclusion, for part one, quantitative methods of wave analysis have been described herein. They appear to be able to make it possible to put wave analysis and wave forecasting on a - 313 - much firmer theoretical and practical basis. It should eventually be possible to analyze records such as those just given accurately, quickly, and quantitatively, by both numerical and physical methods and to relate the power spectra to the storms which produced the waves. - 314 - Iw 614 WEI-H S/T A030"s Odey oy einssory OpOt CETL Byer ‘oT Lavniged ee eee Aayoss ede, 02 eingsorg OSLO O ea acmnoteiay (5ic15)) iii ellie enile 79 Plate X. The Propagation of a Finite Wave Train in Infinitely Deep Water (Continued) HqavatLonsm (5. o)memmouct (C5) (kay Sek coi), eet eenese Plate XI. The Propagation of a Finite Wave Train in Infinitely Deep Water (Continued) Hguations™ Chee) ator ouch (did 4) Tiere ten ben cola 83 Plate XII. Finite Wave Train (Solution) Hegitaiteons) \(oetih) eoheoughin (5/122) llenisimeul nites 85 - 326 = Plate Plate Plate Plate Plate Plate Plate Plate Plate Plate Plate Plate Plate Plate XIII. Finite Wave Train (Properties of the Solution) BquatiLons, (5623) ebhmouehya( 5c SO) eden nt on once XIV. Model Wave Systems with Infinitely Long Crests Equations i(6.1).tthnough (6.ob)P jae — tpi XV. An Infinite Periodic Train of Wave Groups . ‘ Equations (6. 7) throuchm(O.dis) ements os ot XVI. A Finite Train of Regular Wave Groups EquatHions. 6j4 iebhroneh, .(Geli7 jw ete eee) en XVII. Energy Considerations Equattonsi(6.d'6)e through ao. )eqelenuenen oy ae XVIII. The Lebesgue Stieltjes Power Integral Equations, (7.4) -throurhii( 7.10)! ot. aeaee 6 Yenee XIX. Some Examples Equatdions (7.19) through (7126) ae) iene mene XX. The Gaussian Case Equations, (7.27) ethroughaG7.34)5 ss = «6 XXI. The Forecasting Problem for a Sea Surface Represented by the Gaussian Case of the Lebesgue Power Integral with Infinitely Long Crests in the y Direction Money ealosaln (HARE) venereal (eS bo 6 o 6 6 XXII. The Forecasting Problem for a Sea Surface Represented by Infinitely Long Crests, a Gaussian Wave Record, and Winds that Lasts D_ Seconds Over a Fetch s of Length, F. Equations .G7,.oi.) abhrouchun@7cOl5)er ss sls lene XXIII. Short Crested Waves Equations 1 (8.2) Ghrouch s(e.>) aralse) aires XXIV. A Useful Lemma Equations, (8.6)) ,throuch GSell3) ys suis a -amenies XXV. The Initial Value Problem in the y,t Plane for a Disturbance of Finite Duration and Width Equations 4(8..44), throuchis(oeg7)in sneer XXVI. The Initial Value Problem in the y,t Plane for a Disturbance of Finite Duration and Width Equataons (esl 8)ithrenghy (G3.20)) 0. unre semeemte = 327,— Page 92 103 107 110 7, 126 abeye IS) 149 15) iLy/3) Ly? 181 184 Page Plate XXVII. The Initial Value Problem in the y,t Plane for a Disturbance of Finite Duration and Width Equations (8621) through (8524)))3\. «sc omeoe Plate XXVIII. The Initial Value Problem in the y,t Plane for a Disturbance of Finite Duration and Width Equations (82250) through (628)) 9) eR 0 ao Plate XXIX. The Initial Value Problem in the x,y Plane for a Disturbance over a Finite Area Equatlons (Oec9)sthrough' (O.33)" . «2 « 6 « smog Plate XXX. The Initial Value Problem in the y,t Plane for a Wave Train of Finite Width and Finite Duration Bquationsm(s.s4) through (8.36) sense lh vcmmose Plate XXXI. The Initial Value Problem in the y,t Plane for a Wave Train of Finite Width and Finite Duration Equations, (8.37), through “@os46). vee . sken eeRnoS Plate XXII. The Initial Value Problem in the y,t Plane for a Wave Train of Finite Width and Finite Duration Bauatvons "¢os49)) throurhGse53)) . «6 5 « « te noe Plate XXIII. The Initial Value Problem in the y,t Plane for a Wave Train of Finite Width and Finite Duration Hauat tons meoeo4)) chrouch ConoO) mes. ee nell Ponte santuca Plate XXXIV. The Initial Value Problem in the y,t Plane for a Wave Train of Finite Width and Finite Duration Equations ao.ol)e through (OstOcy) «os ve ver tet ten ome Plate XXXV. Interpretation of Results Equations (esos ecarougi Cbe70)*