THE UNIVERSITY OF MICIIGAN INDUSTRY 1ROGRAM OF THE COLLEGE OF ENGINEERING PERIODIC, FINITE-AMPLITUDE, AXISYMMETRIC GRAVITY WAVES Lawrence R. Mack A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the University of Michigan 1958 May, 1958 IP-292

Doctoral Committee: Associate Professor Chia-Shun Yih, Chairman Professor Robert C. F. Bartels Associate Professor Samuel K. Clark Professor Jesse Ormondroyd Professor Victor L. Streeter

TABLE OF CONTENTS Page ACKNOWLEDGMENTS... o. O O O.. o o e o e o. o..... e o e o. ii LIST OF TABLES... o.o,.o o. o...o. ee.....oeoeoo...e v LIST OF ILLUSTRATIONS o.o...s0...0........o. o.ooo...o... vi CHAPTER I INTRODUCTION O.e 0.O D 0.. 0 O *O 0 0 0' * o. 0 *0' * 0 0 a. 1 1a The Scope of the Investigation 2 EHistorical Outline II. THE PROBLEM AND ITS SOLUTION >. iOO-..0.............;.....o o. 4 1. The Governing Equations for Axisymmetric Standing Waves 20 The Solution at General Depth 3. The Solution at the Critical Depths III. ANALYSIS OF THE SOLUTION...e................o................ 22 1. The Surface Profile 2. The Frequency of Oscillation 3. The Pressure and Velocity Distribution 4. The Wave of Maximum Amplitude 5 The Angle at the Crest of a Maximum Wave 6. The Energy of the Wave Motion IV. CONCLUSIONS AND RECOMMENDATIONS................... 47 1. Concluding Remarks 2. Suggestions for Further Study APPENDIX I. THE EXPANSION OF FUNCTIONS IN DINI SERIES....... 51 APPENDIX II. CERTAIN INTEGRALS OF BESSEL FUNCTIONS.......... 60 1l Orthogonality Relations 2. Some Identities APPENDIX III. THE SOLUTION AT GENERAL DEPTH TO THE THIRD APPROXIMATION M-rn'NT^e0@ 64 APPENDIX IV, THE SOLUTION AT THE SECOND-ORDER CRITICAL DEPTHS TO THE SECOND APPROXIMATION...... 68

TABLE OF CONTENTS (C-O' T. ) Page APPENDIX Vo FUNCTIONS OF H APPEARING IN THE ENERGY EXPRESSIONS. o o aoooao o ooaO: * o 0 0 a 0 73 BIBLIOGRAPHY~ o o o o o o v o o o o v o o. o o o o o o o o o o ~ o o o Oo o o o. o o o ~ 0 o 77 BIBL~~~~~~~~~~~~~~~~IOG RAPt'.o.oo'..o0.o.'.'0.......o..o...

LIST OF TABLES Table Page The Eigenvalues Kn O 8 Roots of the Equation, 1- _.t.... o 16 3. - and - for Waves of Approximately the' Maxum Amplitude oo o 4 Oo o aooO o a 0 a 0 30 4 --- at the Second-Order Critical Depths for All = O 01. O.. o ooo... O. o. o.0..o..e...0 0. ooo o.o. 31 50 The Maximum Amplitude at the Second-Order Critical Depths. o..... 0 O... O...Q. * OO.eo 36 60 o<(F)for Various Functions F(r).............e..o 54 7o 5(F) for Various Functions F(r)........,. o 57

LIST OF ILLUSTRATIONS Figure Page Lo (:Co-ordinate System and Geometric Configuration.o... 5 2. The Magnitude | versus H for All =.01, 21 All 3. Configuration of the Free Surface for H = oo...... 25 4. Configuration of the Free Surface for H =.3 —,-o — 25 5. Configuration of the Free Surface for H =,1,,..... 26 60 Configuration of the Free Surface for H = 0 O..... 26 7. The Frequency Correction Factor Gc versus H..o.. 29 8. Maximum Amplitude versus H for H not Critical....,. 35

CHAPTER I INTRODUCTION 1. The Scope of the Investigation This work is a study of finite-amplitude axisymmetric gravity waves in a circular basin of uniform depth. Only periodic, free oscillations of the fluid are considered. The analysis is carried out for a standing wave whose motion to the first approximation is that of the first mode. However, the same procedure may also be used for motion corresponding to another mode. The fluid is assumed to be a non-viscous incompressible liquid. The relative depth of the liquid (that is, the ratio of the depth to the radius of the basin) is not limited a priori to either of the extreme cases of very large or very small values. Rather, the depth is allowed to be completely general. It is seen that at certain discrete values of the re!ative depth a coupled motion can -occur in which a higher mode at a frequency equal to an integral multiple of the primary frequency is of the same order of magnitude as the primary mode. The motion for the depth equal to or very nearly equal to one of these particular depths is investigated by an appropriate modification of the general solution. The main difficulty in -obtaining the solution to the problem is the task of satisfying the two non-linear free-surface boundary conditions. These Conditions, in addition to being non-linear, must be applied at a moving boundary whose position is itself an unknown to be determined, An iteration process is followed in satisfying these conditions.

This problem was selected for study primarily because it.is fundamental in the fields of fluid mechanics and non-linear vibrations. It may, however, have practical application to the phenomenon of seiche or mass oscillations in harbors. It may also be useful in pointing the.way toward the solution of other, not necessarily related, non-linear problems in cylindrical co-ordinates. 2. Historical Outline In the past hundred fifty years many investigations have been devoted to gravity waves. A number of the more important works will be mentioned briefly here; others are listed in the bibliography to this dissertation and in the excellent bibliographical sections of the books by H. F. Thorade(59) 1 and J. J. Stoker(58) It is unfortunate that until very recently finiteamplitude waves in deep water and in shallow water have been treated as separate problems rather than as two aspects of the same problem. The earliest analytical study of progressive gravity waves apparently is that for the case of infinite fliuid depth given by Fo Jo von Gerstner(8) in 1802, and also independently at a later period by W. J. M. Rankine(l8) in 1863, which presents a form of wave motion possessing vorticity, Virtually all subsequent writers, however, have rejected this form, arguing that the wave motion can be generated from rest and hence nbust be irrotationaL, Progressive finite-ampljtude waves in water of infinite depth and in water of large but finite depth were studied by G. G. Stokes(24) in 1847 and Lord Rayleigh(19) in 1876. The existence of such waves when the depth is infinite was proved in 1925 by To. Levi-Civita(l2); the next year Da J. Struik(26) extended this proof to the case of finite depth~ 1 The numbers in-raised parentheses refer to entries in the bibeliography.

-3Periodic waves, progressing without change of form, in shallow water (6) were first indicated by J. Boussinesq in 1877. The name "cnoidal waves" was applied to these by D. J. Korteweg and G. de Vries(10) in 1895. Further study of cnoidal waves was made in 1940 by G. H. Keulegan and G. W. Patterson(41) The solitary wave, which consists of a single intumescence, was first observed by J. Scott Russel 21)(22)(23) in 1838 and subsequently studied by H. Bazin(2) in 1865 and J. Boussinesq(4) in 1871. The solitary wave may be thought of as the limiting case of a wave of infinite wave length and hence (34) belongs in the shallow water class. K. 0. Friedrichs and D. H. Hyers proved the existence of solitary waves in 1954. By replacing the exact non-linear dynamic free-surface boundary condition by another non-linear condition approximating it,2 To V. Davies(2) in 1952 obtained a solution for progressive finite-amplitude gravity waves which is applicable over the entire range of depths. Provided his approximation is valid, both the solitary wave and waves in an infinitely deep fluid are special cases of Davies' solution. Much less study has been devoted to standing waves. The motion of an infinitesimal-amplitude standing.wave is r~thpr easily obtained and is discussed by H. Lamb(l) for several geometrical configurations. The theoret(46) ical and experimental work of J. S. McNown in 1953 should also be noted. To the author's knowledge the only theoretical study of finite-amplitude stand. (49) ing waves prior to the present.work is that of Wo G. Penney and A. I. Price who in 1952 analyzed such waves in a rectangular co-ordinate system. 2 This approximation is of the type, a sin 3 @ (JJ < ). p 6~~~~~

THE PROBLEM AND ITS SOLUTIQN 1. The Governing Equations for Axisymmetri'c Standing Waves The equation governing the irrotational axisymmetric motion of an incompressible non-viscous fluid is, in terms of the velocity potential 4, s'Z O _ 0 3(1) in which 72_,>~ to-1~~ a ~' ai~ ~ (2) is the two-dimensional Laplacian operator, As shown in Figure 1 the origin of the co-ordinate system is on the axis of the cylinder a distance H above the bottom, where H is the mean depth of the fluid The requirement that the velocity normal to the solid boundaries must vanish is expressed by the conditions d; (4) On the free surface, given by t=<(nj t), both kinematic and dynamic boundary conditions must be satisfied. The kinermatic.condition is D in which D is the substantial derivative and F = 0 is the equation of the free surface0 Setting F - and performing the indicated differentiation yields

I Figure 1. Co-ordinate System and Geometric Configuration

as the kinematic surface boundary condition. In the present co-ordinate system the Bernoulli equation takes the form J) +0F[ t) + Cl -~ 2~94 ) ~ Z( i) ~ 3t e(7) in which the reference pressure p is chosen to be the atmospheric pressure acting upon the surface of the liquid For convenience later on the time function of integration F(t) is merged in ~. When a specific form of 4 is assumed, in the next section, it will be pointed out which parts of ~ correspond to the F(t) of equation (7)~ At the surface p equals po and the dynamic surface boundary condition is Since the volume of the fluid.remains. constant, Nig Ar ~ O J (9) where dA is an element of area in a plane normal to the T-axis. Setting dA$=n'A 1 and performing the integration with respect to G, we obtain P0W R -o (10) Equation (1) together with the boundary conditions (5), (4), (6) and (8) and the statement (10) constitute the governing equations of thesystem, The barred quantities appearing in these equations are dintensioznal.

-7Let us now introduce dimensionless independent and dependent variables as follows: 1 l _ t R - ~* (12) C) (dimensional) is the frequency of the oscillation. The depth is made dimensionless by H -H _. (13) H is thus the relative depth parameter (hereafter called merely "depth"). An upper case G is written for the non-dimensional gravitational acceleration to emphasize the fact that, although it takes the place of g in (8), it is a dependent variable which determines the frequency. The dimensionless equations governing the system are thus v 2 + _,'~~~~~ O o(14) o i = - (15) fZ 4 4 - O * (19)

2. The Solution at General Depth Through the use of the method,of separation of variables it is seen that the solution of equation (14) which is finite throughout the fluid region and which satisfies the linear boundary conditions (15) and (16) is =,-4 c (St-Hj) T;(l ) T(k) (20) provided that K assumes the discrete positive eigenvalues Kn for which -T. ( I) t (11, t ). (21) J9(.cr) is a Bessel function of the first kind. The first five eigenvalues are given in Table 1.3 TABLE 1 THE EIGENVALUES Kn n Kn 1 3.83170 59702 2 7.01558 66698 3 10.17346 81351 4 13.32369 19363 5 16.470635 00509 Equations (14), (15), and (16) are linear; hence any linear combination of the eigenfunctions given in (20) will also satisfy (14), (15), and (16). Restricting our attention to periodic solutions, we accordingly choose the following form for c: O= AmPA E~t)K H (22) 5 The first 150 values of Kn are given to ten decimal places (eleven to thirteen significant fizres) in the'British Association Tables (6

-9The n = 0 ternms, in which KO is defined to be zero, are independent -of the space variables and thus correspond to the F(t) which was merged with j in writing the Bernoulli equation in the form (18), These terms make no contribution to the velocities; they are included with solely for c.onvenience. The surface elevation 7(n, )is represented in the form to 7_Vn)ZSZ(j) c + )) a t (23) Because of the long and complicated nature of the expressions obtained for. Amn,. B1 m,)+~m and G, several shorthand notations are defined. below and:used hereafter: (24).A 4M~ 4d Jj'- p The subscript i appears in this work solely as 0 and l. The frequency of an.osillation whose motion is to the first (that isy linear) approximation that of the first axisymmetric mode will be denoted by CO (dimensional). Then the All and B11 terms in (22) correspond to this motion. Since this is a steady-:state oscillation, there is no natural time origin in contrast, for example, to the case of release from rest at some initial configuration. Therefore, the time origin can be chosen arbitrarily. In particular, if the time origin is chosen such that B11 is identically zero, All is the parameter which determines the amplitude of the motion. Its magnitude is allowed to be arbitrary within a certain upper limit This upper limit, corresponding to a breaktng wave, will be dsussed in detail in Section 4 of Chapter III. It is assumed that all other Am, and Bm are of order

-10//) or higher. The order-of-magnitude assumptions may be summarized as A,, = 0~A,,) (25) All other Amn and Bmn (A) or higher. It is further assumed that in satisfying the free-surface boundary conditions (17) and (18) terms of order O(A/) may be neglected in comparison with terms of order 5(A1i) in the first approximation. Similarly, in the second approximation terms of orders O(A~l)and 0'(A/)are retained while terms of order O(AI) or higher are neglected, et cetera. The following procedure is used for satisfying the surface boundary conditions. The approximations are made in order, first, second, third, etc. The assumed forms of $ and 1 in (22) and (23) are substituted into the kinematic surface condition (17). In evaluating the derivatives of A at tZl the functions sinh Kn(Y. +H) and cosh Kn(r,+H) are expanded as ermch ono n erterm the fo r weorer bt Xa trn or re Each non-linear term is of the form XY where both X and Y contain one or more terms of order 6(Al) as well as higher order terms. In order to retain all terms of order i(A,) in the product XY it is necessary to use only those terms of order I(A;) or less in X and in Y. Hence the use of the results of the (j-l)st approximation in the non-linear terms of the j th approximation is permissible.

-11Equation (17) after the sulbstitutiQn of (22) and (23) may be written in the form AMI>o tZ0 6h ai Xt O ~(27) in which each A+z is of the form fe - Kr (n) + fi3 ( 4,, Ad He 24) ( 2o) where each f is known. Because of the orthogonality of the trigonometric functions each + must vanish independently. (29) For all - 1, (29) becomes J~Ji ) ~ F+(4'4A teH) (9> (30) and the functions J (.) are determined in terms of the A and B. For mn mn m =O'f (4Il)B0. )H = - (31) Each function of r occurring in the expression (31) is expanded in a Dini series of Bessel functions. That is, so ~(n) = E o((F) TO (32) where the cQnstants o( are given by )( (F) T (3 The Dini expansions are explained more fully and the Q( Is for the particular functions F(r) of interest here are tabulated in Appendix I. Multiplying

equation (31) by J 30 a, integrating with respect to r from 0 to 1, and noting the orthogonality relations (II-1 and II-3), for B3ssel functions, we obtain explicit expressions for the Boam > 1 in terms of the parameters All and H. Now substituting (22), (23), and (30) into the dynamic surface boundary condition (18), using the value of G from the previous approximation in all terms of order ~ (Al) or higher on the right side of (18), and. applying the same techniques as described above for the kinematic surface condition, we obtain solutions for AM, Bm, and G solely in terms of the parameters All and H. Substitution of Amn and Bmn into the equations (3o) gives 1t(4) in terms of All and H. From (18) a solution for Jo(X) in terms of All, H, and C1, the constant of equation (18), is also obtained. Equation (19) is now applied to determine C1 and hence (P.) in terms solely of the parameters All and H. To show that the time-dependent terms of:Y vanish when integrated in the manner of (19) it is necessary to prove certain indentities. Those identities needed through the third approximation are proved in Appendix II as equations (IIl11) (II -12), and (II 13). The procedure which has been outlined for satisfying the nonlinear free-surface boundary conditions has been carried out through the third approximation. Further approximations do not seem practicable for two reasons. First, the expressions obtained are quite complicated. Secondly, for higher approximations the coefficients o( in the Dini expansions of many more functions F(r) must be computed. Since these coefficients are evaluated by numerical integration, the labor involved is considerable.

-L3The solution to the first approximation is A1,, 41- 1 -_- o A,, Kh, a, 5(4 I6 ~ l -.4) C1 and all the other Amn, Bmn, and are either zero or of higher order than.(<,) Equations (34) are the linear solution for infinitesimalamplitude waves as given in Lamb(ll) and due to Rayleigh(l9) The results of the second approximation, in which the first effects due to the finiteness of the amplitude appear are -,, = -48,, _ = <(35) Atm L ~ (r (36) Cl =) (37) At OA (37) G1 - 4,. K, 1 tl( I. <<E, =1 (.4: 4 This problem was considered as early as 1828 by Pisson?). However,(17) because the theory of Bessel functions had not yet been worked out, his results were not interpreted.

All other Amn, B in, and 3A are either zero or of order higher than ((,A) ~ The quantity (j is a function of H only and is defined as _., (42) I4A The solution to the third approximation, upon which much of the discussion of Chapter III will be based, is presented in Appendix III. In carrying out the oemputations it is found that all B and (_) j(a 2 1), are proportional to B11 and hence, because the time origin was chosen such that Bll 0, are identically zero. It is also found that B are zero, at leapt to order c(5(As), independently of the assumption that Bll is zero. The existence of finite-amplitude axisymmetric standing wayes and the convergence of the iterative process used for obtaining 4 and L are not proved. The existence and convergence theorems for progressive waves (16)(12) (26) (4) all depend essentially on the possibility of transforming the problem to one of steady flow by adopting co-ordinates travelling with the wave. This simplification is in the present case unavailable. The author is not aware of any existence or convergence theorems for finiteamplitude standing waves; Penney and Price(4' P. 268) specifically state, "There seems little likelihood that a proof of the existence of the stationary waves will ever be given." One observation canbe made concerning the thn convergence of the present solution. The factor which is present in th1 a number of the summations of the third approximation, behaves as unity for H very large and as for H very small; thus for small depths this factor will give a greater amplification to the terms for b > of these summations thpn it will for large depths. It can therefore be anticipated that the

convergence of the present solution will be more rapid for large depths than for small. 3, The solution at the Critical Depths Inspection of the second approximation to the solution at general depth, in particular equations (36) and (40) and the definition (42) of, reveals the presence of the factor ) -. _ Z in the denominator. Likewise the third approximation contains the factor -. in the d.no.ninator of certain terms in equations (III-4) and (III-9). These factors are both of the type Ma K4, X ( _ O. 1, 2'' )43 In general the j th approximation will introduce the factor (43) with q = j into the denominator of certain terms which were expected to be of order 0 (As). If there are any values of j,, and H i.or which this factor equals zero, then particular terms in both ~ and ~ become infinite and the general-depth solution must be rejected at those depths, Accordingly, it is desirable to look for the.roots of the equation 0. _ KX 4 _ o *(44) When q = 1,. 1 is a root for all values of H. This root causes no trouble, however, since the factor (43-) iS identically zero when q - -1 and hence we do not divide by it assuming it to be non-zero. Therefore attention can bp restricted to q > 2, for which it is evident that (44) can have no roots when n = O or l.

For ~ > 2, ~X decreaset monotonically from to 1 as H increases from O to oo. Thus only for those such that n will there be roots to equation (44). Since the eigenvalues K. tend ultimately to the form T( +, there will be for each integer q approximately!l(~ A ) discrete values of H, each associated with a particular, for which (44) will be satisfied. The roots of (44) for q = 2,3 are listed in Table 2. In addition the minimum and maximum roots for q = 10 have been computed and are also shown. For future reference we denote the values of H at which there are roots: of (44) with q = 2 as the second-order critical depths and those values of H at which there are roots of (44) with q = 3 as the third-order critical depths, etc. TABLE 2 ROOTS OF THE EQUATION, q 2 H 2 3 o.19811 2 4 0.34698 3 4 o.o84 3 5 0.132 3 6 0.168 3 7 0.207 3 8 0.255 3 9 0.321 -3 10 0.440 o.e 0. o' e. 0 i,'$ o o o Q o o o a a a a o o o D e - o e o o ee 10 12 0o004 10 121 be.761 The physical meaning attached to these critical depths may be seen rather easily. It has been assumed that there is a first mode of

l17order C(A,,)oscillating at frequency 0 and that all other modes and harmonics (that is, all other An, B ) are of order Q(A) or higher. mn' When the depth equals one of the critical depths, the assumption that all other -An, Bmn are of higher order is not valid. In particular, A% and/or will be of order e(A,), where q and t are the q and, assoqLated with the particular critical depth. In fact the condition that All, representing a first mode at frequency X, and tt~ and/or B~, representing an a th mode at frequency Ao, both be of order e(Al) is, to the first approximation, that equation (44) be satisfied. since %AR and/or BQ are of order ~(A,,) when H r He, where He is a critical depth and q and, have those values associated with Hc, and are of order @ (A,') when H is appreciably different from Hc, it is logical to assume that there is some transition range H e ( H < H~ +, where E is some smaul number dependent on Allin which and/or are between order () and order &(4,).5 Thus in order to obtain a solution of the system (14),(15),(16),(17)(i18), and (19) which is valid when H is equal to or is very nearly equal to Hc, we revise the genexaldepth assumptions (25) to the following: *a,, - SAMI) _B, _O (45) All Other Am and Bmn = or highr. To observe how the solution at a critical depth differs in form from the solution when H is not critical the solution can be carried out by 5 The alternative is to assume that bA4e I and/or | H)j has a jump discbntinuity when H = Hc. This alternattive does not seem physically ream sonable.

following the assumptions (45) when the depth equals or very nearly equals either of the two second-order critical depths. The order-of-magnitude assumptions (45) then become B - ~ (46) All other Am and Bmn = ((AI,)or higher. It is understood that 3 = 3 when H R 0.19811 and. = 4 when H, 0.34698. The substitution of 4 (22) and I (23) into the free-surface boundary conditions (17) and (18) and into (19), and the evaluation of the individual,Bmn, and have been carried out in the manner described in the previous section through the second approximation.6 The resuLts of the first approximation are ~All,~ ~ All, ~ 3~ Bi ~ o~ ~(47) a [L 4 _.0 (48) B22~ 0 _ tKP Xoi = O (49) Tj) = A,,K, a,, - (50) 6 When the depth.is approximately equal to one of the j th order critical depths (j 2 3), the method of solution is conceptually identical to that outlined here; however, the labor involved is considerably greater because the solutionr must be carried to the j th approximation in order to obtain the relationship between He and/pr B ap 8nd A1L.

-19~2(E) = g 42 t ot (51) (n) (52) (55) C1 and all other Amn, Bns, and ok are either zero or of order higher than 1 (Al). From (48) and (49) it is seen that if the depth is exactly critical, A,, and Bd -are arbitrary to the first or linear approximation. That is, when H = e., A~ and Ba are linearly independent of All1 However, if the depth is not exactly critical AZ2 and Bdq will be zero. The results of the second.approximation are given in full in Appendix IV. As in the general-depth solution it is found that B? (and indeed all Bn, mv I1) is proportional to Bll and is hence identically zero. The most interesting result, is the relation between A2 and All1 This relation is of the form 42A, +_ 4 (54) in which 0C2 C3, and C4 are functions of H and All given by Ace ~ ~~ I_~~ Soa~~~ _ 4(55) 3 C 4~s (Y', [70 j) 4 Ad Sj 1 c4(56)

C - a' Ln, 03m; 1)|(;l) + 2 ~(e(:II)C3, (57) The quantity C2 + C4 under the radical in equation (54) has been shown (analytically for 2 = 4 and numerically for = 3) to be positive for both second-order critical depths. At first glace A,2 appears to be double-valued. However the possiblility of double-valuedness is quickly dispelled. For any depth nearly but not quite critical (that is, 1C21 very small but non-zero) let All become infinitesimal. By the results of the first approximation A2q must approach zero as All becomes infinitesimal. This requires that the negative sign in equation (54) be taken wh.en 02 > 0. and the positive sign tak.en when C2 < O. Only when C2 is exactly zero is the sign.of Am not uniquely determined. The absolute magnitude of A~R is determined then; however, the motion due to A.l is either in phase with the\ primary All motion -or is 1800~ out of phase. For a depth differing in the slightest amount from the critical depth, the phase between A. and All is also uniquely determined. Thus equation (54) may be written as Ca -Jca +Cq c= _ SC (C >(58)'-C (c ( ) 22 C3

-21 The magnitude j is plotted against H for All =.01 in Figure 2. This figure illustrates the transition of Aa from order 0(A,1) to e(g 1) as H approaches the critical depth. The purpose.of this section has been to point out the existence of the critical depths and to show that the wave motion when the depth is critical or nearly critical can be analyzed by the procedure of the previous section if the order-of-magnitude assumptions (25) for non-critical depths are appropriately modified. This dissertation is primarily concerned with the motion when the depth is non-critical; nevertheless, certain aspects of the critical-depth solution will be discussed in sections2 and 4 of the next chapter. 1.0 0.8 - A23 A23 A24 A24 < o > >o <o A,, A,, All A A2, A,0.4 0.2 0.185 0.190 0.195 0.200 0205 0.210 0.34698 0.19811 0.335 0.340 0.345 ~ 0.3560 H H Figure 2, The Magnitudel versus HI for All =.01

CHAPTER III ANALY$IS OF THE SOLUTION 1i_. The Surface Profile The surface elevation7.(r,t) of a periodic, axisymmetric gravity wave in fluid of general depth, as given to the third approximation in Appendix III, equations (TII-6) through (III-9), may be examined more convenien$ly in the form, (,) Al Kl (I +T1 ca, I t (A,,KA 7f(n) t + (59) + A,,K,&2 (4 ar Zt (,, ~ )3 (n); 4t 3 where the starred functions are defined by A f [ >@)X(n) = (X)- 4s1K,>,Jcl (60), ~Tal,(A,,6') it (a) =,,(y) In the form (59) the effects on ~ of the parameters Alland H have been separated since the J (), (m = 0, 1,2,3), depend only on H. A number of observations may be made illustrating how the charac.teristic~ of a finite-ampJitude wave differ from those of a wave of infinitesimal amplitude. For waves of infinitesimal amplitude, the temporal mean 6_-.o

-23elevation of the surface is identically zero at all radii. There is a nodal circle at r =.628, The surface is horizontal twice during each period, at t LT and t - T The crests and troughs are identical in ped a2. shape; that is 7(,O) = ) (r7Tr). In contrast to the infinitesimal wave, the finite wave does not have these properties. Because of the T term which is independent of time, the temporal mean elevation of the surface at any given radius is in general not zero.7 There is no nodal circle. The surface is never horizontal. Probably the most striking difference between the infinitesimal and the finite wave is the marked alteration in the latter of the shape of the crests and troughs. The crests become higher and narrower and the troughs become broader and shallower as the amplitude is increased. (49) Penney and Price, in their discussion of periodic, finiteamplitude standing waves in rectangular co-ordinates for water of infinite depth, noticed the effects of finite amplitude consistent with their geometrical configuration, equivalent to the effects mentioned above. The narrowing and heightening of the crests and the broadening and flattening of the troughs have been observed both experimentally and analytically over a wide range of relative depths for both standing.and progressive waves of finite amplitude. If we define the amplitude of the surface displacement 7L(r,t) to be 2 0(,0) - (0LO,TT)], and define the quantity N, dependent solely on the parameters All and H, as Y"()0, o) T- h(o)'Fr \ - 1 (61) 7 4, -and hence th~ temporal mean surface elevation, is zero at two particular values of r, which values are functions of H.

then the function - will have an "amplitude' of unity regardless of what values are assigned to All and H. ~ thus indicates the shape of the free surface. This function, evaluated at t = 0 and t =rT, is plotted in Figures 5 through 6 for both finite and infinitesimal waves for four different depths, including the two extremes H = O and H =Wo. For each depth the amplitude parameter A,1 of the finite wave has the value.2 It is understood that when we speak of a finction evaluated K2 K1 when All is zero and/or H is zero or infinity we really mean the limit of that function as All approaches zero and/or H approaches zero or infinity. Figures 3 through 6 clearly show the difference in profile shape between a finite-amplitude wave and an infinitesimal wave, The heightening of the crests and the broadening of the troughs of the finite wave is most pronounced for the small depths; in the limiting case H = 0, the tip of the crest in the center of the basin as given by the non-linear theory is 29.7% higher than that predicted by the linear solution. In calculating the shape of the finite wave for H = Bo and H =.3 the third-order terms made negligible contributions. For the shallower depths, however, the third-order terms were significant, their contributions for H = O being almost half as large as those of the second-order terms. This observation is in keeping with the predition that, if our method is convergent, the convergence will be least rapid for very shallow depths..2. The Frequency of Oscillation The frequency of oscillation X0 of an axisymmetric infinitesimalamplitude first-mode gravity wave in a circular tank of constant depth has been given by previous investigators(9)( () in the dimensional form

-251.6 A, K,2 0 1.2 H= oo A,,K =0.2 0 -0.4' -Q0.8 -1.2 0 0,2 0.4 0.6 0.8 1.0 Figure 3. Configuration of the Free Surface for H = 1.6, 1J 2 L~~ A,,l KA = O0 1.2- 2 -. ~ ~k H. 0.3 A,,K =0.2. N \Z 0.8 -0.4 -0.8 0 - 1.2,,,,,,,.,, 0 0.2 0.4 0.6 0.8 1.O Figure 4. Configuration of the Free Surface for HE =.3

-261.6 AK = 0.2 0.8 t=0'7__*'~'4 - N -0.4 - Figure 5. Configuration of the Free Surface for H.1 1.2 -H=0 A,,K,= 0 -0.8 - 1.2 -, 0 0.2 0.4 0.6 0.8 1.0 Fgrej oiurtooft reuraeSHr

-272z r~ X K, X e, ~~~~~(62) In the non-dimensional notation of this study (62) becomes equation (41) 6 = /,; 41 which is the frequency equation to both the first and second approximations when the depth is non-critical. In the third approximation the frequency equation is G'A,, { + (63) where the correction factor Gc is a function of H only and is given by + i~ t +(\@ S (K) -4~ i12~kL T;P~ O~r(`T + (64) O10 Since8the t 1' and. ) th Since the derivatives with respect to H.of th, t an t = 1,2.. and consequently also those of "t vaish at H = O and H =Oo, it. follows from (64) that = O.when H = 0 and H =,o Gc has been plotted against d.H

H in Figure 7. Since Gc in the range 1 < H < O differs from G1H=1 by less than one-fourth of one percent, only the range O < H < 1 is shown in Figure 7.8 Equation (64) contains the second-order critical-depth factors of the type (43) which cause ~ and ~ to become infinite when H =.19811 3 Ct and H =.34698, respectively. Thus for depths nearly equal to (say within +.002 of) either of these depths, the critical-depth solution must be used. Since (64) is not applicable in the immediate vicinity of the critical depths, the curve in Figure 7 has been drawn smoothly through these depths. The location of the second-order critical depths is indicated in Figure 7 by the vertical dashed. lines. A most interesting feature of Figure 7 is that Gc changes sign; it is positive for small depths and negative for large depths. The value of H at which Gc passes through zero might even be selected as the division between moderately shallow and.moderately deep water. Thus for large depths the frequency of qscillation is decreased.and the period.increased in comparison with those of an infinitesimal-amplitude wave. Conversely, for small depths the frequency is increased.and.the period decreased when cormpared with the values predicted by the linear theory. This result for very large depths is qualitatively the same as that obtained by Penney and Price (49) for standing waves in a rectangular co-ordinate system. If the period of oscillation is represented by ~ (dimensional),9 and the subscript o is used to denote the values of X and r given by the For similar reason only the ranged0 < H < 1 is shown in Figure 8 also, 9 The non-dimensional period is 2a.

'292.4 2.0P 2nd ORDER CRITICAL DEPTHS I I.e 1.2 0.8 0.4 0 -0., 0 0.2 0.4 0.6 0.8 1.0 H,oo Figure 7. The Frequency Correction Factor Gc versus H

-30linear theory, then and will be the relative corrections to the linear theory due to finite amplitude. and have been calculated for the extreme cases H = 0 and. H = o for waves of approximately the maximum amplutude.l These values are shown in Table 3. TABLEI 3 cO-AND'FOR WAVES OF APPRQXIMATELY THE MAXIMUM AMPLUTUDE H AllK_ O.6 + 8.8% -8.1% to.8 - 3,4S +-3.3% When the depth is very nearly equal to either of the secondorder critical depth$, the frequency is given to the second approximration by n4,r' A,,{; YI4 i T K -3' | W /i 0,)+ (65) in which it is understood that 3 = 3 when H R d19811 and = 4 when H ~.34698. Since the sign of Ae:when H = Hc is opposite to that when H = Hc, it is noted from (65) that the sign of the frequency correction when H = H- is opposite to that when H = H+. The predicted corrections c c to the frequency at the second-order critical depths have been computed for an.amplitude of A11 = o0! and are shown in Table 4. 10 The wave of maximum amplitude is discussed in section 4 of this chapter.

TABLE 4 AT THE SECOND-ORDER CRITICAL 00o DEPTHS FOR A =.01.19811- -.05%.19811+ +.l05%.34698- -.01%.34698+ +.01% Although these corrections are very small in magnitude, it should be remembered that the general-depth solution does not predict any correction Intil the third approximation. Since I is linear in Au2 and since the correction term is very small, e will also be linear in All; hence to a suitable scale a plot of I|! versus H for H nearly critical will have the same shape as the curves in Figure 2, if it is assumed that the function of H in 11 square brackets in (65) is constant for values of H sufficiently near Hc. 3. The Pressure and Velocity Distributions The pressure at any point in the fluid may be obtained through the use of the Bernoulli equation. Merging the F(t) in (7) with ~ as was done before writing the dynamic surface boundary condition (8), we obtain the dimensionless pressure equation 4-~= Z+ ~ 2 Ca 3- a)_ )(66) 11 The variation Mf this function with H is much much less than the variation of I|_L I when H is nearly critical. A11

The radial and vertical components of the velocity,.t and / respectively, are obtained by differentiating 4 in the usual manner _j =- --— n. v _= - -' ~ (67) Because other aspects of the solution are of greater interest, no attempt has been made toward detailed analysis of the pressure or velocity distributions. One observation in connection with the velocity, however, should be made. Twice during each period (t = 0 and t = it) the fluid is everywhere momentarily at rest. -This implies that the motion may be generated by giving the free surface the configuration?{(n,) or 7?(f 77-) and releasing the fluid from rest. 4,.The Wave of Maximum Amplitude (1,5) (9) Michell and Havelock have discussed the maximum amplitude of progressive waves in deep water. If the amplitude exceeds this limiting value, breaking will occur at the crests and the waves cannot be propagated with constant form. Penney and Price(49), following a different approach have obtained the maximum amplitude of stationary waves in a rectangular co-ordinate system when the fluid depth is infinite. With their- criterion it can be shown that for axisymmetric standing waves there is a maximum value of A11, this maximum being a function of H. Since the motion is most extreme at the center of the tank, it is assumed that the condition of impending breaking will occur at the tip of the crest at r = 0 at the instant when (O,t) reaches its greatest positive eleva12 tion. 12 (O,t) will in general be maximum when t = 0. However, when the depth is nearly critical and A,/Al < O, there is the possibility that }(o,t) may reach its greates-t value when t is approximately; that is, close to the time when that component of the surface elevation due to A_ makes its greatest contribution. D

-33The criterion limiting the amplitude of the waves is based on the postulate 13 that the fluid cannot withstand tension. If the atmospheric pressure po is zero, the pressure just inside the liquid must be positive Or zero and consequently at the crestl1 e <~ o (68) The dimensionless Euler equation of motion in the z-direction is, in mixed notation, LL6 +' ay-.-~~ -4a~~ ~ IL, (69) -— at a = - G - At the crest = - O, and hence from (68) _ <G,(70) Consequently, the criterion limiting the amplitude of the waves is that the downward acceleration at the crest must not exceed the gravitational acceleration (G in non-dimensional units). Now suppose that po is not zero, but is positive. If is positive at the crest, the liquid is not in tension, but from the equation of motion (69) the downward acceleration in the liquid just below the top of the crest is greater than it is at the crest. This is physically unacceptable. Thus (68) is true at the crest and the criterion (70) applies regardless of weather or not po is zero. At the tip of the crest of a maximum wave the equality sign in (68), and hence in (70), will hold. 13 This criterion may also be obtained(x9) through stability considerations. 14 In the remainder of this section and in the next section the term "crest" refers only to a crest at r = O at the instant of its greatest elevation.

To the first approximation (70) yields -tA.& = n (71) This permits All to become very large when the depth is very small. However, the first approximation to (AlIl ) cannot be expected to be at all reasonable since it is an attempt to use the infinitesimal-amplitude wave solution for predicting the breaking wave. The second and third approximations have also been obtained and all three are plotted in Figure 8.15 No attempt has been made to show the effects of the critical depths in this plot. The maximum amplitude when the depth is critical or nearly critical will be dis= cussed later in this section. The value of (A,,I ) as predicted by the third approximation is about.8 for large depths and about.75 for small depths. It may readily be seen from Figure 8 that the convergence of the successive approximations (if, indeed, they do converge) is much slower for shallow depths. This is in agreement with what was anticipated at the end of section 2, Chapter II In the general-depth solution were assumed valid when the depth is critical, application of (70) would lead to the conclusion that A(4 — K O when the depth is critical. Use of the critical-depth solution in (70) for depths critical or nearly critical results in a less stringent value of (IA- )); however, (A1,, h ) when the depth is critical is considerably smaller than at non-critical depths. By use of 15 The third approximation to (AllK2) a was evaluated for only four values of H, namely oo,.50,.25, andolU.X

-352.8 2.4 - \I ST APPROX. - - 2 ND APPROX. 3 RD APPROX. -- - 2.0 1.6 \ 1.2 0.8 0.4 0 I I I, I, 0 0.2 0.4 0.6 0.8 1.0 H 0co Figure 8. Maximum Amplitude versus H for H not jritical

the relation (58) between Aaq and All, the maximum amplitude when H equals either of the second-order critical depths has been computed to the first approximation -and is shown in Table 5. TABLE 5 THE MAXINEJM AMPLITUDE AT THE SECOND-ORDER CRITICAL DEPTHS H sign of All (ALK)x A,I.19811-.523.19811+ +.428.34698- +.210.34698k+ -249 It is noticed that at either Hc (AllK),ax is smaller when is positive than when this ratio is negative. This is to be expected since when zL is positive the All and A2, components of the motion are in phase and both make their greatest contribution to the downward acceleration simultaneously at t = 0. On the other hand when is negative the two components are out of phase and the greatest downward acceleration during a period will be less than that for the same All when is positive. It follows that (A,,) is larger when 2 is negative than when is A,, positive, both at the same Hc. The component of downward acceleration due to A2 is of order 0(0,) when H = Hc but becomes of order 0(lA,) as IH-Hcj increases. If the criticaldepth solution is substituted into (70) to the first approximation and the resulting equation applied as IH-HcI increases, then the A&z term of that equation becomes of order c(A)-1) but the other terms of order {(Al,) have

-37not been taken into account. Since terms of order are of comparable magnitude with terms of order A) in determining the maximum amplitude, (A,/KL from the first approximation when H R Hc will not approach, as JH-HcJ increases, (A-1fl) from either the first or second approximations as plotted in Figure 8 for H not critical. In order to investigate the manner in which, as IH-HcI increases, (il.) for depths nearly critical approaches the value predicted from the general-depth theory requires that both the general-depth solution and the criticaldepth solution be carried to a sufficiently high order of approximation that the apparent convergence of each has been secured. 50 The Angle at the Crest of a Maximum Wave Stokes(24) showed that, if progressive waves exist having a discontinuity of slope at the crest, the angle enclosed there is 120~. Experimental work(58) has indeed shown that such a wave apparently can (49) exist. Penney and Price recently concluded that the crest of the maximum stable standing wave in rectangular co-ordinates is also pointed, enclosing an angle of 90~. Following the general procedure used by Penney and Price leads to the conclusion that the maximum axisynmmetric standing wave has at r = 0 a crest enclosing an angle of approximately 109.5~. Because the eigenfunctions T all have first derivatives with respect to r which are zero at r = 0, the solution fork(r,t), when taken to any finite order, must necessarily give a wave profile having a horizontal tangent at the crest. However, consideration of the free surface as the isobar, p = Po, does yield a non-zero slope at the crest of a

maximum wave at the instant of its greatest elevation.l6 The equation of the free surface of the maximum wave at the instant t = 0 may be regarded as given by the implicit relation (72) where the curves p(r,z) = C, (C > po), are the isobars in the r - z plane, Assuming p(r,z) to be continuous throughout the fluid, then for an infinitesimal displacement (dr,dz) from the point (r,z) we have Taking the point (r,z) at the tip of the crest, and choosing a displacement (dr,dz) such that the new point is also in the free surface, of d 4 F h2 Ad,,(73) From the previous section ~ is zero at the crest of a maximum wave. From (73) we perceive that e is also zero. Hence the tip of the crest is a singular point. Proceeding to the second order in dr and dz for an infinitesimal displacement from the crest along the free surface, O =y+ 2. on od (4St, (74) Since p is obtained from the velocity potential by the Bernoulli equation (66) 16 For the rest of this section, this instant of greatest elevation is taken as t = 0. Because of the reservation of footnote 12, page 32, it is understood that "t = 0" in this section is not necessarily the same time origin used elsewhere in this paper.

-39and and are identically zero at the crest when t = O, p must satisfy ~l~ -O at the crest when t = 0: (75) At r = 0 the middle term of (75) is indeterminate. However one application of 1' Hospital's rule yields. / @4 _ 4 ~~~ 9(76) Thus at the tip of the crest ~ _ I (77) Since ~ must have the form (22) at least as far as its dependence on r and z is concerned, and since 0' is zero at r = 0, = 0 at r = 0 for all z, All, and t. It follows that not only but also - of, and are zero at r = O, assuming unlimited differentiability of. Since po, C1, and Gz are independent of r, it follows from (66) that and spA JN O - (78) and Q + o =D,(79) Adt an

From (79) it is seen that the middle term of (74) is zero and hence that 2 jckui) + (80) From (77) and (80) @ 9 t i! l i-t (81) at the tip of the crest of the maximum wave. Therefore -J Z i —, (82) The negative root is the one associated with a crest. If the angle between the negative z-axis and the tangent to the surface is denoted by then the total enclosed angle 2~ at the crest of a maximum wave is r2 r toa 4 1~ ~28, (83) This conclusion is valid unless and consequently are zero. That the angle at the crest of the maximum axisymmetric wave should be greater than the 90~ of the maximum two-dimensional standing wave of Penney and Price might be anticipated from the difference in geometry. Let us view both wave types from above. In the latter case there is a "line crest" parallel to the y-axis; as the crest rises toward its greatest elevation, the fluid particles approach it only from the positive and negative x-directions. In the axisymmetric case there is a "point crest" at r = 0; as the crest rises toward its greatest elevation, the fluid

particles approach radially from all angles e. If each system possesses a given amplitude less than the maximum amplitude of either, and if the amplitude of each is gradually increased, it seems logical, because of the geometric difference noted above, for the axisymmetric wave to reach an unstable condition earlier, that is, at a less sharply pointed crest. This is indeed the result just found. At first glanme it appears that (82) is valid for a depression as well as for an elevation at r = 0. However at the bottom of a trough at the instant of its greatest depression the acceleration is upward and hence from (69) d is not zero. From (78) _ is zero there at all times. Thus we conclude from (73) that dz = 0 for a depression and the tangent to the surface at the bottom of the trough is horizontal. 6. The Energy of the Wave Motion In a progressive wave of infinitesimal amplitude the total energy of the motion is one-half kinetic and one-half potential. Similarly in an axisymmetric standing wave, the mean kinetic energy equals the mean potential energy to the first or linear approximation. What is true for an axisymmetric standing wave of finite amplitude? The potential energy V of the wave motion is V= I 6 ff( 4 (84) in which dA is an element of area in a plane normal to the z-axis. Setting dA = rdr dG and performing the integration with respect to G, we obtain V = TT G!.7 ti4hI (85) V 1-ft IT0

If a bar denotes the dimensional energies, then V and T have been made nondimensional by T ___ 9,T (86) The kinetic energy T of the wave motion is ~=tT [( don )~\Lj ( 8 ) y (87) in which dV is an element of volume of the fluid. Upon integration with respect to G, (87) may be written as i re ~( T=ITSZ deTJ(@ )j (88) However, a more convenient form for the evaluation-of T may be obtained by using Green's theorem to transform the triple integration (87) into a double integration -/i a- 2 (89) in which d9 is an element of area in the boundaries of the fluid region, and dn is an increment of length along the normal to d~, positive when directed into the fluid region. Since is zero on the solid boundaries, only the integral over the free surface will make a contribution to (89). Setting dI = rdB ds and performing the 1 integration, T17 - - r f;(90) T~~

-43where C is the curve formed by the intersection of the free surface z = with a plane containing the zaaxis, and ds is an increment of length along C. On C, dz = —- dr, and By the kinematic free-surface boundary condition (17) at z = Thus on C and we finally arrive at the expression -T= f-T [4;7 [ (91) Performing the integrations (85) and (91) on the solution to the third approximation at general depth,. we obtain 77 \/. 4iKr 3 t (I1) [II I Cs) $' (92) ~~ I + a7t QC9 +'

-44and Tfr)4 L S ~~~ IC/94 C.0 -1r0L3t7(93) ___c_, _C, t i C c +1 The C,, (5 < n < 13), are rather long functions of H and are presented in Appendix V. Because the system is conservative and no work is done at the boundaries, the total energy T + V must be constant. This requires that the sums (C5 + CO), (C6 - ClO),(C8 + C02), and (Cg + C13) each be zero. Certain identities, involving integrals of products of Bessel functions, are sufficient to prove that the first three of these sums are each identically zero. These identities have been proved analytically in Appendix II as equations (II-15), (II-16), (II-17), (II-18), and (II-21). C + C13 will also be identically zero if two expressions (II-22)(II-23) are identities. The identity of these expressions has not been proved, but strong evidence for it has been presented in Appendix II. Thus the total energy of the wave is given by -(T+v) LL T(6 1 ) 1 - (C7+ ci) (94) Platzman(50) in his investigation-of the partition of energy in periodic progressive waves of finite amplitude and permanent form in water of infinite depth found that the kinetic energy exceeded the potential

energy. For the wave of maximum amplitude he found the first V 8 term in his expression yielding a value for TLV of 9,931. Representing the temporal mean kinetic and potential energy by TM and VM, respectively, and making use of several of the identities of Appendix II we obtainl7 TJ (6V \vM 6 (6 KI4 OA Al O(n + (95) 1=1 ii=I 2For a wave of approximately maximum amplitude (AllK1 =.8) when H = WO, (95) yields'r_-M =-3o4%. Thus for an acisymmetric standing wave in Vn fluid of infinite depth the mean potential energy exceeds the mean kinetic energy. Although-one would hesitate to predict the sign of TM-VM in advance, this result does seem reasonable since V is always positive while T equals zero twice during each period; consequently both Vmin > TmTmi= 0 and Vmax > TmJ: The differences between the present problem (standing wave in cylindrical co-ordinates) and Platzman's problem (progressive wave in rectangular co-ordinates) cause the algebraic sign of our result for 17 Note that from (92) and (93) TM-VM is of order (A ). This is in qualitative agreement with the result of Rayleigh(.2) who first pointed out that the difference between the kinetic and potential energy of an -oscillatory progressive wave of finite amplitude and permanent form in water of infinite depth is of fourth order.

TM-VM to be opposite to his, both for the case H =0o However, as Starr (55 P. 185) in his article on energy integrals for gravity waves observed, So far as the writer has been able to find there appears to be no simple means for obtaining the magnitude or algebraic sign of..... from general considerations without making use of the detailed solution -to the wave problem. 18 Starr's e is defi~ned as the difference between the kinetic and potential energy of the waves.

CHAPTER IV CONCLUSIONS AND RECOMMENDATIONS 1i Concluding Remarks The exact equations governing the free oscillations of finite-amplitude axisym-metric gravity waves are presented. -Thes'e equations include two types: a -inear group, and two non-linear free-surface boundary conditions. The eigenfunctions are determined. from the linear equationsb. To represent a periodic first-: mode motion a linear combination Of these eigenfunctions is taken. An iteration procedure is followed to. find the coefficients in this combination in terms of an amplitude parameter, All such that the two non-linear boundary ionditions are satisfied.. Because of the complicated nature of the problem neither the existence of this motion nor the convergence of the solution procedure has been proved. No a priori limitations have been made on the depth of the liquid. It is found however, that there are certain discrete depths at which a higher mode at a frequency equal to an integral multiple of the basic frequency is of the same order of magnitude as the first mode. The motiowhen the depth is approximately equal to one of these critical depths is treated by an appropriate modification of the procedure used in obtaining the general.soluti ono -47

Heightening of the crests and broadening of the troughs, when compared to- the linear solution for the surface configuration, result from the analysis and are typical of all finitesamplitude wave solutions available so far, The period-of oscillation, compared to that of infinitesimally small oscillations, is increased for large depths but decreased for small depths~ The maximum amplitude for which an axisymmetric wave will remain stable has been investigatedit is found that the maximum wave has a pointed crest at r - 0 enclosing an angle of approximately 1095~0, The potential energy of the motion is greater than the kinetic energy (at least when the depth is infinite), and the difference between the potential and kinetic energy is proportional to the fourth power of the amplitude. Detailed, results pertaining to the surface configurations the frequency of oscillation, the wave of..maximum ampiitude, and the energy of the wave motion are presented in Chapter III. 2. Suggestions for Further Study Further study in several areas both directly and indirectly related to the present work seems desirable. These areas fall into two classes: those suggested by the results of this work, and more difficult problems toward whose sol-utions this work is but a first step. In this dissertation the main interest has been in a firstmode wave when the depth is nQn-criticalo Mention of the critical depths has been for the purpose of noting their existence and of

showing that the motion when H u:1Ec may be treated by the general procedure outlined in Chapter IIL A detailed study of the coupled motion carrying the solution to higher approximations when H He would undoubtedly reveal many interesting features not noted here. By a judi-ciaus permutation of subscripts the motion of axisymmetric waves of m'odes higher than the first may be obtained.from the general.deplh.solution of this papero However, there will be a different set of critical depths pertaining to the higher modeo.. For odes higherthan thanthe second there will also be the possibility of exciting lower-mode oscillations at frequencies equal to the basic frequency divided by an integer0 For example, a third-mode wave at frequency Co will excite a first mode at frequency ~ 9C when H = 619811 Caution should thus. be exercised in applying the results of this study to higher modes0 An experimental study of the maximum-amplitude axisymmetric wave would be useful in verifying the prediction that such a wave has a crest angle of 109.5~ as well as in gaining a greater understanding.of the mechanism of breaking0 Of conside rable practical interest are those modes -of oscillation which are not axisymmxetric., If -the motion varies -with 0 as well as with zji, and t, there are nodal diameters as well as nodal circles in the linear solution, Which is discussed in some detail by Lamb(l) o Conceptually, a nonrilinear solution for unsymmetric gravity waves in a circular basin is only slightly more difficult than the work of this papero However, a second infinite set of

-50eigenvalaues is introduced by the angular variation and the additional summation over this set makes the problem extremely involved from a cnomputational standpoint. Only free osaillations have been considered in this dissertation. The results obtained may be of use in attempting to solve the more difficult, but very -important, problem of forced vibrations. Both resonant and non-resonant cases should be studied. Very closely related to the forced motion is the problem of determining the motion following release of the fluid from rest with an initial arbitrary axisymmetric configuration of the free surface. This motion, which is easily analyzed if the problem is linearized, will in general be non-periodic.

APPENDIX I TEE EXPANSION OF FUNICTIONS IN DINI SERIES The expansion of an arbitrary function F(r) of the real variable r in the form 11=1 where A,>, 1,J 4 d.......denote the positive zeros in ascending order of magnitude of the function when 7) -1/2 and M is any given constant, was first investigated by Dini(61)o The coefficients in the expansion are given by the formula 0: (IDini noted that the expansion (I-).must be modified by the insertion of an initial term, (), hen M +9 although he ave ts value incorrectly66 597) This initial term is given by e (d,-e(i v+,) 4f2 g FW(r) i (I-3) 19 A different initial term must also be inserted when M + 2)< 0o (66, po597) -51

Of particular interest in this paper is the special case M =-2= 0. Then F(SI)= _X () + i bTn (XtAn) = (1-4) 2 Fln) a4 ) 2 b (+,n) where A are the positive zeros of and the coefficients, bn, are given by n () t(As) l (I-5) It is seen that the A are precisely Kn. It is also seen that, if Ko be defined as zero, the initial termra(r) in (I-4) is the zeroth term of the summation with its coefficient bo given by (I-5) for n = 0. Because there are many different functions F(r) which we wish to expand in Dini series, let us adopt the nomenclature 0( F) for the bn associated with a particular F(r). Then F7) _ 2 o(,,(F T0 (I-6) "~=0 where K.o O and K1, K2, K3..".'..are the positive zeros of J1(K) arranged in ascending order of magnitude and the coefficients 0(,(F) are given by (F) = _. a., ('-7) —.....

-53If F(r) is continuous and has limited total fluctuation in the interval 0. r < 1 and if the integral FleQ).d exists and 0 converges absolutely, then the Din4 series (I-6), is uniformly convergent and uniformly summable in the interval 0 < r < 1 (66, pp. 598-616) All functions F(r) which we encounter, at least through the third approximation, meet these conditions0 Except in the very simplest instances, the integral in (1-7) cannot be evaluated analytically but must be done by numerical means. Those O'(s which have been computed through numerical integration are presented in Table 6. It is believed that the first three decimal places are accurate, while the fourth decimal place, particularly for the higher values of n, is known to be unreliable. The (. whose values may be obtained analytically, were also computed numerically for comparison with the exact values. For the discussion in Chapter III, section 6, of the energy of the wave motion, a number of integrals S(F)are needed, where the operator S( )is defined as 0 It is seen that the o(fs and, I's are not independent but are related by s (041 FS \( eT,( 0(~(F) to ( a S (A Table 7 gives the integrals which have been computed. The first three decimal places are believed accurate, while the fourth decimal place is unreliable.

-54TABLE 6 Wn(F) FOR VARIOUS FUNCTIONS F(r) j2 2 3 F(r):01 Jl 01 exact 6(' s eomputeJ~ 1 s %(F) zero 0.0000 0.1622 0.1622 0.0572 I(F ) |L o Q0000 1. 0000 03523 0. 1761 0.4139 (F ) zero 00001. 4794 -0,3241 0. 3157 3~(F ) zero 0. 0001 0.0070 -0.0174 0.2082 ~q(F ) zero 0.0002 -0.0007 O. 0043 0.0060 0(F F) zero O. 0002 O. 0004 -0.0018 -0.0004 _6(F) zero 0.0003 0.0002 0.0009 0.0004 o~(F) zero 0.0003:0.0004 -0,0005 0,0002 qj(F) zero 0,0007 0.0003 0.00003.0 0004 0O9(F) zero 0,0003 0.0003 -0.0002 0 0003 c('F ) | zero. O 0004 0.0004 0. O01 00004 F (r_ ) | J1 Jll/Klr JO Jo1J03 JolJ04 -Ce (F) O00286 o.0429 zero zero zro (j (F) 0. 1380 0,0811 0.2662 0Q0027 -0,:002 c (F) -0.0185 - O 0775 0. 300 0o2945 o.oo40 ot3(F) |-0.1404 -0.o,o483.0.4254 0.2929 0.3087 V (F) -0,0089 0.0026 0.0076 0.4037 0,2897 E(F ) | 0.0 0017 — 0. 0O 8 0,0076 0 o3919 o((F') -0. 0006 O. 0005 0.0005 -0.0008 0.0076 o(7(F) 0.0003 -0.0003.000002 Oo0006 -0.0008 ~8(F) |-0.00 2 0.0002 0.0004 0.0002 0.0006 oq(F) | 0.0001.0001 0.0003.0004 0.0002 ~ (F) -0.O0001 0.1 0. 000.o4 0. 0005

TABILE 6, CONT T. F(r) 1oto JO1'J017O5 JolJ07 01 09 040(F) | zero zero zero zero zero ql (F) 0 00OO1 0.0000 0.0001 0.0000 0.000 (F P) -0., 0-0003.0002 0.0001 0.0 00 0.0001 Oe3(F) Oo0047 -Oo.004 0.0003 0.0001 o00001 q4(F) o0.3173 0.0052 -, - 0004 0,0003 o.OOO1 5((F ) 0.2882 o, 3230 0.0056 -0. 005 0, 0004 _6d(F) 0 03846 0..2873 0.3272 0.0057 -0.o 0005 q7(F) I 0.0078 0. 3795 0.2868 0. 3303 0,0059 o(F) -0.0007 0.0075 0. 3758 0(F) 0.0007 -0o0007 0 0075 b(F ) 0 o.00003 o.0 o00 -000oo6 F (r') zJ1Jlro Jl111 JllJ13 JllJ14 JllJ15 Q0(F) zero zero zero zero zero. qt (F).0001.0.2436 0~0032 -0.0005.0.0001 qy(F) 00001 0o 0824 0'2604 O. 0054 -0.ooo0008 q3 (F) 0, 0001 -0,3132 0. 0551 0o...2681.0066 qo(F) O.0002 -0. 157 -0 3071 0,0416 0o2716 q$(F) 0.0002 k'6(F) o. oooO o{?(F -0.000 o4

-56TABLE 6, CON'T. F(r) JJ 1J2. -......,.,.... 11...I o, zeroi~ Zero zero zero zero O( (F) -0.0000ooo O.00 o.0000 0.0000 -0.0000 O((F). 0002 -0.0001 0.0000 -0.0000.00000 ~3 (F) -0.0011 0.0003 -0o0001 o0.0001 -0.000 I (F) 0.0073 -0.0013.0.0004 -0.0002 0.0001

-57TABLE 7 S(F) FOR VARIOUS'FUNCTIONS F(r) F(r) 100 S(F) F(r).100 S(F) F(r) 100 S(F) 2.8577 J01J% 0.0009 J01JJ5 0i7564 J ol sJs oljsJ: o,.o o JolJo~.Jo5 ozs 2 J0oJo2 2.1589 0.0Jo 00:oo3 JOJ4Jo o.13 2 0.0219 0.0005 J0J -0.0O11 51 O'olQO3 01co9 010408 2 a, oool6 Js~7 Joi 0Jol4 -006 J oJJo 0o0003 J oJo4Jo oooo8 j~'"~'J 0.o 0008.0 o..004 0 J 06105- 01 0201,0 010j40o9 000 0i26 060003 J.0.9130 0 0 01 0 00 005 jo~ o2.o 0.0.6o3o ~~~~~~~oi.-o8 o i~o3 j6 T 2j o.oo -0.0013 0.0108,~~o ~ oaooo o.oo J01J0 ooJ 0 o -0.0009 JOiJ 21,3602 Jo13.J08 0.0003 JJ.0007108 02 JO1lOJO.J3 1.3263 1Jo03i 0.,0004 JJ5J0 0.0003 Jol302lo0 0.0181 J01J03J010 0.0004 2 J01J06 o,4658 Jo. ~ 5 o.oo01 o8 Jo.Joo4 o.o6oo4 jok..~.0 o.468 J01J0.2J J J 0.5304 JT -0.0015 2-00.6906

.0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~C. 0 0 C, 01 0 0 t 0 0- 0 0 0 fJ"" r"" 1-" c~a r, (1~3O' p, C ct ctH H H H H H H 0 0 0 0 0 0 0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~8 ~ H H-J H H H H HT)-F-J -I - F) O ONj ON H: H H- H H 0-0 0 0 0 N H C H ~Ia H H H OC- O - O - - ~ ON ~ J ~LAJ V'-J 0'CO)'-. CCX) 8~~~~~~~ 0 0 00 0 0 H H 00 0 0 0 0 H 0; 3t-Q' 0 88888 88 ON 0000 -J O Lei0 00 UO 09'8 8 8 8 8 8 H H H H H ~~ ~~ ~~~~~H H Hj H- - - H H H4 H H H - LN H PL I - ri CL )PI ~a I, r "" r" sri.H H H H H H- H, H'0 co) - 0 \ \..' \~Q I 0 co).0 I a I0. 0 0 0 0 0 0 0 ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~~~~0 0 W HP 0 0 0. \JQ01H0 Z0.0 0 H ~~g~ 80 H U 4~~~~)0\'0000 COH C C. C4 C4 CHe- C.ll~~i ICd C.. C~ C..B C- re 0 0 0: C-, Ci C-q C-,.4 C..CC-, ~. ~ ~ H~~~ ~~ ~ ~ ~ H H ~' H H H H H H C-4 C-, C..4 4 C C.,.4 C. -, C- C.. C4. C-4 C..4 H H H H H H~~~ H H -H H'JI -0 RI H0 k() 0 - ON \J1 4 0 0~~~~~~~~~~ O 0 0 0 000 0 0 0 0 C0 0 ON 0 -~~~~~~~~ 0 0 0 0 8 3~~~~~~~~~~~ 0 0 CX \ -~~~~~-'-.0 Co H 0 0~~~~~~~~~~~~~~~~~~~~~~~~L I\ Vo F-J r\C — 4 LA HH H Co IV Hr; lv- 0 43 ON -.

TABLE 7 CON ITO F(r) 100 S(F) F() 100 S(F) F(r) 100 S(F) %o)JALG6 o.o0175 JoJo 0O.4 1 J o08JJ -0.002 JJ1117 -0 0oo30 %J0 -oo Qo08 J2 J O.ol1 1......7 05 JO1 09 ~il o4d11018. 0009 J.J 0.0007 JJo0J12 -O.0001 J04-g11t ~ ~'-0.000o3 JJ0 0 2.1428 Klr 411110Oe0002 Jo3J 08 ~ 005 1 0.6578 J044rjrlo ObooCll I 01 08 J Kjr J 25Ji. O0.0034 J| o.9 O 400 Jo1 a3490 J |Jo' J02 0. 06Jll ~0.0014 4 J03 K1 -o.31506 JozJ l l j Jo3 K~Te -.0o0006. O1jil 1.1189 ~ o.o061 J07J-11 J35~0J04 Jo 8Jl o.ooo4 Jo1J2J0 0-0o 0834 J0 -0.0021 Jo9J -0.0002 JOJ3J -0.4377 6 Ko JoloJ!; o0.0001 Jo1Jo4J1~ -0;0211 O 0004 Joloill JolJlr 1.1189 JocJ07K. 3.,3572 "' 3J0 o 0.0002, JQJ01 13a43572 J J00J2l 0v 3302 31 J 4217o1 JQ1 11J _ Jo.9 KJ r' O1 03 0.6490 1 J07J1 0.04 01 K 0.0001 ~ Jo2j 9 Ji02104

APPENDIX II CERTAIN INTEGRALS OF BESSEL FUNCTIONS Orthogonality Relations Provided that PV (' > O) are the eigenvalues for which 9(,)= it follows from McLachlan(63, P 102-104) that 5)-mZ~~ (11-2) S('T,) 2 TY) (11-3);S (Tl (2)) > (-4) where the oprator S (F) has been defined as S(F) jn F, (J) l I (1-8) o 2. Some Identities The differential equation wr /t wt O (II-5) whtere prfina~~~me. notes differentiation with respect to 37 is satisfied by Tr) for which OT = - m a(~m). Multiply (II-5) byIJ1vvw-, where v,~, and -a are any single-valued and differentiable functions of

j- in the range 0 ~ < I -\v CJ // t- >v><# 1N ~eit9-2 &@ Q _y3 O (II-6) Noting that we may rewrite (II-6) as Mu/tiply by and integrate from 0 t.o?; then( Multiply by dA and integrate from O to 1 - then J s f~l+ _ (119) This result is very generalb -V,:, and ~ have not been restricted to any particular functions of. has been restricted to being D(tn), but K is not necessarily an eigenvalue: If we now require that K be one of the positive eigenvalues,f (mn >I)and perform the inteegrati.on from 0 to 1, then, upon substituting for y and y' and dividing by -K, The special cases of (II-lO) of interest in this paper are obtained by choosing v -,4, from the functions unity,, n, and ~7;,. and by combining certain results obtained directly from. (II-lo0) These identities are ('i I )

rv ~ r -r _4 I,{ a1 0 —4 -,- 10 1t -v dra ~r vVA O 44 b x ~ ~ I _c | I | -I UI, I ru L -- LA - I. C-A FJ LA i-Idv ~ ~ ~ O AFJII

-63From the definitions of P (42), OF (1-7), and S(F) (I-8) and the identity (II-12), TOS1S(7 )I= rp Jo ( l) (11-21) Two expressions of a somewhat different form from the above identities have been verified numerically: 2 5 (r, 71) = 7 (t~{(T,~)-ol nj To I; ( 0, )}, (II-22) and ( II-23) t 70 ~ ( to; ~t ( ( Joa,) 2 1SN(I; o Use of the numerical integrations tabulated in Tables 6 and 7 yields for (II-22) a left side of.02238 and a right side of.02239; for (II-23), a left side of.02041 and a right side of.02041. In each case the first two terms of the infinite sum provide over 99.9% of the total right side. These numerical results strongly indicate the identity of (II-22) and (II-23)o Furthermore, the physical consideration that the total energy of the wave motion must be independent of time requires it.

APPENDIX III THE SOLUTION AT GEN'ERAL DEPTH:TO THE THIRD APPROXIMATION The results of the third approximation to the solution at general depth are: Al = 143 B 41, AI mlJ( ( 1, CPI 0 8P) ( (III-2)'- - Kg,(To p1 13 rI To I T(, \, AAll6 T (III 43 AE+m ~ ('n +a O) )(.Iy>(ol 7} (II - - i (21;~-0 AA (III-4)

PA J I KII~q 0((30 1 h 8A O-n KL p=K, +'1 ()Tz(I)5 2, z -"I So J L 3 t -~~~~~~~15I A ~i~K~t~1U Jcn(, I + -~x ar +-~ 51 t

-664 3 a,'+ 4 4 f 1~:zi r =1

G - i2(1~ ~ T I'joll) KI fZ) 1I ~I All other A., B., and J+,are either zero or of order higher than 0 de(ine) The quantit p is a function o H only and has been defined previously by equation (42) on page 14iLi.

APPENDIX IV THE SOLUTION AT THE SECOND-ORDER CRITICAL DEPTHS TO THE SECOND APPROXIMATION The results of the second approximation to the solution when the depth is equal to or very nearly equal to either of the secondorder critical depths are given below. When H R.19811, - - 3 and when H % 3,34698, Q - 4 in these expressions. All = A,,, 8,& _ o (IV-l) A - A All L-3 (i + J a tt (l z~II } ) (I2) +A = +J c2 C4 (IV-3) C 3 -68

-69-: A A', (O < -n c,~-/) US, e M ( en >,Q+) (D(v 5) (Iv-6) 4ii 6, K, 44 __ 4 (r /, 2) (v-8) "+ tl-g st)cerw

b "~ _(I + - + I + - ice; oHj A rv t, _ - -, r —-,LH LA t v )bR OR 4 — " vo-,N

JT) 4,- A t +, ) Ta.-, 9& K,tA, IA - (IV-12) 16 A. 4 K - ( I~ _(16 ), 5(1 + Ks 0X(:nl (IV15

G = <181 1K + tC(9 -3.i E: W1 t (o (IV-14) in which the functions C2, C3, and C4 of H and All in equations (IV-3) and (IV-4) have been defined previously by equations (55), (56), and (57) on pp...19-20, and the function of, H.has been defined by e;quation (42) on page 14. All.. other Amn, Bmn, and. (r) are either zero or of order higher than 0 (,4).

APPENDIX V FUNCTIONS OF H APPEARING IN THE E1NIERGY.EXPRESSIONS The functions CO of H, (5<n<13), in the expressions for the potential (92) and kinetic (93) energy are: -2 S(TO (:0/ 19 t 7 T ( 1) (V-l) +tctn2R<C S(,,1,) rl,, 8(v-3) c ", I?,. I, sTOa: a 71=l } I 3 - tI "In D- 3 (-3) 32. -3

Pi( oL u i I' rra -^1<, a IF 3, 1 Ji'-t —, r~ )~ ~01 n +1 I_ + + II an A f-I J. -,', +'"' k... ---- /zi~~L1 L LA 4- (,A+)I~A 4 s r o

-75-: (o3-4 T7 L( (v-6) -IOD aI a I ( m 4 S( X o ) _- 3 L God 4 S(:cl Tom) (v-7) 9 -T, o(,- t + 3 E 3" S( TO 3( a \-I -I- V00 3~ r~ f ~ ~ c5 JPI (JII I-Al

-76C_ - rT o 33 (I To T) + 3eS st E w t g S(o TO r 9 S I I,~~, I 5t= 63I- IT K, ~,_3~ - 3;, s(~,:~~)-~*,8 E Kt (,,;K)+.- 15 - (, x, K

BIBLIOGRAPHY ARTICLES OF HISTORICAL INTEREST 1. Airy, G. B., Tides and Waves. Encyc. Metrop., 1845o 2,, Bazin, H., Recherches experimentales sur la propagation des ondes, Memoires divers savants a l'Academie des Sciences, 19, 1865. 3. Bouasse, H., Houle, rides, seiches et marees. Paris: Librairie Delagrave,- 1924. 4. Boussinesq, JO, Theorie de l'intumescence liquide appelee onde solitaire ou de translation se propagent dans un canal rectangulaire. C. R, Acad. Sci. Paris, 72, pp. 755-759, June, 1871. 5. Boussinesq, J., Theorie des ondes et des remous qui se propagent le long d7un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal de vitesses sensiblement pareilles de la surface au fond. Liouville's Jour. Math., 17, p. 55, 1872. 6. Boussinesq, J., Essai sur la theorie des eaux courantes, Memoires divers savants a l'Academie des Sciences, 23,. 1877. 7. Chrystal, Go, An investigation of the seiches of the Loch Earn by the Scottish lake survey. Trans. Roy. Soc. Edinburgh, 45, pp. 361-396, 1906. 8. Gerstner, F. J. von, Theorie der Wellen, Abhandlungen der ko'niglichen bohmischen Gesellschaft der Wissenschaften, 1802. 9, Havelock, T. H., Periodic Irrotational Waves of Finite Height. Proc. Roy. Soc. A, 95, p. 38, 1918. 10. Korteweg, D. J. and de Vries, G., On the Change of Form of Long Waves, Phil. Mag. (5)9 399 p. 442~, 1895. 11. Lamb, H., Hydrodynamicso 6th Ed. New York: Dover Publications, 1945. 12, Levi.Civi ta, T.,~ Determination rigoureuse des ondes permanentes d'ampleur finle. Math. Annalen, 93, ppb 264-314, 1925,~ 13. McCowan, J., On the Solitary Wave, Phil, Mag. (5), 32, pp. 45-58, 1891. -77

ARTICLES OF HISTORICAL INTEREST (CON'T,) 14. McCowan, J., On the Highest Wave of Permanent Type. Phil. Mag. (5), 38, p. 351, 1894. 15. Michell, J. H., The Highest Waves in Water. Phil. Mag. (5), 36, p. 430, 1893. 16. Nekrassov, A. Io., On Steady Waves. Izv. Ivanovo-Voznesensko Politekhn. In-ta, No. 3, 1921. 17o Poisson, S. D.o Sur les petites oscillations de 1'eau contenue dans un cylindre. Ann. de Gergonne, 19, po 225, 1828-29. 18. Rankine, W. J. M., On the Exact Form of Waves near the Surface of Deep Water. Philo Trans. Roy. Soc., 153, p. 127, 1863. 19, Rayleigh, Lord, On Waves, Phil. Mago (5), 1, pp. 257-279, 1876. 20. Rayleigh, Lord, Hydrodynamical Notes —Periodic Waves in Deep Water Advancing Without Change of Type.o Phil. Mag. (6), 21, pp. 177-195, 1911. 21. Russell, S., Report of the Committee on Waveso Report of the 7th meeting of Brit. AssInb Adv, Scio, po A417, 1838o 22. Russell, SO, Report on Waves, British Assn.o Reports, 1844. 23. Russell, S., Report on Waveso Report of the 14th meeting of Brit. Ass'n. Adv. Sci,, p. 311, 1845. 24. Stokes, G. G., On the Theory of Oscillatory Waves. Trans. Cam, Philo Soc., 8, ppo 441-455, 1847. 25. Stokes, G. G., Mathematical and Physical Papers. Supplement to a Paper on the Theory of Oscillatory Waves. Cambridge: University Press,, Vol, 1, pp. 314=326, 1880o 26. Struik, D. Jo, Determination rigoureuse des ondes irrotationnelles periodiques dans un canal a profondeur finie. Math. Annalen, 95, ppo 595-634, 1926.27, Weinstein, A,, Sur la vitesse de propagation de 1 onde solitaire, Reale Acado dei Lincei, Rendiconti, Classe di scienze fisiche, matematiche, naturali, 3, 8, ppo 4631468, 1926. 28. Wilton, Jo R., On Deep Water Waves. Philo Mago (6), 27, p. 385, 1914.

-79MODERN REFERENCES 29, Chappelear, J. E., An Investigation of the Solitary Wave. Trans~ Amer0 Geophys, Union, 37, 6, pp. 726-734, 1956. 30. Daily, J. W. and Stephan, S. C., Jr., The Solitary Wave, MIT Hydrodyn. Lab. Tech, Rep. 8, 1952. 31. Davies, T. Vo, The Theory of Symmetrical Gravity Waves of Finite Amplitude Io Proco Roy. Soc. A, 208, pp. 475-486, 1951. 32. Davies, T. V., Gravity Waves of Finite Amplitude III. Steady, Symmetrical, Periodic Waves in a Channel of Finite Depth. Quart. of App. Math., 10, 1, pp. 57-67, 1952. 33. Dubreuil.-Jacotin, M. L., Sur la determination rigoureuse des ondes permanentes periodiques d'ampleur finie. Jour, de Math. Pures et Appliquees, 13, 9, pp. 217-291, 1934 34, Friedrichs, K. 0 and Hyers, Do Ho, The Existence of Solitary Waves. Communications on Pure and App. Math., 7, pp. 517-550, 1954. 35. Gerber, R., On the Existence of Irrotation, Plane Periodic Flows of an Incompressible Heavy Liquid (in French). CO R. Acad, Sci., 233, 21, pp. 1261-63, 19510 36, Goody, A. J. and Davies, T. V., The Theory of Symmetrical Gravity Waves of Finite Amplitude IVo Steady, Symmetrical, Periodic Waves in a Channel of Finite Depth. Quart. Jour, of Mecho App. Math,, 10, 1, ppo 1-12, 1957. 37. Hunt, J. No, A Note on Gravity Waves. of Finite Amplitude, Quart, Jouro Mech. and Appo Math., 6, 3, pp. 336-343, 1953. 38, Iwasa, Y., Analytical Considerations of Cnoidal and Solitary Waves, Mem, Faculty of Engo., Kyoto Univo. 27, 4, 1955. 39. Kampe de Feriet, J. and Kotik, J,, Surface Waves of Finite Energyo Jouro of Rational Mech, and Analysis, 2, pp. 577-585, 1953. 400 Keller, JO B., The Solitary Wave and Periodic Waves in Shallow Water. Annals of the No Y, Acado of Sci,, 51, 3, ppo 345-350, 1949. 41. Keulegan, G. Ho and Patterson, Go W., Mathematical Theory of Irrotational Translation Waves. U, So Nat'l, Bureau of Stand, Jouro of Res., 24, pp, 47-101, 1940.

-80MODERN REFERENCES (CONw T.) 42~ Kravtchenko, J. and McNown, Jo S., Seiche in Rectangular Ports. Quart, of Appo Math,, 8, 1, pp.: 19-26,. 1955. 43. Kreisel, G., Surface Waves. Quart. of App. Math., 7, 1, pp. 21-44, 1949 446 Longuet-HEiggins, M, S., Mass Transport in Water Wavesa Phil, Trans. Roy. SQoc A,: 245, 903, pp. 535-581, 1953. 45. Longuet-Higgins, M. S., On the Decrease of Velocity with Depth in an Irrotational Water Wave, Proco Cam. Phil. Soc,. 49, 3, 1953. 46. McNown, J. So, Sur 1itentretien des oscillations des eaux portuaires sous luaction de la haute mero Publications Scientifiques et Techniques du Ministere de l'Airg No. 278, Paris, 1953. 47. MeNown, J. So, and Kravtchenko, J., Sur la theorie des ports rectangulaires a profondeur constanteo C. R. Acado Scio, 236, pp. 1531-1533, 19530 48. Packham, Bo A., The Theory of Symmetrical Gravity Waves of Finite Amplitude II. The Solitary Wave. Proc. Roy. Soc. A, 213, ppo 238249, 19520 49. Penney, W. G. and Price, A. I., Some Gravity Wave Problems in the Motion of Perfect Liquids. Part II. Finite Periodic Stationary Gravity Waves in a Perfect Liquid. Phil. Trans. Roy. Soc.. A, 244, 882, pp. 254-284, 1952. 50. Platzman, G. W0,, The Partition of Energy in Periodic Irrotat.ional Waves on the Surface of Deep Water. Jour, Mar. Res., 6, 3, pp e 194-202, 1947. 51. Roseau, M., Contribution a la theorie -des ondes liquides de gravite en profondeur variable. Publications Scientifique$ et Techniques du Ministere de 1]Air, No. 275, Paris, 1952. 52. Starr, Vo P,, A Momentum Integral for Surface Waves in Deep Water, Jour. Mar. Res., 6, 2, pp. 126-135, 1947. 53, Starr, V, P., Momentum and Energy Integrals for Gravity Waves of Finite Height. Jour, Mar. Res., 6, 3, pp. 175-193, 1947. 54Q Starr, V. P. and Platzman, Go Wo. The Transmission of Energy by Gravity Waves of Finite Height. Jour. Mar. Res., 7, 3, pp. 229-238, 19480

881MODERN REERENCES (CONT. ) 550 Stoker, J, J.o Surface Waves in Water of Variable Depth. Quart. of App. Math,, 5, pp. 1-54, 1947, 56, Ursell, Fo, Mass Transport in Gravity Waves. Proc. Cam. Phil. Soc., 49,9 1, pp. 145%150, 1953. 57.. Ursell, F,, The Long-Wave ParadOx in the Theory of Gravity Waves. Proc. Camo Philo Soc.o, 49, 4, pp. 685-694, 19530 BOOKS WITH EXTENSIVE BIBLIOGRAPHIES 58. Stoker, J JO,, Water Waves. New York: Interscience,# 1957. 59. Thorade, H. F,, Probleme der Wasserwellen. Probleme der kosmischen Physik. Bde. 13-14, 1931. MATHEMATICAL REFERENCES 60. British Association for the Advancement of Science. Mathematical Tables, Volume VIo Bessel Functions, Part I, Functions of Orders Zero and Unity. Cambridge: University Press, 1950, 61, Dini, U., Serie di Fourier e Altere rappresentazioni analitiche delle funzioni di una variabile reale. Pisa: Tipografia T. Nistri, 1880. 62~ Fox, L,, A Short Table for Bessel Functions of Integer Orders and Large Arguments. Roy. Soc. Shorter Math. Tables No. 3, Cambridge University Press, 19540 63. McLachlan, N. W., Bessel Functions for Engineers. 2nd Ed,, London: Oxford University Press, 1955. 64. Tables of Bessel~.Clifford Functions of Orders Zero and One. U. So Dept.. of Commerce, Natlo. Bureau Standards App. Matho Series, No.~ 28, 1953. 65. Titchmarch, E. C., Eigenfunction Expansions Associated with SecondOrder Differential Equations. London: Oxford University Press, 1946. 66. Watson, G No., A Treatise on the Theory of Bessel Functions, 2nd Ed,, Cambridge: Cambridge University Press, 19440