A higher order theory for deep water waves

A HIGHER ORDER THEORY FOR DEEP WATER WAVES

Peter 1. Monkmeyer

Assistant Professor

Department of Civil Engineering and

Mathematics Research Center, U. S. Army

University of Wisconsin, Madison, Wisconsin

and

John E. Kutzbach

Research Assistant

Department of Meteorology

University of Wisconsin, Madison, Wisconsin

ABSTRACT

The classical problem of describing the characteristics of deep water waves of finite amplitude is considered. The method of analysis

initially follows that of Nekrasov, but differs in that a non-linear algebraic

equation is derived. This equation is solved to the third, fifth and fifteenth order by means of a digital computer and the data is presented in tabular form. Expressions for the wave speed and wave shape, predicted by the analysis, are compared with the results obtained by Stokes. The highest

wave in water is also discussed.

INTRODUCTION

Recent advances in the development of gravity wave theories have focused increasing attention on the fact that linear theories are inadequate to describe certain free surface phenomena (Wiegel, 1964, pp. 2-3). One

such phenomenon is the deep water wave of finite height, which has been

studied by many researchers over the course of the past century. Among the more important papers which have appeared on this subject are those of

Stokes (1847, 1880), Rayleigh (1876), Wilton (1914) and Levi-Civita (1925). A review of s·ome of these works, as well as others which include the effect

of finite depth, may be found in Oceanographical Engineering (Wiegel, 1964, Chapter 2).

An analysis of deep water waves by Nekrasov (1951), which differs

from that of Stokes, recently appeared in the Russian literature. A rela-tively intensive study of this paper was subsequently conducted by

Milne-Thomson (1960), who included a condensation of some parts of Nekrasov's

paper in the fourth edition of his Theoretical Hydrodynamics. Moreover,

an English translation of the entire work has been prepared by the Mathematics Research Center at the University of Wisconsin.

Sponsored by the Mathematics Research Center, U. S. Army, Madison, Wisconsin

under Contract No. : DA-U-022-0RD-2059. Preliminary computations were supported by the Atmospheric Sciences Division, N. S. F. Grant gp-444.

N e k r a s o v ' s a n a l y s i s l e a d s t o a n o n - l i n e a r i n t e g r a l e q u a t i o n . H o w e v e r , he d i d not o b t a i n a g e n e r a l s o l u t i o n t o the e q u a t i o n , b u t r a t h e r a s o l u t i o n l i m i t e d t o w a v e s of v e r y s m a l l a m p l i t u d e . M o r e o v e r , t h i s s o l u t i o n i s i n m a r k e d d i s a g r e e m e n t w i t h t h a t o b t a i n e d b y S t o k e s . I n an a t t e m p t t o c o o r d i n a t e t h e w o r k of N e k r a s o v and Stokes i n t o one a n a l y s i s , the p r e s e n t s t u d y w i l l b e g i n w i t h t h e f u n d a m e n t a l c o n c e p t s of t h e f o r m e r . The a d v a n t a g e of N e k r a s o v ' s d e v e l o p m e n t i s t h a t i t p r e -s e n t -s and i n i t i a t e -s t h e p r o b l e m i n a r e l a t i v e l y l u c i d manner. H o w e v e r , r a t h e r t h a n d e v e l o p the n o n - l i n e a r i n t e g r a l e q u a t i o n of N e k r a s o v , the p r e s e n t s t u d y w i l l c e n t e r o n an a l g e b r a i c e q u a t i o n . A f t e r t h i s e q u a t i o n has b e e n d e r i v e d , i t w i l l be s h o w n t h a t i t i s e s s e n t i a l l y i d e n t i c a l t o t h a t o b t a i n e d b y Stokes ( 1 8 8 0 ) , t h e r e b y e s t a b l i s h i n g t h e c o n n e c t i o n b e t w e e n t h e t w o t h e o r i e s . W i t h the a i d o f a d i g i t a l c o m p u t e r , t h i r d , f i f t h , and f i f t e e n t h o r d e r s o l u t i o n s of t h e a l g e b r a i c e q u a t i o n have b e e n o b t a i n e d , and w i l l be p r e -s e n t e d i n t a b u l a r f o r m . The a n a l y -s i -s w i l l t h e n be c o m p a r e d w i t h t h o -s e o b t a i n e d b y o t h e r s , i n c l u d i n g S t o k e s . I n p a r t i c u l a r the d e p e n d e n c e o f w a v e s p e e d and w a v e shape o n w a v e h e i g h t w i l l be e x a m i n e d . The d i s -c u s s i o n w i l l be -c o n -c l u d e d b y some o b s e r v a t i o n s o n t h e h i g h e s t w a v e i n w a t e r .

SOLUTION OF THE WAVE PROBLEM

DEVELOPMENT OF THE THEORY

Let us c o n s i d e r a t r a i n of o s c i l l a t o r y w a v e s o f f i n i t e h e i g h t t r a v e l i n g f r o m r i g h t t o l e f t a l o n g t h e f r e e s u r f a c e o f an i n f i n i t e l y d e e p b o d y of w a t e r . The w a v e s have a c o n s t a n t p r o f i l e , w h i c h i s as y e t u n k n o w n . They t r a v e l w i t h a s p e e d , c , w h i c h i s l i k e w i s e u n k n o w n . We s u p e r i m p o s e o v e r the e n t i r e f i e l d a u n i f o r m f l o w m o v i n g f r o m l e f t t o r i g h t w i t h a m a g n i t u d e , c . As a r e s u l t , t h e w a v e f o r m i s b r o u g h t t o r e s t , and b y c h o o s i n g a c o o r d i n a t e s y s t e m w h i c h i s l i k e w i s e at r e s t , the f l o w b e n e a t h the f r e e s u r f a c e i s s e e n t o be s t e a d y . T h i s s t e a d y w a v e w i l l t u r n o u t t o be more amenable t o a n a l y s i s t h a n t h e p r o g r e s s i v e w a v e , w h o s e p r o p e r t i e s are u n d e r c o n s i d e r a t i o n . F i g . 1 shows a segment of t h e w a v e t r a i n , as s e e n i n t h e c o m p l e x z - p l a n e . The y - a x i s i s c h o s e n t o p a s s t h r o u g h the c r e s t of t h e w a v e , C . The w a v e l e n g t h i s L ; the w a v e h e i g h t , m e a s u r e d f r o m t r o u g h t o c r e s t , i s H ; and t h e f l o w f i e l d , e x t e n d i n g o v e r one w a v e l e n g t h , i s b o u n d e d b y AOOBCDAQO, where B C D , t h e u n k n o w n f r e e s u r f a c e , i s g i v e n b y YQ = Yoi^Q ) • The s t i l l w a t e r l e v e l i s l o c a t e d at y = yg , a d i s t a n c e w h i c h r e m a i n s t o be d e t e r m i n e d . The f l o w i s a s s u m e d t o be t w o - d i m e n s i o n a l and i r r o t a t i o n a l

( t h e l a t t e r a s s u m p t i o n i s d i s c u s s e d i n d e t a i l b y Stokes ( 1 8 4 7 ) , i n t h e A p p e n d i x of h i s c l a s s i c a l w o r k ) . T h e r e f o r e , we may d e f i n e a c o m p l e x p o t e n t i a l

w = (}> + 14^ (1) w h e r e ef) i s t h e p o t e n t i a l f u n c t i o n and i|i the s t r e a m f u n c t i o n . As a

c o n s e q u e n c e of the f a c t t h a t t h e c o m p l e x p o t e n t i a l i s a n a l y t i c we o b t a i n t h e C a u c h y - R i e m a n n e q u a t i o n s , w h i c h may be r e l a t e d t o the v e l o c i t y c o m p o n e n t s , u = | i = | i and v = | i = - | ^ ( 2 ) 9x ay ay 8x From t h e s e e x p r e s s i o n s i t may be s h o w n t h a t b o t h t h e p o t e n t i a l f u n c t i o n a n d t h e stream f u n c t i o n s a t i s f y L a p l a c e ' s e q u a t i o n V^<j) = 0 and

V^^J =

0 ( 3 ) On t h e f r e e s u r f a c e , B C D , t h e k i n e m a t i c c o n d i t i o n t h a t the f r e e s u r f a c e i s a s t r e a m l i n e i s g i v e n b y iji = 0 at y = y ^ ( 4 ) and t h e d y n a m i c c o n d i t i o n t h a t the p r e s s u r e i s a c o n s t a n t may be a p p l i e d

t o t h e B e r n o u l l i e q u a t i o n , t o g i v e QQ + Zgy^ = K at y = y ^ ( 5 ) w h e r e q^ i s t h e p a r t i c l e s p e e d at t h e f r e e s u r f a c e and K i s t h e B e r n o u l l i c o n s t a n t . On t h e l o w e r b o u n d a r y , the c o m p l e x v e l o c i t y at A^^ i s e q u i v a l e n t t o t h e s u p e r i m p o s e d speed, c, and t h i s c o n d i t i o n i s t h e r e f o r e g i v e n b y u - i v = c a t y = - < » ( 6 ) A l s o , a d i r e c t c o n s e q u e n c e of t h i s c o n d i t i o n and E q s . 2 and 4, i s t h a t i|j = - o o a t y = - o o ( 7 ) The p r o b l e m , as d e v e l o p e d t o t h i s p o i n t , i s t o o b t a i n t h e s o l u t i o n s

of E q s . 3 s u b j e c t t o t h e b o u n d a r y c o n d i t i o n s , Eqs. 4, 5, 6, and 7 . The f u n d a m e n t a l d i f f i c u l t y w h i c h i m m e d i a t e l y p r e s e n t s i t s e l f i s t h a t t h e l o c a -t i o n of -t h e f r e e s u r f a c e , or upper b o u n d a r y , i s u n k n o w n . I n o r d e r -t o e l i m i n a t e t h i s d i f f i c u l t y i n t h e z - p l a n e , t h e t e c h n i q u e of c o n f o r m a l

m a p p i n g w i l l be e m p l o y e d , so t h a t t h e l o c a t i o n o f t h e f r e e s u r f a c e may be i d e n t i f i e d i n t h e a u x i l i a r y C - p l a n e . The c o n f o r m a l t r a n s f o r m a t i o n t o be u s e d was f i r s t s u g g e s t e d b y N e k r a s o v ( 1 9 5 1 ) , and has a l s o b e e n d i s c u s s e d b y M i l n e - T h o m s o n ( I 9 6 0 ) . It i s g i v e n b y and w h e r e t h e c o n s t a n t c o e f f i c i e n t s , a j , are r e a l , b u t as y e t u n k n o w n . The - p l a n e I s s h o w n i n F i g . 2. As one c a n v e r i f y b y a p p l i c a t i o n of t h e m a p p i n g f u n c t i o n , E q . 8, t h e r e g i o n b o u n d e d b y A30BCDA00 i n t h e z - p l a n e i s m a p p e d i n s i d e t h e u n i t c i r c l e i n t h e ^ - p l a n e s u b j e c t o n l y t o the proper e v a l u a t i o n of t h e c o n s t a n t c o e f f i c i e n t s , a^ . Point A^o appears at t h e o r i g i n , p o i n t C o n the u n i t c i r c l e at x = 0 and p o i n t s B a n d D on t h e u n i t c i r c l e at x = a n d x = - r e s p e c t i v e l y , s e p a r a t e d b y a s l i t . The p r o b l e m t h e n , i s t o o b t a i n v a l u e s of a j s u c h t h a t t h e f r e e s u r f a c e i s m a p p e d o n t o t h e u n i t c i r c l e of t h e ^ - p l a n e . The t r a n s f o r m a t i o n may s u b s e q u e n t l y be u s e d t o l o c a t e t h e f r e e s u r f a c e i n t h e z - p l a n e , and i n c i d e n t a l l y p r o v i d e t h e w a v e s p e e d and t h e o t h e r c h a r a c t e r i s t i c s of t h e w a v e . B e f o r e p r o c e e d i n g t o an a n a l y s i s of t h e b o u n d a r y c o n d i t i o n s i n t h e L - p l a n e , i t i s c o n v e n i e n t t o e x a m i n e some a d d i t i o n a l p r o p e r t i e s of t h e t r a n s f o r m a t i o n . For e x a m p l e , b y s e p a r a t i n g r e a l and i m a g i n a r y parts of t h e m a p p i n g f u n c t i o n and e v a l u a t i n g t h e m o n t h e f r e e s u r f a c e , t h a t i s , on t h e u n i t c i r c l e of t h e £ . - p l a n e , we o b t a i n t h e p a r a m e t r i c e q u a t i o n s t h a t d e s c r i b e t h e f r e e s u r f a c e , y = y f ^ ( x ) , i L , , „ 1 „ 2 1 , 3 z = _ ( l n C + a ^ ^ + - a ^ 4 + - a ^ C -1- . . . ) _m where ^ = r e x p ( i x ) X ₀ - - ^ ( x + a ^ s i n X + - a ^ s i n 2X-I- - a^ s i n 3 x . . . ) ( 9 ) y 0 2 ^ * ^ l ^'^Z^Z ^^'^ 1 ^ 3 3x + . . . ) (10) A l s o b y d i f f e r e n t i a t i n g Eq. 8 we o b t a i n t h e c o m p l e x o p e r a t o r . (11) Since z = z ( 4 ) , E q . 8 i s , i n f a c t , an i n v e r s e t r a n s f o r m a t i o n .

-D E E P W A T E R W A V E S w h e r e and f ( ^ ) - E a.^^ j = 0 ^ (12) a^ = 1. 0 (13) M o r e o v e r , o n t h e f r e e s u r f a c e = e x p ( i x ) so t h a t f ( ^ o ) = E a. e x p ( i j x ) = RQ e x p d G g ) w h e r e t h e m o d u l u s of f { ^ Q ) i s g i v e n b y [ ( E a. cos j x ) % ( E a, siniyd^]^'^ j = 0 J j = 0 '

and t h e argument of ii^^) i s g i v e n b y

(14) ( 1 5 ) ( 1 6 ) = COS E a. cos j x j = 0 ' 0 ( 1 7 ) I n v i e w of t h e m a p p i n g f u n c t i o n g i v e n b y Eq. 8, the b o u n d a r y c o n d i t i o n s , E q s . 4, 5, 6, a n d 7, become 4) = 0 at r = 1 qg + 2gyQ = K at r = 1 u - i v = c a t £ . = 0 ( 1 8 > ( 1 9 > ( 2 0 > l|j = - 00 r e s p e c t i v e l y , i n t h e f , - p l a n e . at r = 0 _{( 2}

E q s . 18 and 21 are p r e c i s e l y the b o u n d a r y c o n d i t i o n s f o r a cloC^' w i s e i r r o t a t i o n a l v o r t e x i n the - p l a n e , so t h a t t h e c o m p l e x p o t e n t i a l f o ' ' t h i s f l o w i s

w = i | ^ l n C ( 2 2 ) 277

By s e p a r a t i n g the r e a l and I m a g i n a r y parts o f t h i s e x p r e s s i o n and r e -a r r -a n g i n g t e r m s , we o b t -a i n \ = ^ c L ( 2 3 ) and r = e x p { ^ ) 2TniJ ( 2 4 ) ( 2 6 ) It i s t h e r e f o r e apparent t h a t x i s t h e n o r m a l i z e d p o t e n t i a l f u n c t i o n , ^ ^ ^ ^ ^ r i s f u n c t i o n a l l y r e l a t e d t o the stream f u n c t i o n . I n o r d e r t o o b t a i n t h e v e l o c i t y c o m p o n e n t s we may u s e Eq. H a d d i t i o n t o E q . 22 t o compute t h e c o m p l e x v e l o c i t y , _ d w _ dw d £ c ( 2 5 ) " " ~ dz " dt, d z ~ HL) I t i s s e e n t h a t t h i s e x p r e s s i o n s a t i s f i e s the t h i r d b o u n d a r y c o n d i t i o n ? E q . 20, s i n c e Eq. 12 shows t h a t f ( t , ) = 1 at C = 0 . At t h e f r e e s u r f a c e Eq. 25 becomes, i n v i e w of E q . 15, " o - % = f ( y - = ^ - - p ( - V a n d t h e r e f o r e the v e l o c i t y c o m p o n e n t s are g i v e n b y By t a k i n g t h e r a t i o o f t h e v e l o c i t y c o m p o n e n t s , i t may a l s o be s e e n ' ^ o r i t a l

QQ i s t h e l o c a l s l o p e angle w h i c h t h e f r e e s u r f a c e makes w i t h a \\orr ^

i n t h e p h y s i c a l z - p l a n e .

I t r e m a i n s t h e n t o d e a l w i t h t h e d y n a m i c b o u n d a r y c o n d i t i o n ^

a l l o t h e r c o n d i t i o n s h a v i n g b e e n s a t i s f i e d by t h e c o m p l e x p o t e n t i a l d e -s c r i b e d i n E q . 2 2 . We t h e r e f o r e -s u b -s t i t u t e Eq-s. 2 7 i n t o E q . 19, -so a-s t o o b t a i n 2 \ + 2gy = K ( 2 8 ) Eq. 2 8 was d e v e l o p e d b y N e k r a s o v , w h o p r o c e e d e d t o r e l a t e t h i s e q u a t i o n t o E q . 15 i n o r d e r t o d e r i v e a n o n - l i n e a r i n t e g r a l e q u a t i o n , f o r w h i c h a g e n e r a l s o l u t i o n has not as y e t b e e n f o u n d . I n t h e p r e s e n t c a s e we w i l l a l s o c o n s i d e r E q . 2 8 . H o w e v e r , t h e e q u a t i o n w i l l be s o l v e d d i r e c t l y , t h o u g h a p p r o x i m a t e l y , b y a l g e b r a i c m e a n s . We may d e f i n e the f o l l o w i n g d i m e n s i o n l e s s t e r m s : ""O 2 ~ L ' ^ Q " ~ T - ~ i r c ' - ^ K ' ^ - ^ ( 2 9 ) w h e r e the w a v e s p e e d f r o m t h e l i n e a r t h e o r y i s g i v e n b y = ( 3 0 ) L V2TT U s i n g t h e s e d i m e n s i o n l e s s t e r m s , we may c o n v e r t E q . 2 8 t o t h e n o r m a l i z e d f o r m . c ' 2 + 2 y ^ R 2 = K ' R 2 ( 3 1 ) w h e r e , CO a YQ =

E

- ^ C O S j x ( 3 2 ) F u r t h e r m o r e , b y e x p a n d i n g t e r m s i n E q . 16, and a p p l y i n g t r i g o n o m e t r i c i d e n t i t i e s , i t m a y be s h o w n t h a t 00 R Q = % + 2

E^

A , COS i x ( 3 3 ) w h e r e

\ • 2

4

" k=o ^ CO ( 3 4 ) ( 3 5 ) By s u b s t i t u t i n g Eqs. 32 t o 35 i n t o E q . 31, t h e problem of t h e d e e p w a t e r w a v e of f i n i t e a m p l i t u d e i s r e d u c e d t o one of f i n d i n g t h e s o l u t i o n of t h e a l g e b r a i c e q u a t i o n , c ' ^ + 2 00 a. E ^ cos ) x / oo •\ V 2 1 °° _V / oo \ V

\"

- K' + cos j x / \ k = o 1

[

= 0

(36) It s h o u l d be n o t e d t h a t t o t h i s p o i n t i n t h e a n a l y s i s no a p p r o x i m a t i o n s h a v e b e e n made. T h e r e f o r e Eq. 3 6 i s an e x a c t r e p r e s e n t a t i o n of t h e p r o b l e m . I n o r d e r t o s o l v e the p r o b l e m f o r a f i n i t e number of c o e f f i c i e n t s , i t w i l l be n e c e s s a r y t o t r u n c a t e the i n f i n i t e t r i g o n o m e t r i c s e r i e s w h i c h appear i n E q . 36. As a c o n s e q u e n c e , w h e r e and A^ + 2 E A, cos j x ( 3 7 )

R2

A„ n E ( 3 8 ) °

k=0 ^

n A ^

TJ

a a ( 3 9 ) ^ k = j ^ ' ^ ^ n a, YQ ^ Z -J-COS JX (40)

(41)

w h e r e i t i s u n d e r s t o o d t h a t a b s o l u t e v a l u e s i g n s are o m i t t e d o n t h e s u b -s c r i p t -s of the term-s o n t h e r i g h t - h a n d -s i d e , a n d f u r t h e r m o r e

= 0

i f

U I

> n

(42)

I t s h o u l d be n o t e d t h a t i n d e r i v i n g Eq. 41 h a r m o n i c s h i g h e r t h a n t h e n t h h a v e b e e n o m i t t e d so t h a t t h e e x p r e s s i o n i s n o t an e x a c t r e p r e s e n t a t i o n of y ' R ^ . •'O

0

The e x p r e s s i o n s f o r

RQ

, Eq.

37,

and y ^ R g , Eq.

41,may

n o w be s u b s t i t u t e d i n t o Eq. 31. By e q u a t i n g t h e c o e f f i c i e n t s of the h a r m o n i c s , we o b t a i n c'^ + 2 Yi ^ \ = ^'K (Oth h a r m o n i c ) (43) k = l " " n a Z • Y ( \ . j + \ + j ) = K'A. ( j t h h a r m o n i c - j =1, 2, . . . , n ) (44) c=l w h e r e a b s o l u t e v a l u e s i g n s are o m i t t e d o n the s u b s c r i p t s , a n d A^ v a n i s h e s i f

U I

> n .

I n v i e w of the f a c t t h a t the u n k n o w n t e r m s i n Eqs. 43 a n d 44 are a l l f u n c t i o n s of t h e h e i g h t of t h e w a v e , i t i s c o n v e n i e n t t o i n c l u d e an e q u a t i o n f o r w a v e h e i g h t . The w a v e h e i g h t i s seen t o be e q u a l t o t h e sum of t h e d i s p l a c e m e n t s of t h e c r e s t and t r o u g h f r o m t h e x - a x i s . T h e r e f o r e , u s i n g Eq. 10,

«=<VX=O + <-VX=. = ^ , | T < ^ = ^ ' ^ ' ' ' - ' ' ^ ' ^^'^

N o r m a l i z i n g w i t h r e s p e c t t o w a v e l e n g t h , t h e d i m e n s i o n l e s s w a v e h e i g h t is H ' = ^

S - L

( j =

1,3,5,

. . . , n )

(46)

i n ( n+2) u n k n o w n s ( c ' , K ' , , 8 2 , . . . , a^) f o r any d e s i r e d v a l u e o f d i m e n -s i o n l e -s -s w a v e h e i g h t , H ' . COMPUTER SOLUTION I n s e t t i n g up t h e e q u a t i o n s f o r c o m p u t e r s o l u t i o n , t h e c o e f f i c i e n t , K' w a s e l i m i n a t e d b e t w e e n t h e f i r s t of Eqs. 44 ( j = 1 ) and e a c h s u c c e s s i v e e q u a t i o n , ( j = 2 , 3, . . . , n ) , t h e r e b y r e d u c i n g Eqs. 4 4 t o ( n - l ) e q u a t i o n s i n ( n - I ) u n k n o w n s ( a^ , a2 aj^_-^), f o r a f i x e d v a l u e o f a^^^. The s i m u l t a n e o u s s o l u t i o n of t h e s e ( n - 1 ) n o n - l i n e a r a l g e b r a i c e q u a t i o n s w a s a c c o m p l i s h e d w i t h t h e a i d o f t h e N e w t o n - R a p h s o n i t e r a t i o n ( H i l d e b r a n d , 1 9 5 6 ) . By t h i s t e c h n i q u e the p r o b l e m was r e d u c e d t o one o f o b t a i n i n g t h e s o l u t i o n of a set of ( n - 1 ) l i n e a r a l g e b r a i c e q u a t i o n s at e a c h i t e r a t i o n . The Jordan r e d u c t i o n m e t h o d was t h e n u s e d t o s o l v e t h e set of l i n e a r e q u a t i o n s f o r a-^, a2 , . . . , a^_ 2 and a^,-^. S u b s t i t u t i o n o f t h e s e t e r m s i n t o E q . 46, E q . 44 ( f i r s t h a r m o n i c ) , and E q . 43, w i t h t h e a i d of E q s . 38 and 39, y i e l d e d v a l u e s f o r H ' , K' and c ' r e s p e c t i v e l y . The e n t i r e p r o -c e d u r e w a s r e p e a t e d f o r v a r i o u s v a l u e s of so t h a t a l l of t h e t e r m s c o u l d be t a b u l a t e d f o r u n i f o r m i n t e n / a l s o f the d i m e n s i o n l e s s w a v e h e i g h t . I n t e r v a l s of 0. 01 were c h o s e n f o r t h e t a b u l a t i o n . The r e s u l t s , as o b t a i n e d f r o m a C D C 1604 c o m p u t e r are g i v e n f o r t h e t h i r d o r d e r ( n = 3 ) , f i f t h o r d e r ( n = 5) and f i f t e e n t h o r d e r ( n = 15) i n T a b l e s 1, 2 and 3. I n v i e w o f t h e r a p i d c o n v e r g e n c e of t h e a j s e r i e s , f o r l o w e r v a l u e s of H ' , t h e c o m p u t a t i o n s i n t h e s e c a s e s were not c a r r i e d t o t h e f i f t e e n t h order, as i s i n d i c a t e d i n T a b l e 3. V a l u e s o f H / L were o b t a i n e d c o r r e c t t o ± 0. 00001, w h i l e a l l o t h e r terms a p p e a r i n g i n the t a b l e s were c o m p u t e d c o r r e c t t o t h e l a s t p l a c e s h o w n . I t s h o u l d be n o t e d t h a t t h i s p r e c i s i o n was o b t a i n e d i n s a t i s f y i n g E q s . 43, 44 and 46, t h e e q u a t i o n s w h i c h were programmed o n t h e c o m p u t e r . H o w e v e r , i n l o o k i n g b a c k t o Eq. 36, w h i c h r e p r e s e n t s an e x a c t e x p r e s s i o n of t h e p r o b l e m , i t i s seen t h a t t h e v a l i d i t y of t h e v a r i o u s a p p r o x i m a t i o n s e m p l o y e d i n t h e s u b s e q u e n t d e v e l o p m e n t i s r e f l e c t e d i n t h e degree t o w h i c h t h e t a b u l a t e d v a l u e s s a t i s f y t h i s g e n e r a l e x p r e s s i o n o f the p r o b l e m . F o r t h e l o w e r v a l u e s of H / L where c o n v e r g e n c e o f t h e a j s e r i e s i s r a p i d , a n d l o w e r order s o l u t i o n s are s u f f i c i e n t , Eq. 36 i s s a t i s f i e d v e r y p r e c i s e l y . H o w e v e r , as H / L i n c r e a s e s , t h e t r u n c a t i o n of t h e t r i g o n o m e t r i c s e r i e s a n d , a l s o t h e o m i s s i o n of h i g h e r h a r m o n i c s i n the d e v e l o p m e n t o f ( y g R g ' r e s u l t i n a l e s s p r e c i s e s a t i s f a c t i o n o f E q . 36.

No a t t e m p t w i l l be made here t o e x a m i n e the c o n v e r g e n c e o f t h e t r i g o n o m e t r i c s e r i e s ( a s g i v e n i n E q . 8) f o r m e d b y t h e c o e f f i c i e n t s . F o r a d i s c u s s i o n o f t h i s q u e s t i o n , r e f e r e n c e may be made t o L e v i - C i v i t a (1925), w h o p r e s e n t e d a c o n v e r g e n c e p r o o f .

T a b l e 1. - Deep w a t e r w a v e c o e f f i c i e n t s - T h i r d o r d e r .

H/L

c '

_K'

^2

a

3

. 142

1. 0979 1.3751

.37171 .28318 . 22320

. 140

1. 0951

1. 3653 .36778 .27696 .21614

. 130

1. 0817

1.3179

.34755 .24651 .18256

. 120

1. 0696

1. 2732 .32641 .21703 .15175

. 110 1. 0587 1. 2314

.30436 .18853 .12366

. 100

1.0487

1.1925

.28139 .16106 .09830

. 090

1. 0396 1. 1568

.25749 .13478 .07576

. 080

1.0314

1.1245

.23261 .10988 .05615

. 070

1.0241

1. 0957 .20672 .08665 .03957

. 060

1.0178

1. 0705 .17979 .06541 .02611

. 050

1. 0124

1.0491

.15183 .04653 .01575

. 040

1.0079

1.0315

.12288 .03040 .00836

. 030

1. 0044 1. 0177

.09304 .01738 .00363

. 020

1. 0020 1.0079

.06247 .00782 .00110

. 010

1.0005

1.0020

.03137 .00197 .00014

. 000

1. 0000 1.0000

.00000 . 00000 .00000

T a b l e 2. - Deep w a t e r w a v e c o e f f i c i e n t s - F i f t h o r d e r .

H/L

c '

_K'

a

2

a

4

a

5

. 142

1.1099

1. 3933 .34066 .25987 .21964 .18964 .16118

. 140

1.1063

1. 3825 .33811 . 25489 .21307 .18215 . 15342

. 130 1.0900

1. 3304 .32413 .23004 . 18173 .14753 . 11848

. 120 1.0756

1. 2816 .30832 .20519 .15253 .11702 .08910

. 110

1. 0628 1. 2366 .29086 .18038 .12534 .09023 .06467

. 100

1. 0513

1. 1954 .27176 .15575 .10026 .06712 .04487

. 090

1.0412

1. 1582 .25100 .13154 .07756 .04772 . 02943

. 080

1. 0323 1. 1250 . 22855 .10806 .05754 .03207 .01799

. 070

1.0246

1. 0958 . 20440 . 08573 .04051 .02009 .01006

. 060

1. 0180

1. 0705 .17861 .06500 .02665 . 01149 .00502

. 050

1.0124

1. 0491 . 15131 . 04640 .01601 .00582 .00215

. 040

1. 0079 1. 0315 .12270 .03035 . 00845 . 00248 .00074

. 030

1. 0045 1. 0177 . 09299 .01737 .00365 .00081 .00018

. 020

1.0020

1. 0079 . 06246 .00782 .00110 .00016 .00002

. 010

1.0005

1. 0020 . 03137 .00197 .00014 .00001 .00000

. 000

1. 0000 1. 0000 .00000 .00000 .00000 .00000 .00000

T a b l e 3, - Deep w a t e r wave c o e f f i c i e n t s - F i f t e e n t h order. H / L c ' K' " l " 2 " 4 . 142 1.1105 1. 3709 . 2 9 8 2 3 . 21496 . 1 8 0 3 9 . 16136 . 14922 . 14064 . 140 1.1064 1. 3629 . 3 0 0 4 6 . 21458 . 17762 . 1 5 6 4 8 . 1 4 2 4 6 . 13221 . 130 1, 0888 1.3162 . 3 0 0 5 5 . 20479 . 16009 . 13278 . 11374 . 0 9 9 3 9 . 120 1. 0742 1. 2712 . 29345 . 1 8 9 7 3 . 13999 . 10939 . 0 8 8 2 6 . 0 7 2 6 5 . 110 1. 0617 1. 2296 . 2 8 1 9 2 . 17149 . 1 1 8 6 0 . 0 8 6 7 6 . 0 6 5 4 9 . 0 5 0 4 4 . 100 1. 0506 1. 1913 . 2 6 6 7 4 , 1 5 1 0 2 . 0 9 6 9 5 . 0 6 5 7 6 . 0 4 6 0 1 . 0 3 2 8 4 . 090 1. 0408 1. 1561 . 24839 . 12924 . 0 7 6 0 9 . 0 4 7 3 0 . 0 3 0 3 1 . 0 1 9 8 1 . 080 1. 0321 1.1240 . 2 2 7 3 2 . 10706 . 0 5 6 9 7 . 0 3 1 9 9 . 0 1 8 5 1 . 0 1 0 9 2 . 070 1. 0245 1. 0954 . 2 0 3 8 8 . 0 8 5 3 4 . 04032 . 02009 . 0 1 0 3 1 . 00540 . 060 1. 0179 1. 0704 . 17842 . 06488 . 0 2 6 6 0 , 0 1 1 5 0 . 0 0 5 1 2 . 0 0 2 3 3 . 050 1. 0124 1.0490 . 15126 . 04635 , 0 1 6 0 0 . 0 0 5 8 2 . 0 0 2 1 8 . 0 0 0 8 3 . 040 1.0079 1, 0315 . 1 2 2 6 8 . 0 3 0 3 5 , 0 0 8 4 5 . 00248 . 0 0 0 7 5 . 00023 . 030 1. 0045 1. 0177 . 0 9 2 9 9 , 0 1 7 3 7 . 00365 . 0 0 0 8 1 . 0 0 0 1 8 . 0 0 0 0 4 . 020 1, 0020 1. 0079 . 06246 . 0 0 7 8 2 . 0 0 1 1 0 , 0 0 0 1 6 . 0 0 0 0 2 . 010 1.0005 1. 0020 . 0 3 1 3 7 . 00197 . 0 0 0 1 4 . 000 L 0000 1.0000 . 0 0 0 0 0 . 0 0 0 0 0 . 0 0 0 0 0 H / L . 142 , 140 . 130 . 120 . 110 . 100 . 090 . 080 . 070 . 060 . 050 . 040 . 030 13406 12415 08799 06062 03938 02376 01313 00653 00286 00107 00032 00007 00001 .12863 . 1 1 7 4 2 07857 .05107 03105 01736 00879 00395 00154 00050 00013 , 12387 , 1 1 1 5 2 , 0 7 0 5 8 , 0 4 3 3 1 , 0 2 4 6 7 , 0 1 2 7 8 , 0 0 5 9 3 , 0 0 2 4 0 ,00083 .00023 00005 10 , 1 1 9 4 5 , 1 0 6 1 4 , 0 6 3 6 4 . 0 3 6 9 1 , 0 1 9 7 0 , 0 0 9 4 7 00402 00147 00045 00011 11 11515 10104 05749 03155 01580 00704 00274 00091 00025 00005 12 11078 09607 05195 02700 01269 00525 00187 00056 00013 13 10619 09107 04687 02310 01020 00393 00128 00035 00007 14 . 10118 08588 04212 01969 00818 00293 00088 0 0 0 2 1 15 . 09537 . 0 8 0 1 6 . 0 3 7 5 1 . 0 1 6 6 4 . 0 0 6 5 0 . 0 0 2 1 7 . 00060 . 0 0 0 1 3

D I S C U S S I O N

COMPARISON W I T H STOKES'S THEORY"

As has a l r e a d y b e e n p o i n t e d o u t , E q , 36 i s an e x a c t r e p r e s e n of t h e p r o b l e m of t h e deep w a t e r w a v e of f i n i t e a m p l i t u d e . A comp^'-^^gj^s of t h i s e q u a t i o n w i t h E q . 13 i n t h e s e c o n d paper b y Stokes ( 1 8 8 0 ) ""^^gices's t h a t t h e y are i d e n t i c a l . * T h i s i s , i n f a c t , not t o o s u r p r i s i n g s i n c e S d e v e l o p m e n t o f E q . 13 i s j u s t as f r e e o f a p p r o x i m a t i o n as i s t h e d e ^ e ment o f E q . 36. J. p r e " A s e c o n d c o n n e c t i o n b e t w e e n S t o k e s ' s d e v e l o p m e n t and t h ^ J ^ . s e n t e d here m a y be c o n s t r u c t e d f r o m t h e m a p p i n g f u n c t i o n , E q . 8 . s e p a r a t i n g r e a l and i m a g i n a r y p a r t s , a n d i n t r o d u c i n g E q s . 23 a n d 2^? o b t a i n L V \ , 2 i T k i | j , . , 2 i T k A ( 4 7 ) — 1 7 e x p ( — - ^ ) s i n ( — / ) " c 211 k ^ c L ' ~ c L k = l a n d ^\: ^ L v \ /2TTMJ, , 2 i T k ^ , ( 4 8 ) — ZJ

- r

e x p

(—7^)

c o s

{—7^)

^ " c 2TT k ^ c L ' ' c L k = l , l i n e s w h i c h , i n c i d e n t a l l y , may be u s e d t o p l o t t h e p o t e n t i a l - a n d s t r e a f i ^ 9 i n t h e z - p l a n e . A c o m p a r i s o n of t h e s e e x p r e s s i o n s w i t h S t o k e s ' s ^ and 10 r e v e a l s t h a t t h e y are a l s o i d e n t i c a l . + If we n o w t u r n o u r a t t e n t i o n t o t h e p r o b l e m o f s o l v i n g E q ^ n -e x a m i n a t i o n of h i s a n a l y s i s r -e v -e a l s t h a t Stok-es w a s f o r c -e d t o ^ o t h e r a s s u m p t i o n , i n a d d i t i o n t o t r u n c a t i n g t h e i n f i n i t e t r i g o n o m e t _ g s e r i e s a n d o m i t t i n g h i g h e r h a r m o n i c s . T h i s a s s u m p t i o n c o n s i s t e d ' ^ ^ t o -o m i t t i n g t h -o s e p r -o d u c t s -of t h e c -o e f f i c i e n t s a j , t h e sum -o f w h -o s e ^ ^ ^ p t i -o n s c r i p t s e x c e e d e d t h e o r d e r o f t h e s o l u t i o n . The b a s i s f o r t h i s a s S ^^-ue was t h a t a j / j i s of o r d e r j . There i s l i t t l e q u e s t i o n t h a t t h i s i s ' ' ' ^ i o n -f o r s m a l l v a l u e s o -f H / L . H o w e v e r , -f o r l a r g e r v a l u e s o -f t h e d i m e ^ ^ S j a n d l e s s w a v e h e i g h t t h e c o e f f i c i e n t s do n o t v a r y g r e a t l y as j i n c r e s - ^ t h e a s s u m p t i o n appears l e s s j u s t i f i e d .

I t i s of c o u r s e w e l l - k n o w n t h a t Stokes d i d not c a r r y h i s a r ^ - -j^<gi4) b e y o n d t h e f i f t h o r d e r . H o w e v e r , i n a s u b s e q u e n t paper, W i l t o n ^ E x p a n d b o t h e q u a t i o n s t o any a r b i t r a r y o r d e r a n d i n E q . 36 r e p l ^ ^ ^ ^ by S t o k e s ' s t e r m ( - k A ) , ) , K' b y C , c ' 2 b y ( l / g ) . x ^ b y 1 , Replace x b y S t o k e s ' s ( - x ) , y b y ( - y ) , aj^ b y ( - k A j , ) , L / 2 * and c b y 1 .

D E E P W A T E R W A V E S

d e v e l o p e d a t w e l f t h o r d e r s o l u t i o n , a l s o e m p l o y i n g a p p r o x i m a t i o n s t o s i m p l i f y h i s c a l c u l a t i o n s . No attempt w i l l be made here t o compare t h e t a b u l a t e d v a l u e s of t h e c o e f f i c i e n t s o b t a i n e d i n t h i s s t u d y w i t h t h e e x -p r e s s i o n s f o r t h e c o e f f i c i e n t s d e r i v e d b y Stokes and W i l t o n , s i n c e a l l t h r e e s t u d i e s are b a s e d on the s o l u t i o n of Eq. 36 and the r e s u l t s f o r lil^® o r d e r s of s o l u t i o n , w h e r e t h e y are a v a i l a b l e , are, u n d e r s t a n d a b l y , q u i t ® s i m i l a r . Rather, a c o m p a r i s o n of t h e p r e d i c t e d w a v e speeds and w a v e s h a p e s w i l l be made i n t h e s u b s e q u e n t s e c t i o n s , w i t h p r i m a r y e m p h a s i s o n t h e s i g n i f i c a n c e of h i g h e r order s o l u t i o n s . WAVE SPEED The e q u a t i o n f o r c o m p u t i n g t h e w a v e s p e e d may be o b t a i n e d b y c o m b i n i n g E q s . 43, 38 and 39 t o g i v e k=0 n a / n \

E

a ^^a k = l ^ \ j = k ^ ^ 1/2 ( 4 9 )

w h i c h may be e v a l u a t e d t o any order. The r e s u l t s f o r t h e t h i r d , f i f t h , ^ , f i f t e e n t h o r d e r s , are g i v e n i n Tables 1, 2 and 3, a n d have a l s o b e e n p l o ^ t ® a g a i n s t w a v e h e i g h t i n F i g . 3. I n order t o compare Eq. 49 w i t h t h e e X ' p r e s s i o n s o b t a i n e d b y Stokes ( 1 8 4 7 ) , f o r t h e t h i r d order. 1 + 2L^ . ( 5 0 ) and f r o m S t o k e s ' s a n a l y s i s ( L e v i - C i v i t a , 1925 and B e a c h E r o s i o n B o a r d , 1 9 4 2 ) , f o r the f i f t h o r d e r L"- 2 L -I 1/2 3 1) t h e s e t w o e q u a t i o n s are a l s o r e p r e s e n t e d i n F i g . 3. An e x a m i n a t i o n of F i g . 3 shows t h a t i n p r e d i c t i n g w a v e s p e e d a f u n c t i o n of w a v e h e i g h t , t h e r e i s , i n g e n e r a l , g o o d agreement b e t w ^ ^ ' ^ t h e r e s u l t s o b t a i n e d b y Stokes and t h o s e of t h e p r e s e n t s t u d y . I n f a C C ? f o r v a l u e s o f the d i m e n s i o n l e s s wave h e i g h t r a n g i n g f r o m 0. 00 t o 0. O ^ t h e c u r v e s are v i r t u a l l y i d e n t i c a l , s u g g e s t i n g t h a t the t h i r d o r d e r dev<s^ X o p -m e n t s of b o t h s t u d i e s ( s e e Table 1) are e n t i r e l y adequate f o r d e t e r -m i n i - ^ ^ t h e w a v e s p e e d i n t h i s range of w a v e h e i g h t .

For v a l u e s o f the d i m e n s i o n l e s s w a v e h e i g h t e x c e e d i n g 0. 0 6 , c u r v e s d i v e r g e t o more t h a n 1% of the d i m e n s i o n l e s s w a v e s p e e d . I n

-che ^ i e w

o f t h e f a c t t h a t v e r y f e w w a v e s w h o s e w a v e h e i g h t e x c e e d e d H / L = 0 . 1 h a v e b e e n o b s e r v e d , e i t h e r i n t h e sea or i n t h e l a b o r a t o r y , t h i s d i v e r -g e n c e i s perhaps o f o n l y l i m i t e d p h y s i c a l s i -g n i f i c a n c e . N e v e r t h e l e s s , i n t h e c o n t i n u i n g s e a r c h f o r an u n d e r s t a n d i n g o f the m e c h a n i s m s w h i c h l i m i t the h e i g h t of a deep w a t e r w a v e , a p r e c i s e e v a l u a t i o n of t h e w a v e s p e e d , w h i c h t a k e s t h e h i g h e r h a r m o n i c s i n t o a c c o u n t , may p r o v e t o be o f a s s i s t a n c e . U n d e r t h e s e c i r c u m s t a n c e s t h e use of t h e f i f t e e n t h o r d e r d e v e l o p m e n t ( T a b l e 3 ) , s h o u l d be c o n s i d e r e d . WAVE SHAPE

I n o r d e r t o d e s c r i b e the wave shape or p r o f i l e o f a d e e p w a t e r w a v e , we seek a r e l a t i o n s h i p b e t w e e n t h e f r e e s u r f a c e c o o r d i n a t e s , x g a n d YQ . The p a r a m e t r i c set of e q u a t i o n s , ""o " " ^ + " l >^ 2 ^ 2 2x +

I

a^ s i n 3x . . . ) ( 9 ) ^0 " " ^ ' ^ 1 ^""^ 2 ^2 ^""^ 2X + - ^ a ^ cos 3x . . . ) (10) r e p r e s e n t s s u c h a r e l a t i o n s h i p . N u m e r i c a l v a l u e s r e l a t i n g XQ a n d y g m a y be o b t a i n e d by s u b s t i t u t i n g a r b i t r a r y v a l u e s of t h e n o r m a l i z e d p o t e n -t i a l f u n c -t i o n , x - T h i s procedure i s adequa-te f o r a g r a p h i c a l r e p r e s e n -t a -t i o n o f w a v e s h a p e . Stokes ( 1 8 8 0 ) , however, s u g g e s t s t h a t t h e t w o e q u a t i o n s m a y be r e d u c e d t o one e x p r e s s i o n , w i t h t h e a i d of L a g r a n g e ' s t h e o r e m ( see W h i t t a k e r and W a t s o n , 1963). I t s h o u l d be n o t e d t h a t t h e r e d u c t i o n b e c o m e s q u i t e l a b o r i o u s f o r h i g h e r order s o l u t i o n s .

F i g . 4 shows t h e shape of a deep w a t e r w a v e , w h o s e h e i g h t i s g i v e n b y H / L = 0 . 1 , w i t h c o m p u t a t i o n s c a r r i e d out t o t h e f i f t e e n t h order. I n o r d e r t o d e t e r m i n e t h e l o c a t i o n of the p r o f i l e , i t i s f i r s t n e c e s s a r y t o d e t e r m i n e t h e e l e v a t i o n of t h e s t i l l w a t e r l e v e l . T h i s may be a c c o m p l i s h e d b y n o t i n g t h a t t h e s t i l l w a t e r l e v e l , yg , i s l o c a t e d s u c h t h a t t h e net area b o u n d e d b y t h e f r e e s u r f a c e and t h e s t i l l w a t e r l e v e l v a n i s h e s . T h e r e f o r e we have L / 2 0 = / ( y , - y ^ ) d x ^ ( 5 2 ) By s u b s t i t u t i n g t h e p a r a m e t r i c p r o f i l e e x p r e s s i o n s , Eqs. 9 and 10, t h i s e q u a t i o n m a y be s o l v e d f o r t h e s t i l l w a t e r e l e v a t i o n , t o g i v e

W i t h the a i d o f t h i s e q u a t i o n , t h e f i f t e e n t h order w a v e p r o f i l e i n F i g . 4 i s o r i e n t e d w i t h r e s p e c t t o t h e s t i l l w a t e r e l e v a t i o n .

For p u r p o s e s of c o m p a r i s o n , F i g . 4 i n c l u d e s , i n a d d i t i o n t o t h e f i f t e e n t h order p r o f i l e , t h e w a v e shapes w h i c h r e s u l t f r o m t h e f i r s t and f i f t h ( S t o k e s ) o r d e r d e v e l o p m e n t s . I n t h e f i r s t o r d e r a n a l y s i s the c l a s s i -c a l l i n e a r t h e o r y o f A i r y (1845) was e m p l o y e d t o g i v e t h e s i n u s o i d a l s h a p e . The f i f t h o r d e r shape ( B e a c h E r o s i o n Board, 1942) was d e r i v e d f r o m the S t o k e s - L e v i - C i v i t a t h e o r y and i s g i v e n b y y^ = Of c o s + {Icx^ +XLa^)^os 2x^ + ( | a ^ + ^ « ^ ) c o s 3 x ^ - j ö ' ^ cos 4x^ + j | | a ' ^ c o s 5x^ ( 5 4 ) w h e r e . = i ( 2 . H ' ) - ^ ( 2 . H - ) ^ y | ^ ( 2 . H ' ) ^ ( 5 5 ) The t w o h i g h e r o r d e r p r o f i l e s s h o w n i n F i g . 4 e x h i b i t t h e s h a l l o w , b r o a d t r o u g h s a n d h i g h , s l i m c r e s t s w h i c h are c h a r a c t e r i s t i c o f f i n i t e a m p l i t u d e w a v e s . T h i s i s seen i n m a r k e d c o n t r a s t t o t h e f i r s t o r d e r s i n u -s o i d a l p r o f i l e w h i c h i -s more a p p r o p r i a t e l y r e -s e r v e d f o r -s m a l l a m p l i t u d e w a v e s . A c o m p a r i s o n of t h e f i f t h and f i f t e e n t h order p r o f i l e s i s a l s o f r u i t f u l , a l t h o u g h t h e d i f f e r e n c e s are l e s s s i g n i f i c a n t t h a n t h o s e i n v o l v i n g t h e s i n u s o i d a l p r o f i l e . The p r i m a r y l i m i t a t i o n of the f i f t h o r d e r e x p r e s s i o n , Eq. 54, i s t h a t t h e w a v e shape i s d e s c r i b e d b y the f i r s t f i v e h a r m o n i c s a l o n e . I n c o n t r a s t t h e f i f t e e n t h order p r o f i l e i s d e s c r i b e d b y t h e f i r s t f i f t e e n h a r m o n i c s . Thus t h e l a t t e r p r o f i l e has a s l i m m e r c r e s t and a b r o a d e r t r o u g h , as i s c l e a r l y s h o w n i n F i g . 4 .

A f u r t h e r r e s u l t o f i n c l u d i n g the h i g h e r h a r m o n i c s i s s e e n i n a c o m p a r i s o n of t h e s m o o t h c h a r a c t e r o f t h e f i f t e e n t h o r d e r p r o f i l e w i t h t h e somewhat w a v y c h a r a c t e r o f the f i f t h order p r o f i l e . I n p a r t i c u l a r t h e r a t h e r f l a t p o r t i o n of t h e f i f t h o r d e r p r o f i l e n e a r XQ / L = ± 0. 4 s h o w s t h e e f f e c t s of u s i n g an i n a d e q u a t e number of h a r m o n i c s . No s u c h f l a t r e g i o n i s s e e n i n t h e f i f t e e n t h o r d e r c a s e . The h i g h e r h a r m o n i c s , t h e r e f o r e , appear t o d e s c r i b e t h e w a v e more p r e c i s e l y . A f u r t h e r w e a k n e s s o f Eq. 54 i s t h a t , f o r the h i g h e r v a l u e s of t h e d i m e n s i o n l e s s w a v e h e i g h t , i t i s i n e x a c t . For e x a m p l e , f o r H / L = 0 . 1 , the c a s e s h o w n i n F i g . 4, t h e d i m e n s i o n l e s s w a v e h e i g h t i s s h y o f t h i s v a l u e b y 1 % . For h i g h e r v a l u e s t h e d i v e r g e n c e f r o m t h e proper v a l u e i n c r e a s e s .

One may summarize t h e n , t h a t f o r many a p p l i c a t i o n s t h e f i f t h o r d e r p r o f i l e i s q u i t e adequate, b u t f o r a more p r e c i s e a n a l y s i s o f t h e p r o b l e m t h e f i f t e e n t h order p r o f i l e i s a b e t t e r r e p r e s e n t a t i o n .

THE HIGHEST WAVE

I n o r d e r t o s t u d y the c h a r a c t e r i s t i c s of the h i g h e s t p o s s i b l e w a v e , i t i s n e c e s s a r y t o add a f u r t h e r r e s t r i c t i o n t o t h o s e i m p o s e d o n t h e d e e p w a t e r w a v e by E q s . 43, 44 and 4 6 . T h i s r e s t r i c t i o n was f i r s t s u g g e s t e d by Stokes i n h i s d i s c u s s i o n of the h i g h e s t w a v e , w h e n he p o s t u l a t e d t h a t a s h a r p peak w o u l d o c c u r at t h e c r e s t of t h i s w a v e . Such a s h a r p peak m u s t r e s u l t i n a s t a g n a t i o n p o i n t and t h e r e f o r e t h e v e l o c i t y w i l l v a n i s h at t h e c r e s t . From the d y n a m i c b o u n d a r y c o n d i t i o n at t h e f r e e s u r f a c e , Eq. 19, we o b t a i n , t h e r e f o r e Zgy^^ = K at r = 1 ;

X

= 0 ( 5 6 ) or, b y s u b s t i t u t i n g t h e p r o f i l e e x p r e s s i o n , Eq. 10, f o r y ^ , n a 2

E

- r

= K' ( 5 7 ) k = l ^ Since t h i s e q u a t i o n i n t r o d u c e s no n e w u n k n o w n s t o t h o s e a l r e a d y a p p e a r i n g i n E q s . 43, 44 and 46, i t s i n c l u s i o n r e s u l t s i n a set of n + 3 e q u a t i o n s w i t h n + 3 u n k n o w n s , w i t h H ' n o w c o n s i d e r e d as u n k n o w n .

The c o m p u t a t i o n was s i m i l a r t o t h a t f o r w a v e s of any h e i g h t , t r i a l v a l u e s o f a^^ b e i n g c h o s e n u n t i l E q . 57 was s a t i s f i e d , a l o n g w i t h E q s . 43, 44 and 4 6 . For the f i f t e e n t h o r d e r the h i g h e s t p o s s i b l e w a v e w a s c o m p u t e d t o have a d i m e n s i o n l e s s w a v e h e i g h t , ( H / L ) f « 0. 1442 ( 58) m a x T h i s compares w e l l w i t h v a l u e s of ( H / L ) » 0.142 ( 59) o b t a i n e d b y M i c h e l l (1893) and ( H / L ) w 0.1418 ( 6 0 ) max o b t a i n e d b y H a v e l o c k ( 1 9 1 9 ) . M i c h e l l and H a v e l o c k b o t h b a s e t h e i r a n a l y s e s o n s e r i e s e x p a n s i o n s about t h e peak, w h i c h c o n v e r g e r e l a t i v e l y r a p i d l y . I n c o n t r a s t t h e s e r i e s o b t a i n e d i n the p r e s e n t a n a l y s i s c o n v e r g e s at a n i n c r e a s i n g l y s l o w rate as t h e d i m e n s i o n l e s s w a v e h e i g h t i n c r e a s e s .

H o w e v e r , t h e c o m p u t a t i o n s s u g g e s t t h a t , as t h e order of t h e a n a l y s i s i s i n c r e a s e d , and more h a r m o n i c s are i n c l u d e d , the c o m p u t e d h e i g h t f o r t h e h i g h e s t w a v e i s r e d u c e d . The m e t h o d of the present a n a l y s i s , i f c a r r i e d t o s u f f i c i e n t l y h i g h o r d e r s , may t h e r e f o r e be e x p e c t e d t o g i v e r e s u l t s a p p r o a c h i n g t h o s e o f M i c h e l l and H a v e l o c k . I n t h e a b s e n c e o f s u c h a c o m p u t a t i o n , E q s . 59 and 60 appear t o be more r e l i a b l e t h a n E q . 5 8 . As a c o n s e q u e n c e . T a b l e s 1, 2 and 3 i n c l u d e v a l u e s o f t h e c o e f f i c i e n t s f o r a maximum d i m e n s i o n l e s s w a v e h e i g h t of 0.142, as s u g g e s t e d b y M i c h e l l . I n c o n c l u d i n g t h i s d i s c u s s i o n of m a x i m u m w a v e h e i g h t , i t s h o u l d a g a i n be p o i n t e d out t h a t v e r y f e w o b s e r v a t i o n s of w a v e s e x c e e d i n g a d i m e n s i o n l e s s w a v e h e i g h t of 0 . 1 have b e e n r e p o r t e d . W h e t h e r o r not t h e t h e o r e t i c a l v a l u e s f o r maximum w a v e h e i g h t d e r i v e d b y M i c h e l l and H a v e l o c k are p h y s i c a l l y r e l i a b l e , i s , as y e t , o p e n t o q u e s t i o n . C O N C L U S I O N A t h e o r e t i c a l s t u d y of deep w a t e r w a v e s h a v i n g a f i n i t e h e i g h t has been u n d e r t a k e n , b a s e d on the w o r k of N e k r a s o v ( 1 9 5 1 ) . By means of a c o n f o r m a l t r a n s f o r m a t i o n the p r o b l e m i s r e d u c e d t o one of o b t a i n i n g a s o l u t i o n t o a n o n - l i n e a r a l g e b r a i c e q u a t i o n , w h i c h i s s h o w n t o be i n agreement w i t h a n e q u a t i o n d e v e l o p e d b y Stokes ( 1 8 8 0 ) . The s o l u t i o n o f t h i s e q u a t i o n , f o r t h e c o e f f i c i e n t s of t h e h a r m o n i c s t h a t d e s c r i b e the s h a p e of t h e f r e e s u r f a c e , i s a c c o m p l i s h e d w i t h t h e a i d of a d i g i t a l c o m p u t e r .

M a c h i n e c o m p u t a t i o n s have b e e n c a r r i e d out t o t h e t h i r d , f i f t h a n d f i f t e e n t h o r d e r a n d the r e s u l t s are s h o w n i n T a b l e s 1, 2 a n d 3. I n e a c h c a s e the c o e f f i c i e n t s have been t a b u l a t e d f o r v a r i o u s v a l u e s of t h e d i m e n s i o n l e s s w a v e h e i g h t . Values o f t h e c o r r e s p o n d i n g d i m e n s i o n l e s s w a v e s p e e d are a l s o i n c l u d e d .

A c o m p a r i s o n of t h e r e s u l t i n g w a v e speeds a n d w a v e s h a p e s of d e e p w a t e r w a v e s , w i t h t h o s e o b t a i n e d b y Stokes, r e v e a l s g e n e r a l l y g o o d agreement. T h i s i s p a r t i c u l a r l y t r u e f o r the l o w e r v a l u e s o f t h e d i m e n -s i o n l e -s -s wave h e i g h t , where e x c e l l e n t agreement i -s o b t a i n e d . H o w e v e r , f o r l a r g e r w a v e h e i g h t s , the i n c l u s i o n of h i g h e r h a r m o n i c s i n t h e f i f t e e n t h o r d e r d e v e l o p m e n t r e s u l t s i n h i g h e r v a l u e s of the w a v e s p e e d t h a n were o b t a i n e d f r o m some o f t h e l o w e r order a n a l y s e s , i n c l u d i n g t h o s e o f S t o k e s . S i m i l a r l y , f o r t h e l a r g e r w a v e h e i g h t s the use of the f i f t e e n t h o r d e r d e v e l -opment i n p r e d i c t i n g w a v e shape r e s u l t s i n b r o a d e r t r o u g h s a n d s l i m m e r peaks t h a n t h o s e p r e d i c t e d b y the l o w e r order a n a l y s e s .

An a p p l i c a t i o n of the f i f t e e n t h order a n a l y s i s t o t h e d e t e r m i n a t i o n of the h i g h e s t w a v e i n w a t e r shows g o o d agreement w i t h t h e r e s u l t s o b -t a i n e d b y M i c h e l l (1893) and H a v e l o c k ( 1 9 1 9 ) .

REFERENCES A i r y , G. B. ( 1 8 4 5 ) . T i d e s and w a v e s : E n c y c l o p . M e t r o p o l . , L o n d o n . Beach E r o s i o n Board ( 1 9 4 2 ) . A s t u d y of p r o g r e s s i v e o s c i l l a t o r y w a v e s i n w a t e r : T e c h . Report N o . 1, U . S. Army C o r p s of E n g i n e e r s , W a s h i n g t o n , D. C . H a v e l o c k , T. H . (1919). P e r i o d i c i r r o t a t i o n a l w a v e s of f i n i t e h e i g h t : Proc. Roy. Soc. L o n d o n , ser. A. v o l . 95, pp. 3 8 - 5 1 .

H i l d e b r a n d , F. B. ( 1 9 5 6 ) . I n t r o d u c t i o n t o n u m e r i c a l a n a l y s i s : M c G r a w - H i l l Book C o . , New York .

L e v i - C i v i t a , T. ( 1 9 2 5 ) . D ë t e r m i n a t i o n r i g o u r e u s e des ondes d ' a m p l e u r f i n i e : M a t h . A n n a l e n , v o l . 93, pp. 264-314. M i c h e l l , J. H . ( 1 8 9 3 ) . On t h e h i g h e s t w a v e s i n w a t e r : P h i l . M a g . , ser. 5, v o l . 36, pp. 4 3 0 - 4 3 7 . M i l n e - T h o m s o n , L . M . ( 1 9 6 0 ) . T h e o r e t i c a l h y d r o d y n a m i c s : F o u r t h e d i t i o n , M a c m i l l a n C o . , N e w Y o r k , pp. 4 2 8 - 4 3 5 . N e k r a s o v , A. I . (1951). The e x a c t t h e o r y of s t e a d y w a v e s o n t h e s u r f a c e of a h e a v y f l u i d : I z d a t . A k a d . N a u k . , SSSR, M o s c o w ( t r a n s l a t e d b y t h e M a t h e m a t i c s Research C e n t e r , U . S. Army, U n i v e r s i t y of W i s c o n s i n ) . R a y l e i g h , L o r d ( 1 8 7 6 ) . On w a v e s : P h i l . M a g . , ser. 5, v o l . 1, p p . 257¬ 279, a l s o S c i e n t i f i c Papers, v o l . 1, C a m b r i d g e U n i v e r s i t y P r e s s , 1899, pp. 2 5 2 - 2 7 1 . Stokes, G. G. ( 1 8 4 7 ) . On t h e t h e o r y of o s c i l l a t o r y w a v e s : T r a n s . C a m b . P h i l . S o c . , V o l . 8, p. 441, a l s o M a t h e m a t i c a l and P h y s i c a l Papers, v o l . 1, C a m b r i d g e U n i v e r s i t y Press, 1880, p p . 197-229= Stokes, G. G. ( 1 8 8 0 ) . Supplement t o a paper on t h e t h e o r y of o s c i l l a t o r y

w a v e s : M a t h e m a t i c a l and P h y s i c a l Papers, v o l . I , C a m b r i d g e U n i v e r s i t y Press, pp. 314-326. W h i t t a k e r , E. T. and W a t s o n , G. N . ( 1 9 6 3 ) . A c o u r s e of m o d e r n a n a l y s i s : 4 t h e d i t i o n , C a m b r i d g e U n i v e r s i t y Press, C a m b r i d g e . W i e g e l , R. L . ( 1 9 6 4 ) . O c e a n o g r a p h l c a l e n g i n e e r i n g : P r e n t i c e - H a l l , I n c . , E n g l e w o o d C l i f f s , N . J. W i l t o n , J. R. ( 1 9 1 4 ) . O n deep w a t e r w a v e s : P h i l . M a g . , ser. 6, v o l . 2 7 , pp. 3 8 5 - 3 9 4 .

APPENDIX- NOTATION A p o i n t o n the b o u n d a r y of t h e w a v e f i e l d at y = - oo A. see E q . 3 5 a^ c o e f f i c i e n t of t h e j t h h a r m o n i c - see E q . 8 B p o i n t o n t h e b o u n d a r y of t h e w a v e f i e l d at the t r o u g h C p o i n t o n t h e b o u n d a r y of t h e w a v e f i e l d at t h e c r e s t c w a v e s p e e d - c / c ^ c f i r s t o r d e r w a v e s p e e d = g L / Z i r J-i D p o i n t o n t h e b o u n d a r y of t h e w a v e f i e l d at t h e t r o u g h f (

4)

see Eq.

12

g a c c e l e r a t i o n due t o g r a v i t y H w a v e h e i g h t H ' = H / L 1 N T T j i n t e g e r w h i c h i d e n t i f i e s t h e j t h h a r m o n i c - see E q . 8 K B e r n o u l l i c o n s t a n t K- . K / c J k i n t e g e r w h i c h i d e n t i f i e s t h e k t h h a r m o n i c L w a v e l e n g t h i i n t e g e r w h i c h i d e n t i f i e s t h e i t h h a r m o n i c n i n t e g e r w h i c h i d e n t i f i e s t h e h i g h e s t h a r m o n i c and t h e o r d e r o f t h e a n a l y s i s q m a g n i t u d e of t h e p a r t i c l e v e l o c i t y at the f r e e s u r f a c e

m o d u l u s of f l ^ p ) - see Eq. 16; a l s o ^c/q^ r r a d i a l c o o r d i n a t e i n t h e 4 - p l a n e u x - c o m p o n e n t of t h e p a r t i c l e v e l o c i t y x - c o m p o n e n t of the p a r t i c l e v e l o c i t y at the f r e e s u r f a c e V y - c o m p o n e n t of the p a r t i c l e v e l o c i t y V p y - c o m p o n e n t of the p a r t i c l e v e l o c i t y at the f r e e s u r f a c e w = (j) + iijj X h o r i z o n t a l c o o r d i n a t e i n t h e z - p l a n e h o r i z o n t a l f r e e s u r f a c e c o o r d i n a t e i n t h e z - p l a n e y v e r t i c a l c o o r d i n a t e i n t h e z - p l a n e v e r t i c a l f r e e s u r f a c e c o o r d i n a t e i n t h e z - p l a n e y s t i l l w a t e r l e v e l i n t h e z - p l a n e s z = X + i y ce see Eq. 55 C = r e x p ( i x ) = e x p ( i x ) , or ^, at t h e f r e e s u r f a c e argument of t{L^)- see E q . 17; a l s o l o c a l s l o p e a n g l e of t h e f r e e s u r f a c e i n t h e z - p l a n e TT = 3. 1415927 (j> p o t e n t i a l f u n c t i o n X t a n g e n t i a l c o o r d i n a t e i n t h e £ , - p l a n e ; a l s o the n o r m a l i z e d p o t e n t i a l f u n c t i o n ijj stream f u n c t i o n