Surface waves on water of variable depth

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UNIVERSITY OF CALIFORNIA >{ARINE PliYSIC/iL LA30R.\T0RY OF THE SCRIPPS INSTITUTION OF OCEANOGRAPHY

SURFACE WAVES ON WATER OF VARIABLE DEPTH Lecture Notes, F a l l Semester, 1950-51

Sponsored by the U.S. Navy O f f i c e o f Naval Research, P r o j e c t NR-083-005, C o n t r a c t N 6 o r i - l l l , Task Order V I ; and Ü. S. Army

Beach E r o s i o n Board, Contract W-499-055-eng 3. C a r l Eclcart

Wave Report No. 100 SIO Reference 51-12

Approved f o r d i s t r i b u t i o n :

August 1951 i

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-L e c t u r e , F a l l Semester, 1950-51 SIO Ref. 51-12

CONTENTS

Page LIST OF SYl-lBOLS _

PREFACE v i 1 . THE GENERAL EQUATIONS, 1

2. THE SHALLOW WATER APPROXIMATION. 6

3. V/ATER OF CONSTANT DEPTH. H 4. C/tNONIC FOmA OF THE SHALLOW WATER EQUATION. 13

5. THE APPROXn-iATION OF RAY THEORY. 15 6. STRAIGHT, PARALLEL CONTOURS. 26 7. THE SOLUTIONS NEAR THE WATER'S EDGE. 29

8. THE SOLUTIONS NEAR THE WATER'S EDGE ( c o n ' t . ) . 44

9. THE SEICHES OF A CIRCULAR LAKE. 51A 10. GENERAL SURVEY OF THE INFLUENCE OF TOPOGRAPHY ON SURFACE V/AVES. 55A

1 1 . THE CASE h ^ {/ ~ c'^ ) . 59 .12. THE CASE (f'/ - e " ° " ( c o n ' t . ) 73

13. THE PROPAGATION OF ENERGY 3Y SURFACE WAVES. 77 14. THE PROPAGATION OF WAVES FROM DEEP TO SHALLOW WATER. 82

15. THE CASE h = ^ It^ . 93 APPENDIX: The Steady Motions. '96

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L e c t u r e , F a l l Semester, 1950-51 SIO Ref. 51-12

LIST OF SYMBOLS

/A, E> wave amplitudes; constants o f i n t e g r a t i o n .

CZ,(S^ see pp. 66, 67.

OJX)t P ^ r ^ i ^ d i f f e r e n t a t i o n - ^ / ^ t + u ^ ^ V^y ^f^t ^ t o t a l energy per cm^.

E,^,Ey components of energy f l o w . F6i.,6,c,x) hypergeometric s e r i e s . Q.^ Jacobi's p o l y n o m i a l . d i f f e r e n t i a l o p e r a t o r , d e f i n e d p. 84. j2 angle o f incidence a f t e r r e f r a c t i o n , r Bessel's f u n c t i o n of order n . \^ k i n e t i c energy d e n s i t y . p. 73: s p r i n g - l e n g t h .

I Laguerre's polynomial of degree n .

L ^ M K d i f f e r e n t i a l o p e r a t o r s : p. 31; p. 48; p. 52; p. 12. |Sl^ Neumann's f u n c t i o n o f order n . p p. 13: P-/a.^ ^ '''^ ; p. 88: 'P- / > ju'^^ Q. wave amplitude,.?, 47; p. 61. r a d i u s . 3 phase f u n c t i o n .

•y p e r i o d o f waves = 2 TT /co ; k i n e t i c energy per cm^. X. see p. 26 f o r d e f i n i t i o n .

X,Y,Z- components of v e c t o r p o t e n t i a l : see p. 97.

a , b components o f wave number v e c t o r : •= c phase v e l o c i t y : cu/k

C„ group v e l o c i t y : cloo/dk a

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a c c e l e r a t i o n of g r a v i t y ; a u x i l i a r y f u n c t i o n d e f i n e d p. 83 water depth.

k l o c a l wave number, radians/cm; V. ^ Z -rr / X n , m i n t e g e r s . ^ excess pressure. ^ excess pressure a t * ^ <^ a u x i l i a r y v a r i a b l e : p. 31; p. 48; p. 52; p. 6 1 . A. J ^ p o l a r coordinates, ^ slope of bottom. i time. LL, (J^, uy components of v e l o c i t y . a u x i l i a r y v a r i a b l e , p. 33. A, Vj h o r i z o n t a l c o o r d i n a t e s . ^ v e r t i c a l c o o r d i n a t e ; p o s i t i v e upward, o c , ^ numerical parameters: p. 62; p. 93. Euler's constant => 0.577... © a u x i l i a r y v a r i a b l e , p. 27.

angle o f incidence i n deep water; a l s o , see p, 65.

<y6 phase constant; i n Appendix: s c a l a r p o t e n t i a l j p o l a r c o o r d i n a t e . ^ co^l <^ ; Z-n /:f< ° deep water wavelength.

A 1 O C £ L 1 wavelength.

y i n t e g e r .

^ d e n s i t y i n g e n e r a l ; p. 53 e t seq., a parameter, c r parameter i n h = C'- (f-cr A ) J •

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L e c t u r e , F a l l Semester, 1950-51 SIO Ref. 51-12

^ p. 62: &^i'cr t) ; p. 93: ^ ^ (-Z^^x). ^ a u x i l i a r y depth c o o r d i n a t e ,

CO frequency, radians/^e^i ° Z-ir/T ~V gamma f u n c t i o n .

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PREFACE

These lectui^e notes a r e , i n one sense, systematic. I t was the i n t e n t i o n t o show t h a t , s t a r t i n g from the l i n e a r i z e d equations o f hydrodynamics, i t i s p o s s i b l e t o g i v e a l o g i c a l l y c o n s i s t e n t treatment o f the propagation of waves i n water o f v a r i a b l e depth. This t h e o r y does n o t , i n a l l r e s p e c t s , conform t o o b s e r v a t i o n ; i t was the i n t e n t i o n t o e x h i b i t the assumptions t h a t u n d e r ] i s the t h e o r y , so t h a t l a t e r . i n v e s t i g a t o r s might a l t e r them w i t h a view t o improving the t h e o r y . The l a t e r p a r t s o f S e c t i o n 7 show i n d e t a i l how the t h e o r y f a i l s when the water depth becomes very s m a l l .

I n another sense, the l e c t u r e s are e c l e c t i c : two phenomena are

emphasized a t the expense o f o t h e r s . The e a r l i e r l e c t u r e s are d i r e c t e d toward the study o f "edge waves," ( S e c t i o n 9 ) . These seem c l o s e l y r e l a t e d to the s u r f beat phenomena described by Munk* and t o recent speculations concerning t h e r e f l e c t i o n of surface waves by deep water**. I n the l a t e r l e c t u r e s , a new wave e q u a t i o n i s d e r i v e d ( S e c t i o n 1 4 ) , t h a t appears able t o account f o r the d i s p e r s i o n and r e f r a c t i o n t h a t occur simultaneously i n water o f moderate depth. This equation i s c o n s i s t e n t w i t h the "energy t h e o r y " p r e v i o u s l y used t o d e a l w i t h t h i s case, b u t appears t o have systèraatic-advantages over the

e a r l i e r t h e o r y .

^MunlCj^W. K., Trans, A.G.uT, 30,""849 (19497,""

•>«-Isaacs, J.D., W i l l i a i u s , E, A., and E c k a r t , C , Trans. ACU, 32, 37 (1951), Y i

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-SURFACE WAVES ON WATER OF VAPJABIE DEPTH

1 , The General Equation.^.

Let t h e zaxis be v e r t i c a l l y upward, t h e x and yaxes i n t h e h o r i -z o n t a l plane; l e t , vr. viy be the t h r e e components o f v e l o c i t y and t h e pressure be - ^ cj p^.x^^^^ ' ^^'^ l i n e a r i s e d equations o f motion are

I C> <•«.>' . 1'

-V- -\- •i-dt •i-)

provided one neglects a l l d e n s i t y g r a d i e n t s .

D i f f e r e n t i a t i n g Eq. ( 1 ) w i t h respect t o t and s u b s t i t u t i n g from Eq. ( 2 ) , we o b t a i n Laplace's equation:

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^i-Lecture, F a l l Semester, 1950-51

I f t h e atmospheric pressure i s c o n s t a n t (say z e r o ) , the equation of the f r e e surface i s

However, t h e f r e e surface must a l s o move w i t h t h e water: t h i s i s expressed by t h e equation

o r , since

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Lefcture, F a l l Semester, 1950-51

and t h e t h i r d o f Eq. ( 2 ) :

The Eq, ( 4 ) and (S) must both be s a t i s f i e d a t the f r e e surface s i n c e p i s p m a l l , we may approximate Eqi ( 4 ) by

and consider Eq. ( 5 ) t o be imposed a t ^ ^ Ö , r a t h e r t h a n a t ^ =^

p

C <^ O t ) / f »

I f t h e bottom o f the body o f water i s impermeable and has t h e e q u a t i o n

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Lecture, F a l l Semester, 1950-51

i t i s necessary t o f o r m u l a t e the c o n d i t i o n o f i m p e r m e a b i l i t y i n mathematical form. The vector

i s normal t o the bottom s u r f a c e , and hence t h e c o n d i t i o n t h a t no water f l o w s through i t i s

By d i f f e r e n t i a t i n g t h i s w i t h respect t o " t and c l i i n i n a t i n g U-^O-, u T hy moans o f Eq, ( 2 ) , we o b t a i n the equation

The mathematical problem i s now one o f a s i n g l e dependent v a r i a b l e 3 : t h a t s o l u t i o n o f Laplace's equation ( E q . ( 3 ) ) which s a t i s f i e s t h e boundary c o n d i t i o n s Eq, ( 5 ) and ( 7 ) i s r e q u i r e d .

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L e c t u r e , F a l l Semester, 1950-^1 ~ 5

i n t i m e , so t h a t Eq. ( 5 ) becomes

S j i _ cO"^ p - O 5a.

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L e c t u r e , F a l l Semester, 1950-51 " ^

2, The Shallow V/ater Approximation,

Since t h e f u n c t i o n p i s t o be w i t h o u t s i n g u l a r i t i e s , one may expand i t i n powers o f ^ :

I f t h i s s e r i e s i s s u b s t i t u t e d i n t d Laplace's equation, one can c o l l e c t tetmé having the same power o f y , and s e p a r a t e l y equate the c o e f f i c i e n t o f each power o f ^ t o zero. I n t h i s way one o b t a i n s the set o f

equations

A

-H O

<2 t c

where t h e a b b r e v i a t i o n

A =

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I f the series i s s u b s t i t u t e d i n t o Eq. (5a) and e v a l u a t e d f o r - Q , only the terms independent o f ^ remain, and these are

- Ub"- ^ o . l o ;

3 f ' f

With Eq. ( 9 ) and ( 1 0 ) , a l l the f u n c t i o n s can be expressed i n terras o f and i t s d e r i v a t i v e s : u s i h g the a b b r e v i a t i o n ^ ~

^^/g-(Note t h a t 2 TV/ It i s the deep water wave-length) we have

f . = ^ ( f

-1 -1 .

e t c

I n t h i s way, one o b t a i n s one s o l u t i o n o f Eq. ( 3 ) and ( 5 a ) f o r every choice o f the f u n c t i o n . I t remains t o determine i n such a way t h a t Eq. ( 7 ) i s also a t i s f i e d . P a r e n t h e t i c a l l y , i t may be

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L e c t u r e , F a l l Semester, 1950-51 ^ 8

remarked t h a t Eq. (4fe) becomes

4c.

so t h a t , except f o r t h e f a c t o r ^ / ƒ ^ , i s t h e h e i g h t o f t h e d i s p l a c e d f r e e surface above i t s u n d i s t u r b e d l e v e l . The problem has thus been reduced t o t h a t o f f i n d i n g t h e shape o f t h e f r e e s u r f a c e ; once t h i s has been found, the d i s t r i b u t i o n o f pressure and v e l o c i t y can be c a l c u l a t e d from Eq, ( 1 1 ) , ( 8 ) and ( 2 ) .

S u b s t i t u t i n g Eq. ( 8 ) i n t o Eq. ( 7 ) , one f i n d s t h a t

I !

^

o x

^

'^^

^ / y

MA.

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L e c t u r e , F a l l Semester, 1950-51

S e t t i n g ' ^ - _ K and u s i n g Eq, ( 1 1 ) , t h i s becomes

U Ö X O x ^ ^ 12.

Eq, ( 1 2 ) i s a p a r t i a l d i f f e r e n t i a l equation o f i n f i n i t e o r d e r , t o be solved f o r . While such equations have been discussed a t i n t e r v a l s since the time o f FJtxler, no systematic method o f s o l v i n g them has been developed. I n t h i s case, we suppose t h a t the dimensionless q u a l i t i e s

are s m a l l f o r a l l % and , I f we then n e g l e c t products o f these s m a l l q u a l i t i e s , we a r r i v e a t t h e approximate d i f f e r e n t i a l e q u a t i o n o f

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L e c t u r e , F a l l Semester, 1950-51 - 10

second o r d e r :

O = -V- K,

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3, Water o f Constant Depth.

I t i s w o r t h c o n s i d e r i n g t h e case o f constant k , since i n t h i s case, we know t h e r i g o r o u s s o l u t i o n .

I n t h i s case, Eq, ( 1 3 ) becomes

vhere 2 r / k - / ^ 2 | ^ i l 2 H water wave-length. Using t h i s r e s u l t i n Eq. ( 1 1 ) i t becomes

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-L e c t u r e , F a l l Semester, 1950-51 - 12

so t h a t

15 =

Now the r i g o r o u s s o l u t i o n i s known t o be

Ocr^oX k ( ? + k ) 16.

where p i s any s o l u t i o n o f Eq. ( 1 4 ) . Since Eq. (15) may be w r i t t e n

^0 O e r J . k k ^ c L i c .2-<xl k k ] ^

i t i s seen t h a t t h e e r r o r s i n Eq. ( 1 5 ) are o f t h e order o f magnitude I ^ i . _ > as was t o be expected.

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4, Canoni.c Form o f t h e Shallow Water Equation. The i n t r o d u c t i o n of t h e new v a r i a b l e

-r^ I '/^ .17.

reduces t h e Eq. (13) t o the f o r m

18.

where

19.

The expression f o r t h e wave number i n water o f constant depth i s U"^ ^- lO^/'uW K / k , Consequently, t h e second term o f Eq. ( 1 9 ) may be expected t o f u n c t i o n as a c o r r e c t i o n f a c t o r t o t h e u s u a l f o r m u l a . To estimate i t s magnitude, l e t - ^ •>^_j ^ being t h e constant slope o f t h e bottom; then Eq. (19) becomes

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~ 14

Taking q-z^ 10 yO -/Ö a~/có ~ 10 sec, t h i s becomösX» t . (.7 ^ ^

Thus, we may n e g l e c t t h i s term everyi^here except v e r y close t o the water's edge. I t appears t h a t i t i s j u s t i f i a b l e t o use the o r d i n a r y formula i n almost a l l , i f not i n a l l , cases o f i n t e r e s t . The s m a l l r e g i o n near the water's edge i s i n t h e s u r f , and we cannot expect our l i n e a r i z e d t h e o r y t o give a good account o f t h i s phenomenon anyway.

I t i s p o s s i b l e t h a t the c o r r e c t i o n term may be i m p o r t a n t a t the edge o f the c o n t i n e n t a l s h e l f , i n the case o f the 3, 6, and 12 hr. components o f t h e t i d e . Moreover, i n some problems, the n e g l e c t o f the c o r r e c t i o n term complicates, r a t h e r than s i m p l i f i e s the a n a l y s i s .

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L e c t u r e , F a l l Semester, 1950-51 _ X5

5. The Approyimatron o f Ray Theory.

A t r a d i t i o n a l way o f o b t a i n i n g an approximate s o l u t i o n o f Eq. (18) leads t o Huyghens» c o n s t r u c t i o n f o r the wave f r o n t s , and t o o t h e r well-known formulae. I t i s assumed t h a t t h e r e are s o l u t i o n s o f the equation t h a t have the forms

y> ^ A C 5 - c o t 3 ^ 20.

the f u n c t i o n s A and 5 being the same i n a l l these s o l u t i o n s and b e i n g r e a l . I f we s u b s t i t u t e the e x p o n e n t i a l form f o r T i n Eq. ( 1 8 ) , t h e r e s u l t i s

Since /A and 5" are b o t h r e a l , t h i s equation i s e q u i v a l e n t t o the two equations

and

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which w i l l be considered successively.

I n Eq. ( 2 1 ) , t h e term A'' A A i s known as the d i f f r a c t i o n term. I t has t h e same dimensions as , and i s somewhat s i m i l a r t o t h e term

Vr''^ ^ k ' ' ^ - i " ( i ^ ) * However, i n s t e a d o f depending on t h e k n o m bottom topography, t h e former depends on t h e unknown amplitude f u n c t i o n . The method o f s o l u t i o n now vuider d i s c u s s i o n i s u s e f u l o n l y when t h e

d i f f r a c t i o n term i s n e g l i g i b l e compared t o k"" , t h e square of the wave nxml tinder these c o n d i t i o n s , we may w i t e

t h i s e q u a t i o n i s known as t h e geometric wave equation, because i t can be s o l v e d g r a p h i c a l l y by a simple method. This method w i l l f i r s t be described and t h e n proven.

Since Eq. (21a) i s a p a r t i a l d i f f e r e n t i a l equation, i t w i l l i n v o l v e an a r b i t r a r y f u n c t i o n o f i n t e g r a t i o n : t h i s may be taken as t h e curve C.^ i n t h e x-y plane on which iS takes on some constant n u m e r i c a l

v a l u e — say the curve 5 := Ck. • The c o n s t r u c t i o n f o r f i n d i n g the curve C : 5 = OL -V- So. (So. being s m a l l ) i s as f o l l o w s : a t p o i n t s o f , one e r e c t s the normals. On these normals, one l o c a t e s p o i n t s a t t h e d i s t a n c e S^^ So. / k(,x^) from . This i s p o s s i b l e

since |< i s a known f u n c t i o n o f ?<- , ^ and (5"a. i s g i v e n . The curve passes through these p o i n t s . Having obtained , one can repeat

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Lecture, F a l l Semester, 1950-51 - 17

the contour Hnes o f the f u n c t i o n S i n succession. I n a d d i t i o n t o obtain¬ i n g the contours, one also o b t a i n s a f a m i l y o f curves t h a t i n t e r s e c t the contours a t r i g h t angles; these are t b e r a y s .

To prove t h i s , l e t X be a p o i n t on and ^6+ T x ^ ^ -t-the p o i n t on where t h e normal t o i n t e r s e c t s t h e l a t t e r . Then because S ( - ^ + SV, ^-v ^ ^C>^ ':J ^ ^ ^ ^ ^ " " ^

a. _21b,

Because the v e c t o r 5 x S h i s normal t o , we have

5 OS

vhere 6 i s a f a c t o r o f p r o p o r t i o n a l i t y . I t i s r e l a t e d t o ÓA by t h e

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Lecture, F a l l Semester, 1950-51 " 18

s u b s t i t u t i n g i n t o Eq. ( 2 l b ) , we have

CL

or by Eq. (21a), S ^ S c^/k, which was t o be shown.

We can t h e r e f o r e consider the f u n c t i o n 5 , which i s Icnovm as the phase f u n c t i o n , or as t h e Hamilton-Jacobi f u n c t i o n , t o have been determined.

Turning t o Eq. ( 2 2 ) , i t may be w r i t t e n

a x

+

d

23.

= O

which has the form o f an e q u a t i o n o f c o n s e r v a t i o n , q u a n t i t i e s

I t suggests t h a t the

I 24.

may be i n t e r p r e t e d as t h e v e c t o r f l o w o f energy i n the w a v e - t r a i n . (The constant o f p r o p o r t i o n a l i t y , I / i j u3, w i l l be j u s t i f i e d beiow.)

To prove t h i s , we r e t u r n t o Eq. ( 1 ) and ( 2 ) ; m u l t i p l y i n g t h e f i r s t by 0 , t h e o t h e r s by u.^ cr and or r e s p e c t i v e l y , and then adding,

I \ one obtains

Ot

Ö 25, Since ^ ^ ( U . ' M - V ^ 3 i s t h e d e n s i t y o f k i n e t i c energy, we i d e n t i f y

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Lecture, F a l l Semester, 1950-51 20

Hence

A

è s

and hence

V/e have had t o make many approximations t o o b t a i n t h i s i n t e r p r e t a t i o n o f F ' i t i s t i i e r e f o r e i m p o r t a n t t o n o t i c e t h a t Eq. ( 2 3 ) was d e r i v e d v a t h -out making these approximations, and has a h i g h e r degree o f v a l i d i t y t h a n our i n t e r p r e t a t i o n .

To summarize these r e s u l t s : i t has been shown t h a t a c e r t a i n approxim-mation, Huyghens» wave-front c o n s t r u c t i o n i s p a r t o f t h e approximate s o l u t i o n

o f the Eq, ( 1 8 ) . Then i t has been shown t h a t t h e "energy f l o w " i s normal t o the wave f r o n t s , and i s p r o p o r t i o n a l t o t h e square o f t h e amplitude f u n c t i o n , and t o the g r a d i e n t o f the phase f u n c t i o n .

This s e c t i o n has been e n t i t l e d "Ray Theory," b u t thus f a r , rays have s c a r c e l y been mentioned. They are d e f i n e d as those curves t h a t i n t e r s e c t a l l o f the curves S = const, a t r i g h t angles: t h e y are the orthogonals o f the f a m i l y o f curves S - const. They d e r i v e t h e i r importance from a v e r y elegant, though somewhat i n o b v i o u s , a n a l y t i c method f o r s o l v i n g Eq. ( 2 l a ) .

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Lecture, F a l l Semester, 1950-51 21

Consider any curve, P , j o i n i n g two p o i n t s A and B i n the x-y plane, and l e t d A 'be i t s element o f l e n g t h . Then we can c a l c u l a t e

the i n t e g r a l

/A ^

I f we keep A and B f i x e d , b u t change the curve T > ^ p t

change i t s v a l u e . Among a l l the p o s s i b l e cuinres, t h e r e w i l l o f t e n be one f o r which X has a smaller v a l u e t h a n any o t h e r ; a l t e r n a t i v e l y , t h e r e may be one f o r which i t has a l a r g e r value than f o r any o t h e r ; these "extremals" w i l l be sho\m t o be the r a y s : t h i s i s Fermat's p r i n c i p l e o f l e a s t t i m e s .

The extreme value o f C= X ) i s r e l a t e d t o the phase f u n c t i o n i n a simple way.

F i r s t , we o b t a i n Euler's d i f f e r e n t i a l e q u a t i o n o f t h e r a y s . For t h i s purpose, suppose*l;hat the e q u a t i o n o f the r a y s i s

where T i s any parameter, such t h a t the p o i n t A corresponds t o T ~ -c and the p o i n t B t o r ^ T • Then

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and hence

X -

k

(A.

cUt: .

J

'A

I f we consider a n e i g h b o r i n g curve j o i n i n g /\ t o X5 ; we may w r i t e

provided t h a t

•^he value o f X. must be, t o a f i r s t approximation, u n a l t e r e d , since t h e curve i s an e x t r e m a l . But, value f o r t h e n e i g h b o r i n g p a t h i s

X

I f we i n t e g r a t e the l a s t ti/o terms by p a r t s , and remember t h a t the v a n i s h f o r z- C,,and , we g e t

A

-^6

•A

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v a n i s h i n o r d e r t h a t the two expressions f o r X be equal: O

^ ~ Ar k a a t j

These are E u l e r ' s d i f f e r e n t i a l equations. By s o l v i n g them, we o b t a i n t h e r a y s ; t h r o u g h any p o i n t , t h e r e i s one r a y i n every d i r e c t i o n , since these equations a r e o f second o r d e r .

Having thus determined what we mean by " r a y s , " we now proceed t o determine t h e f u n c t i o n 5 • b e f o r e , we suppose the one contour 5 -t o be g i v e n , as -the a r b i -t r a r y elemen-t -t h a -t en-ters i n -t o -the s o l u -t i o n o f Eq. ( 2 1 a ) . Then through each p o i n t o f t h i s curve, we c o n s t r u c t t h a t r a y which i n t e r s e c t s t h e contour a t r i g h t angles. Then, i n g e n e r a l , i f yi^ i s any p o i n t , one o f these rays w i l l pass through i t , and w i l l i n t e r s e c t the surface S r o. i n the p o i n t <^ ^ > The r e q u i r e d f u n c t i o n i s now t o be c a l c u l a t e d by the formula

the i n t e g r a t i o n being along the r a y i n q u e s t i o n .

There are v a r i o u s ways o f p r o v i n g t h i s ; perhaps t h e s i m p l e s t i s t o t a k e X ^ v e r y near t o : then t h i s formula reduces t o t h e previous Huyghens' c o n s t r u c t i o n . Repeating t h e c o n s t r u c t i o n f o r successive elements d /) , smd adding, we o b t a i n the r e s u l t j u s t given. However, t h i s argument does n o t make i t c l e a r t h a t the rays must be determined by Euler's

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Lcctüre, F a l l Semester, 1950-51 ^ 24

equations, and i t cannot be considered t o be e n t i r e l y s a t i s f a c t o r y f o r t h i s reason,

A :more s a t i s f a c t o r y p r o o f was given by W. R. Hamilton: c o n s i d e r two p o i n t s B- X'-^ and B'r-jc ^ 6 >t , 5 * These w i l l i n g e n e r a l

l i e on two d i f f e r e n t rays, w l i i c h i n t e r s e c t S - a_ p e r p e n d i c u l a r l y i n t h e points/i-.>: u and A ' - ' x + A ' ^ u -^-6 u • equations o f the two,rays be

and

Nov^, however, S..Y^ 8 ^ are n o t zero a t -c tr and . We can c a l c u l a t e

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O X

by t h e same method as was p r e v i o u s l y used i n d e r i v i n g E u l e r ' s equations:

u.

k-•^6

Because - Y (r ) > a s o l u t i o n o f Euler's e q u a t i o n , the i n t e g r a l J

v a n i s h e s j the- l a s t b r a c k e t also vanishes, since h-K.^

Su^is

a v e c t o r . p a r a l l e l t o S - cx, ,. w h i l e ( ^ \ ...is a v e c t o r normal t o the same curve; t h i s leaves

• i t being-understood t h a t T T C T : ^ on t h e r i g h t . Since- &V- • aiid. are a r b i t r a r y increments, we have

b u t , because o f t h e d e f i n i t i o n o f *u.. . t h i s l e a d s t o " ...

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L e c t u r e , F a l l Semester, 1950-51 « 26

6. S t r a i g h t . P a r a l l e l Contours. (Ray Theory)

I f K and k are constants, the geometric wave equation can be

solved by i n s p e c t i o n : i f we set

t h e equation reduces t o

and

Moreover, one s o l u t i o n o f Eq. (23) i s then A =. constant. I n t h i s s o l u t i o n , the wave-fronts are s t r a i g h t l i n e s , and the amplitude o f t h e waves i s everywhere the same. However, even when k i s constant, Eq, (21a) has s o l u t i o n s t h a t are more complicated.

I f k and k are independent o f j ^ , , we have the case o f s t r a i g h t bottom c o n t o u r s , p a r a l l e l t o t h e y - a x i a . I n t h i s case, we t r y t o f i n d a

f u n c t i o n Y[7^) such t h a t

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This leads t o the equation

a x \ ^

27.

o r

28.

A p a r t i c u l a r s o l u t i o n f o r the amplitude f u n c t i o n can a l s o be foiand: i f A i s a f u n c t i o n o f o n l y , Eq. (23) reduces t o

o r

29.

Now, the v e c t o r

0 £ I

i s normal t o the wave-fronts: consequently, i f X i s t h e angle these make w i t h the y-axis

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Hence

-and hence t h e wave h e i g h t i s p r o p o r t i o n a l t o

30.

Nearshore, c o r s X -^ 1 and we have proven t h e known r e s u l t t h a t the wave h e i g h t i s p r o p o r t i o n a l t o i ^ ' k ' ^ ^

To i l l u s t r a t e these c a l c u l a t i o n s i n more d e t a i l , l e t

where Ix and vT are constants. Then Oo

té'

. . „

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the v e c t o r , 0"^ , uy^ w i t h the energy f l o w i n t h r e e dioBensions. We then show t h a t , t o an adequate approximation.

f ^

% -~ _{

where t h e bar i n d i c a t e s a time average. Taking

we have, by Eq. ( 1 7 ) ,

^ (.,^'^2. A ' ^ S r i C S - c j t ) - (:> t o a rough approximation.

Hence, by Eq. ( 2 ) ,

ƒ a t - 2>x

where t h e dots i n d i c a t e terms i n d t . / i ) y and OA/ox. We have a l r e a d y supposed t h a t such terms are s m a l l , i n d e r i v i n g Eq. (13) and ( 2 1 a ) , so we are perhaps j u s t i f i e d i n n e g l e c t i n g them here. I n t h a t case,

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Lecture, F a l l Semester, 1950-51 ^ 28

makes i t p o s s i b l e t o evaluate the i n t e g r a l :

•33.

v h i c h i s w e l l s u i t e d f o r numerical e v a l u a t i o n , and w i l l be u s e f u l below. The amplitude f u n c t i o n A i s p r o p o r t i o n a l t o \ / ( _ ^ ^ oh^ O) , and t h e height o f t h e waves t o ( 4^ / '-^^^ ^ ^

Note: I f = Ö , t h i s formula becomes i n d e t e r m i n a t e , b u t t h e r e s u l t X - ( / ( T ) iL^ [ ( i ) /O - ^ % ^ ]

e a s i l y obtained.

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LO

O

• 5 1.0 1.5 2.0 25

Note-- For T = 15 min.,

For T- 20 sec,

s -- 0.02,

s -- 0.02 ,

s/4

k

s/4

k

I km.

50 cm.

FIGURE 2.

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7. The S o l u t i o n s Near the Water's Edge

We r e t u r n t o Eq. (13) and consider the case-X'

h -

31*.

which represents a bottom w i t h constant slope A . This w i l l not be j u s t i f i e d f o r l a r g e values o f ^ , since the depth of water w i l l u l t i m a t e l y become g r e a t e r t h a n one wave-length. However, one can o b t a i n u s e f u l conclusions concerning t h e wave-motion near the water's edge, and r e s u l t s t h a t can be used l a t e r i n o t h e r connections.

S u b s t i t u t i n g Eq. (31'") i n t o Eq. ( 1 3 ) , and making the s i m p l i f y i n g assumption t h a t i s independent o f ^ (normal incidence o f the waves) the wave equation becomes D e f i n e the o p e r a t o r 33 ^ f . (V

^ d ^ d_

^ / ^ / x j a /

dx

and expand where C^± O .

This problem has been discussed by J. J. Stoker, Q u a r t e r l y o f A p p l i e d Mathe-m a t i c s 5 31 (1947).

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Then, since

we o b t a i n

I n order t h a t t h i s equation s h a l l be v a l i d , t h e c o e f f i c i e n t o f each power o f X must v a n i s h , so t h a t

c , + c

^

o-3 ^ e ^ t - ( / < / ^ J c . - o

o r

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Lecture, F a l l Semester, 1950-51 - 3 1

This i s an i n f i n i t e s e r i e s f o r ^ , and, as such would be r a t h e r d i f f i c u l t t o use f o r numerical computation. F o r t u n a t e l y , however, t h e f u n c t i o n

(known as Bessel's f u n c t i o n ) has been tabulated,-^f- and we may w r i t e Eq. ( 3 4 )

so t l i a t i s i s simple t o c o n s t r u c t a graph o f t h i s s o l u t i o n . ( F i g . 1 and 2.) U n f o r t u n a t e l y , t h i s i s n o t the most g e n e r a l s o l u t i o n o f t h e Eq. (3?*-), and i t i s r a t h e r d i f f i c u l t t o o b t a i n t h e most g e n e r a l one. I t i s much e a s i e r t o o b t a i n t h e g e n e r a l s o l u t i o n o f t h e e q u a t i o n

I t w i l l be i n s t r u c t i v e t o consider t h i s case; t o o b t a i n the r e s u l t , i t i s neces-s a r y t o neces-set

•«• References: Magnes &" O b e r h e t t i n g e r " S p e c i a l Functions o f Mathematical • Physics," p. 16 (Chelsea, 1949) ( h e r e a f t e r c i t e d as S.F.M.P.).

JahnJce & Ende, "Tables o f F u n c t i o n ^ " p. 128 e t seq. (Dover, 1943).

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where m. i s a constant t h a t must be determined. Since

pr\ X'^ ^ (i^^- i^rx^) X""'^ (KfA)x^'"

we now g e t

whence

C. :r e>

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L e c t u r e , F a l l Semester, 1950-51 " 33

The f i r s t equation could be s a t i s f i e d i f C o , b u t then a l l t h e c's

would vanish and t h e t r i v i a l s o l u t i o n - <^ would r e s u l t . Hence m =: 1 1 - a are t h e o n l y other p o s s i b i l i t i e s . Corresponding t o these, we can determine two

YL L o- f<. ö"C" o-u. t '^ t e e r ,

sets o f c's and get tv/o s o l u t i o n s ; which may be w r i t t e n

I' - J A

^ - J / ' )

36. where

Yi.

Tor systematic reasons t h e a r b i t r a r y constant C ' i s set equal t o

I t i s seen t h a t , when i l - G , these two s o l u t i o n s become i d e n t i c a l , thus ex-p l a i n i n g t h e ex-p e c u l i a r d i f f i c u l t y w i t h Eq. (32'i^). The g e n e r a l s o l u t i o n o f Eq.

(32-iw*-) i s = \ ^ "^T-^ > consequently t h e f u n c t i o n (ö") where

IT

a

_38.

i s a l s o a s o l u t i o n o f t h i s equation;**- i t i s known as Neumann's f u n c t i o n . When p\ -> Q , both numerator and denominator become zero, b u t the f u n c t i o n

can be determined by e v a l u a t i n g the i n d e t e r m i n a t e form:

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L e c t u r e , F a l l Semester, 1950-51 2. TT _n Ï:. Ö From Eq. ( 3 7 ) , Ö —

(

1 1 Hence

vr

-i 0 2-1 ^ ^ +• where \ ( I ) ~ ^ and

1 ^

L

r7 - o

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L e c t u r e , F a l l Semester, 1 9 5 0 - 5 1 - 35

I t i s now c l e a r t h a t t h e two independent s o l u t i o n s o f Eq. (32-«-) are

1^

J (

and we proceed t o consider them i n more d e t a i l .

Figure 1 shows a graph o f these two f u n c t i o n s , p l o t t e d w i t h /-{K% / A as abscissa. I t i s seen t h a t both are o s c i l l a t o r y , and t h a t t h e i r wavelength d i m i n i s h e s w i t h d i m i n i s l i i n g -/} , % approaches zero, J"^ -> / > w h i l e

^4

-^j.'Ori> '

However, has i t s l a s t r o o t so close t o the o r i g i n t h a t i t i s d i f f i c u l t t o r e p r e s e n t t h i s f e a t u r e already on t h e scale o f F i g , 1 . F i g w e 2 i s p l o t t e d w i t h the ';-<-scale enlarged by a f a c t o r o f 100, and shows, among o t h e r s , a graph o f >- f \ l . This graph has i t s s m a l l e s t r o o t a t

o

'^H-Xf /> ~ 0y2'l i r i s e s t o i n f i n i t y between t h i s p o i n t and X = o . The f u n c t i o n 0 ^ , on t h i s sane s c a l e , i s represented by a s t r a i g h t l i n e t h a t has approximately t h e same slope as t h e graph o f |^ ^ f o r l a r g e r values o f

X •

The q u e s t i o n a r i s e s , whether t h e i n f i n i t y o f t h e f u n c t i o n K l can pos-o

s i b l y represent t h e breakers. This must be answered n e g a t i v e l y . This i s most simply seen by a nuiüorical example: l e t the p e r i o d o f the waves be 20 s e c , and the slope o f t h e bottom 2 %\ then K. •=• 5"ö cm. For t h i s example, t h e r e f o r e , the whole graph o f F i g , 2 represents o n l y 125 cm, the water's edge being a t the l e f t , '.he maximum depth o f water, a t t h e r i g h t hand edge, i s o n l y 2.5 cm. The smallcsc r o o t o f NQ occurs 40 cm from the water's edge, and t h e depth here i s o n l y 0.8 cm. I t i s obvious t h a t t h e r e i s l i t t l e p h y s i c a l s i g n i -f i c a n c e t o be a s c r i b e d t o t h e s o l u t i o n i n t h i s range o -f -values. I t i s

(46)

s c a r c e l y necessary t o enter i n t o a l l the reasons: one i s t h a t the wave h e i g h t ? i n which we are i n t e r e s t e d are a l l v e r y much g r e a t e r than the water depths i n -volved. The c o n t r a r y has been assumed i n d e r i v i n g the boundary c o n d i t i o n a t the f r e e s u r f a c e ,

A second example i s o f i n t e r e s t : t h e p e r i o d s o f tsunamis are o f the o r d e r o f 15 minutes. For t h i s p e r i o d and A « 2^, '-^h^ - I k i u , T h u s , i n t h i s case, F i g . 2 covers 2.5 lan, and water depths up t o 50 meters. Even so, the f u n c t i o n -No does not exceed except f o r < 60 meters, where t h e water depth i s l e s s than 1.2 meters. Again, i t i s s c a r c e l y p o s s i b l e t o a s c r i b e any p h y s i c a l s i g n i f i c a n c e t o the d i f f e r e n t mathematical p r o p e r t i e s o f the two s o l u t i o n s a t

y. O . W e s h a l l r e t u r n t o t h i s m a t t e r below.

These examples make i t c l e a r t h a t , f o r waves o f l e s s t h a n 1 minute p e r i o d , on beaches t h a t do not slope v e r y s t e e p l y , we s h a l l be i n t e r e s t e d p r i m a r i l y i n values o f ^ f(_y t h a t are g r e a t e r t h a n 100. For such l a r g e values o f the argument, i t i s known that-^

These equations have been used t o c o n s t r u c t F i g , 3, which extends t h e graphs o f 40.

•Jf^ and "ro l a r g e r values o f X •

I t i s i n t e r e s t i n g t o note t h a t Eq. ( 4 0 ) i s e x a c t l y t h e r e s u l t we should have obtained from the r a y t h e o r y a p p r o x i m a t i o n : f o r , i f

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Moreover, the h e i g h t o f the waves, from Eq. ( 4 0 ) , i s p r o p o r t i o n a l t o 1 / or \ I \\^^ , again i n complete agreement w i t h r a y t h e o r y .

The phase c o n s t a n t s , VUi- , i n Eq. (40) are r a t h e r t y p i c a l , and w i l l recur i n o t h e r a p p l i c a t i o n s . I t may he noted t h a t , f o r l a r g e ,

^ ^ „ ( V ^ J o ^"^^ I ' ^ ^ ^

/ + Jo \ - ^ ( ^ J

/ h ^ ^ These two combinations are p l o t t e d , f o r s m a l l , on F i g . 2.

The d i f f e r e n t behavior o f the two s o l u t i o n s and NQ cannot be dismissed w i t h o u t f u r t h e r i n v e s t i g a t i o n . I t may serve t o c l a r i f y the problem i f i t i s known t h a t these f u n c t i o n s a l s o occur i n o t h e r p h y s i c a l problems; a t y p i c a l one

i s t h a t o f a l t e r n a t i n g e l e c t r o m a g n e t i c f i e l d s . I n t h a t case, the f u n c t i o n NQ represents f i e l d s near a v e r y t h i n w i r e t h a t i s l o c a t e d a t the l o g a r i t h m i c s i n g u l a r i t y and c a r r i e s a c u r r e n t ; the f u n c t i o n JQ represents f i e l d s i n the absence o f such a w i r e . I n t h i s case t h e r e i s a c l e a r p h y s i c a l reason f o r t h e mathematical d i f f e r e n c e between the two s o l u t i o n s .

I n the present case, i t would seem t h a t no p h y s i c a l reason i s t o be ex-pected, since the equations we are s o l v i n g cease t o be v a l i d f o r such s m a l l values o f / . I t i s t h e r e f o r e suggested t h a t we somehow exclude these r e -gions o f very shallow water from c o n s i d e r a t i o n , i n order t o o b t a i n an

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by Vx - f o i ' ; t > X > b u t t h a t , a t 7^ - a v e r t i c a l w a l l e x i s t s . Moreover, suppose the wave-height i s small enough so t h a t the waves do n o t break a g a i n s t t h i s w a l l , b u t are r e f l e c t e d w i t h o u t b r e a k i n g . Then a l l our prev i o u s equations remain prev a l i d f o r > >6 , b u t a new equation enters the p r o b -lem; t h i s equation s t a t e s t h a t no water f l o w s past the p o i n t X - and i s

[X(^yi ;^J;=0' This w i l l be s a t i s f i e d i f and o n l y i f

/

Now, the most g e n e r a l s o l u t i o n f o r i s (A, B c o n s t a n t )

42.

and t h e Eq. ( 4 1 ) w i l l be s a t i s f i e d i f

where C i s an a r b i t r a r y constant, and the accent i n d i c a t e s d i f f e r e n t i a t i o n w i t h r e s p e c t t o the argument o f t h e f u n c t i o n . I f i s l a r g e enough so t h a t Eq. (40) may be used, t h i s becomes

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L e c t u r e , F a l l Semester, IDSO-Sl - 39

The I m p o r t a n t t h i n g t o be noted i s t h a t t h e a d d i t i o n a l boundary c o n d i t i o n , Eq. (41) has reduced the number o f s o l u t i o n s from two t o one; b u t i t has n o t e l i m i n a t e d the Neumann f u n c t i o n from c o n s i d e r a t i o n . I f we w i s h t o l e t become s m a l l , we cannot use Eq, (43) any l o n g e r ; more i m p o r t a n t i s the p h y s i -c a l f a -c t , t h a t i n order t o keep the waves from b r e a k i n g , the -c o n s t a n t C must be made ever s m a l l e r . We can assure t h i s i f we s e t

i f we do t h i s , then, when ^ ö A \ 3-^0, and we do ( r a t h e r a r t i f i -c i a l l y ] e l i m i n a t e the f u n -c t i o n NQ from our -c o n s i d e r a t i o n s .

This example has d e f i n i t e l y excluded s u r f o r the b r e a k i n g o f waves. We consider another example t h a t has one c h a r a c t e r i s t i c i n common w i t h the s u r f : t h e removal o f energy from the o r d e r l y motion o f the wave. I n the case o f s u r f , t h i s energy i s converted i n t o d i s o r d e r l y • motion. We cannot t r e a t t h i s w i t h i n t h e framev;orlc o f our present t h e o r y , b u t , i f we suppose t h e sea w a l l t o be movable, and t h a t i t s motion i s r e s i s t e d by f r i c t i o n , we can t r e a t the removal o f energy from the waves.

Let U be t h e v e l o c i t y o f the w a l l , M i t s mass, R the f r i c t i o n a l constant and F the f o r c e exerted on i t by the waves (M, R, and F are a l l t o be c a l c u l a t e d per u n i t l e n g t h ) : then

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Lecture, F a l l Semester, 1950-51 ^ 40

By analogy t o e l e c t r i c a l and a c o u s t i c a l q u a n t i t i e s , we may c a l l Z the complex impedance o f t h e v m l l .

Now, since the w a l l moves w i t h t h e wave,

and ( t o a c e r t a i n a p p r o x i m a t i o n )

These equations reduce t o the boundary c o n d i t i o n

S u b s t i t u t i n g t h e g e n e r a l s o l u t i o n f o r , we f i n d

' 6 - - c f ï g T / - 7 ' - " : ^ J

44.

45.

the argument i n the Bessel f u n c t i o n s being / 4 ^ throughout. Again, we o b t a i n an a d d i t i o n a l boundary c o n d i t i o n , and t h i s again reduces the number of independent s o l u t i o n s from two t o one b u t , again, we do not e l i m i -nate the Neumann f u n c t i o n .

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L e c t u r e , F a l l Semester, 1950-51 - 4 1

At the expense o f some a l g e b r a , one may w r i t e t h e expression f o r i the form , / /) \'/4 f . ^ - o ^ t - 4 J xn 46. where

/A- =

\J /I Xo 47.

and ;ï^is supposed l a r g e . The equation (46) represents an incoming wave o f amplitude p r o p o r t i o n a l t o /4^', and a r e f l e c t e d wave o f amplitude p r o p o r t i o n a l t o /A^ . The f r a c t i o n o f t h e incoming energy t h a t i s r e f l e c t e d i s

A

These examples are h i g h l y a r t i f i c i a l b u t serve s e v e r a l purposes. One i s the i n t r o d u c t i o n o f t h e q u a n t i t y j O ( j A l ^ ^ ^ ' ^ ^ (c^ ) '^ ^"^^ ^^^^ analogy t o e l e c t r i c a l and a c o u s t i c a l problems (always a dangerous procedure)

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L e c t u r e , F a l l Semester, 1950-51 ~ -^2

t h i s q u a n t i t y w i l l be o f importance i n any.theory of s u r f . , I t may be c a l l e d The value of

the c h a r a c t e r i s t i c jjnpedance o f the surf-zone ."7 "/^^ i s the d i s t a n c e from shore a t which the waves begin t o break.

The other purpose i s t o show t h a t our previous equations have been p h y s i -c a l l y in-complete, and t h a t one a d d i t i o n a l boundary -c o n d i t i o n i s needed. I f i t i s imposed a t some p o i n t o t h e r t h a n X-o o r X r oo , i t may take the g e n e r a l form o f Eq. ( 4 1 ) , or ( 4 4 ) , depending on the p h y s i c a l nature o f t h e problem, However, i t may also be imposed a t >: r. o : t h e n , i n most p h y s i c a l problems, i t takes the form

fcX—^ .-P = 49.

I n our present case, t h i s seems r a t h e r a r t i f i c i a l ( b u t so are the previous p o s s i b i l i t i e s ) ; i f ve accept i t , then we can r e s t r i c t our a t t e n t i o n t o the s o l u t i o n X and exclude h/ . The l a s t p o s s i b i l i t y i s t h a t the boundary con¬ d i t i o n be imposed a t g r e a t d i s t a n c e s , and i s

This says, e s s e n t i a l l y , t h a t t h e r e are no waves r u n n i n g away from shore a t g r e a t d i s t a n c e s . A l l o f the energy o f the incondng waves i s somehow absorbed i n the s u r f zone. Perhaps t h i s equation makes the b e s t sense; t l i i s i s c e r t a i n l y t r u e o f waves whose periods and amplitudes correspond t o o r d i n a r y s w e l l . However, i f we d e a l w i t h very long p e r i o d , low amplitude waves, such as t h e s u r f beat, the t i d e s or t h e seiches o f lakes and harbors, i t i s u n l i k e l y t h a t Eq. (50)

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L e c t u r e , F a l l Semester, 1950-51 43

w i l l be reasonable, since these waves do not break. For such problems, t h e r e f o r e , one o f t h e other boundary c o n d i t i o n s would seem more reasonable. Be-cause o f i t s mathematical s i m p l i c i t y , we \dll choose Eq, (49) i n such cases; t h i s d e f i n i t e l y c o n s t i t u t e s an assumption, however.

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(55)

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8. The_Sc?.utions near the Water's Edge ( c o n t i n u e d )

We 'return t o Huyghen's c o n s t r u c t i o n , b u t continue t o take h = s

Since k"^ - /< / b , i t f o l l o w s t h a t the s e p a r a t i o n between successive phase l i n e s w i l l be " ~

/ s = f a / k ^ $o

and w i l l thus increase w i t h d i s t a n c e from shore. I f t h e i n i t i a l , a r b i t r a r y . , phase l i n e i s not p a r a l l e l t o shore ( i . e . , t o t h e y - a x i s ) the successive phase l i n e s w i l l be more and more i n c l i n e d as they leave the shore.

This can be v e r i f i e d from the geometric wave equation:

K

As b e f o r e , l e t S - X[-^) -h Then

The s u b s t i t u t i o n s

enable one t o evaluate the i n t e g r a l , and o b t a i n

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Since Eq, (52) i s e q u i v a l e n t t o

i t i s e a s i l y seen t h a t the phase l i n e s .5 - X -f- ^ ^ const, are

c y c l o i d s , generated by a c i r c l e o f radius R r o l l i n g on t h e l i n e jx!..- 2R. They t h e r e f o r e have cusps on the l i n e pc ~ 2R, and are tangent t o t h e y - a x i s a t

p o i n t s separated by t h e d i s t a n c e 2 77-R. I t can be shown t h a t t h e rays are s i m i l a r c y c l o i d s , generated by r o l l i n g the c i r c l e o f r a d i u s R on t h e y - a x i s . For convenience, the d i s t a n c e 2 T T R w i l l be c a l l e d the s p r i n g - l e n g t h o f t h e r a y s .

This i n d i c a t e s t h a t waves o r i g i n a t i n g i n shallow water may be r e f r a c t e d u n t i l they r u n p a r a l l e l t o shore, and then turned even more, so t h a t t h e y u l t i m a t e l y r e t u r n t o the beach a t a d i s t a n c e from t h e i r . . p o i n t o f o r i g i n . I f such waves do n o t break, they may repeat t h i s h i s t o r y i n d e f i n i t e l y , u n l e s s , because o f t h e topography (curved shore l i n e , e t c . ) they f i n d an open r o u t e t o sea. (Our present simple assumption t h a t h => s v. does not enable us t o deduce a l l d e t a i l s , and t h i s l a s t remark i s r e a l l y based on an example t h a t w i l l be considered l a t e r . )

The. amplitude f u n c t i o n A can be e a s i l y c a l c u l a t e d (see Eq. ( 2 9 ) ) :

The amplitude o f t h e waves i s p r o p o r t i o n a l t o

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I ^ c t u r ? , F a l l Semester, 1950-51 ™ 46

<2JCI^ bu QJ.T-J

<Ai"t >

0

This expression becomes i n f i n i t e f o r O ö and T T , which values correspond t o X^^f^(i.rxd o . We s h a l l see t h a t these i n f i n i t i e s do not occur i n the more exact p h y s i c a l t h e o r y ; however, the above equation f o r po> can be made reasonably accurate f o r values o f Q not too near the s i n g u l a r p o i n t s , i f the constant phase angle c|:. i s given the r i g h t v a l u e .

Very l i t t l e - a t t e n t i o n has been g i v e n t h i s phenomenon u n t i l r e c e n t l y . Stokes and Lamb"^ considered i t s u f f i c i e n t l y t o bestow the name "edge waves" on i t . However, because t h e r e appeared t o be no o b s e r v a t i o n a l evidence f o r t h e i r e x i s t e n c e , l i t t l e more was done. Munk'^ r e c e n t l y has found t h a t v e r y l o n g p e r i o d (about 4 min.) waves o f low amplitude (about 10 cm.) occur near shore, and has shown t h a t they are generated by groups o f h i g h breakers which occur a t about such time i n t e r v a l s . L a t e r Isaacs^ adduced reasons f o r sup-posing t h a t the phase l i n e s o f t h i s " s u r f beat" are i n c l i n e d to t h e shore. The two data: g e n e r a t i o n near shore and i n c l i n e d phase l i n e s seem t o equate edge waves t o s u r f beat.

When the edge waves are ^examined from the p o i n t o f view o f t h e p h y s i c a l wave equation, the phenomenon .ppears somewhat d i f f e r e n t than d e s c r i b e d above. However, the p r i n c i p a l reason f o r t h i s i s t h a t r a y t h e o r y i s v a l i d o n l y when the d i f f r a c t i o n terras are n e g l i g i b l e . The appearance o f i n f i n i t i e s i n the acpproximate s o l u t i o n i s s u f f i c i e n t evidence t o show t h a t t h i s i s n o t the case

here.u However, we may s t i l l take i t as a working hypothesis t h a t edge waves and s u r f beat are i d e n t i c a l .

M» ™ —^ ^

^ Lamb, Hydrodynamics, p. 447 (N.Y. 1945). 2 W. H. Munk, Trans. A.G.U., 30, 849 (1949).

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- 47

The canonic form o f t h e p h y s i c a l wave equation becomes

The a s s m p t i o n t h a t t h e phase l i n e s a r e i n c l i n e d t o t h e shore takes the form

where b . (We have considered t h e case h =0 i n t h e l a s t s e c t i o n , ) This

reduces t h e p a r t i a l d i f f e r e n t i a l equation t o t h e o r d i n a r y one:

d X

By i n s p e c t i o n o f t h i s e q u a t i o n , we immediately see t h a t , when ^ 2 l < , and d'^C^/cZ/K ^have opposite s i g n s . I n t e r p r e t i n g t l i i s g r a p h i c a l l y , the graph o f must everywhere be concave toward t h e p::-ajd.s: i . e . , t h e f u n c t i o n Q o s c i l l a t e s f o r ^K* When x > Zfion t h e o t h e r hand,

and Q V have the same s i g n , so t h a t i t s graph i s convex t o v a r d t h e PC - a x i s . T y p i c a l f u n c t i o n s having t h i s p r o p e r t y are ^ > s i n h Xp cosh X . Consequently, we may expect t h a t , i n g e n e r a l , Q. w i l l become i n f i n i t e f o r l a r g e values o f 'x- ; o n l y i n s p e c i a l cases w i l l i t behave l i k e

K or I / x and approach zero. Now, since we are i n t e r e s t e d i n waves gen-erated near shore, we s h a l l c e r t a i n l y exclude those s o l u t i o n s f o r which

Qcc-o) ±;-^o ; t h i s gives us one boundary c o n d i t i o n ; t h e o t h e r i s g i v e n by the c o n d i t i o n t h a t t h e waves are so low t h a t t h e y do n o t break; t h i s gives us Q.ia) - f i n i t e .

We proceed t o study t h e p h y s i c a l wave e q u a t i o n i n order t o f i n d these s o l u t i o n s , and the c o n d i t i o n s f o r t h e i r e x i s t e n c e . The Eq, ( 1 3 ) , w i t h

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L e c t u r e , F a l l Semester, 1950-51 - 48 I

becomes

55.

which i s r a t h e r more g e n e r a l than Eq. ( 3 3 * ) . This i s an awkward equation t o study because (Y\ x ^ t u r n s out t o be a t r i n o m i a l . The f u r t h e r s u b s t i t u t i o n

4

- by.

f ^

'J 56.

reduces Eq, (55) t o t h e seemingly more e l a b o r a t e equation (known as Laguerre' e q u a t i o n )

57.

b u t , since

i s a b i n o m i a l , Eq. (57) i s a c t u a l l y much more t r a c t a b l e than Eq. ( 5 5 ) . We e a s i l y o b t a i n t h e one s o l u t i o n

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which i s a g e n e r a l i z a t i o n o f the B^.ssel f u n c t i o n , We a l s o see t h a t we a g a i n have t o do \rith an e x c e p t i o n a l case. However, since we are i n t e r e s t e d o n l y i n those s o l u t i o n s t h a t remain f i n i t e a t the water's edge, we need n o t t r o u b l e t o f i n d the analogue o f Neumann's f u n c t i o n . The s e r i e s o f Eq. (59) i w e l l s u i t e d t o determining the value o f -f- f o r small values o f -zb^x b'^t we w i s h t o f i n d i t s behavior f o r large values. To do t h i s , we note t h a t t h e

f i r s t few terms form an i n c r e a s i n g p r o g r e s s i o n when -2.\:iX. is l a r g e , b u t t h a t l a t e r terms d i m i n i s h , The l a r g e s t term i s t h a t f o r which t h e exponent o f x i s t h e i n t e g e r nearest , where

I f 2i:)y»0^ t h i s i s - 2 b v . Under these c o n d i t i o n s , we can approximate the most i m p o r t a n t terms i n the s e r i e s by s e t t i n g u --o: ' t h i s w i l l make an a p p r e c i a b l e e r r o r i n t h e e a r l y terras, b u t n o t i n t h e l a r g e s t , and r e s u l t s i n

2. b 11

- <2

A c t u a l l y , t h i s crude c a l c u l a t i o n has l e d us t o a q u i t e c o r r e c t r e s u l t : i n g e n e r a l X becomes i n f i n i t e f o r l a r g e , and increases so r a p i d l y t h a t

a l s o becomes i n f i n i t e l i k e e-y.p ( b X.)^

This shows t h a t , i n general, t h e r e i s no s o l u t i o n f o r < ^ t h a t remains : f i n i t e b o t h a t ?c - c and =00. But our argument i s s u b j e c t t o e x c e p t i o n : when oc i s a negative i n t e g e r ( s a y c c ^ - / ^ ) , the c o e f f i c i e n t o f iJ-h^l"^ i s zero when u> n. . The i n f i n i t e series reduces t o a p o l y n o m i a l — c a l l e d the Laguerre polynomial /L (zhy) • There are s e v e r a l n o t a t i o n s f o r these p o l y n o m i a l s : one i s IX

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Lecture, F a l l Semester, 1950-51 " 50

As can be v e r i f i e d by t r i a l , or proven i n o t h e r ways,*

The f i r s t few o f these polynomials are

L^C/O ( , - L , ^ x ) . - X + I

T h e . g r a p h s ^ o f the f u n c t i o n s ^ ( F i g s . 4, 5, 6)

v e r i f y the conclusions we drew concerning t h e f u n c t i o n P: t h e c^^ o s c i l l a t e f o r small values o f -pc , b u t approach zero a s j T n p t o t i c a l l y f o r l a r g e x •

The f u n c t i o n .

* Courant and H i l b e r t , Methoden der mathem. Physik., p. 77 ( B e r l i n , 1924). ^ B r i e f t a b l e s o f ^ C^) are t o be found i n P h y s i c a l Review 45 853

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L e c t u r e , F a l l Semester, 1950-51 - 51

represents s t a n d i n g waves. T h e i r amplitude vanishes on the "nodal l i n e s " determined by the r o o t s o f t h e f a c t o r s ^. ^ and s i n bcj. . Consequently, t h e r e are KX n o d a l l i n e s p a r a l l e l t o shore, v M l e those p e r p e n d i c u l a r t o shore have t h e spacing r r / b = - "Xo • . ' ••. •

The d i s t a n c e /^.^. may a l s o c o n v e n i e n t l y be c a l l e d the l o n g i t u d i n a l wave l e n g t h . One may a l s o have s o l u t i o n s of the form

These r e p r e s e n t r u n n i n g waves, o f l e n g t h A(. , propagated p a r a l l e l t o shore; t h e i r amplitude depends on -p , because o f t h e f a c t o r and vanishes on t h e ^ nodal l i n e s t h a t r u n p a r a l l e l t o shore. ^ '

I f we c o n s i d e r t h e e q u a t i o n o<^-i/t; i t may be w r i t t e n

or

ZTT R - (n^^ ^ ^ 60.

I n words, t h e s p r i n g - l e n g t h o f the rays i s an odd i n t e g r a l m u l t i p l e o f h a l f the l o n g i t u d i n a l wa-\'e-length.

The p e r i o d o f t h e waves i s whence

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L e c t u r e , F a l l Semester, 1950-51 - 5 U

r

and thus depends on > and the i n t e g e r ^ 5 fo]r_every 'Xu^ t h e r e are many p o s s i b l e p e r i o d s , o f which the longest i s 1 . Conversely, f o r every p e r i o d , t h e r e are many p o s s i b l e , o f which the s m a l l e s t i s CjX>T^/^'^» 9'. The Seiches o f a C i r o u l a r Lake

The slow o s c i l l a t i o n s o f a lake or other closed body o f water a r e c a l l e d seiches, These motions are so clow t h a t no turbulence or breaking r e s u l t s . Consequently, we expect them t o be r e f l e c t e d from the shore and t h e r e f o r e t o be represented by those s o l u t i o n s o f the wave equation t h a t are everyv/here

f i n i t e ,

This i s n o t the o n l y analogj-- t o t h e edge waves; since the water deepens toward the center o f t h e l a k e , t h e rays w i l l be curved back toward t h e shore, r e s u l t i n g i n arches, perhaps o f v a r i a b l e s p r i n g l e n g t h , depending on the

topography. I f the l a k e i s c i r c u l a r and i t s bottom has no i r r e g u l a r i t i e s , t h e ray arches w i l l be u n i f o r m i n l e n g t h . Perhaps one can immediately foresee t h a t t h e i r v e r t i c e s must c o i n c i d e w i t h those o f a r e g u l a r polygon; a t any r a t e , t h i s i s one c o n c l u s i o n we s h a l l reach.

Lamb* has t r e a t e d the case

k ^ - n . ' A ^ J x ^ c , ^ 62.

which represents a l a k e o f r a d i u s "R i n a p a r a b o l o i d a l b a s i n f i l l e d t o t h e h e i g h t above i t s lowest p o i n t . Using p o l a r coordinates

t h e p h y s i c a l wave equation (Eq. ( 1 3 ) ) becomes

A 63,

The s i m p l e s t s o l u t i o n s o f t h i s equation have t h e forms

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i-"^> L „ T'"^^ r ^ ' ^ i

64.

since \>(-'^)<i>) =^ ^(^^ ^ i - , i t f o l l o w s t h a t >vucan have any o f the values 0, 1 , 2 b u t no o t h e r s .

The Eq, ( 6 4 ) shows t h a t t h e nodal l i n e s i n c l u d e t h e 2m r a d i i

cL = rr/nt , ^ ' r y ^ t , ^.TA< . e t c . , i n t h e case o f t h e f a c t o r 4 , or, i n the case o f cos , t h e 2m r a d i i o b t a i n e d from these by r o t a t i o n

through the angle rr/xm.

Ve might a l s o have s o l u t i o n s o f t h e form < ^ ( ' ^ ) s i n n. ( 4 - 4 . ) ^os ^^t, where 4^ i s an a r b i t r a r y constant; f o r these t h e 2m nodal r a d i i are again e q u a l l y spaced, and one i s f ^ ^ . A s l i g h t l y d i f f e r e n t type o f s o l u t i o n i s

63a.

This has no nodal r a d i i , b u t represents waves t h a t t r a v e l around t h e l a k e w i t h the angular v e l o c i t y ad/vyi . The waves t h e r e f o r e complete one r e v o l u t i o n i n

K>1, p e r i o d s .

A l l o f these s o l u t i o n s may have o t h e r n o d a l l i n e s : t h e c i r c l e s determined by t h e equation ^ r ^ O r ö , and we now proceed t o d e t e m i u e t h i s f u n c t i o n . I t

must s a t i s f y the 'equation

where /c,, , d e f i n e d by

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i s t h e wave number c a l c u l a t e d f o r the deepest p o i n t o f the l a k e . As always, we f i r s t c a l c u l a t e

— r

This.can be somewhat s i m p l i f i e d : on d e f i n i n g ƒ> by t h e equation

k"/ R ^ I ^ .((° ^-OCj^-^wv 67.

. i t becomes

T h u s , - l e t t i n g •

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where

i s t h e hypergeometric s e r i e s , which w i l l be encountered again l a t e r and then s t u d i e d i n more d e t a i l .

I t might seem t h a t we c o u l d get another s o l u t i o n o f t h e form

CP c? A

-h

e , A.

₎

and t h i s would be t r u e i f yVKj were n o t an i n t e g e r . Being an i n t e g e r , we should be l e d t o conclude t h a t c^^ - C, T ... C ^ . ( ^ 6 , and end w i t h the same s o l u t i o n as g i v e n by Eq. ( 6 8 ) . The Eq. (65) i s thus again an excep-t i o n a l case; b u excep-t , excep-t h e second s o l u excep-t i o n would become i n f i n i excep-t e a excep-t 3 o i . e , , i n t h e c e n t e r o f t h e l a k e . Since we have a l r e a d y concluded t h a t such a s o l u -t i o n i s no-t o f i n -t e r e s -t we a r e spared -the -t r o u b l e o f f i n d i n g i -t .

We must, however, consider the behavior o f t h e s o l u t i o n a t t h e water's edge, a = , The experience o f the previous s e c t i o n leads us t o expect d i f f i c u l t i e s a t t h a t p o i n t , And d i f f i c u l t i e s immediately appear, f o r t h e s e r i e s o f Eq. ( 6 8 ) , i n g e n e r a l , diverges when /\? / il"- >y I . There i s thus no easy way t o c a l c u l a t e c f o r /i, ^ > b u t t h e r e i s t h e j u s t i f i e d sus-p i c i o n t h a t i t w i l l become i n f i n i t e t h e r e .

However, t h e r e are a g a i n s p e c i a l cases i n which t h e s e r i e s \-(<^,^^, ^ i s n o t i n f i n i t e b u t represents a f i n i t e p o l y n o m i a l . These polyno9iial cases occur whenever e i t h e r o>- o r b i s a n e g a t i v e i n t e g e r , say — • Then t h e hypergeometric f u n c t i o n becomes a polynomial degree /a i n . I n these

cases, c e r t a i n l y remains f i n i t e f o r | .

V/e consequently conclude t h a t t h e o n l y s o l u t i o n s t h a t remain f i n i t e every-where are obtained when

2 - ^

in.

n

r

o , I

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' - (Zh-t Zm-h l)iZh + O-I

- i^l •>r /»+/)+ 2 _m. 69. I n these cases'**-70.

%, = ^^^^ - U IR)""ri-n, n+i^^l,jr,^ I, /

We consider a few o f these s o l u t i o n s ! l e t m ^ n ^ <^) " Thus t h i s s o l u t i o n i s a s t a t i c one, and represents t h e case o f a permanent

change i n l e v e l o f t h e l a k e . I f />i ^ n = I , = 8 ^ //^"^ and ^1-2 /„"^//C^ • Consequently, i n t h i s case t h e r e are no nodal

r a d i i , b u t a nodal c i r c l e a t = / ^ / / ^ . I f = /? r Z, u>^ ^ =

and t h e r e are two nodal c i r c l e s a t ^ - i I ± I / - / s ) ^7 ^ . I n g e n e r a l , the , f^) modes have n nodal c i r c l e s , and the normal

f r e q u e n c i e s increase w i t h b o t h ^ and h , although not l i n e a r l y .

This i s s c a r c e l y an exhaustive treatment o f seiches, b u t w i l l serve t o i n d i c a t e t h e i r r e l a t i o n t o edge waves, and t o suggest the s o l u t i o n o f o t h e r problems.

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Lecture, F a l l Semester, 1950-51 55A

10. Oeneral Survey o f the i n f l u e n c e o f Topography on Surface Waves.

Several examples o f s u r f a c e waves on water o f v a r i a b l e depth have now been s t u d i e d . The v a r i e t y o f s o l u t i o n s o f t h e equations i s apparent, and i t becomes d e s i r a b l e t o o b t a i n some i n s i g h t i n t o t h e c o n d i t i o n s under which each phenomenon occurs. This i s best accomplished by a device t h a t makes use of the analogy bet^^een the rays and the m o t i o n o f a p a r t i c l e . I t w i l l

f i r s t be described, and then a p r o o f o f the analogy w i l l be presented. The wave number, , i s a f u n c t i o n o f t h e water depth; i n the cases we are s t u d y i n g

Consequently, the contours, o f a constant depth on a c h a r t o f the sea f l o o r are also the contours o f constant K . We can a l s o c o n s t r u c t a r e l i e f map o f — or imagine i t t o have been c o n s t r u c t e d — so t h a t the equation of i t s surface i s ( t o some s c a l e )

71.

This may be c a l l e d the r e c i p r o c a l r e l i e f map o f the bottom. The plane = O w i l l be c a l l e d t h e datum plane o f the map; a l l o f i t s p o i n t s are below the datura plane. A submarine mcmd or r i d g e w i l l appear as a depression on the r e c i p r o c a l r e l i e f , and a submarine canyon w i l l appear as a r i d g e . The water's edge i s a steep c l i f f t h a t drops o f f t o an i n f i n i t e distance beneath the datimi plane.

I f a b a l l i s allowed t o r o l l w i t h o u t f r i c t i o n on the r e c i p r o c a l r e l i e f map, i t w i l l t r a c e out one o f the p o s s i b l e r a y s , p r o v i d e d t h a t i t has a c e r t a i n

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L e c t u r e , F a l l Semester, 1950-51 - 56,

t o t a l energy. This energy can be imparted t o i t by s t a r t i n g i t from r e s t a t t h e datum l e v e l and a l l o w i n g i t t o r o l l down a chute u n t i l i t reaches t h e surface o f the map.

The p r o o f o f t h i s i s simple: the d i f f e r e n t i a l equations o f the rays are g i v e n on p. 23; t h e parameter T was n o t s p e c i f i c a l l y determined. I f i t i s determined so t h a t

The equations on p. 23 reduce t o

73.

The Eq. (73) are t h e equations of motion o f a sphere r o l l i n g on the surface , w h i l e Eq. (72) s p e c i f i e s t h a t t h e t o t a l energy i s zero, when measured from t h e datum l e v e l .

I t should be noted, however, t h a t t h e t i m e r e q u i r e d f o r t h e sphere t o t r a c e o u t a g i v e n segment o f the r a y i s not equal t o t h e time r e q u i r e d f o r the wave-front t o t r a v e l t h e same d i s t a n c e . I n f a c t , t h e sphere's v e l o c i t y , L L w i l l be i n v e r s e l y p r o p o r t i o n a l t o the wave v e l o c i t y , c •* o j / k .

Since i t i s easy t o v i s u a l i z e t h e r e c i p r o c a l topography, and t h e r o l l i n g o f a sphere, t h i s analogy a f f o r d s an easy and r a p i d method o f a n a l y s i n g wave problems,