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Edge waves in the presence of an irregular coastline Fuller, John David 1975

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EDGE WAV33 IN THE PRESENCE OF AN IRREGULAR COASTLIKE  by  JOHN DAVID FULLER B.Sc, Queen's U n i v e r s i t y , K i n g s t o n , 1973  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE  i n the Department o f Mathematics and t h e I n s t i t u t e o f A p p l i e d Mathematics and S t a t i s t i c s  We accept t h i s t h e s i s as conforming t o the r e q u i r e ^ standard  THE UNIVERSITY OF BRITISH COLUMBIA October  , 1975  In p r e s e n t i n g t h i s  thesis  an advanced degree at the L i b r a r y I  further  for  of  freely  available  this  representatives. thesis for  Department The  It  financial  of  M/iTUEA /r y  requirements  for  this  C ^  e  ("A T-  7  » ,  1*7  that  thesis or  publication  g a i n s h a l l not be allowed without my  T!  for  r e f e r e n c e and study.  i s understood that c o p y i n g or  U n i v e r s i t y o f B r i t i s h Columbia  2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5  t  the  t h a t p e r m i s s i o n for e x t e n s i v e c o p y i n g o f  written permission.  a  of  s c h o l a r l y purposes may be granted by the Head of my Department  by h i s  D  fulfilment  the U n i v e r s i t y of B r i t i s h C o l u m b i a , I agree  s h a l l make it  agree  in p a r t i a l  ABSTRACT  Small i r r e g u l a r i t i e s i n an otherwise s t r a i g h t coast produce c e r t a i n e f f e c t s on l o n g waves on a c o n t i n e n t a l s h e l f .  S t u d i e d here are the gen-  e r a t i o n o f trapped edge waves when a vjrve f r o n t h e deep coean reaches the i r r e g u l a r c o a s t , and. a l t e r a t i o n s i n the propagation c h a r a c t e r i s t i c s o f trapped edge waves, due to the i r r e g u l a r i t i e s . The c o n t i n e n t a l s h e l f i s modelled by a s i n g l e , f l a t - s t e p model, and the i r r e g u l a r i t i e s i n the coast are represented as a s t a t i o n a r y random f u n c t i o n o f d i s t a n c e along the c o a s t , w i t h zero mean.  The wave equations  are thus s t o c h a s t i c d i f f e r e n t i a l equations, w i t h the randomness i n t r o d u c e d through the boundary c o n d i t i o n at the coast. C a l c u l a t i o n s are made f o r the power f l u x i n t o trapped edge waves and a continuous spectrum o f l e a k y modes, both generated by the s c a t t e r i n g o f an i n c i d e n t wave.  Numerical r e s u l t s f o r a s e c t i o n o f the n o r t h e a s t Jap-  anese coast show t h a t t h e r e i s l e s s power t r a n s f e r r e d t o t h e forward t r a v e l l i n g trapped wave than the backward, and l e s s power t o the s c a t t e r e d l e a k y modes than t o e i t h e r the forwai-d o r backward trapped modes. Other c a l c u l a t i o n s show t h a t t h e r e i s a t t e n u a t i o n o f a trapped edge wave, due t o s c a t t e r i n g , t h e r e i s a " t i l t i n g " o f the wave towards the c o a s t , and i n the case o f the same Japanese c o a s t , the longshore components o f the phase and grou~ v e l o c i t i u o nre slight!/"- l e s s than i n the ca.e o f a s t r a i g h t coast. The r / s u i t s are v a l i d f o r wave p e r i o d s -such s h o r t e r than t,he p e r i o d a s s o c i a t e d w i t h the C o r i o l i s parameter  f , and f o r wavelengths much g r e a t e r  than the average s i z e o f the c o a s t a l i r r e g u l a r i t i e s .  - iii  -  TABLE OF CONTENTS  1.  INTRODUCTION..,  i  2.  THE SHALLOW WATER WAVE EQUATIONS  6  3.  EDGE WAVES ON A FLAT CONTINENTAL SHELF WITH A STRAIGHT COAST  4.  SCATTERING BY AN IRREGULAR COASTLINE OF A WAVE INCIDENT FROM THE DEEP OCEAN —  5;.  DETERMINATION OF THE REFLECTION COEFFICIENT  19  SCATTERING BY AN IRREGULAR COASTLINE OF A WAVE INCIDENT FROM THE DEEP OCEAN —  POWER FLUXES INTO DIFFERENT MODES OF  SCATTERED WAVES  6.  11  37  INFLUENCE OF COASTAL IRREGULARITIES ON TRAPPED EDGE WAVES  —  ALTERED DISPERSION RELATION  7.  CONSEQUENCES OF THE ALTERED DISPERSION RELATION  8.  CONCLUDING REMARKS  References  46  56'  w.6'2  63  - iv -  FIGURES  2.1  O r i e n t a t i o n o f axes f o r shallow water equations  6  2.2  Conservation o f mass  7  2.3  S i n g l e step model o f ocean bottom  9  3.1  Ray diagram f o r c>V  15  3.2  D i s p e r s i o n r e l a t i o n f o r l<c<y, *"=10  17  4.1  Branch cut c o n f i g u r a t i o n  27  4.2  Contours o f i n t e g r a t i o n  4.3  Indentations o f path o f i n t e g r a t i o n  4.4  Indentations o f path o f i n t e g r a t i o n  5.1  Power f l u x e s on Japanese coast, f o r <^ =0.413 - p e r i o d = 2 h r . . . . 4 2  5.2  Power f l u x e s on Japanese coast, (J =0.326 - p e r i o d = 1 hr  43  5.3  Power f l u x e s on Japanese coast, co =1.652 - p e r i o d = -J h r  43  5.4  Region o f Japanese coast f o r c a l c u l a t i o n s  44  5.5  Power spectrum f o r Japanese coast o f Figure 5.4  44  7.1  A l t e r e d d i s p e r s i o n r e l a t i o n , lowest mode f o r Japanese coast  60  7.2  A l t e r e d d i s p e r s i o n r e l a t i o n , second mode, f o r Japanese coast  7.3  e - f o l d decay length f o r Japanese coast  ..30 :.-  ...30 30  ....61 6l  ACKNOWLEDGEMENTS  I am indebted to Dr. L.A. Mysak f o r suggesting the t o p i c o f t h i s t h e s i s , and f o r h i s very h e l p f u l c o n s u l t a t i o n and encouragement during the course o f my work.  My thanks a l s o go to Dr. K. K a j i u r a o f the Earthquake  Research I n s t i t u t e o f the U n i v e r s i t y o f Tokyo, who provided me with a great deal o f m a t e r i a l on tsunamis and edge waves on the north east coast of Japan. And thanks go t o my wife f o r w r i t i n g i n the equations by hand. This work was done while I was supported on a N a t i o n a l Research C o u n c i l o f Canada Science  '67 s c h o l a r s h i p .  - 1 -  1.  INTRODUCTION The behaviour o f ocean waves on the c o n t i n e n t a l s h e l f has received  a great d e a l o f a t t e n t i o n during the l a s t two decades. w e l l deserved f o r p r a c t i c a l reasons.  This a t t e n t i o n i s  An improved understanding of the be-  haviour o f tsunamis and other l a r g e waves on the c o n t i n e n t a l s h e l f could l e a d to a b e t t e r tsunami e a r l y warning system or improved harbour design to l e s s e n the d e s t r u c t i v e e f f e c t s o f l a r g e waves.  Researchers at the  Earthquake Research I n s t i t u t e i n Tokyo have devoted much e f f o r t to the study o f tsunamis on the c o n t i n e n t a l s h e l f of Japan —  see, f o r example,  H a t o r i and Takahasi (1964),- H a t o r i (1965a, 1965b, 1967), and Aida (1967,1969). The p e r i o d of a tsunami i s o f the order o f a thousand seconds, a c c o r ding to Bascom (1964, p. 104), which i s v e r i f i e d by f i g u r e s presented by H a t o r i and Takahasi (1964) f o r a p a r t i c u l a r jtsilnami. earth's r o t a t i o n may  So e f f e c t s of the  s a f e l y be ignored i n m i d - l a t i t u d e r e g i o n s , since the  time scale associated with the C o r i o l i s parameter f i s of the order of t e n thousand seconds.  Thus, i n t h i s work, the C o r i o l i s parameter f i s taken  to be zero. When the s i n g l e step model o f a c o n t i n e n t a l s h e l f i s adopted, the a n a l y s i s of plane wave behaviour i s s t r a i g h t f o r w a r d .  Plane waves i n c i d e n t  upon the s h e l f and o r i g i n a t i n g from the deep ocean are r e f r a c t e d at the s h e l f edge i n accordance with S n e l l ' s Law;  p e r f e c t r e f l e c t i o n takes p l a c e  at the coast; and the wave i s r e f r a c t e d again at the s h e l f edge whereupon i t enters the deep ocean.  Because o f p e r f e c t r e f l e c t i o n , the energy f l u x  onto the s h e l f region i s equal to the energy f l u x to the deep ocean from the s h e l f r e g i o n .  Waves o r i g i n a t i n g on the s h e l f may,  cumstances, be trapped on the s h e l f . the deep ocean.  under c e r t a i n  cir-  That i s , there i s no energy f l u x to  These waves, known as "edge waves", obey a c e r t a i n d i s p e r -  s i o n r e l a t i o n which allows only a f i n i t e number o f modes f o r waves of a given frequency.  In t h i s work, there i s an a n a l y s i s of the s i n g l e - s t e p  s t r a i g h t coast model, based on Buchwald and de Szoeke (1973). But when small i r r e g u l a r i t i e s o f an otherwise introduced i n t o the model —  s t r a i g h t coast are  representing i n l e t s , promontories, bays, e t c .  there i s no longer p e r f e c t r e f l e c t i o n at the coast, i n the case of waves o r i g i n a t i n g i n the deep ocean.  As w e l l as the p a r t i a l l y r e f l e c t e d  wave,  some o f the i n c i d e n t energy f l u x i s t r a n s f e r r e d i n t o d i f f u s e l y s c a t t e r e d waves which escape to the deep ocean and i n t o edge waves.  When edge  waves t r a v e l along such a c o a s t l i n e , small i r r e g u l a r i t i e s i n the coast b r i n g about a t t e n u a t i o n of the edge waves and changed phase and group velocities.  An a n a l y s i s of both cases, with numerical  examples, i s p r e -  sented i n t h i s work. The h i s t o r i c a l r o o t s o f t h i s problem have two branches. began with Stokes on a constant  1  One  branch  o r i g i n a l paper i n I846 on fundamental mode edge waves  slope bottom p r o f i l e .  Stokes was  the f i r s t to  recognize  the existence o f waves which t r a v e l p a r a l l e l to the coast and whose energy i s trapped by ocean f l o o r topography i n a r e g i o n near the coast.  After  more that a century, U r s e l l (1952) extended Stokes' work to i n c l u d e the whole spectrum o f p o s s i b l e modes.  There have been f u r t h e r g e n e r a l i z a t i o n s  since U r s e l l * s paper. Snodgrass et a l . (1962) introduced the s i n g l e f l a t step model of the c o n t i n e n t a l s h e l f i n order to i n v e s t i g a t e n o n - r o t a t i o n a l (f=0) edge waves on the C a l i f o r n i a borderland.  I t i s t h i s model which i s used by Buchwald  and de Szoeke (1973), and i n the present work with i r r e g u l a r i t i e s i n the otherwise  straight  coast.  A review a r t i c l e by' LeBlond  and Mysak (1975) on wave motions topo-  g r a p h i c a l l y trapped by depth gradients o f many d i f f e r e n t kinds provides a good overview of the theory o f edge waves. The second branch o f the h i s t o r i c a l roots o f the present problem has a more recent beginning. p e r t u r b a t i o n expansion  Pinsent (1972) employed a second order ordinary  to two r e l a t e d wave problems i n a r o t a t i n g sea  (fjk))  o f a n e a r l y uniform depth bounded by a c o a s t l i n e which i s n e a r l y s t r a i g h t . Pinsent examined the generation o f K e l v i n waves due to the s c a t t e r i n g o f a plane wave i n c i d e n t upon the i r r e g u l a r coast, and the a t t e n u a t i o n o f a K e l v i n wave due to s c a t t e r i n g from the i r r e g u l a r coast.  A K e l v i n wave  i s one which t r a v e l s p a r a l l e l to the coast and whose energy i s trapped against the coast by the r o t a t i o n o f the earth. However, Pinsent's r e s u l t s could not be a p p l i e d t o s i t u a t i o n s i n v o l v i n g extensive c o a s t l i n e s because o f the occurrence o f s e c u l a r terms. Howe and Mysak (1973) g e n e r a l i z e d Pinsent's (1972) work on the s c a t t e r i n g o f an i n c i d e n t plane wave i n order t o deal with extensive c o a s t l i n e s . T h e i r model was a f l a t bottom ocean with an e s s e n t i a l l y s t r a i g h t coast, except f o r small i r r e g u a l r i t i e s .  The i r r e g u l a r i t i e s were considered to  be a random f u n c t i o n o f distance along the coast. governing the wave behaviour  The d i f f e r e n t i a l  thus were s t o c h a s t i c d i f f e r e n t i a l  because o f the s t o c h a s t i c boundary c o n d i t i o n at the coast.  equation  equations,  The method o f  s o l u t i o n was based, on the techniques o f wave propagation i n random media as discussed,by Howe (1971)._  I t was found,: a s i n Pinsent (1972), that a ;  .Kelvin-,-wave i s " generated,_as_ well.as a continuous  spectrum o f wave n o i s e .  Mysak-and Tang (1974) used the same model as Howe and Mysak (1973) and an operator expansion technique t o g e n e r a l i z e Pinsent's (1972) work on the e f f e c t s o f c o a s t a l i r r e g u l a r i t i e s on a K e l v i n wave.  The e f f e c t s o f  -  4 -  the c o a s t a l i r r e g u l a r i t i e s on the wave speed, energy f l u x and  amplitude  of a coherent K e l v i n wave were determined. The present work examines the generation and propagation of edge waves on s i n g l e step c o n t i n e n t a l s h e l f with a coast which i s e s s e n t i a l l y s t r a i g h t except f o r small i r r e g u l a r i t i e s represented as a random f u n c t i o n of distance along the coast.  This work i s , to the author's knowledge,  the f i r s t attempt to introduce the e f f e c t of c o a s t a l i r r e g u l a r i t i e s i n t o the study o f the generation and propagation of edge waves.  I t i s a l s o the  f i r s t attempt to introduce the e f f e c t s of ocean bottom topography i n t o the study of ocean waves i n the presence of a random boundary. I t i s worth emphasizing that throughout t h i s work, r o t a t i o n a l e f f e c t s are disregarded. 'Thus, n e i t h e r K e l v i n waves nor c o n t i n e n t a l s h e l f waves appear i n the a n a l y s i s .  (A c o n t i n e n t a l s h e l f wave i s a wave whose energy  i s trapped on the s h e l f and which r e q u i r e s the presence o f both r o t a t i o n and a s h e l f ) .  A l s o , i t i s assumed t h a t there are no e f f e c t s due to d e n s i t y  s t r a t i f i c a t i o n of the ocean. I'm s e c t i o n 2, the shallow water equations are used to d e r i v e the b a s i c d i f f e r e n t i a l equation f o r t h i s t h e s i s . trapped and leaky modes —  I n s e c t i o n 3> edge waves —  both  on a f l a t c o n t i n e n t a l s h e l f with a s t r a i g h t  coast are examined. In s e c t i o n 4» an expression i s obtained f o r the r e f l e c t i o n c o e f f i c i e n t of the coast.  S e c t i o n 5 deals with the t r a n s f e r of energy from.the i n c i d e n t  wave to the scattered f i e l d ; an expression i s d e r i v e d f o r the power f l u x from the i n c i d e n t wave i n t o the trapped edge waves and i n t o a spectrum of long wave r a d i a t i o n to the deep ocean.  continuous  These q u a n t i t i e s are  c a l c u l a t e d f o r the northeast coast o f Japan. S e c t i o n 6 deals with the a l t e r a t i o n i n the d i s p e r s i o n r e l a t i o n o f  trapped edge waves, brought about by c o a s t a l i r r e g u l a r i t i e s .  In s e c t i o n  expressions are obtained f o r the a l t e r e d phase and group speeds, f o r the e - f o l d decay length of the coherent wave f i e l d , and towards the coast i n the deep ocean region. o f Japan i s used as an example.  f o r the wave " t i l t "  Again, the northeast  coast  However, the e f f e c t s o f c o a s t a l i r r e g -  u l a r i t i e s are i n s i g n i f i c a n t i n t h i s example, because of the l a r g e r a t i o of deep ocean depth to s h e l f depth. C e n t r a l t o the c a l c u l a t i o n s i n sections 4, integral.  5,  6, and  7 i s a single  A l l c a l c u l a t i o n s are v a l i d only f o r waves o f wavelength much  greater than the average s i z e of the  coastal i r r e g u l a r i t i e s .  This  res-  t r i c t i o n i s found to apply to the c a l c u l a t i o n s of Pinsent (1972), Howe and Mysak (1973), and Mysak and Tang (1974) as w e l l .  7>  -  2.  6 -  THE SHALLOW WATER EQUATIONS Regions o f the oceans i n the m i d l a t i t u d e s are considered here, so  the p e r i o d corresponding to the C o r i o l i s parameter f i s about 18 hours. The f i r s t r e s t r i c t i o n t h a t i s made here i s to consider only those water waves whose p e r i o d i s much l e s s than IS hours —  or, i n terms o f the  angular frequency go' o f the waves, i t i s assumed t h a t co'»f assumption o f motion.  .  This  amounts to dropping the C o r i o l i s f o r c e term from the equations S t r a t i f i c a t i o n i s also neglected.  Assuming t h a t v i s c o s i t y i s  n e g l i g i b l e , t h i s leaves only g r a v i t y and the pressure f o r c e . With the co-ordinate axes o r i e n t e d as shown i n f i g u r e  2.1,  k/ccfer J«ff(jce (mean sar-face a{ z=o)  Figure 2.1  :  O r i e n t a t i o n of axes f o r shallow water equations.'  the well-known "shallow water" equations r e l a t i n g the x averaged  1  and y'  depth-  components o f v e l o c i t y u.', v' and the pressure p' are:  o^u  /cH' =  ' a W « n '  l/f  (^p'/^Ti')  =  .  (2.1a)  ~ ll? Up'/^y )  (2.1b)  - 1/P  (2.1c)  1  <  O  -  U  p'/^z') - g  I t i s assumed i n (2.1a,b,c) that the wave amplitude i s much l e s s than the  -  7 -  wavelength, and that the water depth i s much l e s s than the wavelength. Equation (2.1c) may  be immediately  p' = p: +  i n t e g r a t e d to give the pressure:  9P(d>'-2')  (2.2)  where <f> i s the e l e v a t i o n of the water's surface above the mean surface z'=0,  and p ' i s the atmospheric 0  pressure at the surface z'=4>'.  Therefore,  (2.3a) (2.3b)  provided p ' i s constant. 0  Now  consider the incompressible flow o f water through a volume e l e -  ment f i x e d i n space, as p i c t u r e d i n f i g u r e 2.2  .  (Volume. ou{)  s  .  Volume. o( «/«cfer) i « , in f fme oii. 1 (Volume-  (Volume,  oat)  iri)/^  Figure 2.2  Conservation of mass.  -  8 -  I t i s c l e a r from f i g u r e 2.2 t h a t the i n c r e a s e i n f l u i d volume i n the volume element i n time dt i s  - < H ( h - v 4 > ' ) u ' ) / ^ r ~ <^ (( h + 4>')JU- ')/<* ^ '  (2.4)  But the i n c r e a s e i n f l u i d volume can only be taken up by an increase i n the water l e v e l of the volume element.  or,  Therefore,  (2.5)  <H'/<)t' + cK(h+4') u ' ) / ^ ' + d a h - K p V ' ) / c i y = O ,  Throughout the present work, h i s assumed to be constant. assumed t h a t <i>'«K  If i t is  , then (2.5) may be w r i t t e n :  &V/dV + hlbu'faf  + Sis'/wl  --O  (2.6)  D i f f e r e n t i a t i n g (2.6) with respect to t ' g i v e s :  ^f/^t'^h^u'A-x^t'  + ^V/^y'dt'V °  (2.7)  But d i f f e r e n t i a t i o n o f (2.3a) by x« and (2.3b) by y» y i e l d  ^ u ' / ^ t ^ - g ^ ^ / ^ X ^ S W ^ f :  ~9^<*>'/c^'  2  F i n a l l y , s u b s t i t u t i n g (2.8a,b) i n t o (2.7) gives the d i f f e r e n t i a l  ( 2 # 8 a  )  (2.8b)  equation  ^<J>'AH'*- - (SK)  v' 4>'-o a  (2.9)  Throughout the present work, a l l f l u i d motions are assumed to have a given f i x e d angular frequency u>'.  So assuming a time f a c t o r o f  exp(-ico't') i n d>', equation (2.9) becomes  V' 4>' £  +  (to*/3 K ) <P' =0  (2.10)  Equation (2.10) i s the b a s i c d i f f e r e n t i a l equation which w i l l occur i n many s i t u a t i o n s i n t h i s work. Note t h a t when a time f a c t o r o f exp(-icj't') i s assumed, (2.3a,b) give expressions f o r u  and v  1  1  i n terms o f f :  U' = - (?9/CJ'U<^7^-X') -  -  (  i  .  3  /  (2.11a) u  >  '  (  2  .  1  1  b  )  As mentioned i n the i n t r o d u c t i o n , the model o f the ocean bottom used i n t h i s t h e s i s i s a s i n g l e step topography, i l l u s t r a t e d i n f i g u r e 2.3  Figure 2.3 ' '* S i n g l e step model o f ocean bottom.  - 10 -  Both h  2  and h, are assumed to be small enough so that the "shallow  water" approximation" holds f o r the waves considered here.  However, the  reader should keep i n mind that i n the examples discussed l a t e r , these "shallow" regions have depths o f 200 metres and 2,000 metres I The v a r i a b l e s may  now be non-dimensionalized.  z o n t a l plane w i l l be d i v i d e d by the s h e l f width W,  Lengths i n the h o r i and time w i l l be  nbn-  dimensionalized with r e s p e c t to W/s/gh, , the time o f t r a v e l o f a long wave across the s h e l f .  The reader i s reminded that the phase and group  speeds o f a long wave on the s h e l f — equal to /gh,'.  a "shallow water" wave —  Lengths i n the v e r t i c a l d i r e c t i o n —  lacements o f the various waves — the shallow depth h, .  are both  the v e r t i c a l d i s p -  are non-dimensionalized with respect t o  Dimensional v a r i a b l e s are denoted by a prime  (*)  throughout t h i s t h e s i s , and non-dimensionalized v a r i a b l e s are unprimed. In non-dimensional  form, the equations f o r shallow water waves i n  the s h e l f and deep sea regions are, from (2.10):  V ^  + cu  V -^  +(o;y^)4>i.= 0  1  2  where  5C =h /h, > 1. 2  a  z  4>, - O  0<%<L[ ^ _ o o < v j < + ^  (2.12a)  1 <: X < °  (2>12b)  +a  }  -°o<y<  - 11 -  3.  WAVES ON A CONTINENTAL SHELF, WITH A STRAIGHT COAST The mathematical a n a l y s i s of waves i n the presence o f the  irregular  coast i s best understood i f compared to the a n a l y s i s of the s t r a i g h t problem.  The l a t t e r i s now  presented,  coast  based on sections I I and I I I o f the  paper by Buchwald and de Szoeke (1973). The d i f f e r e n t i a l equations have been given above i n (2.12a,b). boundary c o n d i t i o n at the s t r a i g h t coast x=0 penetrate the c o a s t l i n e . to the coast. f o r x=0.  This may  i s t h a t the water does not  be expressed  as zero v e l o c i t y normal  That i s , the x-component of the v e l o c i t y , u, equals  But since u'^lig/ai^Zpi/&x'  The  , the boundary c o n d i t i o n at  zero x=0,  i n non-dimensional form i s :  c^4> y^-X -  o  a t x=0  There are two matching conditions at the s h e l f edge x=l.  (3.1)  F i r s t of  a l l , there must be c o n t i n u i t y i n water l e v e l between the deep and regions —  shallow  that i s  = 4>z  a t x=l  (3.2)  Secondly, there must be c o n t i n u i t y of mass t r a n s p o r t across the s h e l f edge.  This i s expressed  by  h,u,' = h i u a  at  x'=W.  Therefore,  at  x=l  (3.3)  -  12  -  F i n a l l y , wave amplitudes must be f i n i t e everywhere, i n c l u d i n g a t x=f oO. Following Buchwald and de Szoeke, l e t  d)j = A i f - x ) e x p ( i K b - i t ^ t )  j=l,2  •  (3.4)  This represents waves t r a v e l l i n g along the s h e l f , p a r a l l e l to the coast. S u b s t i t u t i n g t h i s form i n t o the d i f f e r e n t i a l equations g i v e s :  ^A./^x  1  + (to*-k*)A.  £ f \ J ^  0<x<l  -- o  -k^Aa = o  (3.5a) (3.5b)  x>l  There are three cases of i n t e r e s t , corresponding to three ranges o f the phase speed c= /k: w  Case 1:  c>y  —  ? c > a , s > c > i , l>c .  leaky modes  In.this, case, as i n the others, j c o n d i t i o n (3.1)  ^M^%  Now,  **>1,  and  c=w/k>Y, so  -o  at x=0 becomes:  a t x=0  co*> *>>V*>k*.  Therefore, o^*-k*>0, and  so  A . --ZC cos(m.-x)  where  m, = 7cj*-k*  , and C i s an undetermined constant.  the s o l u t i o n i n r e g i o n 2,  A*M=  Since oYa*  the deep ocean, i s o f the form  2Acos(rru(-X-0 £) +  -k>0,  - 13 -  where  m= £  -k*  and A and £ are undetermined constants.  The matching:  conditions a r e :  and  ^hJ^%  M./^-x =  a t x=l.  The f i r s t matching c o n d i t i o n y i e l d s :  ZC  cos  m, -- 2 A c o s £  while the second gives:  ZC m, Sin m,  = & A rr\ * sin £  D i v i d i n g the second by the f i r s t  a  a  gives:  hi, "tan nn, =  m,fan £  So £ i s now determined:  £ =  1  7  A l s o , C may be expressed i n terms o f A:  r  - A coa ? - *«.  ~ A  c  s£  So f i n a l l y , the wave f i e l d may be w r i t t e n :  (3.6)  -  14  -  ct>. ^,y,t) = 2 A ^ s i . cos^m.-x) exp fik'i-iwt) 4>»(%,%i.) -•  cos (m»(»-i) + £}exp(-iM-ito*}  0<x<l  (3.7a)  X>1  (3.7b)  The f o l l o w i n g i n t e r p r e t a t i o n of the s o l u t i o n s may Put  m,=o>cose, , mj= /j c o s e w  o f e, and  ©  a  a  are c o n s i s t e n t with  l a t t e r i m p l i e s that k = ( % ) s i n e . <  i  ,, k="sin e, . m, = «*-k z  1t  be made.'  Note that these d e f i n i t i o n s  and with m* =="VV-k%- -.The a  Rewriting the cosine f a c t o r s i n the  s o l u t i o n (3.7a,b) g i v e s :  4**  '  AIH^i>, l"exp^«(-x cose.* * siAe,-t))+exp(i<o(--xcose,*y si'ne.-t^ O<X^l  (3.8a)  1  t  (|>a = A £ex p( ^ f r cos e ••S si * e« - *t -»£V)+ex p(^C-X cos » i + y sIn ©i - * t -£)$ , x >1  (3.8b)  t  I t i s c l e a r t h a t the s o l u t i o n i s a s u p e r p o s i t i o n of two waves  —  one t r a v e l l i n g towards the coast and one away from i t as represented i n the ray diagram i n f i g u r e 3.1  •  The wave's phase speed (and group speed)  i n the deep ocean i s X , i n non-dimensional this i s  K • v/gh! -Jh^/h,  deep ocean).  • v/gh. = J^u  u n i t s ( i n dimensional u n i t s ,  , which i s the long wave speed i n the  The speed i n the shallow r e g i o n on the s h e l f i s 1 ( i . e . /gh~,  i n dimensional u n i t s ) .  The angle o f incidence e  e  i s r e l a t e d to the angle  of r e f r a c t i o n e> by cosin&.==k=( /»)s±ne — i . e . Xsine,=sine* , vrhich i s u  a  S n e l l ' s Law,  since s i n e./sin e»=( shallow speed)/(deep  speed)=Vgh7/,/gh"*=l/tf .  The q u a n t i t y £ i s a phase l a g , due to the passage of the wave back and f o r t h across the s h e l f . To an observer moving along the coast with v e l o c i t y c= /k  , the wave  u  pattern i s frozen.  So i t i s h e l p f u l to think o f the sum of the  and outgoing waves as a standing Wave normal to the coast — and cos(m (x-l)+.e) a  factors —  incoming  the cos(m, x)  and a t r a v e l l i n g wave along the coast, t r a v -  - 15 -  e l l i n g at speed c= /k  , the exp(iky-it)t) factor.  u  Because the wave energy i s not confined to the shelf, waves of this type are called leaky modes.  Figure 3.1 'Ray diagram for  Case 2 —  l^c** —  on.  trapped modes  In t h i s the second case of interest, oo > k * C S n e l l ' s Law does not apa  p l y , since  tfsine,=sin0  t  implies that sine.^l/a —  but k <J/x a y  z  implies  that sin9,>l/& . This case represents a wave originating on the shelf, with t o t a l internal r e f l e c t i o n occurring at the edge of the shelf. I f we put m, =o -k as before, and J>*=k - cuf/a* , then the boundary a  l  8  Z  condition  at x=0, the continuity condition at x=l, and the finiteness condition at x=+ oo imply that  4>, A cos ( m,*) exp (i k"J - icot) :  4> = A cos m, exp(-ICK-D + ikS - i col) a  0<X<1  (3.9a)  x>l  (3.9b)  - 16 -  Again, there i s a standing wave i n the x - d i r e c t i o n , wave i n the y - d i r e c t i o n . deep ocean r e g i o n . other matching  There i s exponential decay o f the wave i n t o the  The waves are edge waves, trapped on the s h e l f .  The  condition,  O^./^TC  implies  and a t r a v e l l i n g  = 3*  ^<t>*A*  ,  .  x=l  that  m, iar> m, = S*JL  With the d e f i n i t i o n s o f m, and 1 , dispersion  r e l a t i o n o)=o(k).  (3.10)  (3.10) represents an i m p l i c i t  So i f the angular frequency, to , i s s p e c i f i e d ,  then the longshore wavenumber, k , the longshore phase speed c^^/k the longshore group v e l o c i t y Cg=do/dk the tangent function,  a l l be determined.  Because o f  there i s an i n f i n i t e number o f d i f f e r e n t modes.  The graph o f the d i s p e r s i o n The c a l c u l a t i o n s  may  , and  r e l a t i o n f o r X =10 2  i s presented i n f i g u r e 3.2  and p l o t t i n g were done with the a i d of a computer.  Note  that f o r a given co , there i s a f i n i t e number of allowable wave numbers, k. In the l e a k y mode case, f o r a given u> , there was  a continuous  spectrum  of allowable \<ravenumbers, k, corresponding to angles of i n c i d e n c e between 0 andfr/2..  But i n t h i s second case, there i s a d i s c r e t e , f i n i t e  of allowable wavenumbers.  spectrum  Edge waves with d i f f e r e n t wavenumbers, f o r a  given w , w i l l be r e f e r r e d to as d i f f e r e n t edge wave modes.  .  -  a)  17 -  f  Figure 3.2 , D i s p e r s i o n r e l a t i o n f o r K c < x , %  Case 3 —  l>c —  2=10.  v i r t u a l modes  o  This case i s p h y s i c a l l y impossible.  The reader can e a s i l y show t h a t  the s o l u t i o n i s <t>.(-x  ct>a  where  r*=k*-a>  a  ^A cosh (rx) exp(-iKy- iuit)  = A tosh r exp (-1 (x-il +  and i = k - w / V l  a  .  mass t r a n s p o r t , i t i s necessary t h a t  - iwt)  I n order t o s a t i s f y c o n t i n u i t y o f  - 18 -  which i s impossible f o r r e a l shore phade speed phase speed  c>l.  c= /k<l a  r  and p o s i t i v e X .  cannot e x i s t .  So waves o f l o n g -  A l l waves must have a longshore  - 19 -  4.  SCATTERING BY AN IRREGULAR COASTLINE OF AM INCIDENT WAVE FROM THE DEEP OCEAN —  DETERMINATION OF THE REFLECTION COEFFICIENT  I t i s assumed that the coast has bays, peninsulas and i n l e t s which are d e v i a t i o n s from an otherwise  s t r a i g h t coast, p a r a l l e l t o the s h e l f  edge, xvhich i s s t i l l assumed to be s t r a i g h t . to be small compared to the s h e l f width.  The d e v i a t i o n s are assumed  The i r r e g u l a r c o a s t l i n e i s  s p e c i f i e d by  (4.D  where  s(y) i s a s t a t i o n a r y , random f u n c t i o n o f y with zero mean.  When  y i s f i x e d , a random f u n c t i o n i s a random v a r i a b l e , with a p r o b a b i l i t y d i s t r i b u t i o n , over an ensemble o f c o a s t l i n e s .  Averages o f q u a n t i t i e s are  taken over the ensemble o f s t a t i s t i c a l l y equivalent c o a s t l i n e s .  This may  appear t o be confusing, since there i s i n r e a l i t y only one coast under consideration.  But i f i t i s assumed t h a t the random f u n c t i o n i s ergodic-,  then ensemble averages are equal to averages over the l e n g t h o f the one r e a l coast.  Ensemble averages are used to s i m p l i f y computations o f aver-  ages over the length o f the coast.  To be s t a t i o n a r y , a random f u n c t i o n  must have s t a t i s t i c a l p r o p e r t i e s independent o f p o s i t i o n , y — the mean over the ensemble, denoted by pendent o f  ^s(y)^  f o r example,  , must be a constant,  inde-  y , and the autocovariance,  R(y,D=<s(y)s(y+Y)>.  must be a f u n c t i o n o f o n l y the l a g , Y . that <s(y)>=0.  (4.2)  I n the present case, i t i s assumed  The small s i z e o f the c o a s t a l i r r e g u l a r i t i e s i s expressed  - 20 -  by r e q u i r i n g  £  —  =  J<s*Ci)>  <^ 1  that i s , the average s i z e d o f the i r r e g u l a r i t i e s should be .small com-  pared to 1, the s h e l f width. The boundary c o n d i t i o n i s , again, zero v e l o c i t y normal to the coast:  u = JJ- ^s/^y  Since  s  on x=s(y)  (4.3)  i s small, the boundary c o n d i t i o n (4*3) may be expanded about  the mean s=0, stopping a t terms i n v o l v i n g £  u = xrScj - u* S -'  S  a  sScj  , and evaluating a t x=0:  on x=0  (4.4)  I t should be noted a t t h i s p o i n t that (4.4) i s not v a l i d i f u*, v* , and u.x-x are l a r g e , o f f s e t t i n g the smallness o f i s o f the order o f  I n f a c t , u*for example  1/A , where A i s the wavelength o f the i n c i d e n t wave.  So f o r very short wavelengths, (4.4) i s i n v a l i d . common t o Pinsent  s .  This i s a shortcoming  (1972), Howe and Mysak (1973), and Mysak and Tang (1974).  The approximation (4.4), which i s a key one i n t h i s t h e s i s , i s v a l i d  only  f o r i n c i d e n t waves o f wavelength much greater than the average s i z e o f the c o a s t a l i r r e g u l a r i t i e s .  I n the opposite  case, when the wavelength  i s much l e s s than the average s i z e o f the i r r e g u l a r i t i e s , the WKB; method may be used. I t i s u s e f u l to d i s t i n g u i s h between the mean wave f i e l d , ^ ) , composed of the i n c i d e n t wave and the p a r t i a l , r e f l e c t i o n , and the s c a t t e r e d wave  - 21 -  field, modes.  $  , which w i l l be shown to be composed o f edge waves and  leaky  The n o t a t i o n < • > represents, as above, an average taken over an  ensemble o f s t a t i s t i c a l l y  equivalent c o a s t l i n e s .  z a t i o n o f the wave f i e l d , corresponding  In a particular  reali-  t o a s i n g l e , given c o a s t l i n e , the  scattered wave f i e l d i s a c o r r e c t i o n t o the mean f i e l d , so t h a t  4> = < 4> > + $  (4.5)  Note that <<*>>=( <*>+ $)= <f<*>> + <$>= <*>>+<$> , implying t h a t <$> =0 . S i m i l a r p a r t i t i o n s apply t o the v e l o c i t y components u. and v . By t a k i n g the ensemble average o f the wave equations  (2.1^2a,b), i t  follows t h a t  v*<<*>,> * co <*,>o > V*<£, e  v*<4w>*^<«0 o =  =o  --O  ,  S i m i l a r l y , the two matching c o n d i t i o n s  <<*>.> = <<*^>  , 3,  ^<<J>.>A-x -- fk+d/d*,  s  &  f o r 0<x<l  f o r x>l  (4.7a,b)  separate:  a t x=l  &/dx  . (4.6a,b)  a t x=l  (4.8a,b)  (4.9a,b)  The c o n d i t i o n o f f i n i t e n e s s a t x=+s»° a p p l i e s separately t o the mean  - 22 and s c a t t e r e d f i e l d s . The coupling between the two f i e l d s occurs i n the s c a t t e r i n g process at the c o a s t l i n e .  The coupling i s introduced i n the mathematical a n a l y s i s  by the boundary c o n d i t i o n (4«4)  a t x=0.  Using the r e l a t i o n s (2.11a,b) i n non-dimensional  —  form  —  the boundary c o n d i t i o n (4.4) may be expressed i n terms o f  s  and <t>,  alone:  = <WS.j ~fc.xx'S- ' - | 4 W S + ^yx'SSy l  on F O  (4.10)  Following Howe and Mysak (1973)* t h i s c o n d i t i o n i s represented  for-  mally by  £<*>. =G.<t>,  where £. , G,, G* so that £<t>, =0 operators  +  G**},-.  are l i n e a r operators.  (4.11)  The operator X  i s non-random,  i s the boundary c o n d i t i o n f o r a s t r a i g h t coast.  G. and Gj. i n v o l v e the random f u n c t i o n  r a t i c a l l y , r e s p e c t i v e l y . So i n the present  X  = &/&-x  G, = s,, V a y - s a ' A a *  s  The  l i n e a r l y and quad-  case,  (4.12a) (4.12b) (4.12c)  -  23 -  I f the mean o f a random l i n e a r operator i s defined t o be the same l i n e a r operator, with c o e f f i c i e n t s i n v o l v i n g of the c o e f f i c i e n t s , then c l e a r l y G =<G> +G , and G, =G, a  t  t  s  replaced by the means  <(G,) =0, but <Gi>^0 since <s*}?4D.  So  .  Now the ensemble average o f (4.11) i s taken: £  <<*>,) =<G.4>.)  <Gi<K>  +  = <G,.<d>.>> + <G,&> (G,<^>) +  = <G,X*,> < G , $ - > +  <G,*&>  +  + <G><4>,) +<G* $.> t  Therefore, £ <4>, > =<G , $, > + <G,z> < *.> + < d $.> (4.13)  since <G.) =0 .  Expanding (4.11) as  + X $, = G,<<fc> + G , $ .  G ^ * , ) + G a $•  = G.<*,> + G . $ , + < G * X * . > §*<4>.> + G * & +  and  s u b t r a c t i n g (4.13) from i t y i e l d s : t  X $. = G,<*> + S*<*>*c.G.$.-<a,$,)] {'G.$.-<'G $.>? +  t  The method o f ' s o l u t i o n  i s t o assume  (4.14)  i s o f a known form, and  to use t h i s form t o make (4.14) a boundary c o n d i t i o n f o r $, , d e s c r i b i n g the  generation o f the random f i e l d by the i n t e r a c t i o n o f the mean f i e l d  with c o a s t a l i r r e g u l a r i t i e s .  Then, together with the f i n i t e n e s s and  -  24 -  matching c o n d i t i o n s , the d i f f e r e n t i a l equations f o r <S> can be solved i n terms o f ($) .  The r e s u l t i n g expression  (4.13) a boundary c o n d i t i o n f o r  for  can then be used to make  <4>,> , i n terms o f <<£.>  only.  Some approximations can be made, keeping i n mind that the object i s to use 4> t o c a l c u l a t e  , not t o c a l c u l a t e <& i t s e l f .  The q u a n t i t i e s  i n c u r l y brackets i n (4.14) describe i n t e r a c t i o n processes o f the scattered f i e l d and the c o a s t a l i r r e g u l a r i t i e s —  i . e . multiple scattering effects.  I t would not be p e r m i s s i b l e to neglect these terms i n any c a l c u l a t i o n o f However, the boundary c o n d i t i o n (4.13) f o r  $ itself.  r e l a t i o n s such as <G, $>  involves  cor-  o f the c o a s t a l i r r e g u l a r i t i e s and the scattered  waves. Let the c o r r e l a t i o n scale o f the c o a s t a l i r r e g u l a r i t i e s s(y) be L . Then very l i t t l e c o n t r i b u t i o n to the c o r r e l a t i o n  w i l l be made by  waves o f cp which have been scattered from p o i n t s on the coast than a distance L from the point on the coast where uated.  greater  <G,$J> i s t o be e v a l -  But f o r s u f f i c i e n t l y small s ( y ) , i . e . for£=</<s*>  small, c o n t r i -  butions t o <t> which are taken a t the point on the coast where  xs  to be evaluated, and which are due to m u l t i p l e s c a t t e r i n g must have been scattered a t l e a s t a distance L away.  So m u l t i p l e s c a t t e r i n g may be ne-  g l e c t e d when c a l c u l a t i n g the c o r r e l a t i o n products i n the boundary c o n d i t i o n (4.13) f o r <4>> . dropped.  That i s , the bracketed terms i n (4*14) may s a f e l y be  Thus, f o r t h i s purpose, the f o l l o w i n g boundary c o n d i t i o n f o r  q> s u f f i c e s :  1$,  = G,<<b> +Q*<<&>  on x=0  (4.15)  A f u r t h e r approximation may be made by keeping only terms up to 0 ( £ * )  -  25 -  i n the boundary c o n d i t i o n (4.13) f o r (<b ) .  I n keeping with t h i s , only  terms up to 0 ( e ) need be kept i n the boundary c o n d i t i o n (4.15) f o r $ , since higher order terms w i l l i n f l u e n c e (cb) , through ( 4 . 1 3 ) , or higher — and  G, i s 0 ( e ) and G  because  a  is0(£*).  to0(£ ) 3  This means that (4.13)  (4.15) may be approximated by  KM  --<G,$> £ &  +<Q*)<ty  (4.16b)  = G,,«b>  Equation (4.16b) i s kno\m as the Born  t  approximation.  (4.12a,b,'c) i n t o (4»l6a,b), and noting t h a t  I n s e r t i n g the expressions <G >=-i( ')  (4.16a)  , gives  s  <<!>,«)  <  s  ^  -  s  &  v  ^  J £  (4.17a,b)  a t x=0  The form to be assumed f o r (<$>) , f o r use i n (4.17b), i s motivated by the leaky modes (3.8a,b), f o r the s t r a i g h t coast problem.  I t i s assumed  t h a t there i s a wave i n c i d e n t on the s h e l f from the deep ocean; that i t i s r e f r a c t e d a t the s h e l f ; that p a r t i a l r e f l e c t i o n takes place a t the coast, which introduces a r e f l e c t i o n c o e f f i c i e n t i n t o the a n a l y s i s ; and t h a t the p a r t i a l l y r e f l e c t e d wave i s r e f r a c t e d out to the deep ocean. So, l o o k i n g at (3.8a,b), i t i s assumed that  = A T e x p if-m.? + l<y-<oO +ATR, exp lfm.n+k!J-«ol) <4>*} A exp i(-m fx-A-fk j-wt-£UAR2expaCrtM^-n + k'j-tjt+ e) r  v  t  where  A  i s the amplitude o f the i n c i d e n t wave,  T  (4.18a) (4.18b)  i s a transmission  -  26 -  c o e f f i c i e n t , R, i s the r e f l e c t i o n c o e f f i c i e n t as observed on the s h e l f , and R* i s the r e f l e c t i o n c o e f f i c i e n t as observed i n the deep ocean. The two matching conditions a t x=l may be used t o express IU and T i n terms o f R. .  T  The c o n d i t i o n  <4>i)=  a t x=l  Lexp(-im,)+- R, expGm,)] = C exp(-i.£.) + R  The c o n d i t i o n  tf ^**)  i m p l i e s that,:  a  exp(ie)]  a t x=l i m p l i e s t h a t :  1  m.T texpf-im,)- R, £xp(inOj] - ^m*. Lexp(-i£.)- Ra exp(i£)1  Solving f o r R  z  n  *  and T i n terms o f R. g i v e s :  X P (  "  ' U a W m ^ R,exp(xovW (m,+ vV<^exp(-ia^ P.« exp(i rv\,)+-exp(-i m.)  (4* 20)  The reader can e a s i l y check, using the formula t h a t f o r a s t r a i g h t coast ( R, =1)  (4.19)  tan£3=(m,tan m, )/( z f n O ,  -the expressions (4.19) and (4.20)  reduce t o :  Ri= 1  and  , T = cos£ /cos m,  which corresponds t o (3.8a) and (3.8b), the s o l u t i o n s f o r a s t r a i g h t c o a s t . Now  i s known except f o r R, .  be used t o f i n d $  The boundary  c o n d i t i o n (4.17b) w i l l  i n terms o f the unknown R, ; then (4.17a) w i l l be used,  with the expression f o r 4> , t o f i n d R,. S u b s t i t u t i n g the expression f o r  i n t o the boundary  c o n d i t i o n (4.17b),  - 27 -  and dropping the amplitude A and the time dependence y i e l d s :  3,* ' T £ i k < l + R.)s,, + m*(i+R.)s} e  I f i t i s assumed that since  and  $, and  <k  have a y-dependence o f  $.= L C A j ^ e ^ ^ A . ^ e ^ ^ e ^ d n  <kLA»,(TJ) e  +A,z(n)e  >  ,  Jt\ - ^A " =^ l- 7V * Jti+^/x ). t  (4.22a)  OXx<l  (4.22b)  d ,x>l n  the branch p o i n t s at", =-(o ( f o r v/^'-co* - Jr^Zi-J^Uo  upon, corresponding to  ) and afn=t°/V  F i r s t the branch p o i n t s 1 = ^/y  To the question o f whether the cuts s t a r t at -  ^/x  by the four diagrams i n f i g u r e 4.1 »  I—  +UI/JC 4Figure 4.1  are  (for considered.  and go i n t o the upper  or lower h a l f planes, there are only four p o s s i b l e answers,  1  t  hold:  The branch cut configuration, must be decided  r  exp(iny), then  s a t i s f y the d i f f e r e n t i a l equations (4.6b") and (4.7b),  the f o l l o w i n g Fourier i n t e g r a l representations  1  (4.21)  a t x=0  i k s  Branch cut c o n f i g u r a t i o n .  represented  - 28 -  P o s s i b i l i t i e s 1 and 2 may be immediately eliminated, reader may check, f o r p o i n t s \r\\>^/x though n =(-i?) . l  ,, J<?~ ^/x  x  s i n c e , as the  - -JC-f)-)*- "Jfo* , even  This problem does not occur i n p o s s i b i l i t i e s 3 and 4.  z  Another c r i t e r i o n must be used to decide between these two. Suppose c o n f i g u r a t i o n number 4 i s chosen. i n t e g r a l (4.22b) i s t o be evaluated  Suppose f u r t h e r that the  by contour i n t e g r a t i o n .  Then i n the  case y>0, the path must be closed above the r e a l a x i s as shown i n f i g u r e 4.2 so t h a t the f a c t o r exp(i^y)  i n the integrand makes the c o n t r i b u t i o n  from the s e m i - c i r c u l a r path go t o zero as the r a d i u s R goes t o i n f i n i t y . S i m i l a r l y , i n the case y<0, the path must be c l o s e d below, as shown i n f i g u r e 4«2 .  Because o f the path segments around the branch cuts (see  f i g u r e 4.2), there w i l l be i n the e v a l u a t i o n o f (4.22b) a term proport i o n a l to exp(-i( /#,)y—iut) f o r y>0 w  e x p ( i ( A )y-icJt) aJ  f o r y<0 .  and a term p r o p o r t i o n a l to  That i s , the choice o f c o n f i g u r a t i o n number 4  leads to the conclusion that there i s a wave t r a v e l l i n g i n the -y d i r e c t i o n i n the r e g i o n  y>0  and i n the +y d i r e c t i o n i n the region y<0 . Put  another way, waves are converging on the point  y=0 .  But t h i s i s i n c l e a r  v i o l a t i o n o f the r a d i a t i o n c o n d i t i o n , which r e q u i r e s waves s c a t t e r e d i n a region (near  y=0  i n t h i s case) to t r a v e l away from, not towards, t h a t  region. S i m i l a r reasoning  a p p l i e d to c o n f i g u r a t i o n number  3  i n f i g u r e 4.1  shows t h a t the r a d i a t i o n c o n d i t i o n i s s a t i s f i e d i n t h i s case. I d e n t i c a l arguments a p p l i e d t o the i n t e g r a l (4.22a) show that the branch cuts o r i g i n a t i n g from n. =±&> have the same o r i e n t a t i o n as those o r i g i n a t i n g from f\ = /v ±<J>  .  Thus, the path o f i n t e g r a t i o n along the r e a l  a x i s has indentations as shown i n f i g u r e 4.3 f o r e i t h e r y>0 o r y<0. I t w i l l be shown l a t e r that the integrands  o f (4.22a,b) have poles  -  on the r e a l a x i s .  The  29  -  path o f i n t e g r a t i o n along the r e a l r\ - a x i s must  be indented below the r e a l a x i s f o r p o s i t i v e poles and above the r e a l a x i s f o r negative poles.  The  argument f o r t h i s i s as f o l l o w s . r\« > 0 .  path i s indented above the r e a l a x i s at a pole i s evaluated  by contour i n t e g r a t i o n f o r  below the r e a l axis — p r o p o r t i o n a l to  y<0 —  Suppose the  Then when (4.22a)  that i s the path i s c l o s e d  there w i l l be a c o n t r i b u t i o n to (4.22a) which i s  exp(i>vy-icot) .  But t h i s means that there i s a scattered  wave t r a v e l l i n g towards the s c a t t e r i n g region near the r a d i a t i o n c o n d i t i o n .  y=0  , i n v i o l a t i o n of  When the path i s indented below the pole i>>0  no v i o l a t i o n o f the r a d i a t i o n c o n d i t i o n occurs.  A s i m i l a r a n a l y s i s shows  that the path o f i n t e g r a t i o n must be indented above poles i> < 0' • The path of i n t e g r a t i o n , with indentations  around branch points  and  poles, i s shown i n f i g u r e . 4 . 4 • The boundedness c o n d i t i o n on lll  > < 0  /8  sAf~ "7?  since =Jn- %  exp(->/rf - ^/a -A +%  1  $  R  c l e a r l y i m p l i e s that  (x-l))-*°°  as x  .  A ,(i7)=0 f o r z  In the region I n I < A W  = - i > / * - n * <% +n = - i  -n*  .  (The f a c t o r  -1  i s obtained from an examination o f the o r i e n t a t i o n of the branch cut at q=  .)  representing  So there i s a f a c t o r i n the integrand  of  expi-lsf^/^  - rf ).-»  component waves of the scattered f i e l d t r a v e l l i n g towards  the coast, where the s c a t t e r i n g occurs.  This i s a v i o l a t i o n of the  radi-  a t i o n c o n d i t i o n unless the amplitudes o f these components are a l l zero that i s unless  A*,(n) = 0  \t\\<^  for  The two matching conditions  .  So  (4.8b) and  A*(n.) = 0  for a l i a  (4.9b) at x=l may  now  F o u r i e r transformed (with respect to y) and used to solve f o r A,i(n.)  i n terms of  AnM  A»(rO.  e x ( + yfif^FU P  The  c o n t i n u i t y c o n d i t i o n at x=l  A ,tW  exp(-Vrf--co* ) = A * a  ( n)  . be  A„(n)  and  implies  (4.23)  —  ,  - 30 -  Figure 4.2  'CJontburs'of i n t e g r a t i o n .  uJ/Y CO  -u)  Figure 4.3  .Indentations o f path o f i n t e g r a t i o n .  es-^ •  Figure 4.4  Indentations o f path o f i n t e g r a t i o n .  - 31  -  The c o n t i n u i t y o f mass transport c o n d i t i o n i m p l i e s  A  < l  (a)^/^-£  + y T  ^-A  1 1  (nl^/^^e  a V  "=  -Wtf-^VAwCrO  (4.24)  S o l v i n g (4.23) and ( 4 . 2 4 ) , we f i n d  M r t  W  (^  r  z Jv-o*  A  ~  f  r  °  (4.24b)  Using (4.24a,b) i n ( 4 . 2 2 a ) , we o b t a i n :  R e c a l l t h a t the boundary c o n d i t i o n has already been c a l c u l a t e d to be:  %, - - T ( i k ( i + R,)^+^(n-R,)53e  ik>}  t  The F o u r i e r transform integrand o f (4.25)  .  The F o u r i e r transform  o f (4.21) with respect to k (n)  In t h i s way,  zi  (4.21)  a t x=0  y  must be equal to the  can be evaluated  o f (4.21) i s :  ^  wY  r °° +  = W T l » + R . U ^ , r K M K r o f 3S e.'  Equating t h i s to the integrand i n (4.25)  J-«o  F (n)  yields:  x  d Y  i n terms o f R,  -  where  =(Jh -uS-  F(n)  - frVn - " V )e  %  32  -  ^ " ° - ( N / V - < ^ + *7"nv  1  W  V ) e  +  S u b s t i t u t i n g t h i s i n t o (4.24a,b), and u s i n g the l a t t e r i n (4.22a,b), i t A i s f i n a l l y p o s s i b l e to w r i t e cp, and  (JV-WTu+iue  1  'JXO  a 11 C\ ^Td+a^ie  .  §  A 4\ i n terms o f  •  R,:  Vv-W  Lco^-knJse F(M  d<\<i\  V x d  (4.26a)  x>l  (4.26b)  The reader i s reminded t h a t these expressions f o r d> do not take i n t o account m u l t i p l e s c a t t e r i n g and are o n l y accurate to 0 (£)_.-  However,  i t i s noteworthy that the expressions (4.26a,b) show t h a t trapped edge waves are generated i n the s c a t t e r i n g process.  For, i f there i s a pole  A at f\ = t^o , say, then the expressions f o r 4> evaluated at the pole i n c l u d e a factor  e x p ( i k y ) e x p ( - i ( r\<>-k)(-y)) = exp(it>y) —  that i s , the pole a t - r ^  corresponds  to a t r a v e l l i n g wave with longshore wavenumber rj .  the poles?  They are l o c a t e d where the denominator o f the integrand i s  zero —  0  Where are  i . e . where:  That i s ,  y^-(e'  7 n  -e  j--^7l -"V(e l  +-e  J  which i m p l i e s  +anhx/^? * - K V o N ^ V  (4.27)  - 33 -  Now consider d i f f e r e n t regions f o r n, on the r e a l a x i s . uP- .  i t i s a l s o true that l i  1  A - ^ / V = - i v/ /^ - h  1  1  0  For 1^^^/^  ,  using \Ay- to = - i Vui"-- K ,  Therefore,  1  1,  and t a n h ( i z ) = i t a n z  , ( 4 . 2 7 ) becomes  (4.28)  >/o -tf -Van v>u^- ^ -- - i K \ A ° V - ^ l  Since the l e f t side o f (4.28 i s r e a l and the r i g h t side i s imaginary, poles are impossible For r?>  i n the regions where  n.*^- °/V  , the square roots i n (4.27) are a l l r e a l and p o s i t i v e , and now  the l e f t side i s p o s i t i v e , while  the r i g h t side i s negative.  are no poles on the r e a l a x i s i n the regions where the regions where V i ^ n < to w  to  on the r e a l a x i s .  -i\A> - n. 1  1  l  x  1  >ui . t  So there Finally, i n  , (4.27) becomes, upon changing v V ~ <-o  z  and using t a n h ( i z ) = i t a n z ,  n/co -^ 4 a o 7 ^ - a 1  ** V i -  l  (4.29)  x  Comparing t h i s to the trapped mode d i s p e r s i o n r e l a t i o n (3.10), i t i s c l e a r that s o l u t i o n s I o f the trapped modes.  to ( 4 . 2 9 ) , f o r a f i x e d to , are the wavenumbers So (4.26a,b) i n c l u d e trapped mode edge waves as  part o f the scattered r a d i a t i o n . Now expression  (4.26a) f o r 4>. , i n terms o f R, , may be used i n the  boundary c o n d i t i o n (4.17a) f o r <<&,) :  <*.«>--<$i,S,)-<$ x»S>-W<4> „)<s*> 1  0 II  ,  The terms o f (4.17a) are now evaluated and r e c a l l i n g that the f a c t o r  A exp(-iwt)  a t x=0  one by one.  (4.17a)  From (4.18a),,  has been dropped from a l l terms,  -  34 -  <<tO n T l - i + fl,^, e. *\ {  a t x=0  (4.30)  a t x=0  (4.31)  Also,  < 4 w > < S ^ - U W C - l + f U e ^ <s*>,  Using ( 4 . 2 6 a ) ,  <$,^S^V-4ivT(ii-R )e^M. o2c'Xiu3 ^Lu -k JUal-<S(Y)Sv)(^)e l  ,  0  )  where  b(ry,oo,jQ  O  (Vk1(V  1  W ^ - s V i f + ( 7 ^ 1 ?  "dndY  qW-^'le  1  s ,  where  i s the autocovariance o f  R(v) = <(s(z)s(z+v))  $  (4.32) (4.33)  / . I  But, using the d e f i n i t i o n o f the power s p e c t r a l d e n s i t y s t a t i o n a r y random f u n c t i o n  a t x=0  o f an ergodic  s } and using the  fact that  s(Y>s,(y)  it  = Vtf><s(Y)s(y)>  = % j R ( Y - y ) = -R» (Y-y)  follows t h a t (4.32) may be w r i t t e n  ( ^ ^ ^ T c i +R ^ e ' ^ i i ^ ^ l ^ - k O - ^ n - O i ^  ,  a t x=0  (4.34:)  a t x=0  (4.35)  The remaining term i n (4.17a) may be w r i t t e n  ($uxS)--T(i+RAe' ^S(n;^)L(o -k T^-^)?rn-^ k  l  n  ,  -  Now ( 4 . 3 0 ) , ( 4 . 3 1 ) ,  35 -  (4.34) and (4.35) may be used t o w r i t e the boun-  dary c o n d i t i o n (4.17a) f o r (<$>,) i n terms o f R, :  iX(- H-R.'U.e ^ ' T d + R.^e^i &«v,«o,»[u*-k»i] [ - ^ ^ ^ ( n - W d i i + V t i T n ? . (-i + R , V  k s  <.s> x  -oO  Keeping terms only t o 0 ( 6*) , and keeping i n mind that  I(;f) = 0 ( 6 " ) ,  the reader can e a s i l y show that:  (4,36)  I t was noted, a f t e r the boundary c o n d i t i o n expansion (4.4)> that the computations i n t h i s s e c t i o n are i n v a l i d f o r waves o f a very short waveThis d i f f i c u l t y appears i n ( 4 . 3 6 ) , f o r although  length.  polynomial  (a? -nk)* <*• co* , § ("aito,*)** l/-o » and  c o r r e c t i o n term p r o p o r t i o n a l to £  2  .  'H^e -, 1  the  l/m.^l/co , making the  I f co i s too l a r g e , then the con-  d i t i o n £ co < < 1 , r e q u i r e d f o r the boundary c o n d i t i o n ( 4 . 4 ) , w i l l be l  violated.  x  I n a d d i t i o n , the magnitude o f the r e f l e c t i o n coefficient,IR< I 1 , which i s p h y s i c a l l y impossible.  w i l l become greater than  The l a t t e r  d i f f i c u l t y i s encountered i n the next s e c t i o n . The r e s u l t (4.36) can be compared t o the expression c o e f f i c i e n t obtained by Howe and Mysak : (1973). ;  f o r the r e f l e c t i o n  R e c a l l t h a t they i n c l u d e d  r o t a t i o n a l e f f e c t s (f)A)) i n t h e i r study o f waves on a f l a t - b o t t o m ocean i n the presence o f an i r r e g u l a r c o a s t l i n e .  If f  i s set equal t o zero  i n the r e f l e c t i o n c o e f f i c i e n t expression o f Howe and Mysak (1973), and if  X  i s set equal t o  1  i n the present  expression  (4.36)  (corresponding  to a f l a t - b o t t o m ocean), then i t can e a s i l y be shown that i n these l i m i t s , (4.36) i s i d e n t i c a l t o the expression  o f Howe and Mysak (1973).  - 36  -  One further point i s worth noting about ( 4 . 3 6 ) . points at r\ = i c o —  There are no branch  the singularities there are removeable.  For i f the  exponentials i n (4.33) are expanded i n a power' series i n w  -uf  close to <^>  factor taken  or  - to , i t i s found, that with the  l/jrt-^  for <a  into account, only even powers of v r\ ~-u^ appear i n the expression for /  S(i^;to,jf). appears.  r  That i s , the square root V _  us- disappears, and only  (n. "-^) 1  - 37 -  5.  SCATTERING BY AN IRREGULAR COASTLINE OF AN INCIDENT WAVE FROM THE DEEP OCEAN — .POWER FLUXES INTO DIFFERENT MODES OF THE SCATTERED WAVE The expression  -(p'){u*)  (p  and u  1  1  are the dimensional  and x - v e l o c i t y ) i s the power f l u x o f the mean f i e l d tion.  pressure  <<t>> i n the -x d i r e c -  When evaluated a t the coast, x=0, the expression i s the work per  u n i t time per u n i t c r o s s - s e c t i o n a l area done by the mean wave f i e l d on the coast.  Since the coast does not move, a l l o f t h i s energy goes i n t o  the s c a t t e r e d waves. -(p')<u') .  Thus, the power f l u x o f the s c a t t e r e d waves i s  A,more u s e f u l .quantity i s the power f l u x averaged over one  c y c l e o f the mean f i e l d , represented  -(p'^u ) 1  .  F i r s t p' and u.' are  evaluated i n terms o f 4>' :  P ^ / W  ,  u«=-(ig/6o')4>;.  However, since these are r e a l q u a n t i t i e s , i t i s necessary  t o take only  the r e a l p a r t s :  p i= |pg<j>» + c.c.  ,  :  u'= - ( i g / o ' ) * , . + c.c.  where "c.c." stands f o r the complex conjugate. the mean f i e l d  Now using (4.18a) f o r  (cj>,>, i t follows t h a t  <P'> ^ f 9 h , AT( i R , U  i k V  +  <u'> ^ M h . ^ ' A T O - M e ^  +c.c.")  1 1  + cc  J  a t x=0  Therefore, a t x=0, representing complex conjugates  by  (5.1a,b)  *  , we have  -  38 -  F i n a l l y , the average over one c y c l e o f the mean f i e l d i s taken:  (5.2)  a t x=0  I t i s now a simple matter to show that the power f l u x o f the i n c i d e n t wave a t the coast —  x=0 , averaged over one c y c l e  simply s e t R,=0  is  gPg " h*A (m.'/o ) | T| * 1  l  t o focus a t t e n t i o n on only the i n c i d e n t wave.  1  There-  f o r e , the p o r t i o n o f the i n c i d e n t power f l u x which goes i n t o the s c a t t e r e d field i s  (l-IR.I*) .  Now,  putting  then, c o r r e c t t o 0 ( * ) , e  Then  = *Vm,(T-T"V  W I T *  Therefore, (5.3)  -  where vl2<m(l)  i s the imaginary  39  -  part of  I .  In order to continue, i t i s necessary i s c l e a r from (4.36) t h a t the determining  to d e t e r m i n e ( I ) f a c t o r i s S (n.;w,a)  integrand, s i n c e the power s p e c t r a l density '(ccf-kn.)  3>(t\-k)  It  i n the polynomial  are both p o s i t i v e d e f i n i t e q u a n t i t i e s .  1  F i r s t of a l l , i t i s c l e a r t h a t % (r\;u>,&) where  and the  .  i s r e a l i n the  regions  (\ >co?- , since? a l l o f the square roots have p o s i t i v e arguments.  In  x  the regions where  ^//^trftLS -  Using the formula  exp(ie) = cos © + i sin©  , and  2  for ^  '  where >"\i =a  pole of '•<£*,  , t h i s can e a s i l y be shown  to reduce to  $(n;ua,»vCgg^ ^ V n ^ V  +an V ^ - n O  '  (5.4)  which i s o b v i o u s l y a r e a l quantity, i n the ranges where However, there i s a c o n t r i b u t i o n t o the imaginary —  from the poles, where  i s a pole l y i n g between  W  A  1  and  the c o n t r i b u t i o n to the i n t e g r a l  »  s a  i  y«  7  1-  1  I .  Then i n order to  Representing v  f ( a ) = (remaining  f a c t o r i n integrand)  i n d e n t a t i o n by a  = % + r exp(ie)  indentation i s  a  g(*\) = ^w -a - t a n Vu* -t\ - frVn?- VV1  ,  ^  ^ LO*-  .  i n t h i s range Suppose evaluate  I , the path of i n t e g r a t i o n must be  indented below the r e a l a x i s f o r nj > 0 . f(n)/g(n) , where  3  p a r t of  >/u>-iy- t a n V w - r\ = K \/rf - ^A" l  0  W  the integrand by > and the f u n c t i o n  , and r e p r e s e n t i n g a p o i n t on  the  (-^e^O) , the c o n t r i b u t i o n from the  - 40 -  Letting  r-»0  gives a c o n t r i b u t i o n to  I  I n s e r t i n g the appropriate expressions f o r  from the pole  f  and  g  n.j  of  g i v e s , f o r the  imaginary c o n t r i b u t i o n  i P ( n i ^ -ti>• \)\  +  (xV^aT » v A v ^ v 7 VansA^F\l-  1__  For poles i n the range  -o^n^  (uf-K*\f• I ( « )  , the path i n d e n t a t i o n s are above  the r e a l l i n e , and the expression f o r  P(^j)  i s i d e n t i c a l , except f o r a  p o s i t i v e s i g n r e p l a c i n g the negative s i g n i n f r o n t .  For a given co ,,  there i s a f i n i t e number o f poles, as can be seen i n f i g u r e 3.2 ;  dispersion relation.  (5.5)  f o r the  So the t o t a l c o n t r i b u t i o n to the s c a t t e r e d f r a c t i o n  o f the power f l u x from the poles i s  -(4/m.)22 ( nj) , which i s simply the p  sum of the power f l u x e s of a l l the trapped edge waves that can e x i s t f o r a given frequency. I t i s worthwhile to check that the s i g n of  -(4/m, )P(qj)  i s positive.  R e f e r r i n g to (5.4), the complicated f a c t o r i n c u r l y brackets i s o b v i o u s l y p o s i t i v e , provided  tan^co -Hj*> 0 1  —  but  tans/cof-tv* = 1  by the d i s p e r s i o n r e l a t i o n , and t h i s i s p o s i t i v e .  -  /V^ -  The power s p e c t r a l den-  s i t y i s a p o s i t i v e quantity, and the f a c t o r on the l e f t of the b i g c u r l y brackets i s i s negative.  -^'/^  when  n.j >0  and  The remaining f a c t o r ,  So, a l t o g e t h e r ,  P(  )  +1>/ ;  ) J  when  (u> -k r^) l  1-  i s negative, which makes  c 0 —  i . e . this factor  i s p o s i t i v e , obviously.  - 41 -  Another c o n t r i b u t i o n to Jj<rn ( I ) rf ^  .  comes from the regions where  I n t h i s region, i t i s a simple matter to show t h a t  S v.i s V ^ V ^ - ^  = = = = = =  __  I t i s immediately apparent that the imaginary p a r t o f (5.6)  (5.6)  i s negative.  The c o n t r i b u t i o n t o the power f l u x i s  The quantity  D  i s a sum over a continuum o f longshore wavenumbers  ' ^ l ^  6  of waves which escape from the s h e l f to the deep ocean. Summarizing,  the p o r t i o n o f the i n c i d e n t power which i s scattered  by the c o a s t l i n e i s made up o f two p a r t s : and waves escaping to the deep ocean.  waves trapped on the s h e l f ,  Represented as a f r a c t i o n o f the  i n c i d e n t power f l u x , t h i s i s  1 - lfU*= -4/ , m  P  (  r  ^  + D  (5.8)  With the a i d of a computer, the r a t h e r complicated expression (5.8) has been c a l c u l a t e d f o r the s p e c i a l case o f the northeast Japanese coast, from about 38°N l a t i t u d e t o about 39.5°N l a t i t u d e , a distance o f 200 km. The s h e l f depth  h, i s taken to be 200 m., and  so that  The s h e l f width i s 21 km.  #*=10.  i s taken to be 2000m.,  F i g u r e s 5.1,5.2,and 5.3 show  the r e s u l t s f o r i n c i d e n t wave periods o f 2 hours, 1 hour and \ hour, corresponding t o to = 0.413, 0.826, 1.652, r e s p e c t i v e l y . "Figure 3.2 shows, that o n l y the lowest mode edge waves can be generated.  The t h i r d case,  0  / ^ ,  -  42 -  to =1.652, i s obviously- i n v a l i d , s i n c e  l-lR,|  must be l e s s than 1.  a  There i s a s e c u l a r i t y i n the expression f o r 1- IR,I*  —  the 0  i s p r o p o r t i o n a l to to* , as noted a t the end o f s e c t i o n 4. sion for  I-IR-I*  i s v a l i d only f o r Cu?^*-  the assumption made i n s e c t i o n 2 that co —where  f  i s the C o r i o l i s parameter.  1 . f  5 termi  So the expres-  The reader w i l l r e c a l l —  i . e . c o » f /(TghT /W)*" t  So taken together, the theory  i s v a l i d i n the region  $/(J$K,/\*/)  ^  For the northeast Japanese coast,  CO  >/«  £ ' =2.60 , and f/(/gh," /W)=.05 ,  l e a v i n g a r a t h e r narrow region o f v a l i d i t y  ...05 ^ c o ^ 2.60 . Total  -5 o.olt  (5.9)  s e a f+e r e e f  porter fluyc  O.OIZ 4  (5  „ 0.00s J  X u  Io a. 1 OOOf <ti B O  — I —  30  60  —i  -  90  naf* of mci'o/ence- «+ s h e l f e-dje. , in otegree-s .  Figure 5.1  'Power f l u x e s on Japanese coast, f o r tu=.413 —  p e r i o d =2 hr.  - 43 -  Figure 5.2  Power f l u x e s on Japanese coast, u) =.826 —  a n g l e - o-f inc.ide.nce-  Figure 5.3  a4- sh&lf ,  period=l h r .  ino/egrees  Power f l u x e s on Japanese coast, o> =1.652 —  period=ghr.  - 44 -  Ocean  map  Figure 5.4  SCaJe — 1:2,000,000  Region o f Japanese coast f o r c a l c u l a t i o n s .  /too  1100  ?00 ^  i  i.  •40  400  .Of Figure 5.5  cycles /k11"»ne.+re  —r.0?  .12. s  Povrer spectrum f o r Japanese coast o f Figure 5.4  - 45  -  I t i s noteworthy t h a t f o r both v a l i d cases, i l l u s t r a t e d i n Figures 5.1  and 5.2,  the quantity  D  range of angles of i n c i d e n c e .  remains f a i r l y constant across the whole This power f l u x o f waves s c a t t e r e d to the  deep ocean i s i n each case l e s s than both the backward and forward tered trapped forward,  edge waves.  i n both cases.  scat-  There i s greater, energy s c a t t e r e d backward than The sharp drop i n the back s c a t t e r and  forward  s c a t t e r i n Figure 5.2 i s due to p e c u l i a r i t i e s o f the power spectrum of the Japanese coast. Figure 5«4 shows the p a r t of the Japanese coast f o r which c a l c u l a t i o n s were done.  Figure 5.5  anese coast of Figure 5.4 f  •  shows the power spectrum  1(f)  o f the Jap-  6.  INFLUENCE OF COASTAL IRREGULARITIES ON EDGE WAVES TRAPPED OM  THE  SHELF  —ALTERED DISPERSION RELATION Sections  4 and  5 discussed  i n c i d e n t from the deep ocean. mode wave has  the s c a t t e r i n g of a long wave which i s In t h i s s e c t i o n i t i s assumed that a trapped  been generated on the  cing mechanism.  s h e l f by s c a t t e r i n g or some other f o r -  In the absence of c o a s t a l i r r e g u l a r i t i e s , the  dispersion  r e l a t i o n f o r such waves i s (3.10), which w i l l be r e f e r r e d to as the order d i s p e r s i o n r e l a t i o n . discussed  The mechanism of generation o f the wave i s not  here, but i t could be due  K a j i u r a (1972.) ).  - It I  s  to an,earthquake, on the s h e l f (e.g.  of the trapped wave.  i s the case, as w i l l be demonstrated i n s e c t i o n 7.  Indeed, t h i s  Other a l t e r a t i o n s to  p h y s i c a l q u a n t i t i e s , such as an a l t e r e d phase speed w i l l a l s o be The  first  discussed  step i s to determine the a l t e r a t i o n due to  c o a s t a l i r r e g u l a r i t i e s o f the d i s p e r s i o n r e l a t i o n GJ = co(k) i n (3.10)  see  expected t h a t c o a s t a l i r r e g u l a r i t i e s w i l l cause  some s c a t t e r i n g and hence attenuation  i n s e c t i o n 7.  zeroeth  the  , implicit-  .  A somewhat d i f f e r e n t approach from that used i n the r e f l e c t i o n coeff i c i e n t problem i s employed here. field  ^  Unlike the former problem, the  w i l l not enter i n t o the c a l c u l a t i o n s here.  The  scattered  object i s to  the a l t e r a t i o n to the d i s p e r s i o n r e l a t i o n of a coherent trapped wave. method employed i s that of Mysak and The mean f i e l d  find The  Tang (1974).  <<t>) again s a t i s f i e s the d i f f e r e n t i a l equations  V\ct>,> + u>*<*.>--o ,  Ooc<l  (4.6a)  V ^ ^ ^ ^ ^ ' O .  x>l  (4.7a)  and the matching  conditions  - 47 -  <<M  <<Pi) ,  at x=l  (4.8a)  <d\> =**<<!>*> ,  a t x=l  (4.9a)  In a d d i t i o n , the s o l u t i o n (<p) must be f i n i t e f o r a l l x  and  y .  The  boundary c o n d i t i o n at the coast i s the same c o n d i t i o n as before —  zero  normal v e l o c i t y —  It i s  although i t i s expressed somewhat d i f f e r e n t l y .  assumed that t h i s boundary c o n d i t i o n can be expressed as  ( B + C ) * , ='o  on x=0  B) i s a d e t e r m i n i s t i c l i n e a r operator,  where  operator. and  ,  so  B=*/bx  and  C  (6.1)  i s a random l i n e a r  Note that i n the absence of c o a s t a l i r r e g u l a r i t i e s ,  B <<t>,>=0 .  C=Ov,  This i s the case of zero x - v e l o c i t y at the coast,  so  .  I t i s assumed that the s o l u t i o n f o r a coherent trapped wave i s of a c e r t a i n form, and t h i s i s s u b s t i t u t e d i n the equations and boundary and matching conditions to determine a d i s p e r s i o n r e l a t i o n .  The  solution i s  assumed to be of the form:  <<J>.> = eosCm-x + c H e ^ , < O . V A e - ^ - ° ^  The time f a c t o r and  m,k,l  The quantity Sfc)  exp(-icot)  (H»l ,  (6.2a)  x>l  (6.2b)  i s again dropped, f o r convenience.  With S  r e a l , t h i s would be the s o l u t i o n f o r a s t r a i g h t coast, S  i s assumed to be small.  and the wave numbers may  =0  C=0.  With c o a s t a l i r r e g u l a r i t i e s ,  be complex.  We  assume that  that the wave i s trapped; a l s o , we take i2oTa(k)>0  and  Re(!)">0 , so  Re(k)>0  so that  a wave t r a v e l l i n g i n the + y - d i r e c t i o n from a source at y=0, say, w i l l be f i n i t e everywhere.  The f i r s t matching c o n d i t i o n (4.8a) i m p l i e s  (6.3)  cos(m+o") = A  and the second (4.9a)  implies  (6.4)  m sin(m+cO = f l A  S u b s t i t u t i n g (6.2a,b) i n t o the d i f f e r e n t i a l equations ( 4 . 6 a ) , (4.7a) and d i v i d i n g (6.4) by (6.3) give the f a m i l i a r r e s u l t s , with one small change:  m tan (rn+cS") = )$*A rr^ + k al  If  l  1  (6.5a)  -- oo*  (6.5b)  + k* =  ( 6  .  5 c  )  <S were zero, (6.5a,b,c) would be the d i s p e r s i o n r e l a t i o n (3.10)  f o r unattenuated trapped modes with a s t r a i g h t coast.  The equations  (6.5a,b,c) would determine a d i s p e r s i o n r e l a t i o n ^ = i o ( k ) known as a f u n c t i o n o f co and w i l l be used t o determine S  k .  i f S  were  The boundary c o n d i t i o n (6.1) a t x=0  .  The f u l l boundary c o n d i t i o n —  zero normal v e l o c i t y a t the coast  —  i s , as i n the r e f l e c t i o n c o e f f i c i e n t problem,  ;U-  Upon expressing  u  and  v  xrs^  i n terms o f <j>  on x=s(y)  (4.3)  and expanding (4.3) at x=0  to 0(e  z  ) , the following expressions for the linear operators  B  and C  are obtained:  8= Vane  ,  C'CsaVs^WiS^-S^d^Ofe*) ,  at x=0  (6.6a)  a t x=0  (6.6b)  Note that once again t h i s i s only v a l i d for long wavelengths — Now suppose there were a deterministic forcing function  i.e.  EbJ^l  F(y) on  the right side of (6.1):  (B+C) <t>,= F(3)  Then assuming (B+C)  ,  on x=0  (6.7)  can be inverted, (6.7) yields:  <<$>>-- <(8+cT'> F  ,  x>0  (6.8)  which implies that  (16+C)")*'<^.)  Writing  (B+C)"  1  as  :  ,  on x=0  (6.9)  (I+B"'cr B'f , using the binomial expansion (since  0=0(6) ) and averaging, gives  < ( B + C ) - ' > ~- ( ( I  + B-'cV  B->  < ( T - (3-C 4 B - C B - ' C ^ '  1  > + 0 (  O  - 8~'<c> + B"' < C B - C > ) B " + 0 ( £ * )  F i n a l l y , t a k i n g the i n v e r s e o f t h i s l a s t expression, and dropping terms o f 0(£*)  and higher g i v e s :  <(B+cr)~  So, s e t t i n g  = 6+<C)-<C8- C> *-<C>B~'<c>  1  ,  (6.10)  J  F(y)=0 ,  C8 + <C> + <c> B-<c> - <C B~'C>)<<*>,) = 0  The l i n e a r operator  <C>  , on x=0  has q u i t e a simple  form.  ^ R ( 0 ) / i y •= 0; f o r a s t a t i o n a r y random f u n c t i o n , and since  (6.11)  Since  t(s^s}=  <s> = 0 and  <s > = <s ) = 0 , 4  M  <CW<s*>^  Now,  B~' i s an i n t e g r a l operator which maps a f u n c t i o n , say f ( y ) ,  defined on the boundary x^O  .  (6.12)  x=0  onto a f u n c t i o n defined on the whole r e g i o n  I n terms o f a Green's f u n c t i o n  8"-Pon=]  G ,  G(*,y;z)^d2.  (6.13)  I t follows t h a t  ^'Ax*  4  ^'/d*  1  G, =O ,  +  Oixd  (6.14a)  ^Gi/a-x* +i G,iA« + V»«Gn.= o ,  x>l  (6.14b)  E G . = ^C,/dx = cffy-'i)  X=0  (6.14c)  a t x=l  (6.14d)  a t x=l  (6.14e)  I  1  u,  G, = Gi. A  G  |  ,  / i %  s  f  tf ^*/^* 1  , ,  ,  As w e l l , there i s the boundedness c o n d i t i o n at condition applies. and x=0 Now  The  <£-function represents  x=+«^ , and the r a d i a t i o n a source o f waves at  y=z,  . introduce  the F o u r i e r transform,  e~  G(*A',a) --\Q(lL,<i\*)  G , of  in<1  G  with respect to y:  (6.15)  dcj  Then (6.14a,b,c,d,e) put r e l a t e d conditions on G :  S~ ^ &  -  C nf-co') G. = o  /b-f  0*oc<l  ,  - Cri - "Vjr*) G, * = o , = e~ * m  G, = G*  ,  on x=0  (6.16c)  ,  at 35*1  (6.l6d)  at x=l  (6.l6e)  ,  The  (6,l6b>)  x>l  x  <^ G), /^x  ' (6.16a)  r a d i a t i o n c o n d i t i o n a p p l i e s to waves generated by the & - f u n c t i o n  source at  y=z,  generation. bounded as  x=0 —  i . e . waves must t r a v e l away from the point o f  In a d d i t i o n , the f i n i t e n e s s c o n d i t i o n ensures that  G^  is  x-s>+°°.  From ( 6 , l 6 a , b ) , i t can be seen that:  Q,=  c„e*  G,--C*,e.  K,ie  , + Cz*e  ,  The branch cuts are o r i e n t e d i n the same way reasons as i n s e c t i o n 4.  In (6.17a,b),  the  Goc^l  (6.17a)  x>l  (6.17b)  as i n s e c t i o n 4,  f o r the same  c^ are f u n c t i o n s o f  and  As i n the r e f l e c t i o n c o e f f i c i e n t problem, the f i n i t e n e s s c o n d i t i o n on  z , G*.  -  forces that  c  2 1  = 0  ©t, = 0  f o r r\*-> for  ^  52 -  , and the r a d i a t i o n c o n d i t i o n ensures .  So  Cw(r\,"a^=0  (6.18)  f a r a l l r\  (6.16c) t o (6.17a) gives the equation  Applying  (6.19)  N/rf-u?' ' C M -Vt^-u^-Cn. - e ~ * l n  The f i r s t matching c o n d i t i o n , (6.l6d) implies  e  c,, + e  c . =c«  The second matching c o n d i t i o n (6.l6e)  ,  v/rf-to'  Vr\ -w*e  Now,  .  1  gives  _vV-^  r  _  .  c,*--frVi-C- '^ 10  c  .  .  (6.21)  t l  (6.19), (6.20) and (6.21) are three l i n e a r equations f o r the three  unknowns  c-,,, c, , c 4  t t  .  C,h>Bh^  c  where  .  c.-VrWu* e  l  (6.20)  «  c  ^ ~ -  F(n) = KJ^J-  I t i s a simple matter to show that  (<^f  -^ V ^ V k  -.--7=4=  (6.22a)  v ^ r  *Vrf-<«>7»* ) e " ( n / F ^ ?  ( 6  -  2 2 c )  ^vV-^V) e " ^ ^] 4  7  1  as i n s e c t i o n 4. I t w i l l be shown that the i n t e g r a l f o r j ~ i n v o l v e s no branch p o i n t s  at i\ = ±<o, as i n the previous problem.  The expression  for  G  may now be  w r i t t e n , by i n v e r t i n g (6.15):  J-oO  That i s , p u t t i n g  d<j(n.) = Cijexp(inz) ,  !d„(t\)e G.( >  W  W  f ^ ^ e - ^ -  + d (rye  je  lt  W  ^ '  )  e  +  ^  i  4  -  l  ,  d^,  ^ ,  Ooc<l  (6.23a)  *>1  (6.23b)  I t should be noted a t t h i s p o i n t t h a t , f o r the same reasons given i n the r e f l e c t i o n c o e f f i c i e n t problem,  r\= A tto  around poles and the branch points for  indentations o f the path o f i n t e g r a t i o n are made below the r e a l a x i s  r p O and above the r e a l axis f o r r ^ O . Now a l l the terms o f (6.11) can be evaluated.  I t has already been  shown that  B= <V<iTC <C>--Ks*>^  i s 0(6*)  ,  a t x=0  ,  a t x=0  x  Therefore,  <C>B~'<C>  , and so i t may be dropped from the c a l -  culations.  The f i r s t t;vo terms o f (6.11) a r e :  (B + < C > ^ < * . > --(-ms-.nS m /a<S*>s^cO 3  +  The d i f f i c u l t term i s  (CB^C > <$,>.  uating  C  CB C g(x,y) ,  y only —  -,  takes  (6.24)  I t should be remembered that i n e v a l -  g(x,y)  and converts  i t to a function of  (Cg)(y); then B"' a p p l i e d t o (Cg)(y) produces a f u n c t i o n o f both  x and y  (B Cg)(x y);  y  (CB"'C g)(y) .  rl  1  only —  and C a p p l i e d to t h i s produces a f u n c t i o n of C a r e f u l l y f o l l o w i n g t h i s step by step procedure  and r e t a i n i n g terms up to Q ( £ ) , we o b t a i n the f o l l o w i n g expression l  f o r t h i s term:  J  Now  -oo  note t h a t  Taking the averaging o p e r a t i o n  < •>  i n s i d e the i n t e g r a l signs and a p p l y i n g  i t to the terms i n the integrand thus y i e l d s :  where the obvious i d e n t i t y § (tiJw,a) = d„(rv) + d^(r\) been used.  The e v a l u a t i o n o f  spectral density  and the property that  <CB"'C><$>  (see (4.33) ) has  uses the d e f i n i t i o n of the power  -  55 -  The c a r e f u l reader w i l l n o t i c e t h a t the s i g n o f the exponential i n the integrand o f the d e f i n i t i o n o f i n section 4 •  I  i s p o s i t i v e here, but was negative  However, there i s i n f a c t no d i f f e r e n c e , s i n c e  I  is a  r e a l q u a n t i t y , and the above s i g n d i f f e r e n c e can only a f f e c t the s i g n o f the imaginary  part of  I  , which i s zero.  So the expression f o r <CB"'C)(<f>,)  now becomes  <CB-^X^>---e >cwJj£^^  (6.25)  lk  I t i s simple t o show, u s i n g ( 6 . 5 b ) , t h a t the polynomial i n c u r l y brackets i s equal to  -(u/'-kri)  .  1  Equating the expression i n (6.25) to (B+<C> )<*,)  yields  m +*n  a!<s >) = - £fa;(o,*)( *-k*ip$(»i-k)dn  '  l  w  (6.26)  «-«>  Since  &  i s small, the approximation  only to 0(€ ) z  tano" -o" may be made.  i n (6.26) and s o l v i n g f o r & y i e l d s ,  Keeping terms  finally:  (6.27)  S( u>,x)(coi-krO §(ri-k)dri z  n>  Note t h a t the i n t e g r a l o f (6.27) and the i n t e g r a l o f the r e f l e c t i o n c o e f f i c i e n t problem (see (4.36) ) a r e i d e n t i c a l except t h a t the parameter k  i s i n the range  (4.36) . at  \=tu),  «yV<k<oJ i n ( 6 . 2 7 ) , but i n t h e range  I t has already been noted t h a t and that p o l e s o f ^(tjw,)^)  S(y\;co,il)  \6<k<u>/V  in  has no branch p o i n t s  l i e i n the ranges CO /^<Y(-<OO 1  Z  .  7.  CONSEQUENCES OF THE ALTERED DISPERSION RELATION . The f i r s t t h i n g to do i s to i n s e r t the expression f o r  d i s p e r s i o n r e l a t i o n (6.5a) .  Since  S i s 0 ( £*),  nY+cmiW<S) = m t a n or,' + m Sec*- rr, • S + 0 L C ) 4  I t i s assumed that  where  m , k , JL  and JL  m,k  & i n t o the  (7.1)  may be w r i t t e n as  rr\ - mo + m ,  (7.2a)  k ' k - + k,  (7.2b)  Jl=JL-»-JL  (7.2c)  are 0 (£) .  I n s e r t i n g (7.2a,b,c) i n t o the expressions  (6.5a.b.c) and equating terms o f 0 ( 1 ) y i e l d s  (7.3a)  m \ + k* = c o *  (7.3b)  k* = ^_lo  (7.3c)  which determine the zeroeth order d i s p e r s i o n r e l a t i o n f o r trapped waves. Equating terms o f 0 (£*")  gives  m,-- - ( - / w J k ,  (7.4a)  k  J , = + (>WJi.)k,  (7.4b) **J>.  (7.4c)  Using (7.4a,b) and the expression (IS.27) f o r & , and f u r t h e r s i m p l i f y i n g  gives,  finally  k, x where  -Isec*  I = I  The £(f\;ui,y)  m  »  o  _ (i-i-<o,*)(u>- t\k^ 1 Cri-tO d n  (7.6)  1  signs of the r e a l and imaginary p a r t s o f , since  (uf-kn) -1  been done i n s e c t i o n 5.  and  _(t\-k)2.0 .  I  depend on those o f  Most of the work has  already  From that work, i t i s known that  ^  _____  "t _ ^ ( ^ . x M ^ - k ^ Y _(rv)<} dr. +_ S(r _>,xHoo*-kn) l(n-k) dlrv  (7.7)  1  u  where the n o t a t i o n  " - ( p o l e s ! " i n d i c a t e s that the parts of the path which  are the indentations the i n d e n t a t i o n s  around the poles are l e f t out of the i n t e g r a t i o n  c o n t r i b u t e only to the imaginary part o f  p o s s i b l e to determine the s i g n of for  V)  l a r g e o j , the  tan^-vf  Re(l)  i n general,  I .  I t i s not  s i n c e , f o r example,  f a c t o r i n the f i r s t term of (7.7) may  s i g n s e v e r a l times over the i n t e r v a l  -to/x<Y{< w^/ir .  —  change  Again, from the  e a r l i e r work, i t i s known that  (7.8)  where (gee  (5.5)  POijV- -ir_L f | +  )  j  f jfV^-yii' +  WA*-^ - )"1 (to'- k r,? ic-v-k) 1  (7.-9)  and >/& -  t a n ^ - n ^ = KV^'-^/u  .  1  As i n the r e f l e c t i o n  coefficient  problem, vflm ( I ) < 0 . How i s the mean f i e l d  a f f e c t e d by these a l t e r a t i o n s t o m,k,,^  due to the c o a s t a l i r r e g u l a r i t i e s ?  To i n v e s t i g a t e t h i s , l e t k, = <* + i £ .  From (7.5) we then have Kzft)j _ UmflVsec^.  *L  /m.  and *m(l)  _  + sec" in.  (7.10a,b)  y'/A  where  R'e(l)  are given by (7.7) and (7.8) r e s p e c t i v e l y .  that  } £ > o , f o r k > 0 . Hence the 0  exp(iky)  term becomes  expfi(k.+«0y  Since  k„> 0  (7.11)  implies the wave i s t r a v e l l i n g i n the  apparent t h a t the o f propagation.  exp(-/9y)  The " e - f o l d "  =  =  +y d i r e c t i o n , i t i s  f a c t o r represents a decay i n the d i r e c t i o n decay length  the wave amplitude decays t o l/e  d  Note  d  ~ ' t h e d i s t a n c e over which  o f i t s value a t y=0 —  k.ft»n»./»i. + S«c*m. +/*//.)  i s , from (7.10)  (7.12)  From (7.4b), i t can be seen t h a t  /  = (k./i.)(=<  4  ip)  Therefore, i n the deep ocean region, the coherent wave takes the form  < d> ) z  e  • e  • e  (7.13)  - 59 -  So i n the deep region, i n s t e a d o f j u s t decay away from the s h e l f , there i s a small wavenumber component i n the x - d i r e c t i o n , with the net r e s u l t being a wave t r a v e l l i n g towards the coast —  i.e.  exp(-i(k /i„ )/3(x-l)-ia>t) . c  This represents a " t i l t i n g " o f the wave towards the coast, as i f the i r r e g u l a r i t i e s on the coast have caught one edge o f tihe wave c r e s t , slovjing down the coast edge o f the wave and making i t t i l t towards the coast. Corresponding to t h i s " t i l t i n g " e f f e c t , the phase and group v e l o c i t i e s are  a f f e c t e d by the c o a s t a l i r r e g u l a r i t i e s , with the i n t r o d u c t i o n o f x-com-  ponents o f v e l o c i t y i n the deep ocean:  (iw/H-fc/u)?)  y W < ^ + k.y)  (7.14b)  I t i s not p o s s i b l e t o say i n general whether the phase and group speeds' are  diminished or augmented by the presence o f c o a s t a l i r r e g u l a r i t i e s ,  because o f the u n c e r t a i n t y i n the s i g n o f oc , which has the complicated factor  Re'(I) .  Summarising, we have seen t h a t when the coherent, i n c i d e n t wave i s an 0(l)  trapped icfge wave, there are a number o f e f f e c t s .  There i s a t t e n -  u a t i o n i n the d i r e c t i o n o f motion, as wave energy i s scattered i n t o i n c o herent trapped and leaky modes o f power f l u x 0(C ) . X  The coherent wave  i n the deep ocean " t i l t s " towards the s h e l f , so t h a t there i s an 0(c ) z  coherent wave power f l u x onto the s h e l f from the deep ocean.  From sec-  t i o n 5, we would expect to have an 0(e ) coherent r e f l e c t i o n term. l  But  t h i s i s not the case, because the form we have chosen f o r the coherent wave does not allow f o r a r e f l e c t i o n term.  The  0(£*)  i n t o the incoherent f i e l d , and i s a leaky mode term.  r e f l e c t e d term goes  - 60 -  Some c a l c u l a t i o n s have been done f o r the northeast coast o f Japan, i n the same region as i n the power f l u x c a l c u l a t i o n s o f s e c t i o n 5.  Figure  7.1 shows the a l t e r a t i o n t o the lowest mode o f the d i s p e r s i o n r e l a t i o n . Figure 7.2 shows the a l t e r a t i o n t o the second mode.  I t can be seen t h a t  the changes are r a t h e r small, i n d i c a t i n g that there i s l i t t l e to the phase and group v e l o c i t i e s .  alteration  Figure 7>3 d i s p l a y s the e - f o l d decay  length as a f u n c t i o n o f the wavelength.  I t can be seen t h a t t h i s decay  length i s q u i t e long, i n d i c a t i n g that attenuation by the c o a s t a l i r r e g ularities i s slight. The negative r e s u l t s o f t h i s s e c t i o n can be a t t r i b u t e d mostly to the largeness o f  V (=10 f o r the northeast Japanese coast) i n the f o r 1  mula (7.5) f o r k, .  o.Cf  Figure 7.1  A l t e r e d d i s p e r s i o n r e l a t i o n , lowest mode f o r Japanese coast.  - 61  -  _  if vM) ' Figure 7.3  e - f o l d decay length f o r Japanese coast.  Note t h a t the minimum decay l e n g t h — f o r periods o f about  3/4 o f an hour.  i . e . the f a s t e s t decay —  occurs  8.  CONCLUDING REMARKS Methods have been presented f o r c a l c u l a t i n g various q u a n t i t i e s  i n t e r e s t i n the  study of ocean waves on a c o n t i n e n t a l  of  s h e l f i n the pre-  sence of a c o a s t l i n e which i s s t r a i g h t except f o r small i r r e g u l a r i t i e s . An  expression f o r the r e f l e c t i o n c o e f f i c i e n t of the  s e c t i o n 4.  The  coast i s found i n  fluxes of power from a wave i n c i d e n t from the deep ocean  i n t o edge waves trapped on the  s h e l f and  i n t o a continuous spectrum of  long, wave)radiation back to the deep ocean are c a l c u l a t e d i n s e c t i o n However, i t i s noted that the c a l c u l a t i o n s are v a l i d only f o r  incident  waves of wavelength considerably greater than the average s i z e of coastal i r r e g u l a r i t i e s —  i . e . f o r €U>«1  .  the  This l i m i t a t i o n a p p l i e s  the papers by Pinsent (1972), Howe and Mysak (1973) and Mysak and (1974) as  5.  to  Tang  well.  Edge waves trapped on the s h e l f have t h e i r propagation characteri s t i c s a l t e r e d by the presence of the c u l a t i n g the a l t e r e d d i s p e r s i o n ermined: and  the  c o a s t a l i r r e g u l a r i t i e s . By  r e l a t i o n , the f o l l o w i n g  the a l t e r e d phase arid group speeds, the  cal-  t h i n g s are  det-  e - f o l d decay length,  " t i l t i n g " of the deep ocean waves towards the coast.  In  the  example o f the northeast coast of Japan, the r e s u l t s are i n s i g n i f i c a n t mainly because  If , the r a t i o of the two 1  depths, i s f a i r l y l a r g e  (^=10).  Further work on edge waves i n the presence of random boundaries might proceed with a more r e a l i s t i c model, such as a constant slope s h e l f which has  random d e v i a t i o n s  f l a t deep ocean.  i n the  s h e l f r e g i o n , and  a sharp drop to  the  -  63 -  REFERENCES I. Aida, 1967, Water Level Oscillations on the Continental Shelf i n the Vicinity of Miyagi-Enoshima, Institute,  4J>,  Bulletin of the Earthquake Research  61-78 .  I. Aida, 1969, On the Edge Waves of the Iturup Tsunami, Earthquake Research Institute,  Bulletin of the  47, 43-54 •  W. Bascom, 1964, Waves and Beaches,  Anchor Books, New York.  V.T. Buchwald & R.A. de Szoeke, 1973, The Response of a Continental Shelf to Travelling Pressure Disturbances, Aust. J. Mar. Freshw. Res., 24, ' 143-158 . T. Hatori, 1965a, On the Alaska Tsunami of March 28, 1964, as Observed along the Coast of Japan, titute,  Bulletin of the Earthquake Research Ins-  4JL, 3994408'.  T. Hatori, 1965b^ On the Aleutian Tsunami of February 4, 1965, as Observed along the Coast of Japan, Bulletin of the Earthquake Research Institute,  41,  773-782 .  T. Hatori & R. Takahasi, 1964, On the Iturup Tsunami of Oct. 13, 1963, as Observed along the Coast of Japan, search Institute,  42,  Bulletin of the Earthquake Re-  543-554 •  M.S. Howe, 1971, Wave Propagation i n Random Media,  J. Fluid Mech., 4J>,  769-783 .  M.S. Howe & L.A. Mysak, 1973>> Scattering of Poincare Waves by an Irregular Coastline,  J. Fluid Mech.,  j>Z,  111-128 .  K. Kajiura, 1972, The Directivity of Energy Radiation of the Tsunami Generated i n the Vicinity of a Continental Shelf, Oceanographical Society of Japan, L.D. Landau. & E.M. Lifschitz,  28,  Journal of the  260-277 .  Fluid Mechanics, Addison-Wesley, 1959, p. 36.  P.H. LeBlond & L.A. Mysak, 1975* Trapped Coastal Waves and their Role in Shelf Dynamics, to appear i n The Sea, vol. 6, Wiley-Interscience Publishers, New York. L.A. Mysak, 1973, Notes on Random Functions and Differential Equations, for lectures in the Department of Mathematics, University of British Columbia. L.A. Mysak & C.L. Tang, 1974* Kelvin Wave Propagation along an Irregular Coastline, J. Fluid Mech., 6^,  241-261 .  H.G. Pinsent, 1972, Kelvin Wave Attenuation along Nearly Straight Boundaries, J. Fluid Mech.,  j>3,  273-2-6 .  F.E. Snodgrass, W.H. Munk, & G.R. Miller, 1962, California's Continental Borderland, Part I, F. Ursell, 1952, 214,  79-97 .  Edge Waves on a Sloping Beach,  Long Period Waves over J. Mar. Res.,  20, 3-30.  Proc. Roy. Soc, A,  

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