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UBC Theses and Dissertations

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 , Kingston, 1973 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the Department of Mathematics and the I n s t i t u t e of Applied 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 to the requ i r e ^ standard THE UNIVERSITY OF BRITISH COLUMBIA October , 1975 In present ing th is thes is in pa r t i a l fu l f i lment of the requirements for an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make it f ree ly ava i l ab le for reference and study. I fur ther agree that permission for extensive copying of th is thes is for scho la r ly purposes may be granted by the Head of my Department or by h is representa t ives . It is understood that copying or p u b l i c a t i o n of th is thes is f o r f i n a n c i a l gain sha l l not be allowed without my wri t ten permiss ion. Department of M/iTUEAy/r T! C ^ The Un ivers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 D a t e ("A T- 7 ,» 1*7 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 long waves on a continental shelf. Studied here are the gen-eration of trapped edge waves when a vjrve fron the deep coean reaches the i r r e g u l a r coast, 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 of trapped edge waves, due to the i r r e g u l a r i t i e s . The continental 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 st a t i o n a r y random function of distance along the coast, with zero mean. The wave equations are thus stochastic d i f f e r e n t i a l equations, with the randomness introduced through the boundary condition at the coast. Calculations are made for the power f l u x i n t o trapped edge waves and a continuous spectrum of leaky modes, both generated by the s c a t t e r i n g of an incident wave. Numerical r e s u l t s f o r a section of the northeast Jap-anese coast show that there i s less power t r a n s f e r r e d to the forward t r a v -e l l i n g trapped wave than the backward, and l e s s power to the scattered leaky modes than to e i t h e r the forwai-d or backward trapped modes. Other c a l c u l a t i o n s show that there i s attenuation of a trapped edge wave, due to s c a t t e r i n g , there i s a " t i l t i n g " of the wave towards the coast, and i n the case of the same Japanese coast, the longshore components of 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 of 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 periods -such shorter than t,he period associated with the C o r i o l i s parameter f, and f o r wavelengths 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 . - i i i -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 11 4. SCATTERING BY AN IRREGULAR COASTLINE OF A WAVE INCIDENT FROM THE DEEP OCEAN — DETERMINATION OF THE REFLECTION COEFFICIENT 19 5;. SCATTERING BY AN IRREGULAR COASTLINE OF A WAVE INCIDENT FROM THE DEEP OCEAN — POWER FLUXES INTO DIFFERENT MODES OF SCATTERED WAVES 37 6. INFLUENCE OF COASTAL IRREGULARITIES ON TRAPPED EDGE WAVES — ALTERED DISPERSION RELATION 46 7. CONSEQUENCES OF THE ALTERED DISPERSION RELATION 56' 8. CONCLUDING REMARKS w.6'2 References 63 - i v -FIGURES 2.1 Orientation of axes for shallow water equations 6 2.2 Conservation of mass 7 2.3 Single step model of ocean bottom 9 3.1 Ray diagram for c>V 15 3.2 Dispersion r e l a t i o n for l<c<y, *"=10 17 4.1 Branch cut configuration 27 4.2 Contours of integration . .30 4.3 Indentations of path of integration :.- . . . 3 0 4.4 Indentations of path of integration 30 5.1 Power fluxes on Japanese coast, for <^  =0.413 - period = 2 hr. . . . 4 2 5.2 Power fluxes on Japanese coast, (J =0.326 - period = 1 hr 43 5.3 Power fluxes on Japanese coast, co =1.652 - period = -J hr 43 5.4 Region of Japanese coast for calculations 44 5.5 Power spectrum for Japanese coast of Figure 5.4 44 7.1 Altered dispersion rela t i o n , lowest mode for Japanese coast 60 7.2 Altered dispersion relation, second mode, for Japanese coast . . . . 6 1 7.3 e-fold decay length for Japanese coast 6 l ACKNOWLEDGEMENTS I am indebted to Dr. L.A. Mysak for suggesting the topic of this thesis, and for his very helpful consultation and encouragement during the course of my work. My thanks also go to Dr. K. Kajiura of the Earthquake Research Institute of the University of Tokyo, who provided me with a great deal of material on tsunamis and edge waves on the north east coast of Japan. And thanks go to my wife for writing i n the equations by hand. This work was done while I was supported on a National Research Council of Canada Science '67 scholarship. - 1 -1. INTRODUCTION The behaviour of ocean waves on the continental shelf has received a great deal of attention during the l a s t two decades. This attention i s well deserved for p r a c t i c a l reasons. An improved understanding of the be-haviour of tsunamis and other large waves on the continental shelf could lead to a better tsunami early warning system or improved harbour design to lessen the destructive effects of large waves. Researchers at the Earthquake Research Institute i n Tokyo have devoted much e f f o r t to the study of tsunamis on the continental shelf of Japan — see, for example, Hatori and Takahasi (1964),- Hatori (1965a, 1965b, 1967), and Aida (1967,1969). The period of a tsunami i s of the order of a thousand seconds, accor-ding to Bascom (1964, p. 104), which i s v e r i f i e d by figures presented by Hatori and Takahasi (1964) for a part i c u l a r jtsilnami. So effects of the earth's rotation may safely be ignored i n mid-latitude regions, since the time scale associated with the C o r i o l i s parameter f i s of the order of ten thousand seconds. Thus, i n this work, the C o r i o l i s parameter f i s taken to be zero. When the single step model of a continental shelf i s adopted, the analysis of plane wave behaviour i s straightforward. Plane waves incident upon the shelf and originating from the deep ocean are refracted at the shelf edge i n accordance with Snell's Law; perfect r e f l e c t i o n takes place at the coast; and the wave i s refracted again at the shelf edge whereupon i t enters the deep ocean. Because of perfect r e f l e c t i o n , the energy f l u x onto the shelf region i s equal to the energy flu x to the deep ocean from the shelf region. Waves originating on the shelf may, under certain c i r -cumstances, be trapped on the shelf. That i s , there i s no energy flux to the deep ocean. These waves, known as "edge waves", obey a certain disper-sion r e l a t i o n which allows only a f i n i t e number of modes for waves of a given frequency. In t h i s work, there i s an analysis of the single-step straight 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 of an otherwise straight coast are introduced into the model — representing i n l e t s , promontories, bays, etc. there i s no longer perfect r e f l e c t i o n at the coast, i n the case of waves originating i n the deep ocean. As well as the p a r t i a l l y reflected wave, some of the incident energy flux i s transferred into d i f f u s e l y scattered waves which escape to the deep ocean and into edge waves. When edge waves travel along such a coastline, small i r r e g u l a r i t i e s i n the coast bring about attenuation of the edge waves and changed phase and group v e l o c i t i e s . An analysis of both cases, with numerical examples, i s pre-sented i n t h i s work. The h i s t o r i c a l roots of this problem have two branches. One branch began with Stokes 1 o r i g i n a l paper i n I846 on fundamental mode edge waves on a constant slope bottom p r o f i l e . Stokes was the f i r s t to recognize the existence of waves which travel 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 region near the coast. After more that a century, U r s e l l (1952) extended Stokes' work to include the whole spectrum of possible modes. There have been further generalizations since Ursell*s paper. Snodgrass et a l . (1962) introduced the single f l a t step model of the continental shelf i n order to investigate non-rotational (f=0) edge waves on the C a l i f o r n i a borderland. It i s this 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-graphically trapped by depth gradients of many different kinds provides a good overview of the theory of edge waves. The second branch of the h i s t o r i c a l roots of the present problem has a more recent beginning. Pinsent (1972) employed a second order ordinary perturbation expansion to two related wave problems i n a rotating sea (fjk)) of a nearly uniform depth bounded by a coastline which i s nearly straight. Pinsent examined the generation of Kelvin waves due to the scattering of a plane wave incident upon the irregular coast, and the attenuation of a Kelvin wave due to scattering from the irregular coast. A Kelvin wave i s one which travels p a r a l l e l to the coast and whose energy i s trapped against the coast by the rotation of the earth. However, Pinsent's results could not be applied to situations i n v o l -ving extensive coastlines because of the occurrence of secular terms. Howe and Mysak (1973) generalized Pinsent's (1972) work on the scattering of an incident plane wave i n order to deal with extensive coastlines. Their model was a f l a t bottom ocean with an essentially straight coast, except for 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 function of distance along the coast. The d i f f e r e n t i a l equation governing the wave behaviour thus were stochastic d i f f e r e n t i a l equations, because of the stochastic boundary condition at the coast. The method of solution was based, on the techniques of wave propagation i n random media as discussed,by Howe (1971)._ I t was found,: as i n Pinsent (1972), that a ; .Kelvin-,-wave is" generated,_as_ well.as a continuous spectrum of wave noise. Mysak-and Tang (1974) used the same model as Howe and Mysak (1973) and an operator expansion technique to generalize Pinsent's (1972) work on the effects of coastal i r r e g u l a r i t i e s on a Kelvin wave. The effects of - 4 -the coastal i r r e g u l a r i t i e s on the wave speed, energy flux and amplitude of a coherent Kelvin wave were determined. The present work examines the generation and propagation of edge waves on single step continental shelf with a coast which i s essentially straight except for small i r r e g u l a r i t i e s represented as a random function of distance along the coast. This work i s , to the author's knowledge, the f i r s t attempt to introduce the effect of coastal i r r e g u l a r i t i e s into the study of the generation and propagation of edge waves. I t i s also the f i r s t attempt to introduce the effects of ocean bottom topography into 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, rotat i o n a l effects are disregarded. 'Thus, neither Kelvin waves nor continental shelf waves appear i n the analysis. (A continental shelf wave i s a wave whose energy i s trapped on the shelf and which requires the presence of both rotation and a sh e l f ) . Also, i t i s assumed that there are no effects due to density s t r a t i f i c a t i o n of the ocean. I'm section 2, the shallow water equations are used to derive the basic d i f f e r e n t i a l equation for t h i s thesis. In section 3> edge waves — both trapped and leaky modes — on a f l a t continental shelf with a straight coast are examined. In section 4» an expression i s obtained for the r e f l e c t i o n coefficient of the coast. Section 5 deals with the transfer of energy from.the incident wave to the scattered f i e l d ; an expression i s derived for the power flux from the incident wave into the trapped edge waves and into a continuous spectrum of long wave radiation to the deep ocean. These quantities are calculated for the northeast coast of Japan. Section 6 deals with the al t e r a t i o n i n the dispersion r e l a t i o n of trapped edge waves, brought about by coastal i r r e g u l a r i t i e s . In section 7> expressions are obtained for the altered phase and group speeds, for the e-fold decay length of the coherent wave f i e l d , and for the wave " t i l t " towards the coast i n the deep ocean region. Again, the northeast coast of Japan i s used as an example. However, the effects of coastal 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 this example, because of the large ratio of deep ocean depth to shelf depth. Central to the calculations i n sections 4, 5, 6, and 7 i s a single integral. A l l calculations are v a l i d only for waves of wavelength much greater than the average size 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 calculations of Pinsent (1972), Howe and Mysak (1973), and Mysak and Tang (1974) as well. - 6 -2. THE SHALLOW WATER EQUATIONS Regions of the oceans i n the midlatitudes are considered here, so the period 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 that i s made here i s to consider only those water waves whose period i s much less than IS hours — or, i n terms of the angular frequency go' of the waves, i t i s assumed that co'»f . This assumption amounts to dropping the C o r i o l i s force term from the equations of motion. S t r a t i f i c a t i o n i s also neglected. Assuming that viscosity i s negligible, t h i s leaves only gravity and the pressure force. With the co-ordinate axes oriented as shown i n figure 2.1, k/ccfer J«ff(jce (mean sar-face a{ z=o) Figure 2.1 : Orientation of axes for shallow water equations.' the well-known "shallow water" equations rel a t i n g the x 1 and y' depth-averaged components of velocity u.', v' and the pressure p' are: o^u /cH' = - l/f (^p'/^Ti') . (2.1a) ' < a W « n ' ~ l l ? U p ' / ^ y 1 ) (2.1b) O = - 1/P U p'/^z') - g (2.1c) It i s assumed i n (2.1a,b,c) that the wave amplitude i s much less than the - 7 -wavelength, and that the water depth i s much less than the wavelength. Equation (2.1c) may be immediately integrated to give the pressure: p' = p: + 9 P(d> ' - 2 ' ) (2.2) where <f> i s the elevation of the water's surface above the mean surface z'=0, and p0' i s the atmospheric pressure at the surface z'=4>'. Therefore, (2.3a) (2.3b) provided p0' i s constant. Now consider the incompressible flow of water through a volume ele-ment fixed i n space, as pictured i n figure 2.2 . Volume. o( «/«cfer) s i « , in f fme oii. 1 (Volume, i r i ) / ^ (Volume. ou{) . (Volume- oat) Figure 2.2 Conservation of mass. - 8 -I t i s clear from figure 2.2 that the increase 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 increase 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. Therefore, or, <H'/<)t' + cK(h+4') u ' ) / ^ ' +dah-KpV')/ciy ,= O (2.5) Throughout the present work, h i s assumed to be constant. I f i t i s assumed that <i>'«K , then (2.5) may be written: &V/dV + hlbu'faf + Sis'/wl --O (2.6) Differentiating (2.6) with respect to t' gives: ^ f / ^ t ' ^ h ^ u ' A - x ^ t ' + ^ V / ^ y ' d t ' V ° (2.7) But d i f f e r e n t i a t i o n of (2.3a) by x« and (2.3b) by y» y i e l d ^ u ' / ^ t ^ - g ^ ^ / ^ X ^ ( 2 # 8 a ) S W ^ f : ~9^<*>'/c^'2 (2.8b) F i n a l l y , substituting (2.8a,b) into (2.7) gives the d i f f e r e n t i a l equation ^<J>'AH'*- - (SK) v' a4>'-o (2.9) Throughout the present work, a l l f l u i d motions are assumed to have a given fixed angular frequency u>'. So assuming a time factor of 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 basic d i f f e r e n t i a l equation which w i l l occur i n many situations i n this work. Note that when a time factor of exp(-icj't') i s assumed, (2.3a,b) give expressions for u 1 and v 1 i n terms of f : U ' = - ( ? 9 / C J ' U < ^ 7 ^ - X ' ) (2.11a) - - ( i . 3 / u > ' ( 2 . 1 1 b ) As mentioned i n the introduction, the model of the ocean bottom used i n th i s thesis i s a single step topography, i l l u s t r a t e d i n figure 2.3 Figure 2.3 ' '* Single step model of ocean bottom. - 10 -Both h 2 and h, are assumed to be small enough so that the "shallow water" approximation" holds for 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 of 200 metres and 2,000 metres I The variables may now be non-dimensionalized. Lengths i n the hor i -zontal plane w i l l be divided by the shelf width W, and time w i l l be nbn-dimensionalized with respect to W/s/gh, , the time of travel of a long wave across the shelf. The reader i s reminded that the phase and group speeds of a long wave on the shelf — a "shallow water" wave — are both equal to /gh,'. Lengths i n the v e r t i c a l direction — the v e r t i c a l disp-lacements of the various waves — are non-dimensionalized with respect to the shallow depth h, . Dimensional variables are denoted by a prime (*) throughout t h i s thesis, and non-dimensionalized variables are unprimed. In non-dimensional form, the equations for shallow water waves i n the shelf and deep sea regions are, from (2.10): V 1 ^ + cuz 4>, - O 0<%<L[ ^_oo<vj<+^ (2.12a) V 2-^ +(o;y^)4>i.= 0 1 <: X <+a° } -°o<y< (2>12b) where 5C2 =ha/h, > 1. - 11 -3. WAVES ON A CONTINENTAL SHELF, WITH A STRAIGHT COAST The mathematical analysis of waves i n the presence of the irregular coast i s best understood i f compared to the analysis of the straight coast problem. The l a t t e r i s now presented, based on sections II and III of 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). The boundary condition at the straight coast x=0 i s that the water does not penetrate the coastline. This may be expressed as zero velocity normal to the coast. That i s , the x-component of the velocity, u, equals zero for x=0. But since u'^lig/ai^Zpi/&x' , the boundary condition at x=0, i n non-dimensional form i s : c^ 4> y^-X - o at x=0 (3.1) There are two matching conditions at the shelf edge x=l. F i r s t of a l l , there must be continuity i n water l e v e l between the deep and shallow regions — that i s = 4>z at x=l (3.2) Secondly, there must be continuity of mass transport across the shelf edge. This i s expressed by h,u,' = h a i u 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, including at x=f oO. Following Buchwald and de Szoeke, l e t d)j = A i f-x)exp(iKb-it^t) • j=l , 2 (3.4) This represents waves t r a v e l l i n g along the shelf, p a r a l l e l to the coast. Substituting t h i s form into the d i f f e r e n t i a l equations gives: ^A./^x 1 + (to*-k*)A. -- o 0<x<l (3.5a) £ f \ J ^ -k^Aa = o x>l (3.5b) There are three cases of interest, corresponding to three ranges of the phase speed c=w/k: ? c > a , s > c > i , l>c . Case 1: c>y — leaky modes In.this, case, as i n the others, jcondition (3.1) at x=0 becomes: ^M^% - o at x=0 Now, **>1, and c=w/k>Y, so 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. Since oYa* - k > 0 , the solution i n region 2, the deep ocean, i s of the form A * M = 2 A c o s ( r r u ( - X - 0 + £ ) - 13 -where m£= -k* and A and £ are undetermined constants. The matching: conditions are: and M./^ -x = ^hJ^% at x=l. The f i r s t matching condition y i e l d s : ZC cos m, -- 2 A cos £ while the second gives: ZC m, Sin m, = & a A rr\a* sin £ Dividing the second by the f i r s t gives: hi, "tan nn, = m , f a n £ So £ i s now determined: £ = 1 7 (3.6) Also, C may be expressed i n terms of A: r - A c ? s £-~ A coa *«. So f i n a l l y , the wave f i e l d may be written: - 14 -ct>. ^,y,t) = 2 A ^ s i . cos^ m.-x) exp fik'i-iwt) 0<x<l (3.7a) 4>»(%,%i.) -• cos (m»(»-i) + £}exp(-iM-ito*} X>1 (3.7b) The following interpretation of the solutions may be made.' Put m,=o>cose, , mj=w/j cose a ,, k="sin e, . Note that these definitions of e, and © a are consistent with m,z = «*-k1t and with ma* =="VV-k%- -.The l a t t e r implies that k=( <%)sine i. Rewriting the cosine factors i n the solution (3.7a,b) gives: 4** ' AIH^i>, l"exp^ «(-x cose.* 1* siAe,-t))+exp(i<o(--xcose,*y si 'ne.-t^ tO<X^l (3.8a) (|>a = A £ex p( ^  f r cos et ••S si * e« - *t -»£V)+ex p(^ C-X cos » i + y sIn ©i - * t -£)$ , x >1 (3.8b) I t i s clear that the solution i s a superposition 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 figure 3.1 • The wave's phase speed (and group speed) i n the deep ocean i s X , i n non-dimensional units ( i n dimensional units, this i s K • v/gh! -Jh^/h, • v/gh. = J^u , which i s the long wave speed i n the deep ocean). The speed i n the shallow region on the shelf i s 1 ( i . e . /gh~, i n dimensional units). The angle of incidence e e i s related to the angle of refraction e> by cosin&.==k=(u/»)s±ne a— i . e . Xsine,=sine* , vrhich i s Snell's Law, since sin e./sin e»=( shallow speed)/(deep speed)=Vgh7/,/gh"*=l/tf . The quantity £ i s a phase lag, due to the passage of the wave back and forth across the shelf. To an observer moving along the coast with velocity c=u/k , the wave pattern i s frozen. So i t i s helpful to think of the sum of the incoming and outgoing waves as a standing Wave normal to the coast — the cos(m, x) and cos(ma(x-l)+.e) factors — and a t r a v e l l i n g wave along the coast, trav-- 15 -elling at speed c=u/k , the exp(iky-it)t) factor. 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 on. Case 2 — l ^ c * * — trapped modes In this the second case of interest, ooa > k * C S n e l l ' s Law does not ap-ply, since tfsine,=sin0t implies that sine.^l/a — but k a y<J/xz implies that sin9,>l/& . This case represents a wave originating on the shelf, with total internal reflection occurring at the edge of the shelf. If we put m,a=ol-k8 as before, and J>*=kZ- cuf/a* , then the boundary 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 = A cos m, exp(-ICK-D + ikS - i col) 0<X<1 x>l (3.9a) (3.9b) - 16 -Again, there i s a standing wave i n the x-direction, and a t r a v e l l i n g wave i n the y-direction. There i s exponential decay of the wave into the deep ocean region. The waves are edge waves, trapped on the shelf. The other matching condition, O^./^TC = 3* ^<t>*A* , . x=l implies that m, iar> m, = S*JL (3.10) With the definitions of m, and 1, (3.10) represents an i m p l i c i t dispersion r e l a t i o n o)=o(k). So i f the angular frequency, to , i s specified, then the longshore wavenumber, k , the longshore phase speed c^^/k , and the longshore group velocity Cg=do/dk may a l l be determined. Because of the tangent function, there i s an i n f i n i t e number of different modes. The graph of the dispersion r e l a t i o n for X2=10 i s presented i n figure 3.2 . The calculations and pl o t t i n g were done with the aid of a computer. Note that for a given co , there i s a f i n i t e number of allowable wave numbers, k. In the leaky mode case, for a given u> , there was a continuous spectrum of allowable \<ravenumbers, k, corresponding to angles of incidence between 0 andfr /2 .. But i n t h i s second case, there i s a discrete, f i n i t e spectrum of allowable wavenumbers. Edge waves with different wavenumbers, for a given w , w i l l be referred to as different edge wave modes. - 17 -a) f Figure 3.2 %, Dispersion r e l a t i o n for K c < x , 2=10. Case 3 — l>c — v i r t u a l modes o This case i s physically impossible. The reader can easily show that the solution i s <t>.(-x ^A cosh (rx) exp(-iKy- iuit) ct>a = A tosh r exp (-1 (x-il + - iwt) where r*=k*-a>a and i l=k a-w/V . In order to satisfy continuity of mass transport, i t i s necessary that - 18 -which i s impossible for r e a l r and positive X . So waves of long-shore phade speed c= a/k<l cannot exist. A l l waves must have a longshore phase speed c>l. - 19 -4. SCATTERING BY AN IRREGULAR COASTLINE OF AM INCIDENT WAVE FROM THE  DEEP OCEAN — DETERMINATION OF THE REFLECTION COEFFICIENT It i s assumed that the coast has bays, peninsulas and i n l e t s which are deviations from an otherwise straight coast, p a r a l l e l to the shelf edge, xvhich i s s t i l l assumed to be straight. The deviations are assumed to be small compared to the shelf width. The irregular coastline i s specified by where s(y) i s a stationary, random function of y with zero mean. When y i s fixed, a random function i s a random variable, with a probability distribution, over an ensemble of coastlines. Averages of quantities are taken over the ensemble of s t a t i s t i c a l l y equivalent coastlines. This may appear to 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 that the random function i s ergodic-, then ensemble averages are equal to averages over the length of the one r e a l coast. Ensemble averages are used to simplify computations of aver-ages over the length of the coast. To be stationary, a random function must have s t a t i s t i c a l properties independent of position, y — for example, the mean over the ensemble, denoted by ^ s ( y ) ^ , must be a constant, inde-pendent of y , and the autocovariance, (4.D R(y,D=<s(y)s(y+Y)>. (4.2) must be a function of only the lag, Y . In the present case, i t i s assumed that <s(y)>=0. The small size of the coastal i r r e g u l a r i t i e s i s expressed - 20 -by requiring £ = J<s*Ci)> <^ 1 — that i s , the average sized of the i r r e g u l a r i t i e s should be .small com-pared to 1, the shelf width. The boundary condition i s , again, zero velocity normal to the coast: u = JJ- ^s/^y on x=s(y) (4.3) Since s i s small, the boundary condition (4*3) may be expanded about the mean s=0, stopping at terms involving £ , and evaluating at x=0: u = xrScj - u* S -' Sa sScj on x=0 (4.4) I t should be noted at this point that (4.4) i s not v a l i d i f u*, v* , and u.x-x are large, o f f s e t t i n g the smallness of s . In fact, u*for example i s of the order of 1/A , where A i s the wavelength of the incident wave. So for very short wavelengths, (4.4) i s i n v a l i d . This i s a shortcoming common to Pinsent (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 thesis, i s v a l i d only for incident waves of wavelength much greater than the average size of the coastal i r r e g u l a r i t i e s . In the opposite case, when the wavelength i s much less than the average size of the i r r e g u l a r i t i e s , the WKB; method may be used. It i s useful to distinguish between the mean wave f i e l d , ^ ) , composed of the incident wave and the partial, r e f l e c t i o n , and the scattered wave - 21 -f i e l d , $ , which w i l l be shown to be composed of edge waves and leaky modes. The notation <•> represents, as above, an average taken over an ensemble of s t a t i s t i c a l l y equivalent coastlines. In a particular r e a l i -zation of the wave f i e l d , corresponding to a single, given coastline, the scattered wave f i e l d i s a correction to the mean f i e l d , so that 4> = < 4> > + $ (4.5) Note that <<*>>=( <*>+ $)= <f<*>> + <$>= <*>>+<$> , implying that <$> =0 . Similar partitions apply to the velocity components u. and v . By taking the ensemble average of the wave equations (2.1^2a,b), i t follows that v*<<*>,> * coe<*,>o > V*<£, =o for 0<x<l . (4.6a,b) v * < 4 w > * ^ < « 0 = o , --O for x>l (4.7a,b) Similarly, the two matching conditions separate: <<*>.> = <<*^> , 3, s & at x=l (4.8a,b) <^<J>.>A-x - fk+d/d*, &/dx at x=l (4.9a,b) The condition of finiteness at x=+s»° applies separately to the mean - 22 -and scattered f i e l d s . The coupling between the two f i e l d s occurs i n the scattering process at the coastline. The coupling i s introduced i n the mathematical analysis by the boundary condition (4«4) at x=0. Using the relations (2.11a,b) i n non-dimensional form — — the boundary condition (4.4) may be expressed i n terms of s and <t>, alone: = <WS.j ~ fc.xx'S - ' - | 4 W S l + ^ yx'SSy on F O (4.10) Following Howe and Mysak (1973)* this condition i s represented f o r -mally by £<*>. =G.<t>, + G**},-. (4.11) where £. , G,, G* are li n e a r operators. The operator X i s non-random, so that £<t>, =0 i s the boundary condition for a straight coast. The operators G. and Gj. involve the random function s l i n e a r l y and quad-r a t i c a l l y , respectively. So i n the present case, X = &/&-x G, = s,, V a y - s a'Aa* (4.12a) (4.12b) (4.12c) - 23 -I f the mean of a random li n e a r operator i s defined to be the same linear operator, with coefficients involving s replaced by the means of the coefficients, then clearly <(G,) =0, but <Gi>^0 since <s*}?4D. So Ga=<Gt> +G t , and G, =G, . Now the ensemble average of (4.11) i s taken: £ <<*>,) =<G.4>.) + <Gi<K> = <G,.<d>.>> + <G,&> +(G,<^>) + <G,*&> = <G,X*,> + <G,$-> + <Gt><4>,) +<G* $.> 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 subtracting (4.13) from i t y i e l d s : t X $. = G,<*> + S*<*>*c.G.$.-<a,$,)]+{'G.$.-<'G t$.>? (4.14) The method of'solution i s to assume i s of a known form, and to use thi s form to make (4.14) a boundary condition for $, , describing the generation of the random f i e l d by the interaction of the mean f i e l d with coastal i r r e g u l a r i t i e s . Then, together with the finiteness and - 24 -matching conditions, the d i f f e r e n t i a l equations for <S> can be solved i n terms of ($) . The resulting expression for can then be used to make (4.13) a boundary condition for <4>,> , i n terms of <<£.> only. Some approximations can be made, keeping i n mind that the object i s to use 4> to calculate , not to calculate <& i t s e l f . The quantities i n curly brackets i n (4.14) describe interaction processes of the scattered f i e l d and the coastal i r r e g u l a r i t i e s — i . e . multiple scattering effects. It would not be permissible to neglect these terms i n any calculation of $ i t s e l f . However, the boundary condition (4.13) for involves cor-relations such as <G, $> of the coastal i r r e g u l a r i t i e s and the scattered waves. Let the correlation scale of the coastal i r r e g u l a r i t i e s s(y) be L . Then very l i t t l e contribution to the correlation w i l l be made by waves of cp which have been scattered from points on the coast greater than a distance L from the point on the coast where <G,$J> i s to be eval-uated. But for s u f f i c i e n t l y small s(y), i . e . for£=</<s*> small, contri-butions to <t> which are taken at the point on the coast where xs to be evaluated, and which are due to multiple scattering must have been scattered at least a distance L away. So multiple scattering may be ne-glected when calculating the correlation products i n the boundary condition (4.13) for <4>> . That i s , the bracketed terms i n (4*14) may safely be dropped. Thus, for t h i s purpose, the following boundary condition for q> suffices: 1$, = G,<<b> +Q*<<&> on x=0 (4.15) A further approximation may be made by keeping only terms up to 0(£*) - 25 -i n the boundary condition (4.13) for (<b ) . In keeping with t h i s , only terms up to 0(e) need be kept i n the boundary condition (4.15) for $ , since higher order terms w i l l influence (cb) , through (4.13), t o 0 ( £ 3 ) or higher — because G, i s 0(e) and G a is0(£*). This means that (4.13) and (4.15) may be approximated by KM --<G,$> +<Q*)<ty (4.16a) £ & = G,,«b> (4.16b) Equation (4.16b) i s kno\m as the Born approximation. Inserting the expressions (4.12a,b,'c) into (4»l6a,b), and noting that <Gt>=-i(s') , gives <<!>,«) < s ^ - s & v ^ J £ at x=0 (4.17a,b) The form to be assumed for (<$>) , for use i n (4.17b), i s motivated by the leaky modes (3.8a,b), for the straight coast problem. I t i s assumed that there i s a wave incident on the shelf from the deep ocean; that i t i s refracted at the shelf; that p a r t i a l r e f l e c t i o n takes place at the coast, which introduces a r e f l e c t i o n coefficient into the analysis; and that the p a r t i a l l y reflected wave i s refracted out to the deep ocean. So, looking at (3.8a,b), i t i s assumed that =ATexp if-m.? + l<y-<oO +ATR, exp lfm.n+k!J-«ol) (4.18a) <4>*} r A exp i(-mtfx-A-fkvj-wt-£UAR2expaCrtM^-n + k'j-tjt+ e) (4.18b) where A i s the amplitude of the incident wave, T i s a transmission - 26 -coefficient, R, i s the r e f l e c t i o n coefficient as observed on the shelf, and R* i s the r e f l e c t i o n coefficient as observed i n the deep ocean. The two matching conditions at x=l may be used to express IU and T i n terms of R. . The condition <4>i)= at x=l implies that,: T Lexp(-im,)+- R, expGm,)] = C exp(-i.£.) + R a exp(ie)] The condition tf1^**) at x=l implies that: m.T texpf-im,)- R, £xp(inOj] - m^*. Lexp(-i£.)- Ra exp(i£)1 Solving for Rz and T i n terms of R. gives: n * X P (" 'UaWm^ R,exp(xovW (m,+ vV<^exp(-ia^ (4.19) P.« exp(i rv\,)+-exp(-i m.) (4* 20) The reader can easily check, using the formula tan£3=(m,tan m, )/( zfnO, that for a straight coast ( R, =1) -the expressions (4.19) and (4.20) reduce to: Ri= 1 and , T = cos£ /cos m, which corresponds to (3.8a) and (3.8b), the solutions for a straight coast. Now i s known except for R, . The boundary condition (4.17b) w i l l be used to f i n d $ i n terms of the unknown R, ; then (4.17a) w i l l be used, with the expression for 4> , to find R,. Substituting the expression for into the boundary condition (4.17b), - 27 -and dropping the amplitude A and the time dependence yields: 3,* 'T£ik<l + R.)s,, + m*(i+R.)s} e i k s at x=0 (4.21) I f i t i s assumed that $, and <k have a y-dependence of exp(iny), then since and satisfy the d i f f e r e n t i a l equations (4.6b") and (4.7b), t the following Fourier integral representations hold: $.= L C A j ^ e ^ ^ A . ^ e ^ ^ e ^ d n , OXx<l < k L A»,(T J) e +A,z(n)e > d n,x>l (4.22a) (4.22b) The branch cut configuration, must be decided upon, corresponding to the branch points at", =-(o (for v/^ '-co* - Jr^Zi-J^Uo ) and afn=t°/V (for Jt\ - ^A1" =^rl-t7V * Jti+^/x ). F i r s t the branch points 1= ^/y are considered. To the question of whether the cuts start at - ^/x and go into the upper or lower half planes, there are only four possible answers, represented by the four diagrams i n figure 4.1 » 1 +UI/JC I— 4-Figure 4.1 Branch cut configuration. - 28 -P o s s i b i l i t i e s 1 and 2 may be immediately eliminated, since, as the reader may check, for points \r\\>^/x ,, J<?~ ^ /xx - -JC-f)-)*- "Jfo* , even though nl=(-i?)z . This problem does not occur i n p o s s i b i l i t i e s 3 and 4. Another c r i t e r i o n must be used to decide between these two. Suppose configuration number 4 i s chosen. Suppose further that the integral (4.22b) i s to be evaluated by contour integration. Then i n the case y>0, the path must be closed above the r e a l axis as shown i n figure 4.2 so that the factor exp(i^y) i n the integrand makes the contribution from the semi-circular path go to zero as the radius R goes to i n f i n i t y . Similarly, i n the case y<0, the path must be closed below, as shown i n figure 4«2 . Because of the path segments around the branch cuts (see figure 4.2), there w i l l be i n the evaluation of (4.22b) a term propor-t i o n a l to exp(-i(w/#,)y—iut) for y>0 and a term proportional to exp(i( a JA )y-icJt) for y<0 . That i s , the choice of configuration 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 direction i n the region y>0 and i n the +y direction 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 clear v i o l a t i o n of the radiation condition, which requires waves scattered i n a region (near y=0 i n t h i s case) to travel away from, not towards, that region. Similar reasoning applied to configuration number 3 i n figure 4.1 shows that the radiation condition i s s a t i s f i e d i n this case. Identical arguments applied to the integral (4.22a) show that the branch cuts originating from n. =±&> have the same orientation as those originating from f\ =±<J>/v . Thus, the path of integration along the r e a l axis has indentations as shown i n figure 4.3 for either y>0 or y<0. It w i l l be shown l a t e r that the integrands of (4.22a,b) have poles - 29 -on the r e a l axis. The path of integration along the r e a l r\ -axis must be indented below the r e a l axis for positive poles and above the r e a l axis for negative poles. The argument for this i s as follows. Suppose the path i s indented above the real axis at a pole r\« > 0 . Then when (4.22a) i s evaluated by contour integration for y<0 — that i s the path i s closed below the r e a l axis — there w i l l be a contribution to (4.22a) which i s proportional to exp(i>vy-icot) . But this means that there i s a scattered wave t r a v e l l i n g towards the scattering region near y=0 , i n v i o l a t i o n of the radiation condition. When the path i s indented below the pole i>>0 no v i o l a t i o n of the radiation condition occurs. A similar analysis shows that the path of integration must be indented above poles i> < 0' • The path of integration, with indentations around branch points and poles, i s shown i n figure.4 . 4 • The boundedness condition on $ R c l e a r l y implies that Az,(i7)=0 for l l l > < 0 / 8 since exp(->/rf - ^/a 1 (x-l))-*°° as x . In the region I n I < W A , sAf~ "7? =Jn- % -A +% = -i>/*-n * <% +n = - i -n* . (The factor -1 i s obtained from an examination of the orientation of the branch cut at q= .) So there i s a factor i n the integrand of expi-lsf^/^ - rf ).-» representing 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 scattering occurs. This i s a v i o l a t i o n of the r a d i -ation condition unless the amplitudes of these components are a l l zero — that i s unless A*,(n) = 0 for \t\\<^ . So A*(n.) = 0 for a l i a . The two matching conditions (4.8b) and (4.9b) at x=l may now be Fourier transformed (with respect to y) and used to solve f o r A„(n) and A,i(n.) i n terms of A»(rO. The continuity condition at x=l implies AnM e x P ( + yfif^FU A , t W exp( -Vr f - -co* ) = A * a ( n) (4.23) - 30 -Figure 4.2 'CJontburs'of integration. -u) uJ/Y CO Figure 4.3 .Indentations of path of integration. es-^ • Figure 4.4 Indentations of path of integration. - 31 -The continuity of mass transport condition implies A < l ( a ) ^ / ^ - £ + y T ^ - A 1 1 ( n l ^ / ^ ^ e a V " = - W t f - ^ V A w C r O (4.24) Solving (4.23) and (4.24), we find M r t W r ( ^ z J v - o * A ~ f r ° (4.24b) Using (4.24a,b) i n (4.22a), we obtain: Recall that the boundary condition has already been calculated to be: %,t - - T ( i k ( i + R,)^+^(n-R,)53eik>} at x=0 (4.21) The Fourier transform of (4.21) with respect to y must be equal to the integrand of (4.25) . In th i s way, kzi(n) can be evaluated i n terms of R, The Fourier transform of (4.21) i s : ^ r+°° w Y = WTl»+R.U^,rKMKrof 3S e.'x d Y Equating t h i s to the integrand i n (4.25) y i e l d s : J-«o F (n) - 32 -where F(n) =(Jh%-uS- - frVn1- " V )e ^" ° - ( N / V - < ^ + *7"nv W V ) e + Substituting t h i s into (4.24a,b), and using the l a t t e r i n (4.22a,b), i t A A i s f i n a l l y possible to write cp, and 4\ i n terms of R,: (JV-WTu+iue 1 ' J X O . • V v - W V x d (4.26a) a 11 C\ § Lco^-knJse d<\<i\ ^ T d + a ^ i e F ( M x>l (4.26b) The reader i s reminded that these expressions for d> do not take into account multiple scattering and are only accurate to 0 (£)_.- However, i t i s noteworthy that the expressions (4.26a,b) show that trapped edge waves are generated i n the scattering process. For, i f there i s a pole A at f\ = t^ o , say, then the expressions for 4> evaluated at the pole include a factor exp(iky)exp(-i( r\<>-k)(-y)) = exp(it>y) — that i s , the pole at-r^ corresponds to a t r a v e l l i n g wave with longshore wavenumber rj0 . Where are the poles? They are located where the denominator of the integrand i s zero — i . e . where: That i s , y ^ - ( e ' 7 n - e j - - ^ 7 l l - " V ( e +-e J which implies +anhx/^? * - K V o N ^ V (4.27) - 33 -Now consider different regions for n, on the rea l axis. For 1^^^/^ , i t i s also true that l i 1 uP- . Therefore, using \Ay- to1, = - i Vui"-- K1 , A 1 - ^/V = - i v/ 0/^ -h 1 and tanh(iz) = i tan z , ( 4 . 27 ) becomes >/ol-tf -Van v>u^- ^  -- - i K\A°V- ^  (4.28) Since the l e f t side of (4.28 i s r e a l and the right side i s imaginary, poles are impossible i n the regions where n.*^- °/V on the r e a l axis. For r?> , the square roots i n (4.27) are a l l r e a l and positive, and now the l e f t side i s positive, while the right side i s negative. So there are no poles on the r e a l axis i n the regions where >uit . Fi n a l l y , i n the regions where wVi l^n x< to 1 , (4.27) becomes, upon changing v V ~ <-oz to -i\A>1- n.1 and using tanh(iz) = i tan z , n/co1-^ 4 a o 7 ^ - a l * * V i x - (4.29) Comparing th i s to the trapped mode dispersion r e l a t i o n (3.10), i t i s clear that solutions I to ( 4 . 2 9 ) , for a fixed to , are the wavenumbers of the trapped modes. So (4.26a,b) include trapped mode edge waves as part of the scattered radiation. Now expression (4.26a) for 4>. , i n terms of R, , may be used i n the boundary condition (4.17a) for <<&,) : <*.«>--<$i,S,)-<$1x»S>-W<4>0„II)<s*> , at x=0 (4.17a) The terms of (4.17a) are now evaluated one by one. From (4.18a),, and r e c a l l i n g that the factor A exp(-iwt) has been dropped from a l l terms, - 34 -<<tO n T l - i + fl,^, e.{*\ at x=0 (4.30) Also, <4w><S^ - U W C - l + f U e ^ <s*>, at x=0 (4.31) Using (4.26a), <$,^S^V-4ivT(ii-R),)e^M.0o2c'Xiu31^Lu l-k JUal-<S(Y)Sv)(^)e(Vk1(V"dndY1 at x=0 (4.32) where b(ry,oo,jQ O W ^ - s V i f + ( 7 ^ 1 ? q W - ^ ' l e / . I (4.33) But, using the d e f i n i t i o n of the power spectral density $ of an ergodic stationary random function s , where R(v) = <(s(z)s(z+v)) i s the autocovariance of s } and using the fact that s(Y>s,(y) = Vtf><s(Y)s(y)> = %jR(Y-y) = -R» (Y-y) i t follows that (4.32) may be written ( ^ ^ ^ T c i + R ^ e ' ^ i i ^ ^ l ^ - k O - ^ n - O i ^ , at x=0 (4.34:) The remaining term i n (4.17a) may be written ($uxS)--T(i+RAe' k^S(n;^)L(o l-k nT^-^)?rn-^ , at x=0 (4.35) - 35 -Now (4 .30), (4.31), (4.34) and (4.35) may be used to write the boun-dary condition (4.17a) for (<$>,) i n terms of R, : iX(- H-R.'U.e ^ ' T d + R.^e^i &«v,«o,»[u*-k»i] [ -^^^(n-Wdi i+Vt iTn?. (-i + R , V k s <.sx> -oO Keeping terms only to 0 ( 6*) , and keeping i n mind that I(;f) =0 (6" ) , the reader can easily show that: (4,36) I t was noted, after the boundary condition expansion (4.4)> that the computations i n this section are i n v a l i d for waves of a very short wave-length. This d i f f i c u l t y appears i n (4.36), for although 'H^e1-, the polynomial (a? -nk)* <*• co* , § ("aito,*)** l/-o » and l/m.^l/co , making the correction term proportional to £ 2 . I f co i s too large, then the con-di t i o n £lcox< < 1 , required for the boundary condition ( 4 . 4 ) , w i l l be violated. In addition, the magnitude of the r e f l e c t i o n coefficient,IR< I w i l l become greater than 1 , which i s physically impossible. The l a t t e r d i f f i c u l t y i s encountered i n the next section. The result (4.36) can be compared to the expression for the r e f l e c t i o n coefficient obtained by Howe and Mysak :;(1973). Recall that they included rotational effects (f)A)) i n their study of waves on a flat-bottom ocean i n the presence of an irregular coastline. I f f i s set equal to zero i n the r e f l e c t i o n coefficient expression of Howe and Mysak (1973), and i f X i s set equal to 1 i n the present expression (4.36) (corresponding to a flat-bottom ocean), then i t can easily 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 to the expression of Howe and Mysak (1973). - 36 -One further point i s worth noting about (4.36). There are no branch points at r\ = i c o — the singularities there are removeable. For i f the exponentials i n (4.33) are expanded in a power' series in w -uf for <a close to <^> or - to , i t i s found, that with the l / j r t - ^ factor taken into account, only even powers of v/r\r~-u^ appear in the expression for S(i^;to,jf). That i s , the square root V _ us- disappears, and only (n.1"-^) appears. - 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 1 and u 1 are the dimensional pressure and x-velocity) i s the power flux of the mean f i e l d <<t>> i n the -x direc-tion. When evaluated at the coast, x=0, the expression i s the work per unit time per unit cross-sectional area done by the mean wave f i e l d on the coast. Since the coast does not move, a l l of this energy goes into the scattered waves. Thus, the power flux of the scattered waves i s -(p')<u') . A,more useful .quantity i s the power flux averaged over one cycle of the mean f i e l d , represented -(p'^u 1) . F i r s t p' and u.' are evaluated i n terms of 4>' : P ^ / W , u«=-(ig/6o')4>;. However, since these are re a l quantities, i t i s necessary to take only the r e a l parts: p:i= |pg<j>» + c.c. , u'= -(ig/o')*,. + c.c. where "c.c." stands for the complex conjugate. Now using (4.18a) for the mean f i e l d (cj>,>, i t follows that <P'> ^ f 9 h , AT( i +R,U i k V+c.c.") <u'> ^ M h . ^ ' A T O - M e ^ 1 1 + c c- J at x=0 (5.1a,b) Therefore, at x=0, representing complex conjugates by * , we have - 38 -F i n a l l y , the average over one cycle of the mean f i e l d i s taken: at x=0 (5.2) I t i s now a simple matter to show that the power flu x of the incident wave at the coast x=0 , averaged over one cycle i s gPg1" h*Al (m.'/o1) | T| * — simply set R,=0 to focus attention on only the incident wave. There-fore, the portion of the incident power flux which goes into the scattered f i e l d i s (l-IR.I*) . Now, putting then, correct to 0 ( e * ) , Then = *Vm , (T-T"V W I T * Therefore, (5.3) - 39 -where vl2<m(l) i s the imaginary part of I . In order to continue, i t i s necessary to d e t e r m i n e ( I ) . I t i s clear from (4.36) that the determining factor i s S (n.;w,a) i n the integrand, since the power spectral density 3>(t\-k) and the polynomial '(ccf-kn.)1 are both positive definite quantities. F i r s t of a l l , i t i s clear that % (r\;u>,&) i s r e a l i n the regions where (\x >co?- , since? a l l of the square roots have positive arguments. In the regions where ^//^trftLS2- , and for ^ ' where >"\i =a pole of '•<£*, Using the formula exp(ie) = cos © + i sin© , this can easily be shown to reduce to $ ( n ; u a , » v C g g ^ ^ V n ^ V +an V ^ - n O ' (5.4) which i s obviously a r e a l quantity, i n the ranges where 0 3 , 1 ^ ^ LO*- . However, there i s a contribution to the imaginary part of I i n this range — from the poles, where >/u>l-iy- tan V w1 - r\i = Ka\/rf - ^ A"1- . Suppose i s a pole l y i n g between W A and » s ay« Then i n order to evaluate the contribution to the integral I , the path of integration must be indented below the r e a l axis for nj > 0 . Representing the integrand by f(n)/g(n) , where g(*\) = w^1 -a7- tan Vu* -t\v - frVn?- WVV- > and the function f(a) = (remaining factor i n integrand) , and representing a point on the indentation by a = % + r exp(ie) (-^e^O) , the contribution from the indentation i s - 40 -Letting r-»0 gives a contribution to I from the pole n.j of Inserting the appropriate expressions for f and g gives, for the imaginary contribution iP(ni^ -ti>• \)\ + 1__ (xV^aT » v A v ^ v 7 VansA^F\l- (uf-K*\f• I ( « ) (5.5) For poles i n the range - o ^ n ^ , the path indentations are above the real l i n e , and the expression for P(^j) i s i d e n t i c a l , except for a positive sign replacing the negative sign i n front. For a given co ,, there i s a f i n i t e number of poles, ; as can be seen i n figure 3.2 for the dispersion re l a t i o n . So the t o t a l contribution to the scattered fraction of the power flux from the poles i s -(4/m.)22p( nj) , which i s simply the sum of the power fluxes of a l l the trapped edge waves that can exist for a given frequency. It i s worthwhile to check that the sign of -(4/m, )P(qj) i s positive. Referring to (5.4), the complicated factor i n curly brackets i s obviously positive, provided tan^co 1 -Hj*> 0 — but tans/cof-tv*1 = - /V^ -by the dispersion rela t i o n , and this i s positive. The power spectral den-s i t y i s a positive quantity, and the factor on the l e f t of the big curly brackets i s -^ '/^  when n.j >0 and +1>;/)J when c 0 — i . e . this factor i s negative. The remaining factor, (u>l-k r^)1- i s positive, obviously. So, altogether, P( ) i s negative, which makes - 41 -Another contribution to Jj<rn (I) comes from the regions where rf ^  . In this region, i t i s a simple matter to show that S v. i s V ^ V ^ - ^ = = = = = = _ _ (5.6) I t i s immediately apparent that the imaginary part of (5.6) i s negative. The contribution to the power flux i s The quantity D i s a sum over a continuum of longshore wavenumbers ' ^ l ^ 6 0 / ^ , of waves which escape from the shelf to the deep ocean. Summarizing, the portion of the incident power which i s scattered by the coastline i s made up of two parts: waves trapped on the shelf, and waves escaping to the deep ocean. Represented as a fr a c t i o n of the incident power flux, t h i s i s 1 - l f U * = - 4 / m , P ( r ^ + D (5.8) With the aid of a computer, the rather complicated expression (5.8) has been calculated for the special case of the northeast Japanese coast, from about 38°N latitude to about 39.5°N latitude, a distance of 200 km. The shelf depth h, i s taken to be 200 m., and i s taken to be 2000m., so that #*=10. The shelf width i s 21 km. Figures 5.1,5.2,and 5.3 show the results for incident wave periods of 2 hours, 1 hour and \ hour, corresponding to to = 0.413, 0.826, 1.652, respectively. "Figure 3.2 shows, that only the lowest mode edge waves can be generated. The t h i r d case, - 42 -to =1.652, i s obviously- i n v a l i d , since l-lR,| a must be less than 1. There i s a secularity i n the expression for 1- IR,I* — the 0 5 termi i s proportional to to* , as noted at the end of section 4. So the expres-sion for I-IR-I* i s v a l i d only for Cu?^*- 1 . The reader w i l l r e c a l l the assumption made i n section 2 that co f — i . e . co t» f /(TghT /W)*" — w h e r e f i s the C o r i o l i s parameter. So taken together, the theory i s v a l i d i n the region $/(J$K,/\*/) ^ CO >/« (5.9) For the northeast Japanese coast, £ ' =2.60 , and f/(/gh," /W)=.05 , leaving a rather narrow region of v a l i d i t y ...05 ^ c o ^ 2.60 . -5 o.olt (5 O.OIZ 4 „ 0.00s J X u I o a . 1 OOOf <ti B O — I — 30 T o t a l s e a f+e reef porter fluyc 60 — i -90 naf* of mci'o/ence- «+ shelf e-dje. , in otegree-s . Figure 5.1 'Power fluxes on Japanese coast, for tu=.413 — period =2 hr. - 43 -Figure 5.2 Power fluxes on Japanese coast, u) =.826 — period=l hr. a n g l e - o-f inc.ide.nce- a4- sh&lf , i n o / e g r e e s Figure 5.3 Power fluxes on Japanese coast, o> =1.652 — period=ghr. - 44 -O c e a n map SCaJe — 1:2,000,000 Figure 5.4 Region of Japanese coast for calculations. /too i i. •4-0 1100 ?00 ^ 400 —r-.0? .12. Figure 5.5 .Of cycles /k11"»ne.+re s Povrer spectrum for Japanese coast of Figure 5.4 - 45 -It i s noteworthy that for 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 remains f a i r l y constant across the whole range of angles of incidence. This power flux of waves scattered to the deep ocean i s i n each case less than both the backward and forward scat-tered trapped edge waves. There i s greater, energy scattered backward than forward, i n both cases. The sharp drop i n the back scatter and forward scatter i n Figure 5.2 i s due to p e c u l i a r i t i e s of the power spectrum of the Japanese coast. Figure 5«4 shows the part of the Japanese coast for which calcula-tions were done. Figure 5.5 shows the power spectrum 1(f) of the Jap-anese coast fof Figure 5.4 • 6. INFLUENCE OF COASTAL IRREGULARITIES ON EDGE WAVES TRAPPED OM THE SHELF  —ALTERED DISPERSION RELATION Sections 4 and 5 discussed the scattering of a long wave which i s incident from the deep ocean. In this section i t i s assumed that a trapped mode wave has been generated on the shelf by scattering or some other f o r -cing mechanism. In the absence of coastal i r r e g u l a r i t i e s , the dispersion r e l a t i o n for such waves i s (3.10), which w i l l be referred to as the zeroeth order dispersion r e l a t i o n . The mechanism of generation of the wave i s not discussed here, but i t could be due to an,earthquake, on the shelf (e.g. see Kajiura (1972.) ). - I t I s expected that coastal i r r e g u l a r i t i e s w i l l cause some scattering and hence attenuation of the trapped wave. Indeed, t h i s i s the case, as w i l l be demonstrated i n section 7. Other alterations to physical quantities, such as an altered phase speed w i l l also be discussed i n section 7. The f i r s t step i s to determine the alteration due to the coastal i r r e g u l a r i t i e s of the dispersion r e l a t i o n GJ = co(k) , i m p l i c i t -i n (3.10) . A somewhat different approach from that used i n the r e f l e c t i o n coef-f i c i e n t problem i s employed here. Unlike the former problem, the scattered f i e l d ^ w i l l not enter into the calculations here. The object i s to f i n d the alteration to the dispersion r e l a t i o n of a coherent trapped wave. The method employed i s that of Mysak and Tang (1974). The mean f i e l d <<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\> =**<<!>*> , at x=l (4.9a) In addition, the solution (<p) must be f i n i t e for a l l x and y . The boundary condition at the coast i s the same condition as before — zero normal velocity — although i t i s expressed somewhat d i f f e r e n t l y . I t i s assumed that this boundary condition can be expressed as ( B + C ) * , ='o , on x=0 (6.1) where B) i s a deterministic linear operator, and C i s a random line a r operator. Note that i n the absence of coastal i r r e g u l a r i t i e s , C=Ov, and so B <<t>,>=0 . This i s the case of zero x-velocity at the coast, so B=*/bx . I t i s assumed that the solution for a coherent trapped wave i s of a certain form, and this i s substituted i n the equations and boundary and matching conditions to determine a dispersion re l a t i o n . The solution i s assumed to be of the form: <<J>.> = eosCm-x + c H e ^ , ( H » l (6.2a) < O . V A e - ^ - ° ^ , x>l (6.2b) The time factor exp(-icot) i s again dropped, for convenience. With S =0 and m,k,l r e a l , this would be the solution for a straight coast, C=0. The quantity S i s assumed to be small. With coastal i r r e g u l a r i t i e s , Sfc) and the wave numbers may be complex. We assume that Re(!)">0 , so that the wave i s trapped; also, we take i2oTa(k)>0 and Re(k)>0 so that a wave t r a v e l l i n g i n the +y-direction 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 condition (4.8a) implies cos(m+o") = A (6.3) and the second (4.9a) implies m sin(m+cO = f l A (6.4) Substituting (6.2a,b) into the d i f f e r e n t i a l equations (4.6a), (4.7a) and dividing (6.4) by (6.3) give the familiar results, with one small change: m tan (rn+cS") = )$*A (6.5a) rr^ + k 1 -- oo* (6.5b) a l l + k* = ( 6 . 5 c ) I f <S were zero, (6.5a,b,c) would be the dispersion r e l a t i o n (3.10) for unattenuated trapped modes with a straight coast. The equations (6.5a,b,c) would determine a dispersion r e l a t i o n ^=io(k) i f S were known as a function of co and k . The boundary condition (6.1) at x=0 w i l l be used to determine S . The f u l l boundary condition — zero normal velocity at the coast — i s , as i n the r e f l e c t i o n coefficient problem, ;U- xrs^ on x=s(y) (4.3) Upon expressing u and v i n terms of <j> and expanding (4.3) at x=0 to 0(ez ) , the following expressions for the linear operators B and C are obtained: 8= Vane , at x=0 (6.6a) C ' C s a V s ^ W i S ^ - S ^ d ^ O f e * ) , at x=0 (6.6b) Note that once again this is only valid for long wavelengths — i.e. EbJ^l Now suppose there were a deterministic forcing function F(y) on the right side of (6.1): (B+C) <t>,= F(3) , on x=0 (6.7) Then assuming (B+C) can be inverted, (6.7) yields: <<$>>-- <(8+cT'> F , x>0 (6.8) which implies that (16+C)")*'<^.) : , on x=0 (6.9) Writing (B+C)"1 as (I+B"'cr B'f , using the binomial expansion (since 0=0(6) ) and averaging, gives <(B+C)-'> ~- ( ( I + B - ' c V 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 , taking the inverse of th i s l a s t expression, and dropping terms of 0(£*) and higher gives: <(B+cr)~ 1 = 6+<C)-<C8-,C>J*-<C>B~'<c> (6.10) So, setting F(y)=0 , C8 + <C> + <c> B-<c> - <C B~'C>)<<*>,) = 0 , on x=0 (6.11) The l i n e a r operator <C> has quite a simple form. Since t(s^s}= ^R(0)/iy •= 0; for a stationary random function, and since <s> = 0 and <s4> = <sM) = 0 , < C W < s * > ^ (6.12) Now, B~' i s an integral operator which maps a function, say f ( y ) , defined on the boundary x=0 onto a function defined on the whole region x^O . In terms of a Green's function G , 8"-Pon=] G ( * , y ; z ) ^ d 2 . (6.13) It follows that ^ ' A x * 4 ^ ' / d * 1 + G , = O , O i x d (6.14a) ^Gi/a-x* +i IG,iA« 1 +u,V»«Gn.= o , x>l (6.14b) E G . = ^ C,/dx = cffy- ' i) , X=0 (6.14c) G, = Gi. , at x=l (6.14d) A G | , / i % s f tf1^*/^* , at x=l (6.14e) As well, there i s the boundedness condition at x=+«^ , and the radiation condition applies. The <£-function represents a source of waves at y=z, and x=0 . Now introduce the Fourier transform, G , of G with respect to y: G(*A',a) --\Q(lL,<i\*) e~ i n < 1 dcj (6.15) Then (6.14a,b,c,d,e) put related conditions on G : S~ ^ - C nf-co') G. = o , 0*oc<l ' (6.16a) & /b-f - Cri x- "Vjr*) G, * = o , x>l (6,l6b>) <^  G), /^x = e ~ m * , on x=0 (6.16c) G, = G* , at 35*1 (6. l6d) , at x=l (6. l6e) The radiation condition applies to waves generated by the & -function source at y=z, x=0 — i . e . waves must travel away from the point of generation. In addition, the finiteness condition ensures that G^ i s bounded as x-s>+°°. From (6, l6a,b), i t can be seen that: Q,= c „ e * K , i e , Goc^l (6.17a) G,--C*,e. + Cz*e , x>l (6.17b) The branch cuts are oriented i n the same way as i n section 4, for the same reasons as i n section 4. In (6.17a,b), the c^ are functions of and z , As i n the r e f l e c t i o n coefficient problem, the finiteness condition on G*. - 52 -forces c 2 1 = 0 for r\*-> , and the radiation condition ensures that ©t, = 0 for ^ . So Cw(r\,"a^=0 far a l l r\ (6.18) Applying (6.16c) to (6.17a) gives the equation N/rf-u?' ' C M -Vt^-u^-Cn. - e~ l n* (6.19) The f i r s t matching condition, (6.l6d) implies e c,, + e c . = c « (6.20) The second matching condition (6.l6e) gives , v / r f - t o ' 1 . . _ v V - ^ r _ . . . Vr\ l-w*e c.-VrWu* e c,*--frVi-C- 1 0'^ c t l (6.21) Now, (6.19), (6.20) and (6.21) are three linear equations for the three unknowns c-,,, c,4, c t t . I t i s a simple matter to show that C ,h>Bh^ ( < ^ f - ^ V ^ V k -.--7=4= (6.22a) c « c ^ ~ - v ^ r ( 6 - 2 2 c ) where F(n) = KJ^J- *Vrf-<«>7»* ) e " ( n / F ^ ? ^ v V - ^ V ) e 4 " 7 ^ 1 ^ ] as i n section 4. It w i l l be shown that the integral for j~ involves no branch points at i\ = ±<o, as i n the previous problem. The expression for G may now be written, by inverting (6.15): J - o O That i s , putting d<j(n.) = Cijexp(inz) , !d„(t\)e + dlt(rye j e d ^ , Ooc<l (6.23a) G . ( W > W f ^ ^ e - ^ - W ^ ' ) e + i ^ 4 - l , ^ , *>1 (6.23b) I t should be noted at this point that, for the same reasons given i n the r e f l e c t i o n coefficient problem, indentations of the path of integration around poles and the branch points r \= t t o A are made below the real axis for rpO and above the r e a l axis for r ^ O . Now a l l the terms of (6.11) can be evaluated. It has already been shown that B= <V<iTC , at x=0 <C>--Ks*>^ x , at x=0 Therefore, <C>B~'<C> i s 0(6*) , and so i t may be dropped from the c a l -culations. The f i r s t t;vo terms of (6.11) are: (B + < C > ^ < * . > --(-ms-.nS+ m3/a<S*>s^cO (6.24) The d i f f i c u l t term i s (CB^C > <$,>. I t should be remembered that i n eval-uating CB-,C g(x,y) , C takes g(x,y) and converts i t to a function of y only — (Cg)(y); then B"' applied to (Cg)(y) produces a function of both x and y (B r lCg)(x 1y); and C applied to this produces a function of y only — (CB"'C g)(y) . Carefully following this step by step procedure and retaining terms up to Q ( £ l) , we obtain the following expression for this term: J - o o Now note that Taking the averaging operation < • > inside the integral signs and applying i t to the terms i n the integrand thus yields: where the obvious identity § (tiJw,a) = d„(rv) + d^(r\) (see (4.33) ) has been used. The evaluation of <CB"'C><$> uses the d e f i n i t i o n of the power spectral density and the property that - 55 -The careful reader w i l l notice that the sign of the exponential i n the integrand of the d e f i n i t i o n of I i s positive here, but was negative i n section 4 • However, there i s i n fact no difference, since I i s a real quantity, and the above sign difference can only affect the sign of the imaginary part of I , which i s zero. So the expression for <CB"'C)(<f>,) now becomes <CB-^X^>---e l k >cwJj£^^ (6.25) I t i s simple to show, using (6.5b), that the polynomial i n curly brackets i s equal to -(u/'-kri)1 . Equating the expression i n (6.25) to (B+<C> )<*,) yields m +*n a!<sl>) = - £fa;(o,*)(w*-k*ip$(»i-k)dn ' (6.26) «-«> Since & i s small, the approximation tano" -o" may be made. Keeping terms only to 0(€z) i n (6.26) and solving for & yi e l d s , f i n a l l y : S(n>u>,x)(coi-krOz§(ri-k)dri (6.27) Note that the integral of (6.27) and the integral of the r e f l e c t i o n coefficient problem (see (4.36) ) are id e n t i c a l except that the parameter k i s i n the range «yV<k<oJ i n (6.27), but i n the range \6<k<u>/V i n (4.36) . I t has already been noted that S(y\;co,il) has no branch points at \=tu), and that poles of ^(tjw,)^) l i e i n the ranges CO1/^<Y(-<OOZ . 7. CONSEQUENCES OF THE ALTERED DISPERSION RELATION . The f i r s t thing to do i s to insert the expression for & into the dispersion 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 C4) (7.1) It i s assumed that m,k and JL may be written as rr\ - mo + m, (7.2a) k ' k - + k , (7.2b) J l = J L - » - J L (7.2c) where m , k , JL are 0 (£) . Inserting (7.2a,b,c) into the expressions (6.5a.b.c) and equating terms of 0(1) yields m \ + k* = c o * (7.3a) k* = (7.3b) ^_lo (7.3c) which determine the zeroeth order dispersion r e l a t i o n for trapped waves. Equating terms of 0 (£*") gives m,-- - ( k - / w J k, (7.4a) J , = + (>WJi.)k, (7.4b) **J>. (7.4c) Using (7.4a,b) and the expression (IS.27) for & , and further simplifying gives, f i n a l l y k, x - I s e c * m o » where I = I _ (i-i-<o,*)(u>1- t\k^  1 Cri-tO d n (7.6) The signs of the r e a l and imaginary parts of I depend on those of £ ( f \ ; u i,y) , since (uf-kn) 1-- and _(t\-k)2.0 . Most of the work has already been done i n section 5. From that work, i t i s known that _____ ^ "t _ ^(^.xM^-k^Y _(rv)<} dr. +_uS(rV)_>,xHoo*-kn)1 l(n-k) dlrv (7.7) where the notation "-(poles!" indicates that the parts of the path which are the indentations around the poles are l e f t out of the integration — the indentations contribute only to the imaginary part of I . I t i s not possible to determine the sign of Re(l) i n general, since, for example, for large o j , the t a n ^ - v f factor i n the f i r s t term of (7.7) may change sign several times over the i n t e r v a l -to/x<Y{< w^/ir . 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- )"1 (to'- k r,? ic-v-k) (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 affected by these alterations to m,k,,^  due to the coastal i r r e g u l a r i t i e s ? To investigate t h i s , l e t k, = <* + i£ . From (7.5) we then have Kz ft) j *L _ UmflVsec^. _ (7.10a,b) /m. + sec" in. y ' / A where R'e(l) and *m(l) are given by (7.7) and (7.8) respectively. Note that } £>o , for k 0 > 0 . Hence the exp(iky) term becomes expfi(k.+«0y (7.11) Since k„> 0 implies the wave i s t r a v e l l i n g i n the +y direction, i t i s apparent that the exp(-/9y) factor represents a decay i n the direction of propagation. The "e-fold" decay length d ~ ' t h e distance over which the wave amplitude decays to l/e of i t s value at y=0 — i s , from (7.10) d= = k.ft»n»./»i. + S«c*m. +/*//.) (7.12) From (7.4b), i t can be seen that / = ( 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, instead of just decay away from the shelf, there i s a small wavenumber component i n the x-direction, with the net result being a wave t r a v e l l i n g towards the coast — i . e . exp(-i(kc/i„ )/3(x-l)-ia>t) . This represents a " t i l t i n g " of 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 of tihe wave crest, slovjing down the coast edge of 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 " effect, the phase and group v e l o c i t i e s are affected by the coastal i r r e g u l a r i t i e s , with the introduction of x-com-ponents of velocity i n the deep ocean: (iw/H-fc/u)?) y W < ^ + k.y) (7.14b) I t i s not possible to say i n general whether the phase and group speeds' are diminished or augmented by the presence of coastal i r r e g u l a r i t i e s , because of the uncertainty i n the sign of oc , which has the complicated factor Re'(I) . Summarising, we have seen that when the coherent, incident wave i s an 0(l) trapped icfge wave, there are a number of effects. There i s atten-uation i n the direction of motion, as wave energy i s scattered into inco-herent trapped and leaky modes of power flux 0(CX) . The coherent wave i n the deep ocean " t i l t s " towards the shelf, so that there i s an 0(cz) coherent wave power flux onto the shelf from the deep ocean. From sec-t i o n 5, we would expect to have an 0(el) coherent r e f l e c t i o n term. But this i s not the case, because the form we have chosen for the coherent wave does not allow for a r e f l e c t i o n term. The 0(£*) reflected term goes into the incoherent f i e l d , and i s a leaky mode term. - 60 -Some calculations have been done for the northeast coast of Japan, i n the same region as i n the power f l u x calculations of section 5. Figure 7.1 shows the alteration to the lowest mode of the dispersion rel a t i o n . Figure 7.2 shows the alteration to the second mode. I t can be seen that the changes are rather small, indicating that there i s l i t t l e a l t eration to the phase and group v e l o c i t i e s . Figure 7>3 displays the e-fold decay length as a function of the wavelength. It can be seen that this decay length i s quite long, indicating that attenuation by the coastal i r r e g -u l a r i t i e s i s sl i g h t . The negative results of this section can be attributed mostly to the largeness of V1(=10 for the northeast Japanese coast) i n the for-mula (7.5) for k, . o.Cf Figure 7.1 Altered dispersion r e l a t i o n , lowest mode for Japanese coast. - 61 -_ if vM) ' Figure 7.3 e-fold decay length for Japanese coast. Note that the minimum decay length — i . e . the fastest decay — occurs for periods of about 3/4 of an hour. 8. CONCLUDING REMARKS Methods have been presented for calculating various quantities of interest i n the study of ocean waves on a continental shelf i n the pre-sence of a coastline which i s straight except for small i r r e g u l a r i t i e s . An expression for the r e f l e c t i o n coefficient of the coast i s found i n section 4. The fluxes of power from a wave incident from the deep ocean into edge waves trapped on the shelf and into a continuous spectrum of long, wave)radiation back to the deep ocean are calculated i n section 5. However, i t i s noted that the calculations are v a l i d only for incident waves of wavelength considerably greater than the average size of the coastal i r r e g u l a r i t i e s — i . e . for €U>«1 . This l i m i t a t i o n applies to the papers by Pinsent (1972), Howe and Mysak (1973) and Mysak and Tang (1974) as well. Edge waves trapped on the shelf have their propagation character-i s t i c s altered by the presence of the coastal i r r e g u l a r i t i e s . By c a l -culating the altered dispersion r e l a t i o n , the following things are det-ermined: the altered phase arid group speeds, the e-fold decay length, and the " t i l t i n g " of the deep ocean waves towards the coast. In the example of the northeast coast of Japan, the results are i n s i g n i f i c a n t mainly because If1 , the ratio of the two depths, i s f a i r l y large (^=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 shelf which has random deviations i n the shelf region, and a sharp drop to the f l a t deep ocean. - 63 -REFERENCES I. Aida, 1967, Water Level Oscillations on the Continental Shelf in the Vicinity of Miyagi-Enoshima, Bulletin of the Earthquake Research Institute, 4J>, 61-78 . I. Aida, 1969, On the Edge Waves of the Iturup Tsunami, Bulletin of the Earthquake Research Institute, 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, Bulletin of the Earthquake Research Ins- titute, 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 Insti- tute, 41, 773-782 . T. Hatori & R. Takahasi, 1964, On the Iturup Tsunami of Oct. 13, 1963, as Observed along the Coast of Japan, Bulletin of the Earthquake Re- search Institute, 42, 543-554 • M.S. Howe, 1971, Wave Propagation in Random Media, J. Fluid Mech., 4J>, 769-783 . M.S. Howe & L.A. Mysak, 1973>> Scattering of Poincare Waves by an Irreg-ular Coastline, J. Fluid Mech., j>Z, 111-128 . K. Kajiura, 1972, The Directivity of Energy Radiation of the Tsunami Generated in the Vicinity of a Continental Shelf, Journal of the  Oceanographical Society of Japan, 28, 260-277 . L.D. Landau. & E.M. Lifschitz, 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 in 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 Bri-tish 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 Boun-daries, J. Fluid Mech., j>3, 273-2-6 . F.E. Snodgrass, W.H. Munk, & G.R. Miller, 1962, Long Period Waves over California's Continental Borderland, Part I, J. Mar. Res., 20, 3-30. F. Ursell, 1952, Edge Waves on a Sloping Beach, Proc. Roy. Soc, A, 214, 79-97 . 

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