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UBC Theses and Dissertations

The wave field on a shelf resulting from point source generation, with application to tsunamis King, David Randall 1978

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TEE HAVE FIELD CN A SHELF .RESULTING FBCM POINT SOOBCE GENERATION, KITH APPLICATION TO TSUNAMIS by DAVID SANCALL KING M. A. t S t a t e U n i v e r s i t y c f C a l i f o r n i a , 1967 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE BEQUIBEMENTS FOB THE DEC-BEE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES (Department of P h y s i c s and I n s t i t u t e of Oceanography) He accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVEBSITY OF BRITISH COLUMBIA September, 1978 Q David R a n d a l l King, 1978 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f The U n i v e r s i t y o f B r i t i s h C o l u m b i a 2 0 7 5 W e s b r o o k P l a c e V a n c o u v e r , C a n a d a V 6 T 1W5 D a t e i i ABSTEACT S t u d i e s were Biade to determine the wave f i e l d of long s u r f a c e g r a v i t y waves generated on a s h e l f by a l o c a l i z e d source, such as might occur i n tsunami generation. Beth a n a l y t i c a l and experimental a s p e c t s of the problem are i n v e s t i g a t e d . An a n a l y t i c a l model i s c o n s t r u c t e d , which examines the f i e l d of s m a l l amplitude shallow water waves propagating on a s h e l f of uniform depth. The s h e l f r e g i o n i s separated frcm a deep water r e g i o n of uniform depth by a l i n e a r v e r t i c a l step. The problem i s s o l v e d on a r o t a t i n g c o o r d i n a t e system, although the more p r e c i s e r e s u l t s are obtained f o r wave f r e q u e n c i e s much gr e a t e r than the i n e r t i a l frequency f o r the case of non-r o t a t i o n . Both exact s o l u t i o n s and asymptotic s o l u t i o n s i n the f a r - f i e l d are found f o r p o i n t source e x c i t a t i o n . The case of z e r o - r o t a t i o n i s i n v e s t i g a t e d f o r both tiae-harmonic and impulse e x c i t a t i o n . The experimental model examines the f i e l d of shallow water waves generated under c o n d i t i o n s chosen t o simulate the a n a l y t i c a l model f o r the case of n o n - r o t a t i o n . The r e s u l t i n g f i e l d i s composed o f c y l i n d r i c a l d i r e c t and r e f l e c t e d waves and of a plane l a t e r a l wave which a r i s e s under c o n d i t i o n s c f t o t a l r e f l e c t i o n . Have speeds are found which are i n agreement with those p r e d i c t e d by ray theory and g e o m e t r i c a l o p t i c s . Wave amplitudes of the d i r e c t and r e f l e c t e d waves behave as c y l i n d r i c a l waves with f r i c t i o n a l damping. The l a t e r a l wave amplitude decays i n accordance with the - 1 . 5 power of the ray o p t i c s pathlength of the l a t e r a l wave i n the deep water, with i i i damping due to f r i c t i o n . Agreement of theory with o b s e r v a t i o n i s o v e r a l l very good. The r e s u l t s are a p p l i e d to the problem of tsunami generation on a s h e l f under simple g e o m e t r i c a l c o n d i t i o n s , A r r i v a l - t i m e s of the v a r i o u s f i e l d c o n s t i t u e n t s at var i o u s p c i n t s along the s h o r e l i n e (the i n n e r edge of the s h e l f ) are c a l c u l a t e d and the l e a d time of the l a t e r a l wave a r r i v a l over the d i r e c t wave i s determined. i v TABLE OF CONTENTS ABSTBACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i i L I S T OF FIGLI EES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v i ACKNOWLEDGEMENTS x i i 1« XKTRODUCTION « • * • * • * • • * • • • • * • * * • • • • • • • » * * « * • * • » • • • * * • » » 1 2. THE MATHEMATICAL MODEL 10 2 . 1 . T i m e - h a r m o n i c Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2 . 1 . 1 . The D i r e c t Wave 13 2 . 1 . 2 . The R e f l e c t e d Have . , 21 2. 1 .3 . The L a t e r a l Save . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2. 1 . 4 . T h e E f f e c t s o f flotation 64 2 . 1 . 5 . Have A r r i v a l - t i m e s 76 2 . 1 . 6 . T h e A p p l i c a b i l i t y o f L i n e a r N o n - d i s p e r s i v e T h e o r y 82 2 . 2 . I m p u l s e E x c i t a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 2 . 3 . D e s c r i p t i o n o f t h e Wave F i e l d . . . . . . . . . . . . . . . . . . . . . 89 3. THE EXPERIMENTAL MODEL 95 3 . 1 . E x p e r i m e n t a l P r o c e d u r e 98 3 . 2 . E x p e r i m e n t a l R e s u l t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 3 . 2 . 1 . The D i r e c t Wave 111 3 . 2 . 2 . The R e f l e c t e d Wave 127 3 . 2 . 3 . The L a t e r a l Wave . 1 3 3 3 . 2 . 4 . The E f f e c t s o f D i s p e r s i o n . . . . . . . . . . . . . . . . . . . . . 146 3 . 2 . 5 . Summary o f t he E x p e r i m e n t a l R e s u l t s . . . . . . . . . . . 151 4. CONCLUSIONS . . . 1 5 4 REFERENCES 157 APPENDIX A : T h e Me thod o f S t e e p e s t D e s c e n t s , 164 APPENDIX B: The Wave Tank ................................ 174 APPENDIX C: The Wave Generator . . 181 APPENDIX D: The Have Absorbers ............................186 APPENDIX E: The Have Detector ............................191 APPENDIX F: E l e c t r o n i c C i r c u i t r y ., 209 APPENDIX G: Data A n a l y s i s ................................216 v i L I S T OF F I G U R E S F i q u r e 1. A) R a y p a t h d i a g r a m f o r t h e l a t e r a l w a v e ....... 4 B) R a y p a t h d i a g r a m f c r l a t e r a l wave a n d t o t a l l y r e f l e c t e d wave . . . . . . . . . . . . . . . . . . 4 F i g u r e 2, S h e l f g e o m e t r y a n d r a y p a t h s f o r t h e d i r e c t a n d r e f l e c t e d w a v e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 F i g u r e 3. C o m p l e x p l a n e o f i n t e g r a t i o n . . . . . . . . . . . . . . . . . . . 19 F i g u r e 4. S h e l f g e o m e t r y u s e d t c f i n d r e f l e c t i o n a n d t r a n s m i s s i o n c o e f f i c i e n t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 F i g u r e 5. T w o - s u r f a c e d R i e m a n n S h e e t 2 8 F i g u r e 6 . , C o m p l e x p l a n e o f i n t e g r a t i o n a n d F i g u r e 7. C o m p l e x p l a n e u s e d t o d e t e r m i n e r ~ 5 — : — > t h e s i g n o f Im>|n - s i n © f o r t h e t w o R i e m a n n S h e e t s . . . . . . . 3 2 F i g u r e 8. C o m p l e x p l a n e o f i n t e g r a t i o n w i t h s t e e p e s t d e s c e n t p a t h a n d b r a n c h p o i n t c o n t o u r .......... , 46 F i g u r e 9. S h e l f g e o m e t r y w i t h r a y p a t h d i a g r a m o f t h e l a t e r a l wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 v i i F i q u r e 10. Time-distance curves f o r the d i r e c t and l a t e r a l waves f c r a s h e l f geometry 77 Fi g u r e 11. Bavefront and r a y diagram f o r the wave f i e l d c o n s t i t u e n t s a r i s i n g f o r c\> © c............... , 92 Figure 12. Wavefront diagram f o r the source l o c a t e d cn the s h e l f edge 94 Fi g u r e 13. Have tank geometry used i n method A ........... 100 Figu r e 14. Have tank regime used.in method A ............. 101 Figu r e 15. Save tank geometry used i n method B ........... 104 Fi g u r e 16. Have tank regime used i n method B 105 Figu r e 17. Recorded wave p r o f i l e s f o r the d i r e c t wave shewn f o r i n c r e a s i n g values of R. ........................ 112 Figure 18. Recorded phase speed as a f u n c t i o n c f d i s t a n c e R 115 Fi g u r e 19. Time-distance curve f o r the d i r e c t wave ....... 116 Fi g u r e 20, A r r i v a l - t i m e o f wave energy as a f u n c t i o n of d i s t a n c e 117 v i i i f i g u r e 2 1 . E n e r g y o f t h e w a v e p a c k e t , m e a s u r e d by d e t e c t o r T ( , a s a f u n c t i o n o f d i s t a n c e f r o m s o u r c e ....... 1 1 9 F i g u r e 2 2 . E n e r g y o f t h e wave p a c k e t , m e a s u r e d b y d e t e c t o r T^, a s a f u n c t i o n o f d i s t a n c e f r o m s o u r c e ....... 120 F i g u r e 2 3 . S q u a r e o f t h e a m p l i t u d e o f t h e l e a d i n g w a v e a s a f u n c t i o n o f E f o r d e t e c t o r T, . . . . . . . . . . . . . . . . . . . . . . . 1 2 4 F i g u r e 24. S q u a r e o f t h e a m p l i t u d e o f t h e l e a d i n g w a v e a s a f u n c t i o n o f R f o r d e t e c t o r T z . . . . . . . . . . . . . . . . . . 125 F i q u r e 2 5 . P r o p o r t i o n o f t h e t o t a l e n e r g y c o n t a i n e d i n t h e l e a d i n q w a v e a s a f i n c t i o n o f B . . . . . . . . . . . . . . . . . . . 126 F i g u r e 2 6 . A r r i v a l - t i m e o f t h e r e f l e c t e d w a v e a s a f u n c t i o n o f R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 F i g u r e 2 7 . A r r i v a l - t i m e o f t h e r e f l e c t e d wave a s a f u n c t i o n o f R' 131 F i q u r e 28. R e f l e c t e d wave a m p l i t u d e a s a f u n c t i o n o f fi' .. 132 F i g u r e 2 9 . H a v e f i e l d o n t h e s h e l f w i t h t h e e m e r g e n c e o f t h e l a t e r a l wave 134 F i g u r e 3 0 . Wave p r o f i l e s o f t h e l a t e r a l a n d r e f l e c t e d w a v e s a f t e r e l e c t r o n i c r e m o v a l c f t h e d i r e c t w a v e ........ 1 3 5 i x F i g u r e 31, Sh e l f geometry and ray path diagram f o r the l a t e r a l wave ......................................... 137 Figure 32. The l a t e r a l wave a r r i v a l - t i m e aa a f u n c t i o n of B ....................................... 138 Fi g u r e 33. A r r i v a l - t i m e of l a t e r a l wave along a s u r f a c e of constant phase 141 Figure 34, L a t e r a l wave amplitude v a r i a t i c n with I pathlength W% . . . . . . . . . . . . . . . . . . . . . 143 Figure 35. V a r i a t i o n of wave amplitude along the l a t e r a l wavefront ........................................ 144 Figure 36. V a r i a t i o n i n l e n g t h 1 as a f u n c t i o n of B ...... 150 Figure 37, Change i n wave packet p r o f i l e due t c d i s p e r s i o n ................................ 152 Figure A-1. Path of s t e e p e s t descent i n the. ccnplex G-plane .......................................... 173 Fi g u r e B-1. Have tank geometry ........................... 175 Fi g u r e B-2. Detector c a r r i a g e mechanism 178 X F i q u r e B-3. apparatus f o r measuring water depth and f r e e s u r f a c e l e v e l 180 Figure C-1. The wave generator . 182 Figur e C-2. Wave generator a c t i v a t i n g c i r c u i t ............ 184 Fiq u r e £-1, B e f l e c t i o n s o f f v a r i o u s wall-absorber c o n f i g u r a t i o n s •.•>.•............,...,....,... ............ 188 Fiq u r e D-2. Wave tank absorber c o n f i g u r a t i o n ............. 190 Fig u r e E-1. ,Wave probe ................................... 192 Fiqure E-2. Wave d e t e c t o r , f r o n t view .................... 193 Fiqur e £-3. Wave d e t e c t o r , s i d e view 194 Fiqure E-4. Schematic of wave d e t e c t o r v o l t a g e r o u t i n g ... 196 Figure E-5. Wide range c a l i b r a t i o n curve f o r d e t e c t o r s T, and T z ...................................... 199 Fig u r e E-6. Fine r e s o l u t i o n c a l i b r a t i o n curve f o r d e t e c t o r T f . . * . , . , . * . . . . . . . . . ' . * . . . . . . . . . . * : . . . . . * 201 Figure E-7. Dynamic c a l i b r a t i o n apparatus ................ 205 x i F i g u r e E-8. Dynamic c a l i b r a t i o n d e t e c t o r response ........ 208 F i g u r e F-1, Detector a m p l i f i e r e l e c t r o n i c s and s i g n a l r o u t i n g ............. * ............................. 210 Figure F-2, A m p l i f i e r network ............................211 Figure F-3. Have t r a c e s of the t o t a l and d i r e c t wave f i e l d s with the d i f f e r e n c e between the two found e l e c t r o n i c a l l y .... ....................................... 2 14 Figure f-4. E l e c t r o n i c s and r e c o r d e r t a b l e ............... 215 Figure G - 1 . , D i g i t i z e d curves of processed data ........... 218 Fi g u r e G-2. Wavefront geometry f o r c y l i n d r i c a l waves ..... 220 Figu r e G-3. P r o f i l e of the wave packet ...................221 x i i ACKNOWLEDGEMENTS I w o u l d l i k e t o t h a n k t h e many p e o p l e who p r o v i d e d t i m e , e q u i p m e n t a n d a d v i c e w h i c h a i d e d i n t h e c o m p l e t i o n o f t h i s t h e s i s . I n p a r t i c l u a r , I w o u l d l i k e t o t h a n k my s u p e r v i s o r . D r . P.H. L e B l o n d f o r h i s c o n s t a n t a v a i l a b i l i t y , i n t e r e s t a n d i n d u l g e n c e , D r . R.W, E u r l i n g a n d D r . L . A . M y s a k f o r t h e i r c o m m e n t s a n d a d v i c e , a n d D r . G.L. P i c k a r d , D i r e c t o r c f t h e I n s t i t u t e o f O c e a n o g r a p h y , f o r h i s c o n s i d e r a t i o n s a n d a i d i n a c q u i s i t i o n o f f i n a n c i a l s u p p o r t . My t h a n k s a l s o go t o D r . W.C. Q u i c k a n d t h e s t a f f o f t h e h y d r a u l i c s l a b o r a t o r y o f t h e D e p a r t m e n t o f C i v i l E n g i n e e r i n g f o r t h e i r a i d a n d e q u i p m e n t w i t h w h i c h t h e e x p e r i m e n t a l p h a s e c f t h i s t h e s i s w a s p e r f o r m e d , t o M r . D. H o l m e s o f t h e D e p a r t m e n t o f E l e c t r i c a l E n g i n e e r i n g f o r h i s a i d i n d e v e l o p i n g t h e wave d e t e c t o r a n d f o r t h e u s e o f t h e e l e c t r o n i c e q u i p m e n t w i t h w h i c h t h e m e a s u r e m e n t s w e r e r e c o r d e d , a n d l a s t l y , t o M i s s G r a c e K a m i t a k a h a r a , who c o n t r i b u t e d o f h e r f r e e t i m e t o a i d i n many p h a s e s o f t h i s p r o d u c t i o n , i n c l u d i n g t h e d a t a a n a l y s i s a n d FMT p r i n t i n g o f t h e t h e s i s . I a l s o e x p r e s s my g r a t i t u d e t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a f o r i t s f i n a n c i a l s u p p o r t t h r o u g h a U n i v e r s i t y F e l l o w s h i p a n d t h r o u g h s e v e r a l t e a c h i n g a s s i s t a n t s h i p s w i t h t h e D e p a r t m e n t o f P h y s i c s . 1 1, I N T R O D U C T I O N When a f i e l d o f l o n g s u r f a c e g r a v i t y w a v e s i s g e n e r a t e d f r o m w i t h i n a l o c a l i z e d r e g i o n o f t h e o c e a n , t h e w a v e s p r o p a g a t e a w a y f r o m t h e s o u r c e r e g i o n w i t h e v e r - w i d e n i n g f r o n t s . T h e f o r m o f t h e s e w a v e s d e p e n d s n o t o n l y u p o n t h e wave s o u r c e , b u t u p o n t h e o c e a n i c b a t h y m e t r y a s w e l l . . T h e s i z e a n d s h a p e o f t h e s o u r c e r e g i o n a s w e l l a s t h e m a g n i t u d e a n d d u r a t i o n o f e x c i t a t i o n d e t e r m i n e t h e i n i t i a l w ave f i e l d . F a c t o r s s u c h a s o c e a n i c b o u n d a r i e s , c o n t i n e n t a l s l o p e s , w a t e r d e p t h a n d t h e e a r t h ' s r o t a t i o n a c t t o i n f l u e n c e t h e f c r m o f t h e p r o p a g a t i n g f i e l d . I f t h e s o u r c e r e g i o n i s l o c a t e d o n t h e c o n t i n e n t a l s h e l f , r e f l e c t i o n s a n d r e f r a c t i o n s r e s u l t a n d t h e d e s c r i p t i o n o f t h e f i e l d b e c o m e s more c o m p l i c a t e d . T h e f i e l d o n t h e s h e l f i s made up o f w a v e s c o m i n g d i r e c t l y f r o m t h e s o u r c e a n d w a v e s r e s u l t i n g f r c m i n t e r a c t i o n o f t h e d i r e c t wave w i t h t h e v a r i o u s b o u n d a r i e s . R e f r a c t e d w a v e s m o v e o f f t h e s h e l f a n d c o n s t i t u t e a f i e l d i n t h e d e e p w a t e r . T h i s t h e s i s e x a m i n e s t h i s w a v e f i e l d a n d a t t e m p t s t o d e t e r m i n e t h e i n f l u e n c e o f s u c h f a c t o r s a s s h e l f g e o m e t r y , r o t a t i o n o f t h e e a r t h a n d mode o f t e m p o r a l e x c i t a t i o n o f t h e s o u r c e . I n g e n e r a l , w a v e p r o p a g a t i o n i s m o s t c o m m o n l y s t u d i e d i n t e r m s o f a s u p e r p o s i t i o n o f p l a n e w a v e s . T h e u n d e r s t a n d i n g o f t h i s t h e o r y i s s u f f i c i e n t t h a t many f e a t u r e s o f w a v e p r o p a g a t i o n i n t h e o c e a n c a n b e a d e q u a t e l y r e p r e s e n t e d u s i n g t h e m e t h o d o f g e o m e t r i c a l o p t i c s . W h i l e t h i s m e t h o d i s a d e q u a t e i n many c a s e s , t h e r e d o e x i s t c o n d i t i o n s u n d e r w h i c h i t f a i l s . F o r a w a v e f i e l d a r i s i n g f r o m a s p a t i a l l y c o n f i n e d s o u r c e , t h e w a v e f r o n t s do n o t a p p e a r p l a n e . E v e n a t l a r g e d i s t a n c e s , t h e a s s u m p t i o n t h a t t h e 2 w a v e f r o n t i s p l a n e i s o n l y a p p r o x i m a t e , a n d i f a s u f f i c i e n t p o r t i o n o f t h e w a v e f r o n t i s o b s e r v e d , i t w i l l b e s e e n t o b e c u r v e d . When s u c h w a v e s i n t e r a c t w i t h a p l a n e b o u n d a r y , t h e d i f f e r e n c e i n g e o m e t r i e s m a k e s t h e u s e o f t h e p l a n e w a ve t h e o r y i n a d e q u a t e : c e r t a i n p h e n o m e n a a r e l e f t u n e x p l a i n e d . S i m p l e g e o m e t r i c a l o p t i c s d o e s n o t , f o r e x a m p l e , a c c o u n t f o r t h e e x i s t e n c e o f a " l a t e r a l w a v e " , w h i c h i f i t d o e s i n d e e d a r i s e m u s t b e a d i f f r a c t i o n e f f e c t . I t i s o f i n t e r e s t t o make an i n -d e p t h e x a m i n a t i o n o f t h i s p r o b l e m . C o n s i d e r a h o m o g e n e o u s , i n v i s c i d , i s o t r o p i c f l u i d w h i c h i s s e p a r a t e d i n t o t w o r e g i o n s , e a c h o f u n i f o r m d e p t h , b y a p l a n e s t e p d i s c o n t i n u i t y . T h i s c o n s t r u c t i o n c a n b e c o n s i d e r e d t o b e a v e r y s i m p l i f i e d r e p r e s e n t a t i o n o f a c o n t i n e n t a l s h e l f o f u n i f o r m d e p t h s e p a r a t e d f r o m a u n i f o r m l y d e e p o c e a n by a v e r y s t e e p c o n t i n e n t a l s l o p e . When a p l a n e w a v e , a r i s i n g f r o m w i t h i n t h e s h e l f r e g i o n , s t r i k e s t h e s t e p t h e r e r e s u l t s a r e f l e c t e d w a v e a n d a r e f r a c t e d w a v e . A c c o r d i n g t o S n e l l ' s l a w , t h e r e f r a c t e d w a v e e n t e r s t h e d e e p w a t e r a t a s t e e p e r a n g l e w i t h r e s p e c t t o t h e n o r m a l t o t h e s t e p t h a n t h e a n g l e o f i n c i d e n c e . F o r i n c i d e n c e a t t h e c r i t i c a l a n g l e , r e f r a c t i o n i s p a r a l l e l t o t h e b o u n d a r y , ' F o r s t i l l g r e a t e r a n g l e s o f i n c i d e n c e , t o t a l r e f l e c t i o n r e s u l t s a n d n o p r o p a g a t i n g w a v e f i e l d i s t r a n s m i t t e d t o t h e d e e p w a t e r . G e o m e t r i c a l o p t i c s g i v e s a n a d e q u a t e i n t e r p r e t a t i o n o f t h e p r o c e s s e s o f r e f l e c t i o n a n d r e f r a c t i o n i n a l l c a s e s e x c e p t t h a t w h i c h a r i s e s w h e n i n c i d e n c e i s a t t h e c r i t i c a l a n g l e . H e r e t h e m e c h a n i s m f a i l s : t h e l a t e r a l w a v e c a n n o t be a c c o u n t e d f o r b y r a y t h e o r y . T h i s i s p a r t i c u l a r l y e v i d e n t when t h e i n c i d e n t w a v e f i e l d a r i s e s f r o m t h e e x c i t a t i o n 3 o f a p o i n t s o u r c e . I n t h i s c a s e , when i n c i d e n c e i s a t t h e c r i t i c a l a n g l e , t h e c r i t i c a l l y r e f r a c t e d w a v e t r a v e l s p a r a l l e l t o t h e b o u n d a r y i n t h e d e e p w a t e r r e g i o n a n d r e r a d i a t e s e n e r q y b a c k o n t o t h e s h e l f b y d i f f r a c t i o n . T h i s r e r a d i a t e d e n e r q y c o n s t i t u t e s a l a t e r a l wave a n d i s i n a c c o r d a n c e w i t h H u y q e n ' s p r i n c i p l e : a l l p o i n t s o n t h e b o u n d a r y a c t a s p o i n t s o u r c e s w h i c h p r o d u c e s e c o n d a r y w a v e l e t s . T h e r e s u l t i n q w a v e f r o n t w i l l b e p l a n e , i n s p i t e o f t h e f a c t t h a t t h e i n c i d e n t w a v e f r c n t i s c y l i n d r i c a l . T h e d i r e c t i o n o f t h e p r o p a q a t i o n o f t h e l a t e r a l w a v e l i e s a l o n q a l i n e m a k i n q a n a n q l e e q u a l t o t h e c r i t i c a l a n q l e , ( $ c , w i t h t h e n o r m a l t o t h e b o u n d a r y . T h i s i s s h o w n i n t h e r a y d i a q r a m i n f i q u r e 1 ( a ) . B a y o p t i c s c a n b e u s e d t o t r a c e t h e p a t h s o f t h e v a r i o u s f i e l d c o n s t i t u e n t s when t h e p a t h l e n g t h s a r e s u f f i c i e n t l y q r e a t c o m p a r e d w i t h t h e w a v e l e n q t h a n d when t h e s o u r c e i s s u f f i c i e n t l y r e m o v e d f r o m t h e b o u n d a r y . U n d e r t h e s e c o n d i t i o n s t h e t r a j e c t o r i e s o f t h e r e f l e c t e d a n d r e f r a c t e d r a y s a r e i n a c c o r d w i t h F e r m a t ' s p r i n c i p l e o f l e a s t p r o p a g a t i o n t i m e . T h u s , o n c e t h e e x i s t e n c e o f t h e l a t e r a l w a v e h a s b e e n a d m i t t e d , t h e v a r i a t i o n o f t h e t o t a l w a v e f i e l d , b o t h i n a m p l i t u d e a n d p h a s e , c a n b e p r e d i c t e d b y s i m p l e q e o m e t r i c a l o p t i c s . F o r i n s t a n c e , i f a p o i n t s o u r c e , S, r a d i a t e s o n a s h e l f , t h e p a t h s o f t h e r e f l e c t e d a n d c r i t i c a l l y r e f r a c t e d r a y s s e e n a t a n o b s e r v a t i o n p o i n t o n t h e s h e l f , P, a r e a s s h o w n i n f i q u r e 1 ( b ) . T h e f i e l d a t p o i n t P r e s u l t s f r o m t h e a r r i v a l o f t w o w a v e s {not i n c l u d i n q t h e d i r e c t w a v e ) , t h e r e f l e c t e d a n d l a t e r a l w a v e s . T h e r e f l e c t e d w a v e f o l l o w s t h e p a t h S BP a n d t h e s p e e d c f p r o p a q a t i o n i s d e t e r m i n e d b y t h e w a t e r d e p t h o n t h e s h e l f , a s s u m i n q t h a t l i n e a r ( A ) D E E P W A T E R L A T E R A L R A Y ( B ) D E E P W A T E R gu re I. A) Ray p a t h d i a g r a m f o r t h e l a t e r a l wave. B) Ray p a t h d i a g r a m f o r the l a t e r a l and t o t a l l y r e f l e c t e d wave. 5 s h a l l o w w a t e r t h e o r y a p p l i e s . T h e c r i t i c a l l y r e f r a c t e d r a y , w h i c h p r o d u c e s t h e l a t e r a l w a v e , ( H e e l a n , 19 53) f o l l o w s t h e p a t h S A C P . T h e s p e e d o f p r o p a g a t i o n o v e r t h e p a t h s e g m e n t s SA a n d CP i s d e t e r m i n e d b y t h e s h e l f d e p t h , w h i l e o v e r t h e s e g m e n t AC t h e d i s t u r b a n c e p r o p a g a t e s w i t h a s p e e d d e t e r m i n e d b y t h e d e p t h o f t h e d e e p w a t e r r e g i o n . S i n c e , b y s h a l l o w w a t e r t h e o r y , t h e w a v e s w i l l p r o p a g a t e more r a p i d l y i n t h e d e e p w a t e r t h a n o n t h e s h e l f , i t i s c l e a r t h a t t h e r e w i l l e x i s t a r e g i o n o n t h e s h e l f w h e r e t h e a r r i v a l o f t h e l a t e r a l w a v e w i l l p r e c e e d t h a t o f e i t h e r t h e d i r e c t o r r e f l e c t e d w a v e , , T h e d i f f r a c t i o n p h e n o m e n o n w h i c h r e s u l t s i n t h e l a t e r a l w a v e i s s o m e t i m e s named d i f f e r e n t l y . I t i s c o m m o n l y r e f e r r e d t o a s a " h e a d w a v e " , a " r e f r a c t i o n a r r i v a l " , a s w e l l a s a " l a t e r a l w a v e " . E a c h n o m e n c l a t u r e d e s c r i b e s some f e a t u r e o f t h e d i s t u r b a n c e . T h e t e r m " h e a d w a v e " m a k e s n o t e o f t h e f a c t t h a t u n d e r t r a n s i e n t c o n d i t i o n s t h e f i r s t r e s p o n s e a t some p o i n t s o f o b s e r v a t i o n w i l l b e d u e t o t h i s w a v e . T h e t e r m " r e f r a c t i o n a r r i v a l " r e f e r s t o t h e r o l e p l a y e d b y t h e r e f r a c t i o n p r o c e s s . T h e t e r m " l a t e r a l w a v e " r e f e r s t o t h e s i d e w a y s o r l a t e r a l p r o p a g a t i o n o f t h e w a v e p a r a l l e l t o t h e b o u n d a r y . H e n c e f o r t h , t h e t e r m " l a t e r a l w a v e " w i l l b e u s e d e x c l u s i v e l y . T h e t e r m " f o r e r u n n e r " i s a l s o a p p l i c a b l e u n d e r c e r t a i n c o n d i t i o n s , s i n c e t h e a r r i v a l o f t h e l a t e r a l wave may p r e c e e d t h a t o f e i t h e r t h e d i r e c t o r r e f l e c t e d w a v e . T h e p u r p o s e o f t h i s t h e s i s i s t o d e v e l o p a m a t h e m a t i c a l m o d e l w h i c h d e s c r i b e s t h e wave f i e l d o n a s h e l f g e n e r a t e d b y a s p a t i a l l y c o n f i n e d s o u r c e l o c a t e d a t a p o i n t o n t h e s h e l f . T h e f i e l d o f l o n g s u r f a c e g r a v i t y w a v e s i s c o n s i d e r e d t o p r o p a g a t e 6 w i t h i n a r o t a t i n g c o o r d i n a t e s y s t e m a n d t h e e f f e c t o f t h i s r o t a t i o n u p o n s u c h p r o p e r t i e s a s w a v e s p e e d a n d e n e r g y i s e x a m i n e d . S o l u t i o n s a r e f o u n d w h i c h d e s c r i b e t h e wave f i e l d i n t e r m s o f i t s i n c i d e n t , r e f l e c t e d a n d l a t e r a l w a v e c o n s t i t u e n t s . T h e i n t e r r e l a t i o n o f t h e s e c o n s t i t u e n t s t h r o u g h o u t t h e s h e l f r e g i o n i s a l s o e x a m i n e d . I n o r d e r t c c o n s t r u c t a t h e o r e t i c a l m o d e l w h i c h i s s o l u b l e , i t i s n e c e s s a r y t o i m p o s e a n u m b e r o f s i m p l i f y i n g c o n d i t i o n s o n t h e p r o b l e m . How c l o s e l y t h e s o l u t i o n s o f t h i s m a t h e m a t i c a l m o d e l r e s e m b l e t h e wave f i e l d a r i s i n g i n t h e r e a l w o r l d o r a l a b o r a t o r y m o d e l r e m a i n s t o be d e t e r m i n e d . To t h i s e n d , a l a b o r a t o r y m o d e l o f a t w o - d e p t h r e g i m e c o n t a i n i n g a s t e p d i s c o n t i n u i t y w a s c o n s t r u c t e d . E x p e r i m e n t s w e r e c o n d u c t e d t o d e t e r m i n e t h e w a v e f i e l d a r i s i n g f r o m a c o n f i n e d s o u r c e l o c a t e d o n t h e s h e l f . T h e s e r e s u l t s a r e c o m p a r e d w i t h t h e s o l u t i o n s o f t h e m a t h e m a t i c a l m o d e l a n d t h e d i f f e r e n c e s d i s c u s s e d . Many o f t h e t e c h n i q u e s u s e d i n s o l v i n g t h e m a t h e m a t i c a l m o d e l a r e s i m i l a r t o t h o s e u s e d by B r e k h o v s k i k h ( 1 9 6 0 ) t o i n v e s t i g a t e t h e f i e l d o f s p h e r i c a l w a v e s u n d e r z e r o - r o t a t i o n . P r o b l e m s i n l a y e r e d m e d i a a r e a l s o d i s c u s s e d by E w i n q ( 1 9 5 7 , c h a p t e r 2 ) , G r a n t ( 1 9 6 5 , c h a p t e r 6) a n d C a q n i a r d ( 1 9 3 9 , c h a p t e r 3 ) . H a d a m a r d ( 1 9 2 3 ) s h o w e d t h a t t h e s o l u t i o n o f t h e wave e q u a t i o n i s d i f f e r e n t f o r e v e n a n d o d d n u m b e r s c f s p a c e d i m e n s i o n s ( W h i t h a m , 1 9 7 4 p . 2 1 9 ) , w i t h t h e s o l u t i o n f o r e v e n d i m e n s i o n s m o r e e a s i l y o b t a i n e d . S t i l l t h e m e t h o d s e m p l o y e d by B r e k h o v s k i k h a r e u s e f u l a n d c a n be r e a d i l y a p p l i e d t o t h e e v e n d i m e n s i o n p r o b l e m . I n t h e e x p e r i m e n t a l i n v e s t i q a t i o n , wave a b s o r b e r c o n f i q u r a t i o n s a n d m a t e r i a l s s i m i l a r t o t h o s e u s e d b y P i t e ( 1 9 7 3 ) a r e e m p l o y e d . 7 An e x a m p l e o f s u c h a f i e l d o f l o n g w a v e s i n t h e o c e a n i s c h o s e n f r o m t h e c l a s s o f w a v e s k n o w n a s t s u n a m i s . M u c h w o r k h a s b e e n d o n e i n t h i s f i e l d a n d a r e c e n t b o o k b y T.S. M u r t y ( 1 9 7 7 ) e n g a g e s i n a c o m p l e t e e x a m i n a t i o n o f t s u n a m i s . T h i s b o o k p r e s e n t s a v e r y c o m p r e h e n s i v e c o v e r a g e o f t h e t h e o r y a n d o b s e r v a t i o n a n d b r i n g s t o g e t h e r t h e v a s t m a j o r i t y o f r e s e a c h w h i c h h a s b e e n d o n e b y a n a l y t i c a l a n d n u m e r i c a l m o d e l l e r s a n d e x p e r i m e n t e r s . C o n s i d e r t h e p r o b l e m o f t s u n a m i s g e n e r a t e d when a s u b m a r i n e e a r t h q u a k e d i s p l a c e s a s i g n i f i c a n t p o r t i o n o f t h e s e a f l o o r o n t h e c o n t i n e n t a l s h e l f . T h e C a u c h y - P o i s s o n p r o b l e m f o r t h e c a s e o f l o n g s u r f a c e g r a v i t y wave h a s b e e n s t u d i e d b y K a j i u r a ( 1 9 6 3 ) , R r a n z e r a n d K e l l e r ( 1 9 5 5 ) , L a M e h a u t e | 1 9 7 1 ) a n d V a n D e m ( 1 9 7 0 ) . K a i j u r a e x a m i n e d t h e l e a d i n g w a v e g e n e r a t e d b y a l o c a l i z e d s o u r c e f o r t h e c a s e o f z e r o - r o t a t i o n . V a n D o r n p e r f o r m e d l a b o r a t o r y e x p e r i m e n t s t o s i m u l a t e o p e n - o c e a n w a v e s r e s u l t i n g f r o m t h e A l a s k a e a r t h q u a k e o f 1 9 6 4 . When t s u n a m i w a v e s g e n e r a t e d o n t h e s h e l f r e a c h t h e c o n t i n e n t a l s l o p e , f i e l d s o f r e f l e c t e d a n d r e f r a c t e d w a v e s r e s u l t . T h e r e f r a c t e d o p e n - o c e a n w a v e s , o f t e n h a v i n g w a v e l e n g t h s o f h u n d r e d s o f k i l o m e t e r s , w h i l e h a v i n g w a v e h e i g h t s o f l e s s t h a n a m e t e r , c a n t r a v e l d i s t a n c e s o f t h o u s a n d s o f k i l o m e t e r s a t s p e e d s o f h u n d r e d s o f k i l o m e t e r s p e r h o u r . When t h e s e w a v e s e n c o u n t e r t h e b o u n d a r i e s c f d i s t a n t i s l a n d s a n d c o n t i n e n t s , t h e i r e n e r g y d e n s i t y may i n c r e a s e c o n s i d e r a b l y , w i t h t h e r e s u l t t h a t t h e w a v e h e i g h t s may b e c o m e v e r y l a r g e . I n s p i t e o f t h e f a c t t h a t much o f t h e wave e n e r g y may be r e f l e c t e d b a c k t o s e a , t h e d e s t r u c t i o n w r o u g h t b y s u c h 8 w a v e s i s w e l l - k n o w n . ( H u r t y , 1 9 7 7 , s e c t i o n 5 . 2 ) . W a v e s p r o p a g a t i n g w i t h i n t h e c o n t i n e n t a l s h e l f r e g i o n c a n g i v e r i s e t o e f f e c t s w h i c h a r e e q u a l l y a s i m p o r t a n t a s t h o s e o f t h e o p e n - s e a t s u n a m i s . T h e s e w a v e s p r o p a g a t e w i t h i n t h e s h e l f r e g i o n , s o m e p r o c e e d i n g d i r e c t l y f r o m t h e s o u r c e t o some c o a s t a l c o m m u n i t y , w h i l e o t h e r s r e f l e c t n o t o n l y o f f t h e s h o r e b u t f r o m t h e c o n t i n e n t a l s l o p e a s w e l l . C o n s e q u e n t l y , m u l t i p l e r e f l e c t i o n s c a n o c c u r w i t h p o s s i b l e a m p l i t u d e r e i n f o r c e m e n t a n d t r a p p i n q ( L o n q u e t - H i g q i n s , 1 9 6 8 ) . T h e w a v e s w h i c h e n c o u n t e r t h e c o n t i n e n t a l s l o p e c a n g e n e r a t e d i f f r a c t i o n a n d s c a t t e r i n q e f f e c t s i n a d d i t i o n t o r e f l e c t e d a n d r e f r a c t e d w a v e s . A l l t h e s e p h e n o m e n a a r e o f i n t e r e s t s i n c e e a c h may i n f l u e n c e t h e o v e r a l l e f f e c t u p o n t h e s h o r e l i n e c o m m u n i t i e s l y i n g i n t h e p a t h o f t h e w a v e s . T h e A l a s k a n e a r t h q u a k e o f 1964 ( C o m m i t t e e o n t h e A l a s k a E a r t h q u a k e , 1 9 7 2 ) , w e l l - k n o w n f o r i t s d e s t r u c t i v e n a t u r e , i s a n e x a m p l e w o r t h e x a m i n i n g . T h i s e a r t h q u a k e r e g i s t e r e d a m a g n i t u d e o f b e t w e e n 8.3 a n d 8.7 o n t h e R i c h t e r s c a l e . I t r e s u l t e d i n a m o v e m e n t o f a n a r e a o f t h e e a r t h ' s s u r f a c e m e a s u r i n g n e a r l y 150 k i l o m e t e r s b y 1 2 0 0 k i l o m e t e r s . T h e m a i n s h o c k l a s t e d some t h r e e o r f o u r m i n u t e s , f o l l o w e d by a s e r i e s o f a f t e r s h o c k s w h i c h o c c u r r e d o v e r t h e n e x t 24 h o u r s . A s a r e s u l t , g r o u n d a n d s t r u c t u r a l d a m a g e r e s u l t e d t h r o u g h o u t a n a r e a o f 1 0 0 , 0 0 0 t o 1 5 0 , 0 0 0 s q u a r e k i l o m e t e r s . I n a d d i t i o n t o p r i m a r y e f f e c t s , s e c o n d a r y d a m a g e r e s u l t e d i n H a w a i i , J a p a n , a n d A n t a r c t i c a when t s u n a m i s r a n q i n q f r o m o n e - t h i r d t o t w o m e t e r s i n h e i q h t a r r i v e d o n t h e i r r e s p e c t i v e s h o r e s . V a n D o r n ( 1 9 7 2 ) c o n c l u d e d t h a t t h e s e w a v e s , w h i c h s p r e a d r a p i d l y a c r o s s t h e P a c i f i c O c e a n , w e r e 9 c a u s e d b y a s u d d e n d i s p l a c e m e n t o f w a t e r i n t h e G u l f o f A l a s k a , w h i c h was i n p a r t d u e t o t h e u p l i f t o f t h o u s a n d s o f s q u a r e m i l e s o f s e a f l o o r , W h i l e o n l y m i n o r d a m a g e r e s u l t e d f r o m t h e o p e n o c e a n w a v e s , t h e w a v e s w i t h i n t h e c o n f i n e s o f t h e c o n t i n e n t a l -s h e l f r e g i o n o f N o r t h A m e r i c a c a u s e d much d a m a g e . T h e s o u r c e a p p a r e n t l y was d i r e c t i o n a l , r a d i a t i n g w a v e s p r e f e r e n t i a l l y s o u t h e a s t w a r d . T h e s e w a v e s a d v a n c e d , s p r e a d i n g d a m a g e i n E r i t i s h C o l u m b i a , W a s h i n g t o n , O r e g o n , C a l i f o r n i a a n d M e x i c o ( S p a e t h a n d B e r k m a n , 1 9 6 4 ) . K a i j u r a ( 1 9 7 0 ) made a s t u d y o f t h e w ave f i e l d g e n e r a t e d b y a s o u r c e o f f i n i t e d i m e n s i o n s . T h i s s o u r c e , c o n s t r u c t e d f r o m a s u p e r p o s i t i o n o f e l e m e n t a r y w a v e s g i v e s r i s e t o a f i e l d w i t h a d i r e c t i o n d e p e n d i n g u p o n t h e m a j o r a n d m i n o r a x e s o f t h e s o u r c e a n d i t s o r i e n t a t i o n . O f t h e t w e l v e d e a t h s r e s u l t i n g f r o m t h e A l a s k a e a r t h q u a k e , t h e t s u n a m i c a u s e d m o r e d e a t h s t h a n a l l o t h e r f a c t o r s c o m b i n e d . M o s t o f t h e d e a t h s r e s u l t e d f r o m a l a c k o f u n d e r s t a n d i n q o f t h e b e h a v i o r o f t s u n a m i s ( M u r t y , 1 9 7 8 , p 2 6 6 ) . T h u s , a n u n d e r s t a n d i n q o f s u c h w a v e s i s n o t m e r e l y a c a d e m i c b u t i s o f some i m m e d i a t e i m p o r t a n c e a s w e l l . W h i l e a l l o f t h e f a c t o r s a f f e c t i n g t h e i m p a c t o f a t s u n a m i c a n n o t b e e x a m i n e d a n a l y t i c a l l y , many f e a t u r e s o f t h e w ave f i e l d a r i s i n g f r o m a n e a r t h q u a k e c a n b e , a n d a b e t t e r u n d e r s t a n d i n g o f t h e o v e r a l l p r o b l e m c a n b e a c h i e v e d . 10 2, THE M A T H E M A T I C A L MODEL Th e f i e l d o f l o n g s u r f a c e g r a v i t y w a v e s g e n e r a t e d b y a s o u r c e c e n t e r e d o n t h e c o n t i n e n t a l s h e l f i s n e c e s s a r i l y c o m p l e x s i n c e b o t h t h e s o u r c e d i s t r i b u t i o n a n d t h e t o p o g r a p h y o f t h e c o n t i n e n t a l m a r g i n a n d d e e p o c e a n a r e n e i t h e r s i m p l e n o r r e g u l a r , T h e s p a t i a l a n d t e m p o r a l d i s t r i b u t i o n o f t h e s o u r c e i s r a r e l y k n o w n a n d e v e n i f i t w e r e , i t i s u n l i k e l y t h a t i t c o u l d b e r e a d i l y m o d e l l e d a n a l y t i c a l l y . L i k e w i s e , f i n d i n g a n a c c u r a t e a n a l y t i c a l m o d e l t o d e s c r i b e t h e t o p o g r a p h i c a l f e a t u r e s w o u l d b e e q u a l l y d i f f i c u l t . I n o r d e r t o c o n s t r u c t a m a t h e m a t i c a l l y s o l u b l e m o d e l w h i c h r e p r e s e n t s t h e p r o b l e m , i t i s n e c e s s a r y t o i m p o s e a n u m b e r o f s i m p l i f y i n q c o n d i t i o n s . I n d o i n q s o , c a r e m u s t b e t a k e n t o r e t a i n t h e e s s e n t i a l f e a t u r e s o f t h e p r o b l e m , a n d t o a s s u r e t h a t a r e a s o n a b l e a n d i n t e r p r e t a b l e s o l u t i o n i s r e a l i z e d . T o t h i s e n d we c o n s t r u c t t h e f o l l o w i n q m o d e l . C o n s i d e r t h e c o n f i q u r a t i c n s h o w n i n f i q u r e 2. L e t a v e r t i c a l s t e p o f d e p t h h z ™ h ( , l o c a t e d a l o n g t h e y - a x i s (x= 0 ) , d i v i d e a n o t h e r w i s e h o r i z o n t a l l y u n b o u n d e d a n d i n f i n i t e d o m a i n i n t o t w o r e q i o n s . L e t t h e d e p t h c f r e q i o n 1 ( t h e s h e l f ) , f o r x>0 , b e u n i f o r m a n d o f m a q n i t u d e h{ a n d t h a t o f r e q i o n 2 < t h e d e e p w a t e r ) , f o r x<0 , be u n i f o r m a n d o f m a q n i t u d e , w h e r e h > h , . T h e s h e l f r e g i o n i s l e f t u n b o u n d e d , s i n c e a b o u n d a r y r e p r e s e n t i n q t h e s h o r e w i l l m a i n l y g i v e r i s e t o r e f l e c t i o n s a n d a d d l i t t l e i n f o r m a t i o n i n d e t e r m i n i n q t h e f i e l d d u e t o t h e s t e p . T h e e f f e c t o f m u l t i p l e r e f l e c t i o n s c a n be h a n d l e d l a t e r , i f s o d e s i r e d , b y i n t r o d u c i n q a n i n n e r b o u n d a r y i n t o t h e s h e l f r e q i o n . L e t t h e d o m a i n r o t a t e u n i f o r m l y a t a n a n q u l a r f r e q u e n c y , f / 2 , a b o u t t h e v e r t i c a l . B a c k u s ( 1 9 6 2 ) , K a i j u r a ( 1 9 5 8 ) a n d L c n q u e t -F i g u r e w a v e s . 2 . S h e l f g e o m e t r y a n d r a y p a t h s f o r t h e d i r e c t a n d r e f l e c t e d 12 H i g g i n s ( 1 9 6 4 ) h a v e made s t u d i e s u s i n g p l a n e w a v e t h e o r y c n t h e e f f e c t o f r o t a t i o n o n l o n g s u r f a c e w a v e s . C o n s i d e r t h e s h e l f a n d d e e p w a t e r r e g i o n s t o c o n t a i n a h o m o g e n e o u s , i n v i s c i d , i n c o m p r e s s i b l e a n d i s o t r o p i c f l u i d ( w a t e r ) . F u r t h e r , a l l o w t h e w a v e s t o b e c f s m a l l a m p l i t u d e c o m p a r e d w i t h b o t h w a t e r d e p t h a n d w a v e l e n g t h , a n d w a v e l e n g t h l a r g e c o m p a r e d w i t h w a t e r d e p t h . T h u s , i t c a n b e a s s u m e d t h a t t h e h o r i z o n t a l f l u i d v e l o c i t i e s a r e n e a r l y c o n s t a n t w i t h d e p t h , t h a t s h a l l o w w a t e r c o n d i t i o n s a r e a p p l i c a b l e a n d t h a t t h e p h a s e s p e e d s i n t h e c a s e o f z e r o -r o t a t i o n c a n b e w r i t t e n 2. 1 Ca - -v| , K < O L e t t h e wave s o u r c e , S , be l o c a t e d a t t h e p o i n t ( x , y ) = ( x c , 0 ) . T h e s p a t i a l d i s t r i b u t i o n o f t h i s s o u r c e i s c o n v e n i e n t l y r e p r e s e n t e d b y a D i r a c d e l t a f u n c t i o n . T h e t e m p o r a l d i s t r i b u t i o n i s e x a m i n e d f o r b o t h t i m e - h a r m o n i c e x c i t a t i o n , w h i c h r e s u l t s i n s t e a d y s t a t e c o n d i t i o n s , a n d i m p u l s e e x c i t a t i o n , w h i c h g i v e s t h e t r a n s i e n t r e s p o n s e . A l t h o u g h t h e t i m e - h a r m o n i c c o n d i t i o n s b e a r l e s s r e s e m b l e n c e t o t h e r e a l p r o b l e m , much i n f o r m a t i o n c a n b e d e r i v e d b y c o n s i d e r i n g t h i s c a s e . , F o r t h e s o u r c e , S , r a d i a t i n g c y l i n d r i c a l w a v e s , l e t t h e w a v e f i e l d b e d e n o t e d a s 13 *li s °U + <V + , * > ° 2.2 Iv- n t , X < 0 2.3 w h e r e ^ = d i r e c t w a v e , * | r = r e f l e c t e d w a v e , • ^ l a t e r a l w a v e , a n d ^ = t r a n s m i t t e d wave. E a c h c o n s t i t u e n t w i l l b e e x a m i n e d i n d i v i d u a l l y . C o n s i d e r t h e t i m e h a r m o n i c c a s e f i r s t . 2 . 1 . T i m e - h a r m o n i c R e g i m e I n f i n d i n g t h e v a r i o u s w a v e c o n s t i t u e n t s i t i s p e r h a p s m o s t r e a s o n a b l e t o e x a m i n e t h e w a v e h a v i n g t h e s i m p l e s t f o r m f i r s t , a n d t h e m o s t c o m p l e x wave l a s t . T o t h i s e n d t h e w a v e s a r e e x a m i n e d i n t h e o r d e r o f d i r e c t w a v e , t h e r e f l e c t e d w a v e a n d t h e l a t e r a l w ave. T h e t r e a t m e n t o f l i n e a r s h a l l o w w a t e r w a v e s u n d e r r o t a t i o n d e v e l o p e d h e r e i n i s w e l l c o v e r e d i n a r e c e n t b e c k o n o c e a n w a v e s b y L e B l o n d a n d M y s a k ( 1 9 7 8 ) . 2 . 1 . 1 . T h e D i r e c t H a v e I f t h e s t e p d i s c o n t i n u i t y l o c a t e d a t x=0 i n f i g u r e 2 i s r e m o v e d ( l e t h 4 = h = h ) # t h e d o m a i n c a n be c o n s i d e r e d t o c o n s i s t o f a u n i f o r m f l u i d o f c o n s t a n t d e p t h h , l y i n g c n a n u n b o u n d e d a n d i n f i n i t e f - p l a n e . T h e s o u r c e i s l o c a t e d b y t h e p o s i t i o n v e c t o r r 6 , a n d t h e o b s e r v a t i o n p o i n t , w i t h r e s p e c t t o t h e s o u r c e , b y t h e v e c t o r R. T h e l i n e a r i z e d E u l e r e q u a t i o n s i n p l a n e p o l a r c o o r d i n a t e s ( R , 9 , z ) a r e w r i t t e n a s 14 B - d i r e c t i o n : U^. - -f v = - ^ 2.4 8 - d i r a c t i o n : \/, -*--fu. - - ~H 2.5 z - d i r e c t i o n : VJ-^ = - ~ P f c - ^ 2.6 w h e r e £ - 2 ^ . s w\<* , l L = a n g u l a r f r e q u e n c y o f t h e e a r t h ' s r o t a t i o n , o s = l a t i t u d e , £ = d e n s i t y , q = a c c e l e r a t i o n d u e t o q r a v i t y , p = p r e s s u r e , u,v,w a r e t h e v e l o c i t y c o m p o n e n t s a l o n q B , & , z d i r e c t i o n s , r e s p e c t i v e l y , a n d d e r i v a t i v e s a r e i n d i c a t e d b y s u b s c r i p t s . A s s u m i n q h y d r o s t a t i c c o n d i t i o n s ( ^ ^ t ^ * ^ , e q u a t i o n 2.6 r e d u c e s t o w h e r e r j ^ s u r f a c e e l e v a t i o n , p= p r e s s u r e i n t h e f l u i d a t d e p t h z , p ^ = p r e s s u r e v a r i a t i o n i m p o s e d u p o n t h e f l u i d . S u b s t i t u t i o n i n t o e q u a t i o n s 2.4 a n d 2.5 q i v e s T h e c o n t i n u i t y e q u a t i o n i s q i v e n a s 2. 8 2. 9 15 U = O 2.10 w h e r e t h e v e l o c i t y "U. - (u^V^W^. T h e k i n e m a t i c b o u n d a r y c o n d i t i o n s a r e 2.11 I m p o s i n g t h e s e c o n d i t i o n s o n t h e p r o b l e m a n d u s i n g - L e i b n i t z * r u l e : >/^ » 2. 12 U(x> U(ic) we o b t a i n t h e i n t e g r a t e d c o n t i n u i t y e g u a t i o n ^ • ^ ^ ^ l ^ - t = ° 2 , 1 3 w h e r e " ^ ^ ^ i s t h e h o r i z o n t a l d i v e r g e n c e o p e r a t o r . F o r w a v e s s u c h a s t s u n a m i w a v e s , t h e w ave h e i g h t , ^ , m a y b e o f t h e o r d e r o f a m e t e r w h i l e t h e d e p t h , h , may r a n g e f r o m h u n d r e d s t o t h o u s a n d s o f m e t e r s . I t c a n t h u s b e r e a s o n a b l y a s s u m e d t h a t « h a n d , w h e n t h i s c o n d i t i o n i s a p p l i e d t o e g u a t i o n 2 . 1 3 , we f i n d , f o r h = c o n s t a n t , 16 W^^  * h i * = * 2. 14 If u and v are eliminated from equations 2.8 and 2 . 9 , and the r e s u l t combined with equation 2. 14, we f i n d , after one time integration, the inhomogeneous wave equation i n the v a r i a b l e , ^ , to be The right hand side of equation 2.15 represents the forcing and can be replaced by a term which represents a time-barmonic point source . To further simplify the problem, assume that the source i s i s o t r o p i c i n order that the s p a t i a l dependence of the source can be represented by the delta function , SCO ,located at 8 =0. Thus, the r i g h t hand side of equation 2.15 can be replaced by the term where cu = angular frequency and the factor of 4TT arises from the properties of the delta function. Let the surface elevation be written i n the form 2.15 2. 16 17 ^(12,0= l C ^ e " l w t 2.17 w h e r e s p e c i f i e s t h e s p a t i a l d e p e n d e n c e o f ^ . T h i s d e p e n d e n c e i s g i v e n by t h e i n h o m o q e n e o u s H e l r a h c l t z e q u a t i o n f o r a u n i t p o i n t s o u r c e l o c a t e d a t E =0 a s f + ^ a < S = - ^ SCO 2. 18 w h e r e 2 . 1 9 T h e s o l u t i o n t o e g u a t i o n 2.18 i s t h e t w o - d i m e n s i o n a l G r e e n ' s f u n c t i o n f o r t h e h o m o g e n e o u s e q u a t i o n , a n d f o r o u t g o i n g w a v e s t h i s r e g u l a r a n d c o n t i n u o u s f u n c t i o n i s q i v e n b y ( M o r s e a n d F e s h b a c h , 1 9 5 3 , p . 8 0 5 - 8 10) 5(0 = LT\ W^C^O 2.20 w h e r e U e()8tl) i s t h e z e r o t h o r d e r H a n k e l f u n c t i o n o f t h e f i r s t k i n d ( W a t s o n , 1 9 6 6 , p . 1 6 7 ) . E q u a t i o n 2.18 i s h o m o g e n e o u s e v e r y w h e r e w i t h t h e e x c e p t i o n o f t h e p o i n t B = 0 . T h e s i n g u l a r i t y a t t h e s o u r c e i s s p e c i f i e d b y 2. 21 1 8 I t i s u s e f u l t o e x p r e s s t h e e q u a t i o n 2.20 i n a n i n t e g r a l r e p r e s e n t a t i o n , S o m m e r f e l d ( 1 9 4 9 , p . 8 8 - 8 9 ) g i v e s s u c h a r e p r e s e n t a t i o n a s CK U (MIL) ^ <- 2. 2 2 T h e c y l i n d r i c a l w a v e f i e l d i s t h u s r e p r e s e n t e d a s a s u p e r p o s i t i o n o f p l a n e w a v e s , s p r e a d i n g o u t w a r d l y i n a l l d i r e c t i o n s . T h e p a t h o f i n t e g r a t i o n , T # i n t h e c o m p l e x © - p l a n e i s s h o w n i n f i g u r e 3, T h e © - p l a n e i s s h o w n w i t h i t s r e a l a x i s d e f i n e d b y 0 a n d i t s i m a g i n a r y a x i s b y © , E x p a n d i n g t h e a n g l e © i n i t s r e a l a n d i m a g i n a r y p a r t s a s 0 - &' i & " 2. 2 3 a n d s u b s t i t u t i n g i n t o t h e e x p o n e n t i n t h e i n t e g r a n d o f e q u a t i o n 2.22 g i v e s Ixe. Ca*© - M<1 «Jwvfc ' s i i ^ © " 4- I Coso'cesk©" 2.2U w h e r e X- i s a s s u m e d t o b e r e a l . B y m a k i n g t h e a p p r o p r i a t e c h o i c e s o f © a n d ©• , t h e r e a l p a r t o f e g u a t i o n 2.24 c a n b e made i n f i n i t e l y n e g a t i v e , r e s u l t i n g i n t h e i n t e g r a n d o f e q u a t i o n 2 , 2 2 a s y m p t o t i c a l l y a p p r o a c h i n g z e r o . F o r t h e l i m i t s o f i n t e g r a t i o n g i v e n i n e g u a t i o n 2 . 2 2 , t h e i n t e g r a n d r e m a i n s b o u n d e d a n d t h e c o n t o u r o f i n t e g r a t i o n y i e l d s a c o n v e r g e n t ure 3. Complex plane of i n t e g r a t i o n . 20 i n t e g r a l . T h e m o s t r a p i d c o n v e r g e n c e i s s e e n t o o c c u r when t h e r e a l p a r t o f & i s c h o s e n a s I n o r d e r t h a t e q u a t i o n 2.22 r e p r e s e n t a s o l u t i o n t o t h e H e l m h o l t z e g u a t i o n , i t i s n e c e s s a r y TT t h a t t h e p a t h o f i n t e g r a t i o n c o n n e c t t h e e n d p o i n t s , — *2.+ a n d - t 0 6 . T h e p a t h o f i n t e g r a t i o n i s r e s t r i c t e d t o t h e s h a d e d r e g i o n s s h o w n i n f i g u r e 3. I t i s a m a t t e r o f c o n v e n i e n c e t h a t t h e p a t h o f i n t e g r a t i o n p a s s t h r o u g h t h e o r i g i n . I f t h e r e a l p a r t o f Xfc. b e c o m e s l a r g e ( a s i n t h e f a r f i e l d ) , t h e e x p o n e n t i a l i n e g u a t i o n 2.22 b e c o m e s v a n i s h i n g l y s u s a l l a t a l l p o i n t s w i t h i n t h e s h a d e d r e g i o n s , w i t h t h e e x c e p t i o n o f t h e i m m e d i a t e n e i g h b o r h o o d o f t h e o r i g i n , i t t h e o r i g i n t h e r e a l p a r t o f t h e a r g u m e n t o f e x p o n e n t i a l i s i d e n t i c a l l y z e r o n o m a t t e r how l a r g e XR. b e c o m e s a n d t h u s t h e s o l e c o n t r i b u t i o n t o t h e i n t e g r a l c o m e s f r o m t h e i m m e d i a t e n e i q h b o r h o o d o f t h e o r i q i n . T h e c o n t o u r o f i n t e g r a t i o n , P, c a n b e d e f o r m e d a t w i l l , p r o v i d e d o n l y t h a t t h e c o n t o u r b e g i n a t a n i n f i n i t e l y r e m o v e d p o i n t c f o n e s h a d e d r e g i o n a n d e n d a t a l i k e w i s e i n f i n i t e l y r e m o v e d p o i n t i n t h e s e c o n d s h a d e d r e g i o n . T h e v a l u e o f t h e i n t e g r a l i s n o t c h a n g e d b y d e f o r m i n q t h e p a t h o f i n t e g r a t i o n p r o v i d e d n o s i n q u l a r i t i e s a r e c r o s s e d d u r i n q d e f o r m a t i o n . T h e a s y m p t o t i c e x p a n s i o n o f H o ^ R ) i s o b t a i n e d b y S a t s c n {1966,p.201). U s i n g S p i e g e l (1968, p.101) t o e v a l u a t e qa&ma f u n c t i o n s t h a t a r i s e , we f i n d 21 . ( I ) X 3-TI u ^ - t - ^ Y x o . e V ' si**. e V + ¥ L T ° ^ ° U ^ A 2 - 2 5 T h u s , t o t h e f i r s t o r d e r , t h e a m p l i t u d e o f t h e i n c i d e n t o r d i r e c t w a v e v a r i e s a s ( X f i ^ 1 , i . e . , i n a c c o r d a n c e w i t h t h e g e o m e t r i c a l o p t i c s a p p r o x i m a t i o n . T h e h i g h e r o r d e r t e r m s i n e g u a t i o n 2 . 2 5 c a n b e t r e a t e d a s c o r r e c t i o n s t c t h i s a p p r o x i m a t i o n . 2 . 1 . 2 . T h e R e f l e c t e d Wave T h e a n a l y t i c a l t e c h n i q u e s e m p l o y e d i n f i n d i n g a n e x p r e s s i o n f o r t h e d i r e c t w a v e ( e q u a t i o n 2 . 2 5 ) c a n l i k e w i s e be a p p l i e d t o f i n d a n e x p r e s s i o n f o r t h e r e f l e c t e d w a v e . B y u s i n g t h e M e t h o d o f I m a g e s , t h e r e f l e c t e d w a v e f i e l d c a n b e c o n s i d e r e d t o a r i s e f r o m a n i m a g e s o u r c e , l o c a t e d a t ( x , y ) = ( - x 0 , 0 ) . T h i s i s s h o w n i n f i q u r e 2. T h e w a v e a m p l i t u d e w i l l be m o d i f i e d b y a n a m p l i t u d e f u n c t i o n ( t h e r e f l e c t i o n c o e f f i c i e n t ) r e p r e s e n t i n g t h e e f f e c t o f t h e b o u n d a r y . , T h e t o t a l w a v e f i e l d o n t h e s h e l f c a n t h u s b e r e p r e s e n t e d i n i n t e g r a l f o r m by t h e e x p r e s s i o n i W Cos + lR(o^e. 2.26 -cut i w h e r e 1f) i s d e f i n e d b y °^ v= s,e , R i s t h e d i s t a n c e f r o m t h e i m a g e s o u r c e , t o t h e o b s e r v a t i o n p o i n t , P, 1R(©^  i s t h e r e f l e c t i o n c o e f f i c i e n t , a n d &( a n d (9^ a r e s h o w n i n f i g u r e 2. T h e 22 second term i n the i n t e g r a n d r e p r e s e n t s the r e f l e c t e d wave. The R e f l e c t i o n C o e f f i c i e n t I t i s neccessary to examine the r e f l e c t i o n c o e f f i c i e n t and to express i t e x p l i c i t l y i n terms of the i n t e g r a t i o n v a r i a b l e , G , and other parameters of the problem. Consider the problem with the geometry shown i n f i g u r e 4. The shallow water equations i n r e c t a n g u l a r c o o r d i n a t e s are 2.27 where the s u b s c r i p t i=1,2 r e f e r s to r e g i o n s 1 and 2, r e s p e c t i v e l y . The terms u and v are v e l o c i t y components i n the x and y d i r e c t i o n s . The i n t e g r a t e d c o n t i n u i t y eguation i s I f equations 2,27 and 2.28 are combined and the time dependency suppressed, the homogeneous Helmholtz equation r e s u l t i n g i s 2. 29 where ^ ' -\ 2.3 0 Choosing plane wave s o l u t i o n s t c equation 2.29 we f i n d REGION 2 Z R E G I O N 1 lz y / / / / / F i g u r e 4. S h e l f g e o m e t r y u s e d t o f i n d r e f l e c t i o n a n d t r a n s m i s s i o n c o e f f i c i e n t s . 24 - \ * 3 f < < o 2.32 where B and T r e p r e s e n t the r e f l e c t i o n and t r a n s m i s s i o n c o e f f i c i e n t s , r e s p e c t i v e l y , The i n c i d e n t wave i s chosen to have u n i t amplitude. The wave v e c t o r ^ = ( f e . K / k ^ i s connected with the angle of i n c i d e n c e © by the equations = k cos © , !a<.^tasvwG 2. 33 The boundary c o n d i t i o n s at x=0 r e q u i r e c o n t i n u i t y of s u r f a c e e l e v a t i o n and mass t r a n s p o r t , i . e. , 2. 34 which of course must be s a t i s f i e d f o r a l l y and t. S o l v i n g eguation 2.27 f o r u { and u^, and using equations 2.31 ,2.32 and 2,34 g i v e s the equations 25 - - W T [ - W ^ . ; ^ , ] e c k ^ 2 , 3 5 In order that these equations hold f o r a l l values of y i t i s necessary t h a t S " ^ 2. 36 Using t h i s r e l a t i o n , the equations i n 2,35 becciae 2.37 These eguations are sol v e d s i m u l t a n e o u s l y f o r R and T. The r e s u l t i n g REFLECTION COEFFICIENT and TRANSMISSION COEFFICIENT are given as 26 T -2.38 In addition to the equations qiven in equation 2,33 we can write k. ( C os t \ ^ h., Svw<S< 2.39 Usinq equations 2.33 and 2 . 39 , alonq with the relation CO •= Inc ^ k v C t 2.40 where c is the phase speed in region 1 and c, is the phase speed in reqion 2, we find Snell's law C fe-, — = = - v\ 2.41 where n is the index of refraction. We can now write equations for the reflection and transmission coefficients in terms of ©,n,f, and o as 27 (£X^ - ^ TCP'S <5 5WvCJ] - [^ v -^swfo - V ^  ^ 2,42 H i t h t h e p r e s e n c e o f a s g i v e n i n e q u a t i o n 2 , 42 , i n t h e i n t e g r a n d o f e q u a t i o n 2 . 26 , i t i s n o l o n q e r p o s s i b l e t o f i n d a s i m p l e e x p r e s s i o n f o r t h e w a v e f i e l d a s was t h e c a s e f o r t h e d i r e c t w a v e . B e c a u s e o f t h e p r e s e n c e o f t h e r a d i c a l , \) h v ~ fw^O i n (^ Cfci r t n e i n t e q r a n d i s d o u b l e - v a l u e d a n d b r a n c h p o i n t s a r e i n t r o d u c e d . I n a d d i t i o n , a n y z e r o s o f t h e d e n o m i n a t o r o f (Rt©) i n t r o d u c e s i m p l e p o l e s . I f t h e i n t e q r a t i o n p a t h e n c o u n t e r s a n y b r a n c h c u t s o r p o l e s d u r i n g d e f o r m a t i o n , t h e c o n t r i b u t i o n s d u e t o t h e i n t e g r a t i o n s a r o u n d t h e c u t s a n d t h e r e s i d u e s o f t h e F c l e s m u s t b e i n c l u d e d . B i e m a n n S h e e t s I t i s n e c e s s a r y t o r e n d e r t h e f u n c t i o n s i n g l e - v a l u e d . To a c c o m p l i s h t h i s we f o r m a t w o - s h e e t e d B i e m a n n s u r f a c e ( M a t h e w s a n d H a l k e r , 1965,p. 448) o f t h e c o m p l e x & - p l a n e ( f i g u r e 5) . On e a c h s h e e t t h e f u n c t i o n fR.C«0 w i l l be s i n g l e - v a l u e d . A s s u m e t h e B i e m a n n s u r f a c e i s f o r m e d i n s u c h a way t h a t o n t h e u p p e r B i e m a n n s h e e t t h e i m a g i n a r y p a r t o f t h e r a d i c a l i s p o s i t i v e , i . e . , W*vv\v-Sv*?"09 > O . o n t h e l o w e r s h e e t , l e t t h e i m a g i n a r y p a r t o f t h e r a d i c a l be n e g a t i v e , i . e . , ^v* / \ \J L e t t h e t w o s h e e t b e j o i n e d w h e r e t h e i m a g i n a r y p a r t o f t h e F i g u r e 5. T w o - s u r f a c e d R i e m a n n S h e e t . 2 9 r a d i c a l i s i d e n t i c a l l y z e r o , i . e . , W 5 v^-Svw1© = a. T h i s i s t h e s t a n d a r d s e l e c t i o n when t h e t i m e d e p e n d e n c y i s o f t h e f o r m - (.cut e. . A l s o , i t i s c o n v e n i e n t t o h a v e t h e b r a n c h c u t s f o l l o w t h e l i n e s j o i n i n g t h e t w o s h e e t s , t h e c u t s t h e r e b y p r o v i d i n g t h e means o f p a s s i n g f r o m o n e s h e e t t o t h e o t h e r . W h i l e t h e s e l e c t i o n o f c u t s i s a r b i t r a r y , t h e s e l e c t i o n made d o e s d e t e r m i n e t h e d i s p o s i t i o n o f t h e r e g i o n s i n t h e c o m p l e x p l a n e i n w h i c h cW Jv^-Sv*1©' i s p o s i t i v e o r n e g a t i v e . T h e p o s i t i o n s o f t h e b r a n c h p o i n t s a r e g i v e n b y t h e s o l u t i o n s t o \]yf- s (9 - o . T h u s , t h e b r a n c h p o i n t s , B ( a n d B ^ , a r e l o c a t e d a t 6 - ± A v c?\n A 2 . 4 3 w h e r e n i s r e a l a n d l e s s t h a n u n i t y . T h e b r a n c h p o i n t s l i e o n t h e r e a l © - a x i s , i n t h e i n t e r v a l -17/z<<9 <r^fx-B r a n c h C u t S e l e c t i o n D e f i n e t h e b r a n c h c u t by t h e e g u a t i o n ( B r e k h o v s k i k h , 1 9 6 0 , p . 2 5 2 ) V- - 2.44 w h e r e x i s r e a l a n d p o s i t i v e a n d l i e s i n t h e i n t e r v a l (0 , 9*0 • T h e b r a n c h p o i n t s B, a n d B z a n d b r a n c h c u t s a r e s h o w n i n f i g u r e 6. I t i s s e e n t h a t t h e c u t r u n s f r o m t h e b r a n c h p o i n t ( x=0) a l o n g t h e r e a l $ - a x i s t o t h e o r i g i n ( x =n) a n d f r o m t h e o r i g i n z a l o n g t h e i m a g i n a r y $ - a x i s t o t o o , i . e . , Svw ® "t a s x -* °© . I t i s n e c e s s a r y t o v e r i f y t h a t t h e i m a g i n a r y p a r t o f t h e r a d i c a l i s p o s i t i v e e v e r y w h e r e o n t h e u p p e r R i e m a n n S h e e t f o r 3 0 F i g u r e 6 . C o m p l e x p l a n e o f i n t e g r a t i o n a n d s t e e p e s t d e s c e n t p a t h . 31 t h e g i v e n c h o i c e o f b r a n c h c u t s ( M a t h e w s a n d H a l k e r , 1 9 6 5 , p . 4 5 0 ) . fieferring t o f i g u r e 7, d e f i n e t h e f o l l o w i n g : W - "S vvs. C3 - \ ft - Svw<s 1 e ** ^ 2 . 4 5 w h e r e oC a n d |>» a r e r e a l a n d s o s e l e c t e d a s t o make <k-o ,^-o when t h e Sv*.# i s r e a l a n d i t s m a g n i t u d e l e s s t h a n n o n t h e u p p e r B i e m a n n s h e e t . T h e n , w h e r e t h e s i g n c f t h e s g u a r e r o o t h a s b e e n c h o s e n p o s i t i v e , i . e . , ^ K V - SY*" 1© ^ ° f o r " - ^ < ^ y v ^ < > ^ . T h i s r e g u i r e s t h a t <*+^ =o when - s *\*G < >v . T h e v a l u e s o f a n d | i w i t h r e s p e c t t o t h e b r a n c h c u t s a r e s h o w n i n f i g u r e 7. I t i s c l e a r t h a t t h e sum oC+|^ i s e v e r y w h e r e i n t h e i n t e r v a l 0 t o 2 T T . U s i n g t h e i d e n t i t y e - C o ^ ) + i w ^ 2.47 we s e e t h a t t h e i m a g i n a r y p a r t i s p o s i t i v e i n a l l q u a d r a n t s . T h u s , t h e s e l e c t i o n o f t h e c u t s i s c o n s i s t e n t w i t h t h e a s s u m p t i o n t h a t ^YVN^V^-SYW 1© > 0 e v e r y w h e r e o n t h e u p p e r B i e m a n n s h e e t . U s i n g t h e n o t a t i o n s h o w n i n f i q u r e 4 , t h e r a d i c a l c a n b e w r i t t e n a s F i g u r e 7. C o m p l e x p l a n e u s e d t o d e t e r m i n e t h e s i g n o f I m v n 2 - s i n 3 c 3 f o r t h e two R i e m a n n S h e e t s . 33 r r ^ , tew \|V\-Svw l© - V\CaS<S, - — C O S © , - 2 . 4 8 E x p a n d i n g i n t o i t s r e a l a n d i m a g i n a r y p a r t s , i , e , , _ te.,^-v-v W'i'K ' a n < * s u b s t i t u t i n g i n t o e g u a t i o n 2.32 g i v e s C " < * 2 ^ 9 I n o r d e r t o s a t i s f y t h e r a d i a t i o n c o n d i t i o n f o r o u t g o i n g s a v e s a s x ->-<», i t i s n e c e s s a r y t h a t t h e i m a g i n a r y p a r t o f fe!)t,i.e. , k.," , b e c h o s e n p o s i t i v e . T h u s , f r o m e g u a t i o n 2.48 we s e e t h a t we m u s t c h o o s e VW^YNV-SY*' 1® ' >o a n d t h e i n t e g r a t i o n p a t h m u s t b e g i n a n d e n d o n t h e u p p e r E i e m a n n s h e e t . iJ2£lic§£i°Il o f t h e M e t h o d o f S t e e p e s t D e s c e n t s at d i s t a n c e s f r o m t h e s o u r c e w h i c h a r e l a r g e c o m p a r e d w i t h t h e w a v e l e n g t h , t h e f i e l d c a n be f o u n d u s i n g t h e m e t h o d o f s t e e p e s t d e s c e n t s . U s i n g t h i s t e c h n i q u e { s e e a p p e n d i x a) , t h e p a t h o f i n t e g r a t i o n , P, i s d e f o r m e d t o t h e s t e e p e s t d e s c e n t p a t h , P . T h e e g u a t i o n f o r t h e s t e e p e s t d e s c e n t p a t h i n t h e c o m p l e x $ - p l a n e i s g i v e n a s Ce^ (e'-G^ c o s U = 1 2.50 T h i s p a t h r u n s f r o m G ^ - ^ ^ O ^ ' i a o t o t h e i m a g i n a r y a x i s , w h e r e i t e n c o u n t e r s t h e b r a n c h c u t . T h e p a t h i s t h e n d e f o r m e d a r o u n d t h e c u t a n d c o n t i n u e s t o t h e p o i n t & =0^ o n t h e r e a l - a x i s , w h i c h i t o c r o s s e s a t an a n g l e o f 45 . F r o m t h i s p o i n t i t p a s s e s o n t o t h e 34 p o i n t G - 4^. V©^-»•<*> . The p o i n t $-<9j, i n t h i s case i s the saddle p o i n t ( f i g u r e 6), When d e f o r c i n g the i n t e g r a t i o n path, \ , to the s t e e p e s t descent path ,("*, care must be taken to note any c o n t r i b u t i o n a r i s i n g from a branch p o i n t , From f i g u r e 6 i t i s seen t h a t when the angle of i n c i d e n c e i s l e s s than the c r i t i c a l r e f r a c t i o n a ngle, i . e . , © Z < ( 9 C , where © c~- Ak/csvn v\ , the s t e e p e s t descent path , [ , does not i n t e r s e c t the branch p o i n t and no c o n t r i b u t i o n r e s u l t s when the path i s deformed around the cut. Thus, the e n t i r e path (except f o r a s m a l l p o r t i o n , shown with the dashed l i n e i n f i g u r e 6, which makes l i t t l e c r no c o n t r i b u t i o n to the i n t e g r a l ) may be c o n s i d e r e d t c l i e cn the upper Riemann sheet. The s t e e p e s t descent path c r o s s e s the r e a l i ©•-axis at (S- = © t , T h i s c r o s s i n g i s d i s c u s s e d i n appendix A. Poles of the Integrand To f i n d the p o l e s of the i n t e g r a n d i n eguation 2.26, we s e t the denominator of the r e f l e c t i o n c o e f f i c i e n t egual to zero and s o l v e f o r &. The eguation we must solve i s Y V H C G S S ^ + Sw^p 'J + t^ - S v v J O y -lt> S v v ^ ~ \ ^ o 2.51 where G-S^ i s the value of & at the pole. Rearranging eguation 2.50 and sguaring both s i d e s we f i n d v ? - s w ? ® ? w4co^<Sp wtC^-^vv^6p+lC^V\Xvf-0Svw<!)pcos©p 2.52 The l e f t - h a n d s i d e of t h i s eguation can be r e w r i t t e n as v\xSvUl©p •v-wCe*'©^- Sw?Op • D i v i d i n g the eguation by a>st©P r e s u l t s 35 i n a q u a d r a t i c equation i n iovw CJ 2. 53 which has the s o l u t i o n 2. 54 For the case of % s m a l l , the expression f o r 8^ can expanded i n a M a c l a u r i n expansion about % . T h i s r e s u l t s i n be 2.55 When the method of s t e e p e s t descents i s used to e v a l u a t e the i n t e q r a l r e p r e s e n t i n q the r e f l e c t e d wave i n equation 2,26, the main c o n t r i b u t i o n to the i n t e q r a l comes from the immediate neighborhood of the saddle p o i n t , which l i e s cn the r e a l S-a x i s . Usinq the expansion (equation 2.23) o f $ i n t o i t s r e a l and imaqinary p a r t s , the e x p r e s s i o n f o r tan O can be expanded as 2. 56 In the neiqhborhood of the 0- - a x i s , where v a l u e s of 0- are s m a l l , equation 2.56 takes the approximate form 36 -to^O f o ^ o ' ^ i »" ( V ©' 2.57 Equating the r e a l and imaginary parts of eguations 2.55 and 2.57 l o c a t e s the pole p o s i t i o n s i n the complex $ -plane. These r e s u l t s give 58 The l o c a t i o n of the pole s P ( and P^, f o r n r e a l and l e s s than u n i t y , depends upon eguation 2.51 being s a t i s f i e d . Expanding r ~ r — T T " 1 \\n -sxn S i n t o r e a l and imaginary p a r t s r e s u l t s i n the reguirement t h a t c o s O p be ne g a t i v e . T h i s means t h a t no poles l i e i n the r e g i o n 1<SP\ <-1'^ . . The poles P( and P z that do a r i s e are shown i n f i g u r e 6 . I t i s c l e a r t h a t when the i n t e g r a t i o n path, P, i s deformed to the st e e p e s t descent path , T , no c o n t r i b u t i o n s a r i s e from p o l e s being crossed. Let us e s t a b l i s h where poles P, and are l o c a t e d on the Biemann s u r f a c e , i . e . ,whether the poles are on the upper or lower sheet. T h i s r e q u i r e s t h a t the s i g n o f &vs\]w l-£vv?(9^ be determined f o r each pole. Eguation 2.51 can be rearranged to read J hv -«i VwX© j, - - v\ Cos <S ^  (i-v^) ^ 2. 59 Using the r e l a t i o n s 3 7 Co<> CS - COS ©' cosU <9" W ( 5 ' S u v V ' S V N * C S ~ Sv**©' C w ^ o ' ' +- v Cos <S* Cj" 2 , 6 0 e g u a t i o n 2.59 c a n b e r e w r i t t e n a s + S v v - © i ^ U O p " + w t y w © p svwU©p'3 2.61 F o r s m a l l v a l u e s o f © f , t h i s b e c o m e s T h u s , V f v \ l - $ v w ^ ^ Op + 2.63 S u b s t i t u t i n g f o r 0^', t h e e x p r e s s i o n g i v e n i n e g u a t i o n 2 . 5 8 , g i v e s U s i n g t h i s e x p r e s s i o n , we c a n d e t e r m i n e by i t s s i g n u p o n w h i c h B i e m a n n s h e e t e a c h p o l e i s l o c a t e d . T h e p e r t i n e n t i n f o r m a t i o n f o r e a c h p o l e i s g i v e n b e l o w : 38 T h u s , p o l e P, l i e s o n t h e u p p e r B i e m a n n s h e e t w h i l e p o l e F t l i e s o n t h e l o w e r B i e m a n n s h e e t . T h e s t e e p e s t d e s c e n t p a t h c a n b e d e f o r m e d a l o n g t h e c u t w i t h o u t e n c i r c l i n g t h e b r a n c h p o i n t . He c a n now p r o c e e d t o e v a l u a t e t h e r e f l e c t e d w a ve i n i t s i n t e g r a l f o r m b y a p p l y i n g t h e m e t h o d o f s t e e p e s t d e s c e n t s . F o r t h e a n g l e o f i n c i d e n c e , S ^ l e s s t h a n t h e c r i t i c a l a n g l e , CL a n d f o r 'itd'^l , t h e r e f l e c t e d wave i s g i v e n a s r ™ - c c C ^ - ^ + ^ r K c ^ r j _ A . ^yv-j} 2 . 6 5 w h e r e (RCQ^ i s t h e v a l u e o f t h e r e f l e c t i o n c o e f f i c i e n t e v a l u a t e d a t <& - \ , a n d C©^ ) i s t h e v a l u e o f t h e s e c o n d d e r i v a t i v e o f t h e r e f l e c t i o n c o e f f i c i e n t w i t h r e s p e c t t o (9- , e v a l u a t e d a t (9-- <5 T , T h e r e f l e c t e d w a v e s o e x p r e s s e d i s t h e s a d d l e p o i n t c o n t r i b u t i o n . A c c o r d i n g t o B r e k h o v s k i k h <1S60,p.255), t h e d e f o r m a t i o n o f V1 i n t o f 1 1 i m p l i e s t h a t t h e s a m e f i e l d i s r e p r e s e n t e d by s e t s o f p l a n e w a v e s c h o s e n i n d i f f e r e n t w a y s . O n c e t h e s t e e p e s t d e s c e n t p a t h i s c h o s e n , t h e f i e l d s e e n a t t h e p o i n t o f o b s e r v a t i o n , P , a p p e a r s t o b e o f p l a n e w a v e s h a v i n g t h e s a m e p h a s e , i . e . , t h e s a m e p h a s e a s a w a v e r e f l e c t e d f r c m t h e b o u n d a r y a t t h e a n g l e Qt. T h u s , t h e f i e l d a t P i s c o m p o s e d p r i n c i p a l l y o f p l a n e w a v e s r e f l e c t i n g f r o m t h e b o u n d a r y a t a n g l e s c l o s e t o (9 ,^ t h e a n g l e o f i n c i d e n c e o r r e f l e c t i o n a s .39 c o n s t r u c t e d .by g e o m e t r i c a l o p t i c s . T h e s a d d l e - p o i n t c o n t r i b u t i o n a a y a l s o b e i n t e r p r e t e d a s a l o c a l p l a n e - w a v e f i e l d p r o p a g a t i n g a l o n g B w i t h v a r y i n g a m p l i t u d e . T h e f i r s t t e r m i n e g u a t i o n 2.6 5 i s t h e p r i n c i p a l t e r m a n d g i v e s t h e r e f l e c t e d w a v e i n t h e g e o m e t r i c a l o p t i c s a p p r o x i m a t i o n . I n t h i s c a s e , t h e r e f l e c t i o n c o e f f i c i e n t f o r t h e c y l i n d r i c a l wave i s t h e same a s t h a t f o r a p l a n e w a v e . T h e a d d i t i o n a l t e r m s i n e g u a t i o n 2 . 6 5 , t h o s e w h i c h v a r y a s » c a n be t r e a t e d a s c o r r e c t i o n t e r m s . T h e i r i m p o r t a n c e a r i s e s i n a n u m b e r o f c a s e s ; i n p a r t i c u l a r , when t h e d i s t a n c e s o f S a n d P f r o m t h e b o u n d a r y a r e s m a l l c o m p a r e d w i t h t h e w a v e l e n g t h , t h e d i r e c t a n d r e f l e c t e d g e o m e t r i c a l o p t i c s w a v e s t e n d m o r e a n d m o r e t o c a n c e l o n e a n o t h e r . When t h e d e s t r u c t i v e i n t e r f e r e n c e i s c o m p l e t e , t h e f i e l d w i l l be e n t i r e l y d e t e r m i n e d b y t h e c o r r e c t i o n t e r m s . T h e w a v e a m p l i t u d e w i l l t h e n d e c r e a s e w i t h d i s t a n c e b e t w e e n S a n d P a s C K ^ ) E g u a t i o n 2.65 i s i n a p p l i c a b l e w h e n Ql-* fic , i . e . , when t h e a n g l e o f i n c i d e n c e a p p r o a c h e s t h e c r i t i c a l a n g l e . F r o m t h e e x p r e s s i o n f o r (R(©) g i v e n i n e g u a t i o n 2 . 4 5 , we f i n d 40 r u ' X \ , SwvOCoSO . f -i f 2.66 When we allow CS^C^ i n equation 2.66 we see t h a t l ^CO^ - ^ 0 0 . I f we take the second d e r i v a t i v e of (6.(0) with r e s p e c t to 0- , there aqain are terms c o n t a i n i n g 0 W~-\v*S& i n the denominator. Thus, as <*t-^ (9c, tfcCS,?)-*''0. The problem here l i e s i n the f a c t t h a t i n ap p l y i n g the method of s t e e p e s t descents , £i<&) i s assumed to be a slowly v a r y i n g f u n c t i o n . T h i s i s c l e a r l y not the case when (JS^ -XJ^ , and the method of s t e e p e s t descents cannot be a p p l i e d under t h i s c o n d i t i o n . I t i s necessary t c f u r t h e r analyze the f i e l d i n the range of i n c i d e n t angles c l o s e to the c r i t i c a l angle. Be f l e c t i o n near the C r i t i c a l Angle That p a r t o f eguation 2.26 which r e p r e s e n t s the r e f l e c t e d wave can be w r i t t e n , with the time dependence suppressed, as C C H G . 1 Cos CG-Or^ % = \ ^ - C < s > ) e cXCS 2.67 He can examine the f i e l d when i n c i d e n c e i s near the c r i t i c a l angle by i n t r o d u c i n g a new v a r i a b l e , | i , where j*> i s smal l and 41 d e f i n e d b y | i = ® c ~ © 2.68 i F o r ><K. l a r g e , i t i s p o s s i b l e t c c h o o s e t h e p a t h o f i n t e g r a t i o n o v e r >^ s o t h a t o n l y s m a l l v a l u e s o f >^ c o n t r i b u t e t o t h e i n t e g r a l ( B r e k h o v s k i k h , 1 9 6 0 , p . 2 8 2 ) . T o t r a n s f o r m t h e i n t e g r a n d o f e g u a t i o n 2 . 6 7 , w r i t e t h e e x p o n e n t i a l a s C.Kfl' Cos l®-<&0 CHO.' [.COS CQx-C9CiCto^A -svwCvO^ftl C " 1 1 2.69 T h e r e f l e c t i o n c o e f f i c i e n t ( e g u a t i o n 2.42) c a n be e x p a n d e d a b o u t Sw^© , w h i c h b e c o m e s s m a l l a s &-*> 0 t . T h u s , n e g l e c t i n g s e c o n d o r d e r t e r m s i n \JvO--'ivw'1® we f i n d 2. 70 C A S O -I n a p p l y i n g t h e m e t h o d o f s t e e p e s t d e s c e n t s , we c a n r e p l a c e 0 by ©^ a n d f u r t h e r , a s s u m i n g t h a t 42 ~ 5 7 1 ^ ' 2.71 we can r e p l a c e Q L by (S t. T h i s i s done everywhere except i n the r a d i c a l , s i n c e \J \vJ}-<«. © as The r a d i c a l can be r e w r i t t e n i n terms of R, as i i S v u^t C o s 1 © - Sw>0 O i ^ ® c With StV>|i we f i n d 2.72 I t i s necessary i n s e l e c t i n g t h e s i g n of the r a d i c a l to s a t i s f y the c o n d i t i o n W > O • T h i s r e g u i r e s choosing the negative s i g n . T h i s approximation holds everywhere except when (SC-»T^/1_ (_y\-% . T h i s c o n d i t i o n i s e a s i l y avoided i n the a n a l y s i s h e r e i n . Making these s u b s t i t u t i o n s i n equation 2.70 g i v e s 43 2.73 T h e r a d i c a l i n e q u a t i o n 2.7.3 c a n b e r e w r i t t e n f o r s m a l l a s c - WO 2.74 T h u s , e q u a t i o n 2.67 b e c o m e s n—x ui g i - v \ ITT _ 1 [cos C<v«0 w s p . - svw (<v<0^"] ( ^ f / ^ ; , 2.75 U s i n q t h e i d e n t i t y ( B r e k h o v s k i k h , 1 9 6 0 , p . . 2 8 3 ) 44 rx--Me u , ^ M 1 t I p t = \ & - . - f l ^ 2 . 7 6 where ^ 0 i s an a r b i t r a r y r e a l q u a n t i t y , -7/^  <-pu < \ , and D^ CO a r e p a r a b o l i c c y l i n d e r f u n c t i o n s (Ab ramowi tz and S t e q u n , 1964 , p. 689) and s e t t i n g e g u a t i o n 2 . 7 5 c a n be e x p r e s s e d i n two i n t e g r a l s o f the f o r m shown i n e q u a t i o n 2 . 7 6 . T h u s , we f i n d V* -C- r~ l V 2 . 7 8 where t h e p o s i t i v e s i g n i n s i d e t h e b r a c k e t r e f e r s t o t h e c a s e where G-^ Q^  and t h e n e g a t i v e s i g n r e f e r s to t h e c a s e where C5,">®. The e x p r e s s i o n f o r t h e r e f l e c t e d wave g i v e n i n e g u a t i o n 2 . 7 8 i s s i m i l a r t o t h a t f o u n d by B r e k h o v s k i k h ( 1 9 6 0 , p . 2 8 6 ) . B o t h e x p r e s s i o n s i n v o l v e t h e same p a r a b o l i c c y l i n d e r f u n c t i o n s . U s i n q t h e i d e n t i t y ( L e b e d e v , 1 9 7 2 , s e c t i o n 10.3) 45 ^ - " ^ ^ V^\±iA 2.79 we f i n d f o r ® x < S t 2 , 1 r 1" t. +<• r : r,DjJ-rt)D 2. 80 T h i s i s the complete e x p r e s s i o n f o r the r e f l e c t e d wave f i e l d on the s h e l f when i n c i d e n c e i s a t an angle l e s s than the c r i t i c a l angle. The asymptotic technique we have used l e a d s to a p a r a b o l i c c y l i n d e r f u n c t i o n r e p r e s e n t a t i o n of the f i e l d which y i e l d s continuous and f i n i t e v a l u e s f o r the f i e l d as the angle of i n c i d e n c e approaches the c r i t i c a l angle. The case where ©1"><SC i s handled i n the next s e c t i o n . 2.1.3. The L a t e r a l Save When the angle of i n c i d e n c e exceeds the c r i t i c a l angle, the path of i n t e g r a t i o n deforms to the steepest-descent path as shown i n f i g u r e 8. In order that i n t e g r a t i o n can be completed, i t i s necessary that the ends of the paths r and f be connected, a t o B and C to D. L e t C and D l i e on the upper n ' Biemann sheet and be connected. That p o r t i o n of 1 t r a v e r s e d i n going from B to the cut must l i e on the lower Biemann sheet. In order t o connect the end p o i n t s A and B, which now l i e on d i f f e r e n t s h e e t s , i t i s necessary t o use the path segments AE, EF (along the branch cut) and FB, with no c o n t r i b u t i o n a r i s i n g from AE and FB. 47 Hhen the integration path encircles the branch point i n a positive sense, the integral around the cut gives r i s e to a branch point contribution in the form of a l a t e r a l wave. If t h i s i s done, eguation 2.67 w i l l consist of two parts, the saddle point contribution (the refl e c t e d wave) and the branch point contribution (the l a t e r a l wave). The i n t e g r a l around the cut can be written V ' j ^ V 1 d e +Lj^i®\ftf *G 2 , 8 1 where l£($^is the r e f l e c t i o n c o e f f i c i e n t along the l e f t edge of the cut and Stasis that along the r i g h t edge of the cut. Since the difference between the r e f l e c t i o n coefficents on the two sides of the cut i s only i n the sign of the r a d i c a l , these integ r a l s can be combined and the l a t e r a l wave expressed as CHR. Cos C©-<0 2. 82 where I. 83 Define the steepest descent path , | , in such a way that i t 4 8 g o e s f r o m ©' ~ ©c a l o n g t h e l i n e o n w h i c h t h e r e a l p a r t c f t h e e x p o n e n t i n e g u a t i o n 2,82 d e c r e a s e s m o s t r a p i d l y . T h i s i s t h e l i n e o n w h i c h (c«-T>R»'cax(o-«J,S] i s c o n s t a n t . By s e t t i n g © - < 9 C a n d u s i n g t h e i d e n t i t y i n e q u a t i o n 2 . 6 0 , we f i n d t h e e q u a t i o n c f t h e p a t h t o b e O S -©0 CJOIU - Ceo. (CPC - O^) 2.84 II T h e s t e e p e s t d e s c e n t p a t h , Y , i s s h e w n i n f i g u r e 8. I t i s n e c e s s a r y t h a t W [*5 ^ (& b e p o s i t i v e , i . e . , Swx ( 8 ' - O *vvx^ > 0 2.85 No p r o b l e m i s e n c o u n t e r e d w h e n t h e p a t h a l o n g t h e c u t i s d e f c r m e d t o t h e s t e e p e s t d e s c e n t p a t h , s i n c e n o s i n g u l a r i t i e s a r e c r o s s e d . He c a n r e w r i t e e q u a t i o n 2.82 a s V" S i n c e , f o r l a r q e , o n l y s m a l l v a l u e s o f ©• c o n t r i b u t e t o t h e i n t e q r a l , we c a n e x p r e s s t h e i n t e q r a l i n t e r m s o f 6>" a n d s e t Q-1 - <9C e v e r y w h e r e i n t h e i n t e g r a n d w i t h t h e e x c e p t i o n o f -i T h e e x p o n e n t i a l i n t h e i n t e g r a n d c a n be w r i t t e n a s Ws\^ (G^ 'hv*Wo',=XfcV' SvUCe -^cQ. A l s o , dL®-«tdL©" . U s i n g e q u a t i o n " it 2 . 6 1 we c a n e x p a n d t h e r a d i c a l , a n d a l l o w i n q t h a t Sv^U©- e a n d CO">U©" - I , we f i n d 49 In order t h a t Vv^ v\v-Sw^ cTVo we r e q u i r e t h a t With Sv**©t-h and <4i.fit-. J i _ u s i n g the above s u b s t i t u t i o n s , equation 2.86 becomes 00 I n t r o d u c i n g a change o f v a r i a b l e , such that \J~"$""= x and using the i d e n t i t y o eguation 2.88 becomes The expr e s s i o n f o r the l a t e r a l wave given i n eguation 2.89 can be w r i t t e n i n a d i f f e r e n t form. Using the r e l a t i o n s from f i g u r e s 5 0 2 a n d 9, we c a n w r i t e w h e r e Y0-=. L 0cc/v<3 t a n d ^ - -L,G*.© t . O s i n g t h e s e r e l a t i o n s we f i n d « "K I wL x + L 0 -*-L,3 2.92 e X L v ceo.©c T h u s , we c a n w r i t e t h e e x p r e s s i o n f o r t h e l a t e r a l w a v e a s 2. 90 2. 91 SI Z A x / / / / / •h. -h //////// F i g u r e 9. S h e l f g e o m e t r y w i t h r a y p a t h d i a g r a m o f t h e l a t e r a l w a v e . 52 O w i n g t o t h e c o n t i n u o u s l e a k a g e o f e n e r g y a l o n g t h e l a t e r a l w a v e p a t h , t h e a m p l i t u d e o f t h e l a t e r a l w a v e d e c a y s m e r e r a p i d l y w i t h d i s t a n c e t h a n t h a t o f e i t h e r t h e d i r e c t o r r e f l e c t e d w a v e s . A t l a r g e d i s t a n c e s f r o m t h e s o u r c e , L ^ v , a n d t h e l a t e r a l w a v e a m p l i t u d e v a r i e s a s r . R e c a l l i n g t h a t , t o t h e f i r s t o r d e r , t h e d i r e c t w a ve a m p l i t u d e { e g u a t i o n 2 .25) v a r i e s a s R , i t i s c l e a r t h a t a t l a r g e d i s t a n c e s f r o m t h e s o u r c e t h e l a t e r a l wave c o n s t i t u t e s a d i f f r a c t i o n e f f e c t w h i c h i s g e n e r a l l y w e a k e r t h a n t h a t o f e i t h e r t h e d i r e c t o r r e f l e c t e d c o n s t i t u e n t s . U n d e r s t e a d y s t a t e c o n d i t i o n s d e t e c t i o n o f t h e l a t e r a l w a v e w o u l d t h u s b e d i f f i c u l t i n t h e f a r f i e l d . The e x p r e s s i o n f o r t h e l a t e r a l w a v e g i v e n i n e g u a t i o n 2 . 9 3 i s i n v a l i d f o r Y\-=» I a n d f o r L1- js o . T h e l a t e r a l w a v e p a t h s e g m e n t a l o n g t h e s h e l f e d g e , L ^ , a p p r o a c h e s z e r o when t h e o b s e r v a t i o n p o i n t P l i e s i n t h e v i c i n i t y o f t h e a n g l e o f t o t a l r e f l e c t i o n , ® c. T h i s i s t h e b o u n d a r y o f t h e d o m a i n o f e x i s t e n c e o f t h e l a t e r a l w a v e . E g u a t i o n 2 . 9 3 i s s i n g u l a r t h e r e b e c a u s e t h e s a d d l e p c i n t i n t h e i n t e g r a n d o f e g u a t i o n 2 . 8 6 a p p r o a c h e s t h e b r a n c h p o i n t . T h u s , i t i s n e c e s s a r y t o e x a m i n e t h e l a t e r a l w a v e i n m o r e d e t a i l w hen i n c i d e n c e a p p r o a c h e s t h e c r i t i c a l a n g l e . T h i s i s a c c o m p l i s h e d b y m e a n s o f p a r a b o l i c c y l i n d e r f u n c t i o n s . T h e c a s e o f M-» \ i s n o t o f 5 3 i n t e r e s t here. C r i t i c a l Angle R e f l e c t i o n To i n v e s t i g a t e the case when the angle o f i n c i d e n c e approaches the c r i t i c a l angle, we can proceed i n a manner s i m i l a r t o t h a t a p p l i e d to the r e f l e c t e d wave. By i n t r o d u c i n g a new v a r i a b l e /I {eguation 2.68) and r e p l a c i n g © by Gz and (5 cby <&c everywhere except i n ^ vvv- S w»\s , the f u n c t i o n V ( © ) i n the in t e g r a n d of eguation 2 .83 reduces to N M - z — — T T — 2.94 Eguation 2 .88 then becomes J C i ( , | v T I C t f t ^ f O / V c \ - r - e . . d f c 2 . 9 5 6 Using equation 2 . 7 6 , we f i n d the e x p r e s s i o n f o r the l a t e r a l wave, i n terms of p a r a b o l i c c y l i n d e r f u n c t i o n s , t o be where ^ and VJ are def inded by equation 2. 77. For HR* l a r g e , and "j) can be replaced with i t s 4 asymptotic expansion (Abramowitz and Stegun,1964,p.689) which i s 54 To the f i r s t order t h i s g i v e s 2.97 From eguation 2.77 we d e r i v e the r e l a t i o n s f Using these e x p r e s s i o n s , we f i n d <1H«.' cos ( ( Sv-Oj 3/ T h i s e x p r e s s i o n , along with equations 2.92 and 2.93 all o w s us to express the l a t e r a l wave as 2. 100 where which i s v a l i d everywhere i n the f a r f i e l d region of the s h e l f . T h i s e x p r e s s i o n f o r the l a t e r a l wave i s the same as t h a t given by equation 2.93 except f o r the f a c t o r H^). The l a t e r a l wave (equation 2. 100) can be examined more 55 c l o s e l y by looking at the behavior of F(^") as a function of the angle of incidence, ® x . The asymptotic expansion of D.y^C'Vj-H'^ given i n Abramowitz and Stegun (1964, p. 289) i s I ~ :—Tv, + which results i n -x-v- • ••• - 2.102 Thus, for <SL s u f f i c i e n t l y f ar from © c , ^ =\J ^ xP.'Sw^!^*] In th i s case F ^ 0 1 \ and ^ L , as given i n eguation 2.100, becomes the same as that given by equation 2.93. Expanding t^^V'V^ i n powers of Yj we fin d 56 i'. \ (4 3 T h i s r e s u l t s i n . ^  u J . K J 2 . 1 0 4 T a b l e s o f p a r a b o l i c c y l i n d e r f u n c t i o n s i n A b r a m o w i t z a n d S t e g u n ( 1 9 6 4 , p . 7 0 2 - 7 2 0 ) a l l o w t a b u l a t i o n o f t h e l a t e r a l w a v e f i e l d f o r a l l a n g l e s o f i n c i d e n c e g r e a t e r t h a n o r e q u a l t o t h e c r i t i c a l a n g l e . I n f i g u r e 9 , t h e d o m a i n o v e r w h i c h t h e l a t e r a l w a v e c a n e x i s t i s s h o w n . C l e a r l y , f o r i n c i d e n c e a t a n g l e s l e s s t h a n , no l a t e r a l w a v e a r i s e s , W h i l e i t m i g h t b e e x p e c t e d t h a t t h e a m p l i t u d e o f t h e l a t e r a l wave w o u l d b e c o m e z e r o a t ($ x = \ , s u c h i s n o t t h e c a s e . F r o m e g u a t i o n 2 . 1 0 4 , f o r r-/ > ^JfT V, •v^-^o, ' V ) ' " * ' p ( y * ^ | . T h u s , e g u a t i o n 2 , 1 0 0 b e c o m e s 57 V- T ( v e t , J T ^ V ( * L y -2.105 From equation 2.92, we f i n d }<L a - "HQ-* Cos (QT -(O Tu 2. 106 s i n c e tos Dsinq equation 2.77 we thus f i n d 2. 107 Osinq equation 2.107 i n equation 2.105, we f i n d the l a t e r a l wave f o r (9C t c be Thus, we see t h a t t h e amplitude of the l a t e r a l wave does not become zero a t the boundary of the re q i o n o f i t s e x i s t e n c e . Equation 2.78 qives the complete e x p r e s s i o n f o r the 58 r e f l e c t e d wave. Depending upon the sign chosen i n the brackets, the e x p r e s s i o n r e p r e s e n t s the f i e l d f o r i n c i d e n c e a t angles g r e a t e r or l e s s than the c r i t i c a l angle. For i n c i d e n c e at an angle g r e a t e r than the c r i t i c a l angle, eguation 2.78 g i v e s the f i e l d i n c l u d i n g the l a t e r a l wave c o n t r i b u t i o n ( t h i s i s shown i n eguations 2.109 and 2.110). The f a c t t h a t the amplitude of the l a t e r a l wave i s not zero at the boundary of the domain of i t s exi s t e n c e causes no concern s i n c e the amplitude of the f i e l d (eguation 2.78) i s continuous t h e r e . The r e f l e c t e d wave f o r i n c i d e n c e g r e a t e r than the c r i t i c a l angle To f i n d the r e f l e c t e d wave f o r the case 0 1 >© C , we s e l e c t the negative s i g n i n eguation 2.78. In a d d i t i o n , we must take i n t o account t h a t t h i s e x p r e s s i o n i n c l u d e s the l a t e r a l wave e f f e c t and must remove t h i s c o n s t i t u e n t i n order to o b t a i n the r e f l e c t e d wave only. From eguation 2.78, s e l e c t i n g the negative s i g n , we f i n d 2.109 Removing the l a t e r a l wave (eguation 2.96) from eguation 2.109 r e s u l t s i n the expre s s i o n 59 2!1 ~ i 2 1 10 which i s i d e n t i c a l to equation 2.78 i n the case where t \ < ® L . T h i s e x p r e s s i o n i n turn reduces d i r e c t l y to the e x p r e s s i o n qiven i n equation 2.80 and qives the r e f l e c t e d wave f i e l d f o r any <&u c l c s e to Gc. By expandinq T}(/(.-v^-iv^), we can examine more c l o s e l y the r e f l e c t e d wave f i e l d f o r i n c i d e n c e near the c r i t i c a l anqle. Expandinq i n a power s e r i e s i n vj , we f i n d 2. I l l with t h i s , the expr e s s i o n f o r the r e f l e c t e d wave becomes 60 F o r (9 t-=® c/ 2 . 1 1 3 T h e f i r s t t e r m i n t h i s e x p r e s s i o n i s m u l t i p l i e d b y t h e c o e f f i c e n t o f r e f l e c t i o n f o r p l a n e w a v e s i n c i d e n t a t C3 L-© t,i. e. , b y t h e t e r m T h u s , t h e g e o m e t r i c a l o p t i c s a p p r o x i m a t i o n 2. 114 i s a p p r o p r i a t e i n t h e f a r f i e l d . T h e c o r r e c t i o n t e r m { t h e s e c o n d t e r m i n t h e c u r l y b r a c k e t s i n e g u a t i o n 2.112) we n o t e d e c r e a s e s a s CKR.') . T h i s t e r m d e c r e a s e s m o re s l o w l y t h a n i t s c o u n t e r p a r t i n e g u a t i o n 2 . 6 5 , w h i c h d e c r e a s e s a s (Kfl'l . E x p a n d i n g Dv W"^ ^ i n p o w e r s o f , we f i n d 61 2. 115 and : 7<-/\ v J h - ^ ^ e 2. 116 Substitution of eguations 2.115 and 2.116 into eguation 2.80 gives For the case when ®X<QC , write c ] V w * (0V-C9j! =J^ vv*C.0t-fl)0 • Using t h i s change, the r e f l e c t e d wave i s given by 62 V 2. 118 When ,v^ ">"> | , we r e t a i n only the f i r s t two terms i n the c u r l y b r a c k e t s . These are the f i r s t two terms i n the power s e r i e s expansion i n vTw1— W ( 9 - \) Svwlfi c S v v s C t e t - © ^ of the r e f l e c t i o n c o e f f i c e n t (see eguation 2.73). Thus, f o r ^ - ^ - c ^ « \ we o b t a i n the g e o m e t r i c a l o p t i c s approximation. For the case when <St"? © t , we once again use the r e l a t i o n given i n eguation 2.79. T h i s g i v e s two terms, one which r e p r e s e n t s the r e f l e c t e d wave and one which r e p r e s e n t s the l a t e r a l wave. From eguation 2.80 we f i n d 2. 119 Using the expansions given i n equations 2.99, 2.101 and 2.115 we f i n d 6 3 5 . w h e r e Ctof) C U .^C-h v i - ^ . T h e f i r s t p a r t o f t h i s e q u a t i o n i s t h e s a m e a s e g u a t i o n 2 . 1 1 7 , w h i l e t h e s e c o n d p a r t i s t h e s a m e a s e q u a t i o n 2 . 1 0 0 . T h u s , t h e f i e l d o n t h e s h e l f h a s b e e n e x a m i n e d f o r a l l a n g l e s o f i n c i d e n c e , i n c l u d i n g t h o s e n e a r t h e c r i t i c a l a n g l e . T h u s a c o m p l e t e e x p r e s s i o n f o r t h e f i e l d i s o b t a i n e d . S u m m a r y Th e d e v e l o p m e n t o f t h e l a s t t w o s e c t i o n s h a s r e s u l t e d i n e x p r e s s i o n s f o r t h e r e f l e c t e d a n d l a t e r a l w a v e s f o u n d b y d i f f e r e n t m e t h o d s . , S o m e o f t h e s e a r e m o r e a c c u r a t e o r m o r e g e n e r a l t h a n o t h e r s . a s y m p t o t i c e x p r e s s i o n s f o r t h e r e f l e c t e d a n d l a t e r a l w a v e s i n t h e f a r - f i e l d a r e f o u n d ( e q u a t i o n s 2 , 6 5 a n d 2,93) u n d e r t h e a s s u m p t i o n o f g e o m e t r i c a l o p t i c s . U n f o r t u n a t e l y , t h e s e e x p r e s s i o n s a r e i n v a l i d w hen i n c i d e n c e a p p r o a c h e s t h e c r i t i c a l a n g l e . H e r e g e n e r a l e x p r e s s i o n s a r e f o u n d i n t e r m s o f p a r a b o l i c c y l i n d e r f u n c t i o n s . A c o m p l e t e e x p r e s s i o n f c r t h e f i e l d ( t h e r e f l e c t e d a n d l a t e r a l w a v e s ) , v a l i d f o r a l l a n g l e s o f i n c i d e n c e , i s g i v e n i n e q u a t i o n 2 . 7 8 . A c o m p l e t e e x p r e s s i o n f o r t h e l a t e r a l w a v e i n t e r m s o f p a r a b o l i c c y l i n d e r f u n c t i o n s i s g i v e n i n e g u a t i o n 2 . 9 6 . The p a r a b o l i c c y l i n d e r f u n c t i o n s i n equations 2.78 and 2.96 can be evaluated by r e p l a c i n g them with the a p p r o p r i a t e asymptotic expansions. Thus, the expr e s s i o n s f o r the r e f l e c t e d and l a t e r a l waves w i l l be expressed i n asymptotic form. In p a r t i c u l a r , the r e f l e c t e d wave i s found f o r i n c i d e n c e a t angles gr e a t e r than the c r i t i c a l angle i n equation 2.112 ( f o r 0^ ^ c l o s e to 6 C) , i n equation 2.113 ( f o r <5t= Gt) , and i n equation 2.117 (f o r <9V s u f f i c i e n t l y f a r from (9 C). For i n c i d e n c e at anqles l e s s than the c r i t i c a l a n g l e , the r e f l e c t e d wave i s qiven by equation 2.118. In a l l cases, the f i r s t order terms y i e l d the r e s u l t s found by q e o m e t r i c a l o p t i c s . The asymptotic form of the l a t e r a l wave i s found by expandinq the p a r a b o l i c c y l i n d e r f u n c t i o n s i n eguation 2.96 (or i n the f a r - f i e l d by expanding equation 2,100). For <9t s u f f i c i e n t l y f a r from ® c , the l a t e r a l wave i s given by equation 2.93 (the g e o m e t r i c a l o p t i c s s o l u t i o n ) . Eguation 2.108 i s used f o r <V>0^ . 2.1.4. The E f f e c t s o f Rotation The D i r e c t Wave From equation 2.25 (the asymptotic form o f the d i r e c t wave f i e l d ) , t h e d i r e c t wave i n the f a r - f i e l d can be found i n accordance with the q e o m e t r i c a l o p t i c s approximation. R e t a i n i n q the f i r s t order term t h i s becomes 2.121 The d i s p e r s i o n r e l a t i o n i s now w r i t t e n 65 3^ 2 * 1 2 2 JC/ where = /co . Two p o i n t s are immediately apparent from 2.122. F i r s t , under r o t a t i o n {£/0), the waves are d i s p e r s i v e . Second, the C o r i o l i s f o r c e decreases the wave number f o r a given wave frequency or c o n v e r s e l y , i n c r e a s e s the wave frequency f o r a given wave number. The phase and group speeds of the d i r e c t wave are found t o be and 2. 123 2. 124 r e s p e c t i v e l y . From these e q u a t i o n s , i t i s seen t h a t r o t a t i o n r e s u l t s i n an i n c r e a s e i n the phase speed and a decrease i n the group speed. In eguation 2.121, X appears both i n the amplitude f u n c t i o n and the e x p o n e n t i a l . The f a c t o r >C appearing i n the amplitude may be w r i t t e n 66 X 2. 125 T h e e f f e c t o f t h e r o t a t i o n i s t h u s t o i n c r e a s e t h e w a v e a m p l i t u d e ' ^ h e p r e s e n c e o f K i n t h e e x p o n e n t i a l a g a i n s h o w s t h a t r o t a t i o n r e s u l t s i n a n i n c r e a s e i n t h e p h a s e s p e e d o f t h e w a v e . S i n c e no w o r k r e s u l t s f r o m r o t a t i o n , i t i s r e q u i r e d t h a t t h e e n e r g y f l u x p e r u n i t a r e a r e m a i n c o n s t a n t . F o r a w a v e u n d e r z e r o - r o t a t i o n h a v i n g t h e f o r m 2. 126 t h e e n e r g y f l u x p e r u n i t a r e a , £ , i s 2, 127 w h e r e c ^ = c = Igh. U n d e r r o t a t i o n , " ^ l ^ l l i i r ) . T h e e n e r g y f l u x , £ , may now b e w r i t t e n a n d 2. 128 w h i c h t o o r d e r (r a g r e e s w i t h 2 . 1 2 7 . E g u a t i o n s 2.8 t o 2.10 c a n b e w r i t t e n a s 67 ^ - £v/ = - 3 ^ 2, 1 2 9 \>tv - o 2.130 where i t i s assumed t h a t d e r i v a t i v e s i n S are zero and where (u,v) i s the v e l o c i t y i n the (E,$) d i r e c t i o n . I f we write i . (vtt t -ouO V 0 <L 2.132 ^ « ^ C M B. - w t ) and s o l v e 2.130 to 2.132 f o r u and v, we f i n d co . -f 2.133 These ex p r e s s i o n s can be used t o f i n d the k i n e t i c and p o t e n t i a l energy d e n s i t i e s per unit h o r i z o n t a l area. The k i n e t i c energy d e n s i t y (K. E, D,) i s given as * 2.134 The p o t e n t i a l energy d e n s i t y (P.E. D.) i s 68 P.-&.D. ~- kt^V ~ 4 f^Hol^ 2.135 I t i s seen by comparing 2.134 and 2.135 that the k i n e t i c energy d e n s i t y i s g r e a t e r by a f a c t o r of (1 + £/1-<E, ) than the p o t e n t i a l energy d e n s i t y when under r o t a t i o n . In the case of z e r o - r o t a t i o n there i s e g u i p a r t i t i o n of energy. From eguation 2.133, the v e l o c i t y component i n the ©• -d i r e c t i o n , v , can be expressed i n terms of the r a d i a l v e l o c i t y component, u. T h i s r e s u l t s i n the r e l a t i o n which shows t h a t the C o r i o l i s f o r c e produces p a r t i c l e motion i n the water which i s t r a n s v e r s e t o the wave. The path t h a t the water p a r t i c l e s f o l l o w i s an e l l i p s e with semi-major a x i s given fay lu 0\ and semi-minor a x i s given by fclwd. I t i s o f i n t e r e s t t o examine the e f f e c t of r o t a t i o n upon some s p e c i f i c waves. In p a r t i c u l a r , l e t us look a t tsunami waves r e s u l t i n g from the Alaskan earthquake of 1964. Hwang and Divoky (1964, p. 191) s t a t e that the s u r f a c e e l e v a t i o n at any p o i n t showed short (~10-minutes) waves superposed on a l a r g e r and l o n g e r (~1.5-hour) system. Let us examine t h i s system of waves o i n the r e g i o n of Vancouver,B.C., i . e . , a t a l a t i t u d e of 4S N, The value of ? •= 2 - H-sin^, where << i s the l a t i t u d e , v a r i e s from zero at the eguator to one c y c l e every 12 hours at the p c l e . At Vancouver, the value of ir i s about one c y c l e per 15.9 hours. The short tsunami waves ( 10 minutes) r e s u l t i n a value 69 -1 c f €. o f a b o u t 10 w h i c h i n t u r n r e s u l t s i n r o t a t i o n a l e f f e c t s o f t h e o r d e r o f 10 , e x c e p t i n v 0 w h i c h i s o f o r d e r t. T h e - 1 l a r g e r w a v e s ( 1.5 h o u r s ) y i e l d a n 6 s l i g h t l y l e s s t h a n 10 , -1 w h i c h r e s u l t s i n r o t a t i o n a l e f f e c t s o f a b o u t 10 . T h u s , a t t h i s l a t i t u d e , t h e e f f e c t o f r o t a t i o n u p o n t h e s e t s u n a m i w a v e s i s n e g l i g i b l e . T h e R e f l e c t e d g a v e F r o m t h e f i r s t o r d e r t e r m i n e g u a t i o n 2.6 5, t h e e x p r e s s i o n f o r t h e r e f l e c t e d w a ve i n t h e f a r - f i e l d may be w r i t t e n ~ . W £ 2 . 1 3 7 T h e e f f e c t o f r o t a t i o n o n t h e w a v e s p e e d s o f t h e r e f l e c t e d wave i s t h e s a m e a s t h a t o f t h e d i r e c t w a v e . L i k e w i s e , t h e a m p l i t u d e f u n c t i o n c o n t a i n s t h e f a c t o r X a n d t h e c h a n g e i n a m p l i t u d e d u e t o t h i s t e r m r e s u l t s i n t h e same c h a n g e s e e n i n t h e d i r e c t w a v e . B u t , t h e a m p l i t u d e f u n c t i o n o f t h e r e f l e c t e d w a v e a l s o i n c l u d e s t h e f a c t o r ( R . { 6 \ ) , t h e r e f l e c t i o n c o e f f i c i e n t e v a l u a t e d a t © = (5V. W i t h s o m e r e a r r a n g i n g t h i s i s g i v e n b y e g u a t i o n 2 .42 a s flSU©0= ^X'Sw,Q>^ * ^  feQ-^^v"^ 2 . 1 3 8 E x p a n d i n g t h i s a n d r e t a i n i n g t e r m s t o o r d e r <c g i v e s 2. 139 w h e r e {£.{<&,) i s t h e r e f l e c t i o n c o e f f i c i e n t e v a l u a t e d a t $ = ©, i n t h e c a s e o f z e r o r o t a t i o n . F r o m e g u a t i o n 2 . 1 3 9 we s e e t h a t 70 r o t a t i o n r e s u l t s i n an i n c r e a s e i n the magnitude of r e f l e c t i o n Thus, r o t a t i o n produces a r e f l e c t e d wave of amplitude g r e a t e r than or egual to t h a t produced under zero r o t a t i o n . The i n c r e a s e problem i n t h i s e x p r e s s i o n blowing-up, s i n c e zeros of the denominator correspond to the c r o s s i n g of poles and i t was shown in s e c t i o n 2.1,2 (p.21) that no poles are c r o s s e d . B e c a l l f u r t h e r t h a t the asymptotic e x p r e s s i o n f o r the r e f l e c t e d wave (2.65) was i n v a l i d f o r <8t-^  ® c . I t i s of i n t e r e s t to determine i f r o t a t i o n causes a change i n the p r o p o r t i o n o f energy c a r r i e d away from the boundary by the r e f l e c t e d and t r a n s m i t t e d waves. Consider a wave of u n i t amplitude i n c i d e n t upon the boundary at angle ©• ( f i q u r e 4). For the case of zero r o t a t i o n , the p r o p o r t i o n of i n c i d e n t wave enerqy contained i n the r e f l e c t e d wave i s c o e f f i c i e n t as w e l l as a chanqe i n phase. The modulus of (Q, (<Jt) to the order i s found to be ,t- x. i s p r o p c r t x o n a l to tr . The denominator of the fe term causes no £ r iRol Co 0*-(9 2. 141 Under r o t a t i o n , t h i s p r o p o r t i o n becomes (usinq equation 2.124) 71 2 . 1 4 2 T h e r a t i o o f e q u a t i o n 2 . 1 4 2 t o e q u a t i o n 2 . 1 4 1 g i v e s , t o o r d e r <c T h u s , u n d e r r o t a t i o n , t h e p r o p o r t i o n o f e n e r g y c a r r i e d b y t h e r e f l e c t e d wave i s g r e a t e r t h a n o r e g u a l t o t h e p r o p o r t i o n c a r r i e d b y t h e r e f l e c t e d w a v e u n d e r z e r o - r o t a t i o n . T h e t r a n s m i s s i o n c o e f f i c i e n t g i v e n b y e g u a t i o n 2 . 4 2 , may be r e w r i t t e n a s TteV- 2. 144 T h i s c a n b e a p p r o x i m a t e d t o t h e o r d e r (: by w h e r e H6(©)is t h e t r a n s m i s s i o n c o e f f i c i e n t f o r z e r o - r o t a t i o n . T h u s , l i k e t h e r e f l e c t i o n c o e f f i c i e n t , r o t a t i o n c a n r e s u l t i n a n i n c r e a s e i n t h e m a g n i t u d e o f t h e t r a n s m i s s i o n c o e f f i c i e n t a s w e l l a s t o c a u s e a p h a s e s h i f t . T h e m o d u l u s o f \Ud t o o r d e r £-i s 72 2,146 and |T(®M = Tito i + 6 -—• = 2. 147 The p r o p o r t i o n o f i n c i d e n t wave energy contained i n the t r a n s m i t t e d wave i s , r e s p e c t i v e l y , f o r the case of r o t a t i o n and z e r o - r o t a t i o n ano It. = l T 0 \ Coz cos e,  t l 1 c 0 l cose 2.148 2. 149 where Co, a n c * Co t are the group speeds i n the shallow and deep wafer r e s p e c t i v e l y , f o r z e r o - r o t a t i o n , C ,^ and C g v are the group speeds i n the shallow and deep water under r o t a t i o n , © and Or are the angles of i n c i d e n c e and ©, and © l f are the angles of r e f r a c t i o n f o r z e r o - r o t a t i o n and r o t a t i o n , r e s p e c t i v e l y . Using equation 2.124 and S n e l l ' s law, with © = Gr, the r a t i o o f equation 2.148 to equation 2.149 (using equation 2.147) i s found to be 73 + e _ _ r 2.150 1\* ^ » -T N. _ Thus, under r o t a t i o n , the p r o p o r t i o n of enerqy c a r r i e d by the t r a n s m i t t e d wave i s g r e a t e r than or egual t o the p r o p o r t i o n c a r r i e d by the t r a n s m i t t e d wave under z e r o - r o t a t i o n . Equation 2,145, usinq Ov^cesS =- W C a^fi -\)v\v~ s +v^Cos©-+ J""h-S*v?© can be w r i t t e n U Cfe) ; -, • 1 I + c£ _ -,; which agrees with eguation 2.37, The energy f l u x c a r r i e d onto the boundary must equal the enerqy f l u x l e a v i n q the boundary. Thus, f o r a r e f l e c t e d wave of amplitude 11*1 and a t r a n s m i t t e d wave of amplitude \lfl , the f o l l o w i n q equation must be s a t i s f i e d ; 1 C 5 , cos® - llfc\ZC5, cos© + l lT\ ZC^ i coss, 2 , 1 5 2 where c ^ and c<^ are the qroup v e l o c i t i e s on the s h e l f and i n the deep water, r e s p e c t i v e l y and ©^ i s the anqle of r e f r a c t i o n . Equations 2.138 and 2.144 can be r e w r i t t e n as 74 2. 153 V X C I A O V © +Oc©0 - i & G - i ^ S *v.Q 2.154 f r o m t h e s e we f i n d Vlf = • — VvHvvc^CS 4-c*wC9,^ 1' A - e H i - ^ M Sws V© r 1 r 1 Z W i t h c^ (=Jgh, (1-€-) a n d C c ^ ^ g h ^ { 1-€: ) > we c a n r e a r r a n g e e g u a t i o n 2.152, f i n d i n g V\ O r © "tw 1 (v\ cwJ& + ( J A C ^ +- (T Cl - v \ v ) \ 0 J w h i c h v e r i f i e s e n e r g y c o n s e r v a t i o n . T h e L a t e r a l H a v e E g u a t i o n 2.93 c a n be u t i l i z e d i n e x a m i n i n g t h e e f f e c t s o f r o t a t i o n o n t h e l a t e r a l w a v e i n t h e f a r - f i e l d ( i C L j » 1 ) . R e w r i t i n g 2.93 i n t e r m s o f (z we f i n d c (L 0 + L, -v A L 0 -out ] 75 T h e p r e s e n c e o f X i n t h e e x p o n e n t i a l r e s u l t s i n c h a n g e s i n t h e wave s p e e d s s i m i l a r t o t h o s e f o u n d f o r t h e i n c i d e n t w a v e . L i k e w i s e t h e w a v e n u m b e r a n d w a v e l e n g t h a r e s i m i l a r l y a f f e c t e d , -X T h e f a c t o r i n t h e wave a m p l i t u d e c a n b e w r i t t e n X -x _ r<* ,J/ 2.156 "L T h u s , t h i s f a c t o r , d u e t c r o t a t i o n w i l l i n c r e a s e t h e w a v e a m p l i t u d e r e l a t i v e t o t h e c a s e o f z e r o - r o t a t i o n b y a f a c t o r o f ( 1 + ^ 6 ) . T h e o t h e r f a c t o r i n t h e wave a m p l i t u d e w h i c h i s d e p e n d e n t u p o n r o t a t i o n i s { n - i f c - \ j l - n ^ ) , A p p r o x i m a t i n g t h i s f a c t o r t o o r d e r 6 f i n d s (v\-C6 U-*x J ^ v\ ( ^ l - ^ t e — — ) 2. 157 T h e p r e s e n c e o f t h i s f a c t o r r e s u l t s i n a d e c r e a s e i n wave a m p l i t u d e a n d a p h a s e s h i f t . T h e m o d u l u s o f 2 . 1 5 7 i s a p p r o x i m a t e l y 2. 158 I n c o n c l u d i n g , t h e C o r i o l i s f o r c e 1) i n c r e a s e s t h e p h a s e 76 s p e e d , 2) i n c r e a s e s t h e w a v e l e n g t h f o r f i x e d f r e g u e n c y a n d 3) d e c r e a s e s t h e g r o u p s p e e d f o r a l l t h r e e c o n s t i t u e n t s . , T h e w a ve a m p l i t u d e i n c r e a s e s f o r t h e d i r e c t a n d r e f l e c t e d w a v e s , b u t may i n c r e a s e o r d e c r e a s e f o r t h e l a t e r a l w a v e . T h e f a c t t h a t t h e s e 7. c h a n g e s a r e o f o r d e r 6 s h o w s t h a t , f o r t s u n a m i w a v e s , t h e e f f e c t s o f r o t a t i o n c a n be n e g l e c t e d , 2 , 1 , 5 , Wave a r r i v a l - t i m e s I t w as m e n t i o n e d e a r l i e r t h a t , i n c e r t a i n r e g i o n s o n t h e s h e l f , t h e a r r i v a l o f t h e l a t e r a l w a v e p r e c e e d s t h a t c f a n y o t h e r w a v e . T h i s w i l l n o t b e o b v i o u s i n t h e c a s e o f t i m e -h a r m o n i c e x c i t a t i o n b e c a u s e t h e w a v e f i e l d w i l l e x i s t i n i t s s t e a d y s t a t e f o r m ; t h e f i e l d i s p r e s e n t t h r o u g h o u t t h e e n t i r e s p a t i a l d o m a i n f o r a l l t i m e . R e c a l l i n g t h a t t h e a m p l i t u d e o f t h e -3/Z l a t e r a l w a v e d i m i n i s h e s a s r i n t h e f a r - f i e l d , w h i l e t h e d i r e c t a n d r e f l e c t e d w a v e s d e c r e a s e a s r r e s u l t s i n t h e p r e s e n c e o f t h e l a t e r a l w a v e b e i n g o b s c u r e d b y t h e l a r g e r d i r e c t a n d r e f l e c t e d w a v e s . U n d e r t r a n s i e n t c o n d i t i o n s , t h e l a t e r a l w a v e i s e v i d e n c e d b y i t s e a r l i e r a r r i v a l - t i m e . T h e f a c t t h a t a s e g m e n t o f t h e g e o m e t r i c a l o p t i c s p a t h f o l l o w e d b y t h e l a t e r a l w a v e i s l o c a t e d i n t h e d e e p w a t e r ( s e e s e g m e n t L o f f i g u r e s 9 a n d 10) w h e r e i t t r a v e l s w i t h a s p e e d g r e a t e r t h a n t h a t o f w a v e s p r o p a g a t i n g o n t h e s h e l f , r e s u l t s i n t h e p o s s i b i l i t y t h a t i t may a r r i v e f i r s t . I t i s o f i n t e r e s t t o d e t e r m i n e i n w h i c h r e g i o n s o f t h e s h e l f t h i s o c c u r s a s w e l l a s t o d e t e r m i n e t h e a r r i v a l - t i m e s c f a l l o f t h e c o n s t i t u e n t s . T h e e f f e c t s o f r o t a t i o n h a v e b e e n s h o w n , i n t h e c a s e o f t s u n a m i w a v e s , t o b e s m a l l . L i t t l e w i l l be a d d e d t o t h e p r e s e n t d i s c u s s i o n b y i n c l u d i n g r o t a t i o n a n d t h e a n a l y s i s h e r e i n d e a l s 77 F i g u r e 10. T i m e - d i s t a n c e c u r v e s f o r t h e d i r e c t a n d l a t e r a l w a v e s f o r a s h e l f g e o m e t r y . 7 8 c n l y w i t h t h e c a s e c f z e r o - r o t a t i o n . T h e g e o m e t r i c a l o p t i c s p a t h f o r t h e l a t e r a l w a v e i s s h o w n i n f i g u r e 9. T h e d i r e c t wave p a t h i s a s t r a i g h t l i n e f r o m t h e s o u r c e , S, t o t h e p o i n t o f o b s e r v a t i o n , P. To e n h a n c e t h e d e v e l o p m e n t a n d t o a r r i v e a t m o r e g e n e r a l r e s u l t s , f i g u r e 9 i s r e a r r a n g e d t o f o r m f i g u r e 1 0 . T h i s i s d o n e b y r o t a t i n g t h e x - y a x e s o f f i g u r e 9 t h r o u g h a n a n g l e o f ^ - ^ i • A d j a c e n t t o t h e r a y p a t h d i a g r a m i s s h o w n a t i m e - d i s t a n c e c u r v e w h i c h i l l u s t r a t e s t h e f i r s t r e s p o n s e a r r i v i n g a t d i s t a n c e B. T h e b r e a k i n t h e c u r v e o c c u r s a t t h e " c r o s s o v e r - p o i n t " , t h e d i s t a n c e b e y o n d w h i c h t h e l a t e r a l w a v e a r r i v e s f i r s t a n d b e f o r e w h i c h t h e d i r e c t w a v e a r r i v e s f i r s t . T o d e v e l o p e x p r e s s i o n s f o r t h e a r r i v a l - t i m e s a n d c r o s s o v e r -d i s t a n c e , e s t a b l i s h t h e f o l l o w i n g g e o m e t r y : l e t t h e w a t e r d e p t h on t h e s h e l f a n d i n t h e d e e p w a t e r be h , a n d h £ , r e s p e c t i v e l y . U s i n g s h a l l o w w a t e r t h e o r y , t h e c o r r e s p o n d i n g p h a s e s p e e d s a r e c,= ( g h , ) a n d c%= ( g h v ) . L e t t h e s o u r c e , S, a n d o b s e r v a t i o n p o i n t , P, be l o c a t e d p e r p e n d i c u l a r d i s t a n c e s f r o m t h e s h e l f e d g e c f d , a n d d t , r e s p e c t i v e l y . F r o m f i g u r e 1 0 , we o b s e r v e t h e f o l l o w i n g r e l a t i o n s h i p s : <4% U 2. 1 5 9 1L dU 2 , 160 79 R ^ 2 . 1 6 1 T h e t i m e r e q u i r e d f o r t h e d i r e c t w a v e t o qo f r o m S t o P i s X T J " C , 2 . 1 6 2 T h e t i m e f o r t h e l a t e r a l w a v e t o r e a c h P i s Li d , L ~ C X C , O L © 1 C , c n 2 . 1 6 3 F o r i n c i d e n c e a t t h e c r i t i c a l a n q l e , ® v = C, S v w © x • — - i . vv Cervfc,. = V_Z1 2.164 O s i n q e q u a t i o n s 2.161 a n d 2.164, e q u a t i o n 2.163 b e c o m e s E q u a t i n q e q u a t i o n s 2,162 a n d 2.165, we d e t e r m i n e t h e d i s t a n c e B a t w h i c h t h e d i r e c t a n d l a t e r a l w a v e s a r r i v e s i m u l t a n e o u s l y . T h i s r e s u l t s i n a n e x p r e s s i o n q i v i n q t h e c r o s s o v e r d i s t a n c e a s 80 R c - C ^ + d O - E " C , < " 2. 166 A t i m e - d i s t a n c e r e l a t i o n s h o w i n g t h e f i r s t r e s p o n s e w i t h d i s t a n c e i s s h o w n i n t h e t o p p o r t i o n o f f i g u r e 1 0 . T h e f i r s t s e g m e n t o f t h i s c u r v e , f o r R < R t , i s g i v e n by e g u a t i o n 2 . 1 5 9 , w h i c h s h o w s t h e f i r s t r e s p o n s e i s d u e t o t h e d i r e c t w a v e . T h e s e c o n d s e g m e n t , R"> 8 C , i s g i v e n b y e g u a t i o n 2. 1 6 2 , w h i c h s h o w s t h a t t h e f i r s t r e s p o n s e i s d u e t o t h e l a t e r a l w a v e . T h e i n t e r c e p t o f t h e l a t e r a l wave c u r v e w i t h t h e t i m e - a x i s , t , i s f o u n d b y s e t t i n g R = 0 . D o i n g t h i s , we f i n d t 0 - 2 . 1 6 7 T h e c h a n g e f r o m t h e d o t t e d p o r t i o n o f t h e l a t e r a l w a v e c u r v e t o t h e c l a s h e d p o r t i o n d e f i n e s t h e e d g e o f t h e r e g i o n o f e x i s t e n c e o f t h e l a t e r a l w a v e . T h a t i s n o l a t e r a l w a v e s a r e p r e s e n t f o r R <R0, s i n c e r e f l e c t e d w a v e s a r r i v i n g a t t h a t p o i n t v i a t h e i r g e o m e t r i c a l o p t i c s p a t h a r e w a v e s r e f l e c t i n g a t a n g l e s l e s s t h a n t h e c r i t i c a l a n g l e , i . e. , <9,] <• <SC . S i n c e o n l y i n c i d e n c e a t <9^>©c g i v e s r i s e t o t h e l a t e r a l w a v e , l a t e r a l w a v e s a r e p r e s e n t o n l y i n t h e r e g i o n R > R 0 . D u r i n g t h e e x p e r i m e n t a l i n v e s t i g a t i o n , d a t a f r o m w h i c h t h e v a r i o u s t r a v e l - t i m e s a n d c r o s s o v e r - d i s t a n c e w e r e d e t e r m i n e d w e r e t a k e n w i t h t h e s o u r c e a n d p o i n t o f o b s e r v a t i o n l o c a t e d e g u i d i s t a n t l y f r o m t h e s h e l f e d g e . C o m p a r i s o n o f t h e 81 e x p e r i m e n t a l f i n d i n g s w i t h t h e o r y i s e n h a n c e d b y d e v e l o p i n g h e r e e x p r e s s i o n s f o r t h e t r a v e l - t i m e s a n d c r o s s o v e r d i s t a n c e f o r t h e c a s e d,=d_=d, ©, = \ a n d B=y. E g u a t i o n s 2. 1 6 2 , 2. 1 6 5 , 2. 166 a n d 2 . 1 6 7 r e d u c e t o 2. 168 2. 169 4c 1* \{ C x - C, 2.170 4, _ 8LA [ C j - C , 1 V © - — • 2. 171 A s a g e o p h y s i c a l e x a m p l e , c o n s i d e r a s h o r t t s u n a m i w a v e { ~ 1 5 - m i n u t e p e r i o d ) g e n e r a t e d a t t h e m i d p o i n t o f a 100 k m - w i d e s h e l f w h i c h l i e s a d j a c e n t t o a d e e p w a t e r r e g i o n . L e t t h e d e p t h o n t h e s h e l f a n d i n t h e d e e p w a t e r be 150 m a n d 4 km, r e s p e c t i v e l y . F o r a n o b s e r v a t i o n p o i n t , w h i c h l i k e t h e s o u r c e , l i e s a d i s t a n c e 50 km f r o m t h e s h e l f e d g e , t h e c r o s s o v e r d i s t a n c e i s c a l c u l a t e d t o be a b o u t 1 2 5 km d o w n s h e l f f r c m t h e s o u r c e . T h e w a v e s c h o s e n w o u l d h a v e a s p e e d o f a b o u t 140 k m / h r a n d a w a v e l e n g t h o f a b o u t 3 5 km. T h u s , c r o s s o v e r w o u l d o c c u r a t a d i s t a n c e o f a b o u t 3.6 w a v e l e n g t h s . A l o n g t s u n a m i s a v e ( ~ 1 . 5 h r . - p e r i o d ) w o u l d h a v e a w a v e l e n g t h o f n e a r l y 2 0 0 km a n d c r o s s o v e r w o u l d o c c u r w i t h i n t h e p a s s a g e o f o n e w a v e . F o r c o m m u n i t i e s l y i n g a l o n g t h e s h o r e ( 1 0 0 k m f r o m t h e s h e l f e d g e ) t h e p o i n t a t w h i c h t h e l a t e r a l a n d d i r e c t w a v e s w o u l d 82 a r r i v e s i m u l t a n e o u s l y (the cr o s s o v e r - d i s t a n c e ) would be l o c a t e d at a d i s t a n c e R=175km from the source. I t would take these waves 1.25 hr to t r a v e r s e t h i s d i s t a n c e . For shore communities l y i n q f u r t h e r from the source, the a r r i v a l of the l a t e r a l wave would preceed t h a t of the d i r e c t wave. A community l o c a t e d 300km from the source would r e c o r d the a r r i v a l of the l a t e r a l wave at time t=1.47 hr a f t e r g e n e r a t i o n , whereas the d i r e c t wave would a r r i v e a f t e r a time t=2. 14hr. Thus, the a r r i v a l o f the l a t e r a l wave would preceed that of the d i r e c t wave by t=0.67hr. For a community l o c a t e d 500km from the sou r c e , the l a t e r a l wave w i l l a r r i v e a f t e r time t=1.75hr, while the d i r e c t wave w i l l a r r i v e a f t e r time t=3.57hr, a time d i f f e r e n c e of t=1.82hr. We can see t h a t the d i f f e r e n c e i n a r r i v a l - t i m e o f the d i r e c t and l a t e r a l waves can become s i g n i f i c a n t even over r e l a t i v e l y s h o r t d i s t a n c e s i n the ocean. These d i f f e r e n c e s i n a r r i v a l - t i m e s could be used t o warn and prepare l o c a l p o p u l a t i o n s to take evasive a c t i o n s i n p r e p a r a t i o n f o r the oncoming d i r e c t and r e f l e c t e d tsunami waves. 2,1.6. The A p p l i c a b i l i t y of L i n e a r Non-dispersive Theory The a n a l y s i s contained h e r e i n has been based on l i n e a r non-d i s p e r s i v e theory. An examination i s made t o e s t a b l i s h i f the use of t h i s model i s a p p r o p r i a t e and i n p a r t i c u l a r , i f i t a p p l i e s to the case of tsunami waves. Backus (1962) examines the e f f e c t of r o t a t i o n on s m a l l amplitude ocean waves observed at l a r g e d i s t a n c e s from the source. Murty (1977,p.137) d i s c u s s e s the a p p r o p r i a t e model equations f o r tsunamis propagating on the c o n t i n e n t a l s h e l f . Small amplitude, shallow water waves s a t i s f y the 8 3 i n e q u a l i t i e s . 2 .172 w h i c h i m p l y l i n e a r i t y a n d n c n - d i s p e r s i v e n e s s , r e s p e c t i v e l y , H o w e v e r , t h e t y p e o f w a v e w h i c h e x i s t s d e p e n d s c r i t i c a l l y c n t h e r a t i o o f t h e s e t w o p a r a m e t e r s . T h e r e s u l t i n g p a r a m e t e r , w h i c h i s u s e d t o d e t e r m i n e t h e a p p r o p r i a t e m o d e l t o be u s e d ( a c c o r d i n g t o n c c l i n e a r d i s p e r s i v e e f f e c t s ) , i s t h e U r s e l l n u m b e r , d e f i n e d b y { f o r e x a m p l e , s e e Hammack a n d S e g u r , 1 9 7 8 ) F o r t h e c a s e U<M, t h e s i t u a t i o n i s o n e i n w h i c h t h e S t o k e s w a v e l i m i t e x i s t s a n d n o n l i n e a r i t y i s e v e n l e s s i m p o r t a n t t h a n d i s p e r s i o n ; f o r 0 = 0 ( 1 ) , t h e K o r t e w e g - d e V r i e s e g u a t i o n i s a p p l i c a b l e ; f o r U V > 1 , t h e h y d r a u l i c l i m i t a p p e a r s w i t h t h e p o s s i b i l i t y o f s h o c k w a v e s a r i s i n g , W h i l e t h e r e s e e m s t o b e n o d i s a g r e e m e n t w i t h t h e s e r e s u l t s , t h e r e i s d i s a g r e e m e n t w i t h how t o s e l e c t t h e l e n g t h s c a l e u s e d t o c h a r a c t e r i z e a n o n - m o n o c h r o m a t i c wave t r a i n a n d how t o i n t e r p r e t t h e ' o r d e r u n i t y ' , Hammack a n d S e g u r ( 1 9 7 8 ) h a v e r e s o l v e d t h i s p r o b l e m by c o n s t r u c t i n g a d i m e n s i o n l e s s U r s e l l n u m b e r , 0 0 , b a s e d u p o n t h e o v e r a l l d i m e n s i o n s o f t h e i n i t i a l d a t a . U 0 i s g i v e n a s U = 2. 173 84 O e ^ = - l V / l 2 . 1 7 4 w h e r e L d e n o t e s t h e c h a r a c t e r i s t i c l e n g t h o f t h e i n i t i a l d a t a , h i s t h e w a t e r d e p t h a n d V i s t h e d i m e n s i o n l e s s v o l u m e o f t h e d i s t u r b a n c e , g i v e n by V - ~ t 2 . 1 7 5 w h e r e * a ' i s a c h a r a c t e r i s t i c wave a m p l i t u d e . A p p l i c a t i o n o f e q u a t i o n s 2 . 1 7 4 a n d 2 . 1 7 5 r e s u l t s i n t h e d e t e r m i n a t i o n o f t h e a p p r o p r i a t e m o d e l e g u a t i o n s . Hammack a n d S e g u r ( 1 9 7 8 ) h a v e e x a m i n e d t h e i n f l u e n c e o f i n i t i a l c o n d i t i o n s o n t h e c h o i c e o f > a n d t h e p o s s i b l e c h a n g e s i n r e g i m e a s t h e w a v e s p r o p a g a t e a w a y f r o m t h e s o u r c e . T h e i r f i n d i n g s i n d i c a t e t h a t l i n e a r n o n d i s p e r s i v e t h e o r y i s a p p l i c a b l e f o r t h e l e a d i n g w a v e o f a l o n g t s u n a m i ( X ~ 1 0 0 m i l e s ) i n t h e d e e p o c e a n up t o a d i s t a n c e o f 4 0 , 0 0 0 m i l e s ( i . e . , e f f e c t i v e l y e v e r y w h e r e ) a n d o n c o n t i n e n t a l s h e l v e s f o r p r o p a g a t i o n d i s t a n c e s u p t o 2 0 0 m i l e s ( i . e . , e f f e c t i v e l y o v e r t h e w i d t h o f m o s t s h e l v e s ) . F o r s h o r t e r t s u n a m i s (>'-40 m i l e s ) , t h e l e a d i n g w a v e may b e d i s p e r s i v e b e f o r e t h e w i d t h o f t h e s h e l f i s c r o s s e d , i n w h i c h c a s e t h e a p p l i c a t i o n o f t h e K o r t e w e g - d e V r i e s e q u a t i o n may b e i n o r d e r . D i r e c t w a v e s , p r o p a q a t i n q a l o n q t h e s h e l f may t r a v e l d i s t a n c e s s u b s t a n t i a l l y q r e a t e r t h a n 200 m i l e s , a n d i n t h e s e c a s e s t h e a p p l i c a b i l i t y o f t h e m o d e l m u s t b e e x a m i n e d . T h e l a t e r a l w a v e , b y c o n t r a s t , t r a v e l s much o f i t s d i s t a n c e a l o n q 85 the s h e l f edge, i n t h e deep water, and the l i n e a r ncndis p e r s i v e r e s u l t s obtained are d i r e c t l y a p p l i c a b l e , as they are f o r the deep ocean saves. 2.2. Impulse E x c i t a t i o n Have f i e l d s generated i n the ocean are never d e s c r i b a b l e by a s i n g l e time-harmonic. T y p i c a l l y , e x c i t a t i o n occurs i n a r e l a t i v e l y s h o r t i n t e r v a l o f time and the i n i t i a l f i e l d i s cc ir posed of waves of many wavelengths. .Consider the t r a n s i e n t f i e l d response to impulse e x c i t a t i o n f o r the case of zero-r o t a t i o n . As i t was s t a t e d i n s e c t i o n 2. 1. 5, under t h i s c o n d i t i o n the l a t e r a l wave a r r i v e s at some p o i n t s on the s h e l f before any other wave. Let the r i g h t hand s i d e of equation 2.15 be w r i t t e n with the time-dependency i n the form of an i m p u l s e , f u n c t i o n , i . e . . The response to t h i s e x c i t a t i o n can be determined f o r a r b i t r a r y times, but f o r the present i n v e s t i g a t i o n , c n l y the l e a d i n g edge values of the f i e l d are considered. These responses are recovered from time-harmonic s o l u t i o n s i n the f a r - f i e l d by recourse to the asymptotic p r o p e r t i e s {Watsons Lemma) c f the Laplace transform. Consider the Laplace transform 2. 176 fBracewell,1965,p.219) of 86 2. 177 where "^((1,0=0 f o r t < t a , t 0 being the time a r r i v a l c f the l e a d i n g edge a t d i s t a n c e R. Eguation 2. 176 can be r e w r i t t e n as 00 ^ - S t , \ , s t \cat^~ e j c x+t^^x 2 ,17s where T = t - t 0 . I f Be^.s$ i s p o s i t i v e and s u f f i c i e n t l y l a r g e , the i n t e g r a l converges and the main c o n t r i b u t i o n to the i n t e g r a l comes from the neighborhood o f t =0, i . e . , at the l e a d i n g edge. Expanding (R,t) about "c=0 r e s u l t s i n an expansion of the form oo 2.179 ivi s o where Re\«<-}>-1 and the c o e f f i c i e n t s C t a are f u n c t i o n s of R only. S u b s t i t u t i n g eguation 2.179 i n 2.178 we f i n d to to et. C %T «-l <M-V* W - . o 2. 180 where, i n a d d i t i o n to i n t e r c h a n g i n g the order of summation and i n t e g r a t i o n , we have used E u l e r ' s i n t e g r a l of the second kind to 87 repre s e n t the gamma f u n c t i o n (Gradshteyn and Byshik,1971). The c o e f f i c i e n t s i n eguations 2. 179 and 2.180 are the same. The time-harmonic s o l u t i o n f o r the d i r e c t wave (from eguation 2.25) i n the f a r - f i e l d i s w r i t t e n —-z e k £ » \ 2 ' 1 8 1 where k=^/c,, c , being the phase speed on the s h e l f . Eguation 2.181 can be reaaranged t c the form where, we r e c a l l \ (1/2)={TT. Eguation 2.182 can be put i n the form of 2.180 by r e p l a c i n g -iuD with s and l e t t i n g ou+m+1=1/2. T h i s r e s u l t s i n an expression (from eguation 2. 179) f o r the l e a d i n g edge of the d i r e c t wave being The source f o r t h i s time-dependent f i e l d i s S ( t ) . Since 'vit)(B,t) i s r e a l , then *1 (R,t) -0 f o r t *~ E/c, . The wave d e s c r i b e d by eguation 2.183 i s a c y l i n d r i c a l wave which o r i g i n a t e s a t the source (B=0) at time t=0. The l e a d i n g edge a r r i v e s a t the point of o b s e r v a t i o n a f t e r time t= B/c,. , The time-harmonic s o l u t i o n f o r the r e f l e c t e d wave (from eguation 2.65) i n the f a r - f i e l d i s 88 fcR1^ 2.184 w h e r e T h e t r a n s i e n t s o l u t i o n f o r t h e l e a d i n g e d g e o f t h e r e f l e c t e d w a v e ( f o u n d i n t h e s a m e way a s t h e t r a n s i e n t s o l u t i o n f o r t h e d i r e c t wave) i s nrca,o - fcio ft, T ^ T ^ , * * R i - les T h i s w a v e i s a c y l i n d r i c a l w a v e w h i c h a p p e a r s t o a r i s e f r o m a n i m a g e s o u r c e S ( R =0) a n d a r r i v e s a t t h e p o i n t o f o b s e r v a t i o n P ( I , a f t e r t i m e t = R / c ( , R b e i n g t h e d i s t a n c e f r o m S t o P. S i n c e R / > B, t h e r e f l e c t e d w a v e w i l l a r r i v e a f t e r t h e d i r e c t w a v e . T h e t i m e - h a r m o n i c s o l u t i o n f o r t h e l a t e r a l wave ( f r o m e g u a t i o n 2 . 93) i n t h e f a r - f i e l d i s r I \A(u0+L,-v w L a V " V s l .1 -\ ,.-n- / - i —I 3 . 0 WIT e L ' ^ r . , V 3 / T " 2.186 T h i s i s r e w r i t t e n a s 89 sL- 4IE-i—_——rc^uao ceo 2. 187 w h e r e we h a v e u s e d n = c , / c t , j 1 (3/2) = a n d 0 ( L a ) i s t h e H e a v i s i d e u n i t f u n c t i o n . I t i s c l e a r f r o m f i g u r e s 1 (a) a n d 9 t h a t 0 when 1^=0. I f e q u a t i o n 2.187 i s c o m p a r e d w i t h 2.179 ( s u b s t i t u t i n g s f o r - i w ) , e g u a t i o n 2.172 y i e l d s t h e t r a n s i e n t s o l u t i o n f o r t h e l e a d i n g e d g e o f t h e l a t e r a l w a v e i n t h e f o r m ~ ^ U C U t ^ - T + - 2.188 T h e l e a d i n g e d g e a r r i v e s a f t e r t i m e t = ( L 0 + L ( ) / c ( • L a / c t , w h i c h a t c e r t a i n p o i n t s o f o b s e r v a t i o n p r e c e e d s a n y o t h e r wave ( s e e s e c t i o n 2. 1. 5 o n c r o s s o v e r - d i s t a n c e s a n d a r r i v a l - t i m e s ) . T h e w a v e f r o n t o f t h e l a t e r a l w a v e i s p l a n e , a s s h o w n i n f i g u r e s 1 ( a ) a n d 9. 2.3. D e s c r i p t i o n o f t h e H a v e F i e l d T h e d e v e l o p m e n t made t h u s f a r i n c h a p t e r 2 h a s r e s u l t e d i n n u m e r o u s e x p r e s s i o n s d e s c r i b i n g t h e d i f f e r e n t w a v e c o n s t i t u e n t s , u n d e r a v a r i e t y o f c o n d i t i o n s . H h i l e t h e s e e x p r e s s i o n s p r o v i d e a d e g u a t e i n f o r m a t i o n a s t o t h e b e h a v i o r o f t h e w a v e s , i t i s f e l t t h a t a d e s c r i p t i o n o n a m o r e p h y s i c a l l e v e l i s a p p r o p r i a t e . S i n c e t h e e f f e c t s o f r o t a t i o n h a v e b e e n d i s c u s s e d i n s e c t i o n 2.1.4, t h i s d i s c u s s i o n w i l l f o c u s o n w a v e s i n t h e f a r - f i e l d , u n d e r z e r o - r o t a t i o n , a s f o u n d f r o m g e o m e t r i c a l o p t i c s . T h e d i r e c t w a v e , g i v e n i n e g u a t i o n 2.183, h a s t h e f o r m o f a c y l i n d r i c a l w a v e w h i c h o r i g i n a t e s a t t h e s o u r c e , S ( f i g u r e 2 ) , 90 a t t i m e t = 0 . T h e s p e e d o f p r o p a g a t i o n o f t h i s w a v e , c, = (gh 1, , i s t h e s p e e d o f p r o p a g a t i o n o n t h e s h e l f , a s d e t e r m i n e d f r c m l i n e a r s h a l l o w w a t e r c o n d i t i o n s , , T h e w a ve a r r i v e s a t a n o b s e r v a t i o n p o i n t P { a t d i s t a n c e B f r o m t h e s o u r c e ) , a f t e r t i m e t = B / c , , t h e t i m e i t t a k e s t h e wave t o t r a v e r s e t h e d i s t a n c e R a t t h e s p e e d -ML c , . T h e d i r e c t wave a m p l i t u d e d i m i n i s h e s a s B T h e r e f l e c t e d w a v e , a s g i v e n i n e g u a t i o n 2 . 1 6 5 , i s a n a l o g o u s i n d e s c r i p t i o n t o t h e d i r e c t wave. I n t h i s c a s e , t h e w a v e a p p e a r s t o a r i s e f r o m a n i m a g e s o u r c e , s', a t t i m e t = 0 . T h i s w a v e a l s o p r o p a g a t e s w i t h a s p e e d c, , a n d a f t e r t i m e t = B / c , , r e a c h e s an o b s e r v a t i o n p o i n t P ( a t d i s t a n c e H f r o m S ) . S i n c e B ' > B , t h e a r r i v a l o f t h e r e f l e c t e d w a v e w i l l f e l l o w t h e a r r i v a l o f t h e d i r e c t w a v e . T h e a m p l i t u d e o f t h e r e f l e c t e d w a v e i s a f u n c t i o n n o t o n l y o f t h e d i s t a n c e f r o m t h e s o u r c e , b u t o f t h e r e f l e c t i o n c o e f f i c i e n t a s w e l l . T h e v a l u e o f t h i s a m p l i t u d e f u n c t i o n c a n be c a l c u l a t e d , g i v e n t h e i n d e x o f r e f r a c t i o n a s s o c i a t e d w i t h t h e s t e p a n d t h e a n g l e o f i n c i d e n c e , f r o m t h e e x p r e s s i o n g i v e n i n e q u a t i o n 2 . 1 8 4 . F o r t\> # c, t o t a l r e f l e c t i o n r e s u l t s a n d A s t h e r e f l e c t e d wave p r o p a q a t e s a w a y f r o m t h e s o u r c e , i t s a m p l i t u d e d i m i n i s h e s a s T h e l a t e r a l wave" i s g i v e n b y e q u a t i o n 2 . 1 8 8 . T h e e x i s t e n c e o f t h i s wave d e p e n d s u p o n t h e a b i l i t y o f t h e m e d i u m t o s u p p o r t w a v e s a t t w o s p e e d s o f p r o p a g a t i o n . I n c o n t r a s t t o t h e d i r e c t a n d r e f l e c t e d w a v e s , w h i c h a r e c y l i n d r i c a l i n s h a p e , t h e l a t e r a l w a v e f r o n t i s p l a n e . F r o m e q u a t i o n s 2 . 1 8 6 a n d 2 . 9 2 , t h e s u r f a c e c f c o n s t a n t p h a s e i s g i v e n b y 9 1 le-CU-t-L. + h L j - H [ r ? w s © c - K * + * 0 ^ Cos <& t] c Convtcvwt- 2 . 1 8 9 w h i c h i s t h e e g u a t i o n o f a s t r a i g h t l i n e i n t h e x y - p l a n e , B r e k h o v s k i k h ( 1 9 6 0 , p . 2 7 7 ) p r e s e n t s a d i a g r a m s h o w i n g t h e r e l a t i o n s h i p o f t h e v a r i o u s c o m p o n e n t s o f t h e f i e l d . A s i m i l a r d i a g r a m i s s h o w n i n f i g u r e 1 1 . F r o m t h i s f i g u r e , t h e d i r e c t w a v e (1) a n d t h e r e f l e c t e d wave (2) a r e r e p r e s e n t e d a s c y l i n d r i c a l f r o n t s o r i g i n a t i n g a t S a n d s ' , r e s p e c t i v e l y . T h e l a t e r a l w a v e (3) i s s h o w n a s a p l a n e w a v e f r o n t , a t t a c h i n g w i t h t h e b o u n d a r y ( p o i n t A, f i g u r e 11) t o t h e d i s t u r b a n c e ( 4 ) , w h i c h p r o p a g a t e s a l o n g t h e s h e l f e d g e i n t h e d e e p w a t e r , a n d a t t a c h i n g o n t h e s h e l f t a n g e n t i a l l y w i t h t h e r e f l e c t e d w a v e f r o n t ( p o i n t B, f i g u r e 1 1 ) . P o i n t B l i e s o n a r a d i u s f r o m S w h i c h m a k e s a n a n g l e &-c w i t h t h e n o r m a l t o t h e b o u n d a r y . T h e a m p l i t u d e o f t h e l a t e r a l w a v e i n c r e a s e s a l o n g t h e w a v e f r o n t f r o m t h e b o u n d a r y t o i t s p o i n t o f t a n g e n c y w i t h t h e r e f l e c t e d w a v e . T h i s i s c l e a r f r o m e g u a t i o n 2 . 1 8 6 a n d f i g u r e 9, s i n c e r e a c h i n g a n y p o i n t a l o n g t h e w a v e f r o n t , f r o m p o i n t A t o p o i n t B o f f i g u r e 1 1 , r e q u i r e s t h e p a t h l e n g t h L ^ be s h o r t e n e d . . T h e d i s t u r b a n c e (4) i n f i q u r e 11 i s t h e c a u s e o f t h e l a t e r a l w a v e . As t h i s d i s t u r b a n c e p r o p a g a t e s a l o n g t h e b o u n d a r y , i t r e r a d i a t e s e n e r g y b a c k o n t o t h e s h e l f . I t i s a c o n s e q u e n c e o f t h i s c o n t i n u o u s r e r a d i a t i o n t h a t t h e a m p l i t u d e o f t h e l a t e r a l w a v e d e c a y s m o r e r a p i d l y w i t h d i s t a n c e f r o m t h e s o u r c e t h a n e i t h e r t h e d i r e c t o r r e f l e c t e d w a v e s . F o r t h e d i s t u r b a n c e (4) p r o p a g a t i n g w i t h a w a v e l e n g t h i n t h e d e e p w a t e r , i n o r d e r . F i g u r e I I . W a v e f r o n t a n d r a y d i a g r a m f o r t h e w a v e f i e l d c o n s t i t u e n t s a r i s i n g f o r © t > <9C . 9 3 t h a t c o n t i n u i t y be p r e s e r v e d a c r o s s t h e b o u n d a r y , i t i s n e c e s s a r y t h a t t h e l a t e r a l w a v e h a v e w a v e l e n g t h >, = > 1 s i n c \ , w h e r e G c i s t h e c r i t i c a l a n g l e . T h i s s h o w s t h a t t h e d i r e c t i o n o f p r o p a q a t i o n o f t h e l a t e r a l w a v e i s a l o n g a l i n e w h i c h m a k e s a n a n q l e ® c w i t h t h e n o r m a l t o t h e b o u n d a r y . T h i s p r o c e s s i s d i s c u s s e d i n d e t a i l i n B r e k h o v s k i k h { 1 9 6 0 , p . 2 7 6 ) , An i n t e r e s t i n q s i t u a t i o n a r i s e s when t h e s o u r c e i s p o s i t i o n e d o n t h e b o u n d a r y (x=0 i n f i q u r e 2 ) . I n t h i s c a s e , ( J o n e s , 1 9 6 4 , p. 6 6 1 ) B = R / = L o = 0. T h e w a v e f r o n t s w h i c h a r i s e f r o m t h i s c o n f i q u r a t i o n a r e s h o w n i n f i q u r e 1 2 . The d i r e c t wave ( 1 ) , l o c a t e d i n t h e s h e l f r e q i o n , i s i n t h e f o r m o f a s e m i - c i r c u l a r w a v e f r o n t w h i c h p r o p a q a t e s a d i s t a n c e B = c , t i n t i m e t f r o m i t s g e n e r a t i o n . T h e r e f l e c t e d w a v e f r e n t (V) c o i n c i d e s w i t h t h a t o f t h e d i r e c t w a v e . T h e r e f r a c t e d w a v e ( 2 ) , i n t h e d e e p w a t e r , i s i n t h e f o r m o f a s e m i - c i r c u l a r w a v e f r o n t , w h i c h p r o p a q a t e s a d i s t a n c e R - ^ c ^ t i n t i m e f a f t e r i t s g e n e r a t i o n . S i n c e c ^ c, , i t f o l l o w s t h a t B l > B ) , I n o r d e r t o p r o v i d e c o n t i n u i t y a c r o s s t h e b o u n d a r y , i t i s n e c e s s a r y t h a t t h e l a t e r a l w a v e (3) b e p r e s e n t . T h i s w a v e , w h i c h i s i n t h e f o r m o f a p l a n e w a v e f r o n t , a t t a c h e s w i t h (2) a t t h e b o u n d a r y ( p o i n t A, f i q u r e 12) a n d w i t h (1 ) a t p o i n t B. F o r t h e s i t u a t i o n s h o w n i n f i q u r e 1 2 , a n y p o i n t s o f o b s e r v a t i o n l o c a t e d i n t h e r e q i o n b o u n d e d b y t h e s t e p , c u r v e (1) a n d c u r v e (3) w i l l b e r e a c h e d f i r s t b y t h e l a t e r a l w a v e . a F i g u r e 12. W a v e f r o n t d i a g r a m f o r t h e s o u r c e l o c a t e d o n t h e s h e l f e d g e . 95 3. T h e E x p e r i m e n t a l M o d e l I n c h a p t e r 2 , a n u m b e r o f c o n d i t i o n s a r e i m p o s e d o n t h e m a t h e m a t i c a l m o d e l i n o r d e r t o f a c i l i t a t e t h e f i n d i c g o f s o l u t i o n s w h i c h r e p r e s e n t t h e t h e v a r i o u s f i e l d c o n s t i t u e n t s . T h e s e i m p o s i t i o n s u n a v o i d a b l y r e s u l t i n seme l o s s o f i n f o r m a t i o n , a n d i t i s r e a s o n a b l e t o e x p e c t t h a t t h e w a v e f i e l d f o u n d b y u s i n g t h e m a t h e m a t i c a l m o d e l w i l l d i f f e r f r o m t h e f i e l d o b s e r v e d i n a r e a l m o d e l . I t i s o f i n t e r e s t t o d e t e r m i n e t h e s e d i f f e r e n c e s a n d t h e i r c a u s e s . T h i s i n f o r m a t i o n w i l l l e a d t o a b e t t e r u n d e r s t a n d i n g o f t h e wave f i e l d a n d w i l l a l s o p o i n t o u t s o m e o f t h e l i m i t a t i o n s a n d d e f i c i e n c i e s t h a t r e s u l t f r o m t h e u s e o f t h e m a t h e m a t i c a l m o d e l . I n o r d e r t o make t h e s e d e t e r m i n a t i o n s , a l a b o r a t o r y m o d e l r e p r e s e n t a t i v e c f t h e m a t h e m a t i c a l m o d e l w a s e s t a b l i s h e d a n d t h e r e s u l t s o f m e a s u r e m e n t s made o n i t a r e e x a m i n e d . S u c h f i e l d p r o p e r t i e s a s e n e r g y d e n s i t y v a r i a t i o n w i t h p o s i t i o n , t h e p h a s e a n d e n e r g y s p e e d s o f t h e v a r i o u s f i e l d c o n s t i t u e n t s a n d wave p r o f i l e s a r e d e t e r m i n e d . T h e s e f i n d i n g s a r e c o m p a r e d w i t h t h e e x p e c t e d w ave p r o p e r t i e s a s d e t e r m i n e d b y t h e m a t h e m a t i c a l m o d e l f o r t h e c a s e c f z e r o - r o t a t i o n . T h e l a b o r a t o r y m o d e l u s e d i n t h e e x p e r i m e n t w a s c o n s t r u c t e d w i t h i n a w a v e t a n k l o c a t e d i n t h e h y d r a u l i c s l a b o r a t o r y o f t h e D e p a r t m e n t o f C i v i l E n g i n e e r i n g , T h e t a n k i s r e c t a n g u l a r i n s h a p e a n d m e a s u r e s 5.0m i n l e n g t h , 2,2m i n w i d t h a n d 0.6m i n d e p t h , A d e t a i l e d d e s c r i p t i o n o f t h e w a v e t a n k i s p r e s e n t e d i n a p p e n d i x B. T h e b o t t o m a r e a o f t h e w a v e t a n k w a s d i v i d e d i n t o t w o d o m a i n s : a p e r m a n e n t s h e l f r e g i o n a n d a r e g i o n w i t h a v e r t i c a l l y p o s i t i o n a b l e b o t t o m . When t h e w a v e t a n k was f i l l e d 96 w i t h w a t e r , t h e w a t e r d e p t h i n t h e r e q i o n w i t h t h e a d j u s t a b l e b o t t o m c o u l d be v a r i e d , w h i l e t h a t o v e r t h e s h e l f r e q i o n r e m a i n e d f i x e d . B y p o s i t i o n i n g t h e a d j u s t a b l e b o t t o m t o a l e v e l l o w e r t h a n t h a t o f t h e s h e l f b o t t o m , a g e o m e t r y s i m i l a r t c t h a t c f a s h e l f r e g i o n l y i n g a d j a c e n t t o a d e e p w a t e r r e g i o n was o b t a i n e d . T h e t w o r e g i o n s w e r e s e p a r a t e d by a v e r t i c a l , l i n e a r s t e p d i s c o n t i n u i t y . One s u c h c o n f i g u r a t i o n ( f i g u r e B - 1 ) g a v e r i s e t o a l a b o r a t o r y m o d e l r e p r e s e n t a t i v e o f t h e g e o m e t r y s h o w n i n f i g u r e 2. I n t h i s c a s e t h e s t e p c o r r e s p o n d s t o t h e b o u n d a r y l o c a t e d a t x=0. By p l a c i n g t h e a d j u s t a b l e b o t t o m a t t h e same d e p t h a s t h e s h e l f b o t t o m , t h e w a t e r d e p t h t h r o u g h o u t t h e w a v e t a n k c o u l d b e made u n i f o r m . B o t h t h e s e c o n f i g u r a t i o n s w e r e u s e d i n t h e e x p e r i m e n t a l p r o c e d u r e . T h e s i d e a n d e n d w a l l s o f t h e w a v e t a n k p r e s e n t b o u n d a r i e s t h a t do n o t c o n f o r m t o t h e g e o m e t r y s h o w n i n f i g u r e 2 . T h e b o u n d a r i e s p r e s e n t i n t h e l a b o r a t o r y m o d e l r e s u l t i n r e f l e c t i o n s t h a t d c n o t o c c u r i n t h e m a t h e m a t i c a l m o d e l . fihile t h e s e b o u n d a r i e s c o u l d n o t b e r e m o v e d , t h e y c o u l d b e made t o a p p e a r " f a r - d i s t a n t " f r o m t h e s h e l f e d g e , t h e w a v e s o u r c e a n d a n y r e l e v a n t p o i n t s o f o b s e r v a t i o n w i t h i n t h e t a n k . T h i s was a c c o m p l i s h e d b y l i n i n g t h e i n n e r s u r f a c e o f t h e t a n k w a l l s w i t h a wave a b s o r b e r . By s h a p i n g a g o o d w a v e a b s o r b i n g m a t e r i a l i n t o a n a p p r o p r i a t e a b s o r b i n g g e o m e t r y , a n y w a v e s i n c i d e n t u p o n t h e t a n k w a l l s w e r e e f f e c t i v e l y a b s o r b e d a n d t h e w a l l - r e f l e c t e d f i e l d w a s r e d u c e d t o a n i n s i g n i f i c a n t l e v e l a n d c o u l d be n e g l e c t e d . T h e w a v e a b s o r b e r s a r e d i s c u s s e d i n a p p e n d i x D. I n t h e m a t h e m a t i c a l m o d e l , t h e i n c i d e n t w a v e f i e l d a r i s e s f r o m a n i s o t r o p i c p o i n t s o u r c e l o c a t e d o n t h e s h e l f . T h e w a v e s 97 f r o m t h i s s o u r c e p r e s e n t c y l i n d r i c a l w a v e f r o n t s , a s d e s c r i b e d i n c h a p t e r 2. I n t h e l a b o r a t o r y m o d e l i t was n o t p o s s i b l e t o g e n e r a t e t h e i n c i d e n t f i e l d f r o m a p o i n t , b u t i t was p o s s i b l e t o r e d u c e t h e s i z e o f t h e s o u r c e s o t h a t i t s l a r g e s t d i m e n s i o n was much s m a l l e r t h a n t h e o t h e r r e l e v a n t d i m e n s i o n s c o n s i d e r e d i n t h e e x p e r i m e n t . T h i s r e s u l t e d i n w a v e s i n t h e f a r - f i e l d w h i c h a p p e a r e d t o a r i s e f r o m a p o i n t s o u r c e . T h e w a v e g e n e r a t o r u s e d i n t h e e x p e r i m e n t was d e s i g n e d t o f u l f i l l t h e r e q u i r e m e n t s t h a t i t b e s m a l l a n d i s o t r o p i c a n d i s d e s c r i b e d i n d e t a i l i n a p p e n d i x C, T h e l e a f s y s t e m t h a t g e n e r a t e s t h e d i r e c t w a v e h a s s u f f i c i e n t r a d i a l s y m m e t r y a n d m o v e s w i t h s u f f i c i e n t s i m u l t a n e i t y t h a t t h e r e q u i r e m e n t o f i s c t r o p y i s s a t i s f i e d . T h u s , t h e i n c i d e n t w a ve f r o n t g e n e r a t e d i n t h e e x p e r i m e n t a l m o d e l i s c o n s i d e r e d t o b e c y l i n d r i c a l . T h e m e a s u r e m e n t o f t h e f i e l d a t v a r i o u s p o i n t s i n t h e w a v e t a n k w a s a c c o m p l i s h e d t h r o u g h t h e u s e o f t h r e e c a p a c i t a n c e - t y p e d e t e c t o r s . T h e s e d e t e c t o r s , d e s c r i b e d i n a p p e n d i x E , r e s p o n d t o c h a n g e s i n t h e f r e e s u r f a c e e l e v a t i o n w i t h a l i n e a r l y r e l a t e d c h a n g e i n i n s t r u m e n t c a p a c i t a n c e . T h i s c h a n g e i n c a p a c i t a n c e i s e l e c t r o n i c a l l y c o n v e r t e d i n t o a d. c. v o l t a g e s i g n a l w h i c h i s r e c o r d e d . O f t h e t h r e e d e t e c t o r s , t w o w e r e m o v a b l e a n d w e r e u s e d t o m e a s u r e t h e f r e e s u r f a c e c h a n g e s a t d i f f e r e n t p o s i t i o n s i n t h e t a n k . T h e s e d e t e c t o r s w e r e s u s p e n d e d f r o m a t r u s s a n d c a r r i a g e w h i c h r o d e o n a r a i l r u n n i n g t h e f u l l l e n g t h o f t h e t a n k . T h i s m e c h a n i s m w a s d e s i g n e d t o a l l o w t h e d e t e c t o r s t h r e e d e g r e e s o f f r e e d o m a n d t o b e p o s i t i o n e d a t a n y p o i n t o f i n t e r e s t w i t h i n t h e t a n k . T h e t h i r d d e t e c t o r w a s s t a t i o n a r y a n d w a s u s e d t o m o n i t o r t h e d i r e c t wave f i e l d a n d t h u s a c t e d a s a c o n t r o l o n 98 t h e s o u r c e . A f u l l d e s c r i p t i o n o f t h e d e t e c t o r m o u n t i n g s i s g i v e n i n a p p e n d i x B. T h e o u t p u t s i g n a l s f r o m t h e d e t e c t o r s , w h i c h r e p r e s e n t c h a n g e s i n f r e e s u r f a c e e l e v a t i o n , w e r e r o u t e d t h r o u g h a n e l e c t r o n i c n e t w o r k w h i c h , i n a d d i t i o n t o h a n d l i n g s i g n a l l e v e l i n g a n d a m p l i f i c a t i o n , p e r f o r m s e l e c t r o n i c s u m m i n g a n d / o r d i f f e r e n c i n g o f t h e s i g n a l s f r o m t h e v a r i o u s d e t e c t o r s . I n t h i s w a y , r e s u l t a n t s i g n a l s r e p r e s e n t a t i v e o f t h e v a r i o u s f i e l d c o n s t i t u e n t s w e r e o b t a i n e d . F o r i n s t a n c e , i f t h e s i g n a l r e p r e s e n t a t i v e o f t h e d i r e c t w a v e s e e n a t a p o i n t P i s s u b t r a c t e d f r o m t h e s i g n a l r e p r e s e n t a t i v e o f t h e t o t a l w a v e f i e l d s e e n a t P, t h e r e s u l t a n t s i g n a l i s o n e w h i c h i s r e p r e s e n t a t i v e o f t h e w a v e f i e l d a r i s i n g f r o m t h e i n t e r a c t i o n o f t h e d i r e c t w a v e w i t h t h e s h e l f e d g e , i . e . , t h e sum o f t h e e d g e -r e f l e c t e d a n d l a t e r a l w a v e s . T h e s i g n a l s w e r e r e c o r d e d o n a s i x -c h a n n e l C l e v i t e - B r u s h r e c o r d e r . T h e s i g n a l h a n d l i n g a n d r e c o r d i n g t e c h n i g u e a r e d i s c u s s e d i n a p p e n d i x F. A d d i t i o n a l i n f o r m a t i o n p e r t a i n i n g t o t h e w a v e f i e l d w a s e x t r a c t e d f r o m t h e r e c o r d e d d a t a b y means o f a d i g i t i z e r w h i c h was u s e d i n c o n j u n t i o n w i t h a p r o g r a m m a b l e c a l c u l a t o r a n d p l o t t e r . T h i s e q u i p m e n t a n d p r o c e s s a r e d i s c u s s e d i n a p p e n d i x G, 3. 1. E x p e r i m e n t a l P r o c e d u r e I n o r d e r t o o b t a i n a n a d e q u a t e d e s c r i p t i o n o f t h e wave f i e l d , i t i s n e c e s s a r y t o d e t e r m i n e a d e q u a t e l y t h e c o n s t i t u e n t s c o m p r i s i n q t h e f i e l d t h r o u g h o u t t h e r e q i o n o r r e q i o n s o f i n t e r e s t . I h e n a d e t e c t o r s e n s e s a c h a n g e i n s u r f a c e e l e v a t i o n a t some p o i n t o f o b s e r v a t i o n , i t m e a s u r e s t h e t o t a l d i s t u r b a n c e a t t h a t p o i n t . I t c a n n o t d i f f e r e n t i a t e b e t w e e n t h a t p a r t o f t h e 99 d i s t u r b a n c e c r e a t e d b y t h e d i r e c t w a v e a n d t h a t p a r t c r e a t e d b y t h e d i s t u r b a n c e a r i s i n g f r c m t h e i n t e r a c t i o n o f t h e d i r e c t w a v e w i t h t h e s h e l f e d g e . T o f u r t h e r c o m p l i c a t e m a t t e r s , t h e v a r i o u s w a v e c o n s t i t u e n t s e x i s t i n t h e f o r m o f wave p a c k e t s c f some f i n i t e l e n g t h a n d i n g e n e r a l , d i f f e r e n t p o r t i o n s o f d i f f e r e n t p a c k e t s a r r i v e a t t h e p o i n t o f o b s e r v a t i o n c o n c u r r e n t l y . T h e d i r e c t a n d r e f l e c t e d w a v e s t r a v e l a t a s p e e d d e t e r m i n e d by t h e w a v e d e p t h o n t h e s h e l f . T h e d i s t a n c e w h i c h e a c h w a v e t r a v e l s d e p e n d s u p o n t h e g e o m e t r i c a l o p t i c s p a t h t h a t t h e wave f o l l o w s . T h e l a t e r a l w a v e p r o p a g a t e s o v e r a g e o m e t r i c a l o p t i c s p a t h w h i c h i s l o c a t e d p a r t l y o n t h e s h e l f a n d p a r t l y i n t h e d e e p w a t e r z o n e . T h e p r o p a g a t i o n s p e e d o v e r t h e v a r i o u s p o r t i o n s c f t h e g e o m e t r i c a l o p t i c s p a t h f o r t h i s wave d e p e n d s u p o n t h e w a t e r d e p t h i n t h e r e g i o n i n w h i c h t h a t p o r t i o n c f t h e p a t h i s l o c a t e d . T h i s i s s h o w n i n f i g u r e 1b. I t was n e c e s s a r y t o d e v i s e a m e a n s b y w h i c h t h e d i r e c t w a v e f i e l d a n d t h e f i e l d a r i s i n g f r o m t h e s h e l f e d g e c o u l d be s e p a r a t e d . T o t h i s e n d , t w o m e t h o d s w e r e e m p l o y e d w h i c h r e l y u p o n c h a n g e s i n wave t a n k g e o m e t r y a s w e l l a s r e c o r d i n g a n d a n a l y s i n g t e c h n i g u e s , B o t h m e t h c d s u s e t h e s a m e w a v e s o u r c e u n i t , wave d e t e c t o r s , e l e c t r o n i c s a n d r e c o r d i n g e g u i p m e n t . T h e y d i f f e r i n t h e g e o m e t r y o f t h e s h e l f c o n f i g u r a t i o n , t h e l o c a t i o n o f t h e s o u r c e a n d t h e p l a c e m e n t o f t h e d e t e c t o r s . B o t h m e t h o d s r e l y u p o n t h e i n t r o d u c t i o n a n d r e m o v a l o f a d e e p w a t e r z o n e , w h i l e t h e s h e l f r e g i o n i s l e f t i n t a c t . M e t h o d A T h e w a ve t a n k a r r a n g e m e n t u s e d i n m e t h o d A i s s h e w n i n f i g u r e 13 a n d f i g u r e 14 ( w h e r e t h e wave t a n k i s v i e w e d f r o m t h e 100 Z A •h, • >-- h '/////////////////. Y A D E E P W A T E R S'(X.,YJ r 3 S H E L F X F i g u r e 1 3 . Wave t a n k g e o m e t r y u s e d i n m e t h o d A . ;u re 1 4 . Wave t a n k r e g i m e u s e d i n m e t h o d A . 101a. 102 opposite end). A d e t a i l e d d e s c r i p t i o n of the wave tank i s given i n appendix B, Two g e o m e t r i c a l arrangements were i n t r o d u c e d d u r i n g the use of method A, In the f i r s t , by e l e v a t i n g the movable bottom i n the deep-water zone, (x <0), so t h a t i t l a y at a l e v e l egual to t h a t of the s h e l f bottom, the water depth throughout the wave tank was made uniform and equal to the depth on the s h e l f , h , . The wave source was l o c a t e d near one end of the tank at the p o s i t i o n (x,y)= (x 0 , y 0 ), The f i x e d d e t e c t o r , T 3 , was p o s i t i o n e d a t the po i n t ( x , y ) = ( x % , y % ) , a d i s t a n c e r 3 from the source, The two p o s i t i o n a b l e d e t e c t o r s , T, and T z , were p o s i t i o n e d at the p o i n t s (x, y) = (x,, y, } and (x,y) = (JJ Z ,yz) , r e s p e c t i v e l y . Detector T, l a y a d i s t a n c e r , from the source, while T v was at a d i s t a n c e r ^ from the source. The d i s t a n c e s e p a r a t i n g T, and T^ was maintained at a constant value,D. Osing t h i s geometry, d e t e c t o r s T,/T z, and T 3 measured the d i r e c t wave f i e l d a t the d i s t a n c e s r , , r l , and r 3 from the source, r e s p e c t i v e l y . In the second arrangement, the p l e x i g l a s s sheets t h a t form the bottom i n the r e g i o n x<0 were lowered a d i s t a n c e h t-h, . T h i s c r e a t e d a s h e l f edge along the l i n e x=0 which acted to separate a deep water r e g i o n o f depth (for x<0) from a s h e l f r e g i o n of depth h | (for x>0). When the source was a c t i v a t e d , the d e t e c t o r T j saw a f i e l d which i n i t i a l l y was composed of a d i r e c t wave, l i k e t h a t observed i n method A, at a subsequent time, r e f l e c t i o n s from the s h e l f edge reached d e t e c t o r T 3 , but onl y the f i r s t s i g n a l r ecorded by T 3 was used as a c o n t r o l cn the experiment. .Detectors T, and T^ _ measured the f i e l d composed o f the d i r e c t wave , the r e f l e c t e d wave and the l a t e r a l wave 103 c o n s t i t u e n t s . The data obtained by usinq these two arrangements were d i g i t i z e d and the d i f f e r e n c e o f the two s e t s cf data used to p rovide a measure of the r e f l e c t e d and l a t e r a l wave combination (see f i q u r e G-1). T h i s technique i s d e s c r i b e d i n appendix G. I n t h i s way the phase speed , wave amplitude and energy, and the wave shape f o r the v a r i o u s wave components at v a r i o u s p o s i t i o n s i n the tank could be determined. Performing a s y s t e m a t i c s e t of measurements throughout the r e g i o n or r e g i o n s of i n t e r e s t , made i t p o s s i b l e t o determine the behavior and v a r i a t i o n of the wave f i e l d s with p o s i t i o n . Method B The wave tank arrangement used i n method B i s shewn i n f i g u r e s 15 and 16 . Some c a p i l l a r y waves (which damp out g u i c k l y ) are seen t o a r i s e from the corner. I f we allow the wave tank to be d i v i d e d i n t o f o u r guadrants, guadrants 1,3, and 4 always c o n s t i t u t e a s h e l f r e g i o n with water depth h|. Quadrant 2 (x<0,y>0) i s a region i n which the bottom may be r a i s e d or lowered. Hhen the bottom i s e l e v a t e d to a l e v e l egual to t h a t of the s h e l f bottom, the depth o f water throughout the wave tank i s uniform and of depth h,. I f the bottom of t h i s r e g i o n i s lowered a d i s t a n c e t ^ - h ^ , a s h e l f edge i s cre a t e d along the l i n e x=0 f o r y>0. T h i s edge separates the deep water region o f depth hx from the s h e l f r e g i o n o f depth h ( . For t h i s arrangement, the source i s p o s i t i o n e d at the poin t (x,y)=(x 0,0)« D e t e c t o r T 3 i s placed at the p c i n t (x,y)= (x $ ,0) , a d i s t a n c e r 3 from the s o u r c e , and d e t e c t o r s T, and T ^  are p o s i t i o n e d at the p o i n t s (x,y)=(x,,y,) and <x,y)= (xlf y^) which l i e at d i s t a n c e s r , and r ^ from the source, r e s p e c t i v e l y . Y D E E P WATER SHELF o S(Xo,o) T3(x3 v,) o =—>-o — r 3 2 • F i g u r e 15. Wave t a n k g e o m e t r y u s e d i n m e t h o d B g u r e 1 6 . Wave t a n k r e g i m e u s e d i n m e t h o d B. 105 a. 106 The experimental procedure followed i n method B was to f i r s t e s t a b l i s h a regime wherein the water was o f uniform depth h, throughout the wave tank. Detectors T, and Tz were p o s i t i o n e d symmetrically with r e s p e c t t o the source. Thus, with the source,S, l o c a t e d at the point (x,y}=(x o , 0 ), d e t e c t o r T ( was l o c a t e d at (x,y)=(x,,y () and de t e c t o r 1z p o s i t i o n e d a t (x,y)= (x, ,-y ( ) , D e t e c t o r T j was placed a t the point ( x , y ) = ( x 3 , y 3 ) . T h i s geometry i s shown i n f i g u r e 12. With the d e t e c t o r s i n p o s i t i o n , the source was a c t i v a t e d and the surface - v a r i a t i o n seen by the v a r i o u s d e t e c t o r s r e c o r d e d . Since the water depth throughout the t a r k was uniform, a l l d e t e c t o r s recorded only the d i r e c t wave f i e l d at t h e i r r e s p e c t i v e p o s i t i o n s . By examining the propagation time f o r the waves to reach T ( and Tx, adjustment of the p o s i t i o n o f e i t h e r T ( and/or T z c o u l d be made so t h a t the two d e t e c t o r s were t r u l y e q u i d i s t a n t from the source. With t h i s accomplished, the source was again a c t i v a t e d and the e l e c t r o n i c gains o f the s i g n a l s from the two d e t e c t o r s a d j u s t e d so t h a t the recorded s i g n a l s from T, and T t were the same. This was accomplished by d i f f e r e n c i n g the s i g n a l s from T, and T z e l e c t r o n i c a l l y and a d j u s t i n g the r e s p e c t i v e gains so that the r e s u l t a n t s i g n a l was zero. While t h i s was never completely accomplished, s i n c e some d i f f e r e n c e s i n the propagation paths to T, and T ^  always e x i s t , the r e s u l t a n t s i g n a l could be minimized t o the p o i n t where i t was i n c o n s e q u e n t i a l . Upon s a t i s f a c t o r y p o s i t i o n i n g o f the d e t e c t o r s , the bottom i n quadrant 2 was lowered a d i s t a n c e h -h^. With a l l other 107 parameters and instrument p o s i t i o n i n g l e f t unchanged, the source was again a c t i v a t e d . Detector T 3 recorded the d i r e c t f i e l d once again and t h i s was compared with the previous run. The f i e l d seen by T, now was composed of the r e f l e c t e d and l a t e r a l wave components i n a d d i t i o n to the d i r e c t wave. The f i e l d sensed by 1% was the same as that sensed i n the previous run. i . e, , the d i r e c t wave f i e l d o n l y . In a d d i t i o n to r e c o r d i n g the s i g n a l s from T, and T 2 , the two s i g n a l s were d i f f e r e n c e d e l e c t r o n i c a l l y and the r e s u l t a n t s i g n a l was then a measure of the f i e l d due to the sum of the r e f l e c t e d and l a t e r a l waves (see f i g u r e F-3). Thus, a d i r e c t measure of the d i r e c t wave and the r e f l e c t e d p l u s l a t e r a l waves was achieved. Hhen the f i e l d measurements had been completed f o r one p o s i t i o n of T ( and T ^ , i . e. , f o r one d i s t a n c e B from the source, the d e t e c t o r s were moved to a new p o s i t i o n and the process repeated, By c o n t i n u i n g t h i s process, data were ob t a i n e d throughout the region or r e g i o n s cf i n t e r e s t and the v a r i o u s f i e l d p r o p e r t i e s were determined as a f u n c t i o n of p o s i t i o n . H h i l e the use of method B provided a d i r e c t measure of the wave f i e l d a r i s i n g from the i n t e r a c t i o n of the d i r e c t wave with the s h e l f edge, the s p a t i a l range of i n v e s t i g a t i o n was r e s t r i c t e d . S i n c e the wave source was l o c a t e d near the mid-point of the wave tank, the.range of p o s i t i o n s , r e l a t i v e to the s o u r c e , t h a t d e t e c t o r s T ( and T^ _ c o u l d assume was r e s t r i c t e d to a d i s t a n c e of about o n e - h a l f t h e l e n g t h of the wave tank. Method A by c o n t r a s t allowed examination of the f i e l d t o d i s t a n c e s twice as f a r from the wave source, but the f i e l d a r i s i n g from the s h e l f edge c o u l d not be found d i r e c t l y . I t was necessary t o 108 d i g i t i z e the wave p r o f i l e s and to process the data with a c a l c u l a t o r . To determine the accuracy of the r e s u l t s found by using method A, the data recorded by T, and T^ i n method E were d i g i t i z e d and processed and compared over the common range o f i n v e s t i g a t i o n with those obtained u s i n g method A. The d i f f e r e n c e i n r e s u l t s found from using these two methods was w i t h i n the acccuracy of the d i g i t i z a t i o n process. The e r r o r i n c u r r e d i n smoothing i s l e s s than 0.5% (H.P. P e r i f e r a l Manual), while t h a t c f t r a c i n g the curve was no more than 3-H%. A p p l i c a t i o n of the Experimental Method In conducting any experimental i n v e s t i g a t i o n , c e r t a i n procedures and programs are f o l l o w e d . I t was f i r s t determined which experimental method was to be a p p l i e d and what r e g i o n or r e g i o n s of i n t e r e s t were to be i n v e s t i g a t e d . Once the regime to be used was e s t a b l i s h e d i n the wave tank, a process was begun by which the experiment was made ready to be conducted. A procedure of monitoring the v a r i o u s p h y s i c a l parameters was undertaken throughout the d u r a t i o n of the experimental process.. Most experimental runs were conducted w i t h i n the p e r i o d of one working day, but u s u a l l y w i t h i n a matter of hours. An experimental day was begun by e s t a b l i s h i n g the wave tank geometry and p o s i t i o n i n g of a l l the a p p a r a t i to be used i n the experiment. The wave tank was f i l l e d with f r e s h water to a p r e s c r i b e d depth and measurements made of the water depth throughout the wave tank. T h i s assured us that the u n i f o r m i t y of depth throughout the tank was maintained. Hater temperatures were recorded throughout the water column once the system had reached e g u i l i b r i u m . Thermal s t r a t i f i c a t i o n d i d not become a 109 problem over the d u r a t i o n of an i n v e s t i g a t i o n , which was u s u a l l y a time span of a few hours. L i k e w i s e , the mean temperature change of the water dur i n g t h i s period proved s m a l l and could be negl e c t e d . The wave d e t e c t o r s were c a l i b r a t e d each day p r i o r to conducting the measurements. The i n t e r n a l gain and d . c , voltage l e v e l of each instrument was checked and ad j u s t e d i f necessary. T h i s was done both at the instrument l e v e l and i n the a m p l i f i e r e l e c t r o n i c s . A l l o p e r a t i o n a l a m p l i f i e r gains were checked, and with the use of a s i g n a l generator, a l l s i x channels of the C l e v i t e - B r u s h r e c o r d e r were checked and any adjustments t h a t were necessary were made. The v a r i o u s d e t e c t o r s were p o s i t i o n e d i n the water column and adjusted so t h a t they were immersed to the proper depth and c o r r e c t l y o r i e n t e d . Once t h i s was accomplished, voltage l e v e l adjustments were made so that zero output voltage of a d e t e c t o r corresponded t o the e g u i l i b r i u m f r e e s u r f a c e a t r e s t . The p o s i t i o n s of the movable d e t e c t o r s along the tank l e n g t h were measured by means of a s c a l e running the l e n g t h of the tank. T h i s measure of the c a r r i a g e p o s i t i o n was c o r r e l a t e d with the d i s t a n c e of the d e t e c t o r from the source. The p o s i t i o n of the c o n t r o l d e t e c t o r was checked d a i l y . The p o s i t i o n of the source mechanism was noted and the a c t i o n of the s o l e n o i d was t e s t e d . The motion t h a t t h i s imparts to the l e a f system was adjus t e d so t h a t the source behavior was c o n s i s t e n t from day to day. The s i g n a l recorded by the f i x e d d e t e c t o r was examined and compared with p r e v i o u s runs. Very l i t t l e problem was encountered regarding the r e p r o d u c i b i l i t y of 110 t h e w a v e p a c k e t g e n e r a t e d by t h e s o u r c e . I t w a s o n l y w h e n t h e s o u r c e a n d / o r t h e f i x e d d e t e c t o r w e r e r e p o s i t i o n e d t h a t a new c a l i b r a t i o n w a s n e e d e d . O n c e a l l o f t h e s e p r e - e x p e r i m e n t a l s t e p s w e r e t a k e n , a q u i c k r e c h e c k o n s u c h t h i n g s a s w a t e r d e p t h , w a t e r t e m p e r a t u r e a n d o u t p u t d . c . v o l t a g e l e v e l s w a s made. W i t h t h i s d o n e , t h e s y s t e m was now o p e r a t i o n a l a n d t h e t a k i n g o f m e a s u r e m e n t s c o u l d p r o c e e d . E a c h d a y a s y s t e m a t i c p r o g r a m o f m e a s u r e m e n t s was e s t a b l i s h e d . T h e p o s i t i o n s a t w h i c h m e a s u r e m e n t s w e r e t o b e m ade, a s w e l l a s c h a n g e s i n w a v e t a n k g e o m e t r y , w e r e p l a n n e d . W i t h t h i s p r o g r a m i n h a n d , m e a s u r e m e n t s w e r e b e g u n . W i t h t h e d e t e c t o r s i n p o s i t i o n t o make t h e f i r s t m e a s u r e m e n t , t h e s o u r c e w a s a c t i v a t e d a n d t h e s i g n a l s r e c o r d e d . E a c h m e a s u r e m e n t was r e p e a t e d f o r a t l e a s t t w o C l e v i t e - B r u s h r e c o r d e r c h a r t s p e e d s . W i t h t h e r e c o r d e r o p e r a t i n g a t a s l o w s p e e d ( 5 m m / s e c ) , a n o v e r a l l v i e w o f t h e w a v e p a c k e t a r r i v i n g a t t h e p o i n t o f o b s e r v a t i o n was o b t a i n e d . A m o r e d e t a i l e d r e c o r d i n q was t h e n made a t a h i q h e r c h a r t s p e e d { 2 5 m m / s e c ) . I n a d d i t i o n t o r e c o r d i n g t h e d e t e c t o r o u t p u t s w i t h t h e B r u s h r e c o r d e r , e a c h d e t e c t o r s i g n a l w a s m o n i t o r e d o n a d i g i t a l v o l t m e t e r . T h i s p r o v i d e d a n a c c u r a t e s u r v e i l l a n c e o f t h e v a r i a t i o n i n d . c . v o l t a g e l e v e l s a t t h e r e c o r d e r i n p u t s . A n y s m a l l a d j u s t m e n t s t h a t w e r e n e c e s s a r y w e r e r e a d i l y m ade. O n c e s u f f i c i e n t d a t a f o r a p a r t i c u l a r p o s i t i o n h a d s a t i s f a c t o r i l y b e e n r e c o r d e d , t h e m o v a b l e d e t e c t o r s w e r e r e p o s i t i o n e d . A n y n e c e s s a r y a d j u s t m e n t s w e r e made a n d t h e d a t a t a k i n g p r o c e s s was r e p e a t e d . T h e p r o c e d u r e w a s c o n t i n u e d o v e r a s u f f i c i e n t r a n q e o f o b s e r v a t i o n p o i n t s t h a t t h e d a t a a c c r u e d 111 would g i v e an accurate d e s c r i p t i o n of the f i e l d throughout the r e g i o n of i n t e r e s t . During the experiment, water depths and water temperatures were recorded. I t was seldom found necessary to make any adjustments, and the c o n d i t i o n s under which the experiment was conducted were maintained as reasonably c o n s t a n t . In order to assure o u r s e l v e s o f the r e p r o d u c i b i l i t y of the experiment, Beasurements were made from time t o time a t p o s i t i o n s of previous measurements. Comparison of these data r e v e a l e d t h a t such f a c t o r s as instrument gains and response, the p h y s i c a l parameters of the wave tank and the source c o n f i g u r a t i o n , and wave genera t i o n remained s a t i s f a c t o r i l y constant. 3 . 2 . Experimental R e s u l t s fl d e s c r i p t i o n of the wave f i e l d generated by the experimental model i s provided by determining the wave c o n s t i t u e n t s which comprise the f i e l d . Since the d i r e c t wave f i e l d i s the c o n s t i t u e n t most e a s i l y generated and measured, i t i s examined f i r s t . 3 . 2.1. The D i r e c t Wave In determining the f i e l d of the d i r e c t wave, both methods A and B of s e c t i o n 3.1 are used. While t h i s i s not necessary, the r e s u l t s are complementary. The data used i n a n a l y z i n g the d i r e c t wave f i e l d are obtained over a wide range of R, the d i s t a n c e from the source to the o b s e r v a t i o n p o i n t . One s e r i e s of wave p r o f i l e s which were recorded f o r i n c r e a s i n g v a l u e s of R i s shown i n f i g u r e 17. Due to the a m p l i f i e r e l e c t r o n i c s , a l l wave amplitudes should be m u l t i p l i e d by a f a c t o r of -1 to resemble the waves i n the tank. A l s o , the a m p l i f i e r gains were changed as F i g u r e 17- Recorded wave p r o f i l e s f o r the d i r e c t wave, shown f o r i n c r e a s i n g v a l u e s o f R. 113 R was i n c r e a s e d , s o some d i s c r e p a n c y i n t h e c h a n g e i n w a v e a m p l i t u d e i s o b s e r v e d ( t h i s c a n be c o m p e n s a t e d f o r when u s i n g t h e d a t a f o r t h e v a r i o u s c a l c u l a t i o n s p e r f o r m e d ) * I t i s f r o m w a v e p r o f i l e s s u c h a s t h e s e t h a t t h e v a r i o u s p r o p e r t i e s c f t h e d i r e c t w a ve f i e l d , s u c h a s wave s p e e d s a n d w a v e e n e r g y , a r e d e t e r m i n e d . She P h a s e S j i e e d T h e p h a s e s p e e d i s d e t e r m i n e d i n t w o w a y s , T h e f i r s t u s e s t h e m e t h o d d e s c r i b e d i n " m e t h o d A" o f s e c t i o n 3 . 1 , i n w h i c h t h e p r o p a g a t i o n - t i m e s f o r t h e w a v e s t o r e a c h d e t e c t o r s T, a n d T^ a n d t h e t i m e f o r t h e wave t o p r o p a g a t e f r o m T, t o T 2 a r e m e a s u r e d . F o r t h e l a t t e r m e a s u r e m e n t , t h e s e p a r a t i o n d i s t a n c e b e t w e e n T, a n d T ^ i s m a i n t a i n e d a t 0.3 m. T h e a r r i v a l - t i m e o f t h e w a v e a t a d e t e c t o r i s t a k e n a s a p o i n t o f c o n s t a n t p h a s e i n t h e w a v e p a c k e t . T h e p o i n t s e l e c t e d i s t h e f i r s t p o i n t o f z e r o - e l e v a t i o n w h i c h o c c u r s o n t h e s a v e t r a c e f o l l o w i n g t h e a r r i v a l o f t h e l e a d i n g e d g e o f t h e w a v e p a c k e t . T h e p o i n t m a r k e d "A" on t h e f i r s t w a v e t r a c e o f f i g u r e 17 i s s u c h a p o i n t . T h i s p o i n t i s r e a d i l y d e t e r m i n e d s i n c e t h e s l o p e o f t h e t r a c e i s a maximum t h e r e a n d t h e l e v e l o f z e r o - e l e v a t i o n i s e a s i l y d e f i n e d . T h u s , t h e p h a s e s p e e d a t a n y p o i n t o f o b s e r v a t i o n c a n be d e t e r m i n e d f r o m t h e r a t i o o f t h e s e p a r a t i o n d i s t a n c e b e t w e e n T, a n d T^ (0.3m) a n d t h e p r o p a g a t i o n t i m e f o r t h e wave t o g o f r o m T, t o I 2 ( A . t ) . T h e p h a s e s p e e d a t d i s t a n c e R i s g i v e n b y 114 P A t C fD 3 » 1 where R i s taken as the mean d i s t a n c e of d e t e c t o r s T , and T^ from the wave source, Values of phase speed, c p ( i ) , found a t values of R from 0 . 3 t o 1,8m a r e shown i n f i g u r e 18, These waves, propagating i n water of 0.0128m depth, r e s u l t i n an — - i - i average phase speed of c p=0,35Q±0,0064 msec , where 0.0064 msec i s one standard d e v i a t i o n . T h i s i s shown as the s o l i d l i n e (with e r r o r bars) i n f i g u r e 18. The t r a v e l time f o r the wave t o propagate from the wave source to the d e t e c t o r (a d i s t a n c e R) i s p l o t t e d , f o r both d e t e c t o r s T ( and T z, as a f u n c t i o n c f R, i n f i g u r e 19. Superimposed upon these data i s a phase speed curve f o r c =0,354 -t i msec , the value c a l c u l a t e d f o r h =0.0128m and g=9,80m/sec, usi n g l i n e a r shallow water theory. T h i s value of. phase speed f a l l s w i t h i n the range cf the average phase speed, c p , found by measurement. The Enercij Propagation Speed The speed of energy propagation i s found by determining the propagation time of the c e n t r o i d of the p o t e n t i a l energy c f the wave. Using the methods d e s c r i b e d i n appendix G, the c e n t r o i d of the p o t e n t i a l energy i s c a l c u l a t e d f o r the l e a d i n g wave i n the packet and f o r the whole wave packet. The r e s u l t s of these c a l c u l a t i o n s are shown, i n the form of a r r i v a l - t i m e of the c e n t r o i d as a f u n c t i o n of d i s t a n c e , i n f i g u r e 20. I t i s observed t h a t very l i t t l e d i f f e r e n c e r e s u l t s from using the c e n t r o i d of 0.6 n0 4 10 u 0 2 - 1 —o—o—o—o—G- T5 O — o — Q — O 0 —O 0 ~~ oS i b R ( m e t e r s ) F i g u r e 1 8 . R e c o r d e d p h a s e s p e e d as a f u n c t i o n o f d i s t a n c e R. 1.5 11k 2 4 6 t(seconds) F i g u r e 1 9 . T i m e - d i s t a n c e c u r v e f o r t h e d i r e c t w a v e . 117 R(meters) F i g u r e 2 0 . A r r i v a l - t i m e o f w a v e e n e r g y a s a f u n c t i o n o f d i s t a n c e . 1 1 8 t h € l e a d i n g w a v e o n l y , a s c o m p a r e d w i t h t h e c e n t r o i d f o u n d f o r t h e w h o l e p a c k e t . T h e a v e r a g e e n e r g y s p e e d o v e r a r a n g e i n R _ -\ f r o m 0,3 t o 2 ,1m i s f o u n d t c be c =0. 34 4"t0. 0 0 8 8 m s e c , w h e r e 0 . 0 0 8 8 m s e c i s o n e s t a n d a r d d e v i a t i o n . A c u r v e f o r t h i s a v e r a g e e n e r g y s p e e d i s s u p e r i m p o s e d u p o n t h e d a t a i n f i g u r e 2 0 ( s h o w n a s a s o l i d l i n e ) . A c u r v e r e p r e s e n t i n g t h e p h a s e s p e e d a s - i c a l c u l a t e d f r o m t h e w a t e r d e p t h ( c p = 0 , 3 5 4 msec ) i s a l s o s h o w n ( t h e d a s h e d l i n e ) . T h e d i f f e r e n c e b e t w e e n c f r a n d c ^ i s d i s c u s s e d i n s e c t i o n 3 . 2 . 4 . H a v e A m p l i t u d e a n d E n e r a j j T h e wave p r o f i l e s a s r e c o r d e d o n t h e c h a r t r e c o r d i n g s c a n be u s e d t o a s c e r t a i n d i r e c t l y t h e v a r i a t i o n i n wave a m p l i t u d e w i t h p o s i t i o n . A r o u g h m e a s u r e o f w a v e e n e r g y c a n b e d e t e r m i n e d f r c m t h e a m p l i t u d e o b s e r v e d ( s u c h a n e x a m i n a t i o n i s made l a t e r ) , b u t i t i s d e e m e d p r e f e r a b l e t o d e t e r m i n e t h e w a v e e n e r g y by means c f i n t e g r a t i o n o v e r t h e w a v e p a c k e t u s i n g t h e m e t h o d s d e s c r i b e d i n a p p e n d i x G. T h e w ave p o t e n t i a l e n e r g y p e r u n i t a r e a , £ ( R ) , f o r t h e w h o l e w a v e p a c k e t i s c a l c u l a t e d o v e r a r a n g e o f p o s i t i o n s , R, f r o m t h e d a t a r e c o r d e d b y d e t e c t o r s T , a n d T^. T h e r e s u l t s ( n o r m a l i z e d t o t h e p o t e n t i a l e n e r g y p e r u n i t a r e a r e c o r d e d a t a d i s t a n c e o f o n e m e t e r , 6 0 ) a r e s h o w n i n f i g u r e s 21 a n d 2 2 . F o r s n a i l e n o u g h w a v e s ( L o n g u e t - H i g g i n s , 1 9 7 4 , 1 9 7 5 ) t h e d i f f e r e n c e b e t w e e n t h e k i n e t i c a n d p o t e n t i a l e n e r g y i s c f t h e o r d e r o f t h e f o u r t h p o w e r o f t h e w a v e s l o p e a n d o n e may s a f e l y c h a r a c t e r i z e t h e t o t a l e n e r g y a s t w i c e t h e p o t e n t i a l e n e r g y . T h e m a t h e m a t i c a l m o d e l o f c h a p t e r 2 p r e d i c t s t h a t t h e -I e n e r g y f l u x o f t h e d i r e c t wave s h o u l d v a r y a s R . F r o m t h e s h a p e o f t h e c u r v e s i n f i g u r e s 21 a n d 2 2 , t h i s i s c l e a r l y n o t t h e F i g u r e 2 1 . E n e r g y o f t h e w a v e p a c k e t , m e a s u r e d b y d e t e c t o r , a s f u n c t i o n o f d i s t a n c e f r o m s o u r c e . 5.0 n R (meters) F i g u r e - 22. E n e r g y o f t h e w a v e p a c k e t , m e a s u r e d b y d e t e c t o r , a s f u n c t i o n o f d i s t a n c e f r o m s o u r c e . 121 case. A p o s s i b l e e x p l a n a t i o n of t h i s discrepancy i s a f f o r d e d by c o n s i d e r i n g 1 the e f f e c t s of f r i c t i o n (Harleman,1963). To t h i s end, we i n t r o d u c e a l i n e a r f r i c t i o n term i n t o the momentum equations. Onder r a d i a l symmetry, the momentum equation i n the d i r e c t i o n of propaqation i s 3*2 where t i s the f r a c t i o n a l constant (with dimension of (time) ). The i n t e q r a t e d c o n t i n u i t y equation i s 3.3 Assume a s o l u t i o n of the form \ ~ e 3. 4 S u b s t i t u t i o n i n 3,3 r e s u l t s i n 3.5 S u b s t i t u t i o n i n 3.2 q i v e s the d i f f e r e n t i a l equation 1_ 3.6 which has the s o l u t i o n 122 In the f a r - f i e l d the asymptotic form of eguation 3.7 becomes 1 +^  Using the binomial expansion magnitude of u frcm Equation 3.8 i s io , JT7c%^ \ ^ l L + HLVi ( £ y . The I . 1 7^  ^  3,9 where «c = /2 ^ gh and i s c a l l e d the f r i c t i o n a l decay c o e f f i c i e n t . By i n c l u d i n g a l i n e a r f r i c t i o n term i n the eguations of motion, the wave amplitude decays not as R , as p r e d i c t e d i n chapter 2, but as R e , where the e x p o n e n t i a l f a c t o r r e s u l t s from f r i c t i o n . The wave energy £. v a r i e s i n accordance with C ~ - T ^ ^ 3.10 -1 and not as R as p r e d i c t e d i n chapter 2. Applying l e a s t sguares f i t s t o the data of f i g u r e s 21 and -I ^ -1 22 r e s u l t s r n values o f of 0.69m (T =0. 49 sec ) and 0.78m (t.=0.55 sec ), r e s p e c t i v e l y . Curves f o r t h e s e l e a s t sguares f i t s are shown with the s o l i d l i n e s i n f i g u r e s 21 and 22. These r e s u l t s are i n re a s o n a b l e agreement with these found by P i t e (1S73) i n h i s study of f r i c t i c n a l l y damped waves. 123 Energy. C a l c u l a t e d Ergm the Average Amj 11 tude o j the Leading Have A c o n s i d e r a b l e e f f o r t i s r e q u i r e d t o determine the enerqy d e n s i t y v a r i a t i o n i n the above manner and i f a more d i r e c t method can be found t h a t y i e l d s s a t i s f y i n g r e s u l t s , i t would be p r e f e r a b l e i n many cases. Since most of the enerqy of the wave packet i s c o n c e n t r a t e d i n the l e a d i n q wave (see f i g u r e 25), we c a l c u l a t e the energy based on the wave amplitude taken d i r e c t l y frcm the c h a r t r e c o r d . The amplitude used i n the c a l c u l a t i o n s i s taken t o be c n e - h a l f of the wave he i g h t , H, of the l e a d i n g wave, i . e . , o n e -half of the d i f f e r e n c e i n e l e v a t i o n of the h i g h e s t poin t cn the c r e s t and the lowest point i n the trough (see f i g u r e 17). T h i s average amplitude i s given as ^ = ^ /2 and the square of the amplitude ^ g i v e s a q u a n t i t y p r o p o r t i o n a l to the energy of the l e a d i n g wave. The r e s u l t s of such c a l c u l a t i o n s based on data recorded by d e t e c t o r s T, and are shown as a f u n c t i o n of p o s i t i o n i n f i g u r e s 23 and 24, r e s p e c t i v e l y . These r e s u l t s have been normalized t o the amplitude squared *|0 found at a d i s t a n c e of one meter. Least sguares f i t s are made to the -1 data of f i g u r e s 23 and 24 with r e s u l t i n g v a l u e s of «*.=<),75m and 0.71m , r e s p e c t i v e l y . The l e a s t sguares curves are shown with the s o l i d l i n e i n both f i g u r e 23 and 24. T h i s method of determining the v a r i a t i o n i n p o t e n t i a l energy d e n s i t y i s much simple r and the r e s u l t s reasonably good. I t was s t a t e d p r e v i o u s l y that nearly a l l of the energy i n the wave packet remains i n the l e a d i n g wave of the packet as i t propagates away from the source. C a l c u l a t i o n s (using the methods of appendix G) of the r a t i o of energy i n the l e a d i n g wave to the t o t a l energy i n the wave packet (E L'/E T) are made from the data 124 1 0 - . 0.2 0.5 1.0 2.0 R(meters) F i g u r e 2 3 . Sq u a r e o f t h e a m p l i t u d e o f t h e l e a d i n g w a v e a s a f u n c t i o n o f R f o r d e t e c t o r T , . 1%S 5.0 i R (meters) F i g u r e 2 4 . S q u a r e o f t h e a m p l i t u d e o f t h e l e a d i n g w a v e a s a f u n c t i o n o f R f o r d e t e c t o r T _ . 1.0- Q—o—Q Q_ E; 0.5-0 0.5 1.0 1.5 R(meters) F i g u r e 2 5 . P r o p o r t i o n o f t h e t o t a l e n e r g y c o n t a i n e d i n t h e l e a d i n g w a v e a s a f u n c t i o n o f R. 2.0 •2.5 127 and the r e s u l t s p l o t t e d In f i g u r e 25, 3.2.2. The R e f l e c t e d Wave In chapter 2, i t was determined that the r e f l e c t e d wave i s a c y l i n d r i c a l l y spreading d i s t u r b a n c e which appears to emanate from the image source, S , at time t=0 (see f i g u r e 2). Equation , "~% 2.180 p r e d i c t s that the wave amplitude w i l l decay as < R ) and w i l l be modified by [R_{ © ) , the r e f l e c t i o n c o e f f i c i e n t . A comparison of equations 2.178 and 2.180 l e a d s to the e x p e c t a t i o n t h a t the r e f l e c t e d wave (except f o r the e f f e c t of the amplitude f u n c t i o n , K ( 0 ) ) w i l l behave i n much the same way as the d i r e c t wave. In order t o examine the r e f l e c t e d wave qenerated i n the experiment, i t i s necessary t c i s o l a t e i t from the r e s t of the wave f i e l d . To accomplish t h i s , methods A and B ( s e c t i o n 3.1) were used. The ranqes over which each of these methods c o u l d be a p p l i e d i s d i s c u s s e d i n s e c t i o n 3.1. fThe wave f i e l d r e v e a l e d by the use of e i t h e r o f these methods i s a combination of the r e f l e c t e d and the l a t e r a l wave (an example of the waveform y i e l d e d by method A i s shown i n f i q u r e F-3, while the waveform qiven by usinq method B i s shown i n f i q u r e G-1), These two waves are d i s t i n q u i s h e d from one another by usinq the f a c t t h a t they propaqate with d i f f e r e n t speeds. There i s a s u f f i c i e n t l y l a r q e ranqe of values of R, the d i s t a n c e from the source t o the p o i n t of o b s e r v a t i o n , over which these two waves are we l l - s e p a r a t e d and d i s t i n c t . The ranqe cf R over which measurements are made depends upon the wave tank geometry; i n a d d i t i o n to the placement of the source and the d e t e c t o r s , the qeometry of the step at the s h e l f edqe and the water depth play a r o l e . Consider 128 the f o l l o w i n g two experimental setups: (1) t h i s geometry has v a l u e s h,=1,28 cm and h-, = 2. 19 cm, with a r e s u l t i n g index of r e f r a c t i o n n=0,76, The r e s u l t i n g o c r i t i c a l angle has a v a l u e <9C=49.8 . F o r o b s e r v a t i o n s of the wave f i e l d made along a l i n e which l i e s a c o n s t a n t d i s t a n c e from the edge of the s h e l f (0. 15m) , where the source i s a l s o l o c a t e d 0.15m from the edge, any r e f l e c t e d part of the wave f i e l d seen at d i s t a n c e s from the source g r e a t e r than R=0.35m r e s u l t s from the r e f l e c t i o n a t a n g l e s g r e a t e r than the c r i t i c a l angle, (2) i n t h i s arrangement, h =1.28 cm and h =12.7 cm and o n=0,32 which r e s u l t s i n a c r i t i c a l angle <9C=18.5 • In t h i s case,any r e f l e c t e d waves seen a t d i s t a n c e s R>0.Im are waves which have been t o t a l l y r e f l e c t e d . In c o n s i d e r i n g how c l o s e to the source t o measure the f i e l d , c o n s i d e r a t i o n was given to t s a t i s f y i n g the requirement t h a t "X.RV>1« For waves of wavelength > =0.15m, we r e q u i r e that R >0.25m. For waves i n case (1) t h i s means R >0.2m and i n case (2) R > 0. 1m. T h i s means th a t the s m a l l e s t value of R at which measurements can l e g i t i m a t e l y be made i s near the value beyond which t o t a l r e f l e c t i o n o c c u r s . For t h i s reason, no measurements were made t o determine the r e f l e c t i o n c o e f f i c i e n t over the ranqe o f anqles of i n c i d e n c e l e s s than the c r i t i c a l anqle. The Phase Sjaeed and A r r i v a l - t i m e The a r r i v a l - t i m e of the r e f l e c t e d wave (determined i n a l i k e manner as that a p p l i e d t o f o r the d i r e c t wave) was found as a f u n c t i o n of the p o s i t i o n of the o b s e r v a t i o n p o i n t , B. The r e s u l t s of these measurements are shown i n f i q u r e 26. The g e o m e t r i c a l o p t i c s path f o l l o w e d by the r e f l e c t e d wave a c t u a l l y F i g u r e 2 6 . A r r i v a l - t i m e o f t h e r e f l e c t e d w a v e a s a f u n c t i o n o f R. 130 t r a v e r s e s the path l e n g t h R' r a t h e r than R. Using the phase speed c, = Jgh] and c a l c u l a t i n g R' from R by using the s h e l f geometry, a curve of the expected a r r i v a l - t i m e of the wave i s p l o t t e d as a f u n c t i o n of R, T h i s curve i s shown by the s o l i d l i n e i n f i g u r e 26. In a d d i t i o n , a t i m e - d i s t a n c e curve f o r the d i r e c t wave, c a l c u l a t e d from c, = (gh,' , i s a l s o shown (dashed l i n e ) i n f i g u r e 26. As would be expected the a r r i v a l times of the d i r e c t and r e f l e c t e d waves approach one another a s y m p t o t i c a l l y f o r l a r g e values of R. A second p l o t cf the a r r i v a l - t i m e data f o r the r e f l e c t e d wave i s made, i n t h i s case p l o t t i n g i t as a f u n c t i o n of B ' ( c a l c u l a t e d frcm measured values of R). These data are shown i n f i g u r e 27, Using c (=JghJ , a curve of the expected a r r i v a l - t i m e of the r e f l e c t e d wave i s c a l c u l a t e d as a f u n c t i o n of R T h i s curve ( s o l i d l i n e ) i s shown i n f i g u r e 27 and i s i n good agreement with the experimental data. The average phase speed c a l c u l a t e d from these data g i v e s c=0. 34810. 0071 m/sec., where 0.0071 i s one standard d e v i a t i o n . The value of c, = fgh| =0, 354 m/sec, (see p. 114) f a l l s w i t h i n the range of values found e x p e r i m e n t a l l y f o r c. Save Amplitude Values of the r e f l e c t e d wave amplitude obtained frcm the experimental data are shown as a f u n c t i o n of R', the g e o m e t r i c a l o p t i c s d i s t a n c e t h a t the r e f l e c t e d wave t r a v e l s i n t r a v e r s i n q the d i s t a n c e between the source and the point of o b s e r v a t i o n , i n f i g u r e 28. These amplitudes have been normalized to the r e f l e c t e d wave amplitude a t R=1.0m ( R' = 1.044IH). From equation 2.180, we would a n t i c i p a t e the s l o p e cf the amplitude versus d i s t a n c e R ' i n f i g u r e 28 to be -1/2. C l e a r l y , the data do not o ~~! 1 1 1 r— 0.3 0.5 0.7 1.0 2.0 R'(meters) F i g u r e 2 8 . R e f l e c t e d w a v e a m p l i t u d e a s a f u n c t i o n o f R. 13 3 d e f i n e a s t r a i g h t l i n e , Once a g a i n , c o n s i d e r i n g the e f f e c t s of f r i c t i o n to be i n t h e form of a l i n e a r f r i c t i o n term i n the equations of motion (as was done i n the case of the d i r e c t wave), the r e f l e c t e d wave amplitude i s found t o vary as n ~ r= e 3 1 1 where ot, i s the f r i c t i o n a l decay c o e f f i c i e n t i n water of depth h,. Applyinq l e a s t squares f i t t o the data i n f i q u r e 28 r e s u l t s i n a value of<^,=0,78m , A curve f o r t h i s f i t i s shown ( s o l i d l i n e ) i n f i q u r e 28. The value of the f r i c t i o n a l decay c o e f f i c i e n t found from these data i s i n aqreement with the values found f o r the d i r e c t wave («C=0.75m ' and 0,71m"1) on paqe 122. 3.2.3 The L a t e r a l Have The experimental methods j A and B of s e c t i o n 3.1) and the methods of a n a l y s i s a p p l i e d t o determining the l a t e r a l wave are s i n i l a r to those used to f i n d the d i r e c t and r e f l e c t e d waves. The l a t e r a l wave i s r e c o q n i z e d by i t s a r r i v a l - t i m e , and over a l a r g e p o r t i o n of the wavetank i t i s the f i r s t d i s t u r b a n c e to reach the d e t e c t o r . An example of the t o t a l wave f i e l d , with the emergence of the l a t e r a l wave with i n c r e a s i n g values of fi, i s shown i n the sequence of wave t r a c e s i n f i q u r e 29. The arrows mark the a r r i v a l - t i m e of the l a t e r a l wave (point A, f i q u r e 17), Fiqure 30 shows a seguence of wave p r o f i l e s c f the l a t e r a l and r e f l e c t e d waves a f t e r the d i r e c t wave had been removed e l e c t r o n i c a l l y . As was the case f o r the t r a c e s shown i n f i q u r e 17, the wave amplitudes should be m u l t i p l i e d by a f a c t o r of -1 F i g u r e 29. T h e e m e r g e n c e o f t h e l a t e r a l w a v e f r o m t h e w a v e p a c k e t m e a s u r e d o n t h e s h e l f , s h o w n f o r R i n c r e a s i n g . F i g u r e 3 0 . Wave p r o f i l e s o f t h e l a t e r a l and r e f l e c t e d w a v e s a f t e r e l e c t r o n i c r e m o v a l o f t h e d i r e c t w a v e . 13 6 t o p u t t h e m i n p r o p e r p e r s p e c t i v e w i t h t h e w a v e s s e e n i n t h e t a n k . T h e a m p l i f i e r g a i n w a s a l s o c h a n g e d w i t h i n c r e a s i n g v a l u e s o f E a n d t h i s m u s t be t a k e n i n t o a c c o u n t when c o m p a r i n g w a v e a m p l i t u d e s . To e n h a n c e t h e f o l l o w i n g d i s c u s s i o n , a d i a g r a m o f t h e s h e l f r e g i o n , i n c l u d i n g t h e g e o m e t r i c a l o p t i c s p a t h o f t h e l a t e r a l w a v e , t h e l a t e r a l w a v e f r o n t a n d o t h e r p e r t i n e n t p o i n t s , i s s h o w n i n f i g u r e 3 1 . Wave A r r i v a l - t i m e T h e e x p e r i m e n t a l d a t a f o r t h e t i m e o f a r r i v a l o f t h e l a t e r a l w a v e a r e s h o w n a s a f u n c t i o n o f d i s t a n c e 8 ( t h e d i s t a n c e SP i n f i g u r e 3 1 ) i n f i g u r e 3 2 , T h e d a t a a r e t h e r e s u l t o f m e a s u r e m e n t s made a t v a r i o u s d i s t a n c e s B f r o m t h e s o u r c e a l o n g t h e l i n e A A ' i n f i g u r e 3 1 , U s i n g w a v e s p e e d s c , a n d c z f o u n d f r o m l i n e a r s h a l l o w w a t e r t h e o r y i n e g u a t i o n 2. 1 6 3 , a t h e o r e t i c a l c u r v e o f t h e a r r i v a l - t i m e o f t h e l a t e r a l w a v e w i t h d i s t a n c e i s g e n e r a t e d . T h i s c u r v e i s s h o w n ( s o l i d l i n e ) i n f i g u r e 3 2 . T h e a v e r a g e s p e e d o f t h e l a t e r a l w a v e o v e r a r a n g e o f B f r o m 0.8 t o 2,1m i s f o u n d t o b e c = 0 . 4 0 3 * 0 . 0 1 8 m / s e c . U s i n g e g u a t i o n 2 . 1 6 3 , w i t h c, = 0 , 3 5 4 m / s e c a n d c z = 0 . 4 6 3 m / s e c , t h e s p e e d a t 1.45m ( t h e m i d p o i n t o f t h e r a n g e o v e r w h i c h t h e d a t a w as t a k e n ) i s f o u n d t o b e 0 . 3 9 4 m / s e c . T h i s v a l u e f a l l s w i t h i n t h e r a n g e f o u n d e x p e r i m e n t a l l y . A t i m e - d i s t a n c e c u r v e f o r t h e d i r e c t w a v e , c a l c u l a t e d u s i n g e g u a t i o n 2 . 1 6 2 , i s a l s o p l o t t e d ( d a s h e d l i n e ) i n f i g u r e 3 2 , T h e c r o s s i n g o f t h e t w o c a l c u l a t e d c u r v e s f i x e s t h e t h e o r e t i c a l c r o s s o v e r d i s t a n c e , t h e d i s t a n c e B c b e y o n d w h i c h t h e l a t e r a l wave a r r i v e s f i r s t a n d b e f o r e w h i c h t h e d i r e c t w a v e a r r i v e s f i r s t . E q u a t i o n 2 . 1 6 6 g i v e s t h e c r o s s o v e r -d i s t a n c e a s 0.82m. The l a t e r a l s a v e i n t h e e x p e r i m e n t a r r i v e d a t F i g u r e 3.1. S h e l f g e o m e t r y and r a y p a t h d i a g r a m f o r t h e l a t e r a l w a v e . F i g u r e 3 2 . T h e l a t e r a l w a v e a r r i v a l - t i m e a s a f u n c t i o n o f R. 00 139 the p o i n t B=0.8m at time t =2.16 s e c , while the d i r e c t wave ( f i g u r e 19) w i l l a r r i v e a t time t =2.27 s e c , so the c r o s s o v e r -d i s t a n c e determined e x p e r i m e n t a l l y i s l e s s than t h a t p r e d i c t e d by theory. Eguation 2. 163 gives the p r e d i c t e d a r r i v a l - t i m e of the l a t e r a l wave at the c r o s s o v e r - p o i n t as t=2.275 sec. The l a t e r a l wave a r r i v a l - t i m e found e x p e r i m e n t a l l y i s 1.1555 l e s s than t h a t p r e d i c t e d by theory. A comparable decrease i n the cros s o v e r d i s t a n c e would be expected. The val u e of E at which data f o r the l a t e r a l wave was f i r s t used was a t B=0.8m. Th i s i s very c l o s e t o the p r e d i c t e d c r o s s o v e r - d i s t a n c e of B=0.62m and u n f o r t u n a t e l y , the l a c k o f experimental data i n the v i c i n i t y around t h i s value of B prevents a more accurate d e t e r m i n a t i o n of B C . For B<0.8m, the l a t e r a l and r e f l e c t e d wave packets are not s u f f i c i e n t l y separated (and are not seen as d i s t i n c t waves) to allow an accurate e s t i m a t i o n of e i t h e r the a r r i v a l - t i m e or the amplitude of the l a t e r a l wave. In chapter 2, the mathematical model p r e d i c t e d the l a t e r a l wavefront t o be plane (eguation 2.189). An experimental i n v e s t i g a t i o n was made to determine the sit ape of the wavefront (tc see i f i t i s indeed plane) and the v a r i a t i o n i n amplitude along the wavefront. To make these d e t e r m i n a t i o n s , measurements of the l a t e r a l wave were made along the l i n e BB' i n f i g u r e 31. The angle t h i s l i n e makes with the s h e l f edge was chosen, as a matter of experimental convenience, to be 45 and not egual to o the c r i t i c a l angle (in t h i s case 6 C=49.8 ). A l i n e making an angle © c with the s h e l f edge i s shown i n f i g u r e 31 as BC'. This l i n e d e f i n e s a l i n e of constant phase of the l a t e r a l wavefront. The a r r i v a l - t i m e o f the l a t e r a l wave (determined by a point 140 cf constant phase on the p r o f i l e of the wave) was measured along t the l i n e BB , from the s h e l f edge to a point 0.35m frcm the edge. The r e s u l t s of these measurements are shown i n f i g u r e 33. The l i n e BB ' i s not a l i n e o f constant phase (BC^ i s a l i n e of constant phase) and the d i s t a n c e the l a t e r a l wave must t r a v e l t o reach p o i n t s on BB' i s l e s s than t h a t r e g u i r e d t c reach BCR (except at p o i n t B, where the t r a v e l - t i m e s are t h e same). T h i s d i f f e r e n c e i n propagation time i n c r e a s e s as the po i n t o f ob s e r v a t i o n (on the l i n e BB') i s removed f u r t h e r from the s h e l f edge. A curve of the p r e d i c t e d a r r i v a l - t i m e of the wave along the l i n e BB' i s c a l c u l a t e d using l i n e a r shallow water theory (eguation 2.160). T h i s curve i s shown (the s o l i d l i n e ) i n f i g u r e 33. The l a t e r a l wave i s seen to c o n s i s t e n t l y a r r i v e p r i o r to the time p r e d i c t e d by theory. The average d i f f e r e n c e i s c a l c u l a t e d t c be about 2%. A curve showing the p r e d i c t e d a r r i v a l - t i m e of the wave at po i n t s on the l i n e BC ' i s a l s o shown i n f i g u r e 33 (dashed l i n e ) . The time r e g u i r e d (from t h e o r e t i c a l c o n s i d e r a t i o n s ) f o r the l a t e r a l wave to t r a v e r s e the d i s t a n c e frcm B ' to C ' ( f i g u r e 31) i s c a l c u l a t e d to be t=0.12 s e c . T h i s i s a l s o the time d i f f e r e n c e between curves (1) and (2) at Lz=0.308m i n f i g u r e 33. In the experiment, t h i s p o i n t i s where the maximum of A t occurs. S p a t i a l l y , the s e p a r a t i o n of p o i n t s B' and C' i s A. L, = c,flkt= 0.042m, The pathlength L ( , from the s h e l f edge to c' (on the l i n e of constant phase) i s 0, 584m. The percentage d i f f e r e n c e i n pathlength t r a v e r s e d by the wave i n I f <y going to po i n t B i n s t e a d c f C i s about l/o. From the f i n d i n g s of t h i s experiment, i t i s concluded t h a t the l a t e r a l wavefront can be d e s c r i b e d as plane. —I 1 1 , — 0.5 1.0 LOmet ers) F i g u r e 33. A r r i v a l - t i m e o f t h e l a t e r a l w a v e a l o n g a s u r f a c e o f c o n s t a n t p h a s e . 142 Wave A m p l i t u d e E g u a t i o n 2 . 9 3 p r e d i c t s t h a t t h e a m p l i t u d e o f t h e l a t e r a l -K w a v e w i l l d e c a y a s ( L ^ ) . Wave a m p l i t u d e s w e r e m e a s u r e d a l o n g t h e l i n e AA' i n f i g u r e 3 1 a n d a r e s h o w n , n o r m a l i z e d t o t h e a m p l i t u d e a t R=1m, i n f i g u r e 3 4 . I t i s s e e n t h a t t h e s e d a t a d o n o t c o n f o r m t o a c u r v e o f s l o p e - 3 / 2 , a n d o n c e a g a i n a l i n e a r f r i c t i o n t e r m i s i n t r o d u c e d i n t o t h e e q u a t i o n s o f m o t i o n , F r o m t h e s e e q u a t i o n s i t i s f o u n d t h a t t h e a m p l i t u d e o f t h e l a t e r a l wave ( f o r w a v e s o b s e r v e d a l o n q t h e l i n e AA ) w i l l d e c a y a s \ ~ L t £ 3.12 w h e r e t<x i s t h e f r i c t i o n a l d e c a y c o e f f i c i e n t i n t h e d e e p w a t e r ( d e p t h h l ) . A p p l y i n q a l e a s t s q u a r e s f i t t o t h e d a t a s h o w n i n f i g u r e 34 r e s u l t s i n a v a l u e o f <*.v = 0.32m . A c u r v e f o r t h i s v a l u e o f i s s h o w n a s a s o l i d l i n e i n f i g u r e 3 4 . T h e v a l u e o f oC f o u n d i n t h i s c a s e h a s a v a l u e «<.v =0,41 , w h e r e tC, (p(, = G.78m ') i s t h e f r i c t i o n a l d e c a y c o e f f i c i e n t f o u n d i n t h e e x a m i n a t i o n o f t h e r e f l e c t e d w a v e . S i n c e t h e p r o p a g a t i o n p a t h L z i s a l o n g t h e d e e p w a t e r s i d e o f t h e s t e p , i t i s e x p e c t e d ( P i t e , 1 9 7 3 , f i g u r e 17) t h a t a t t e n u a t i o n d u e t o f r i c t i o n i n t h e d e e p w a t e r w i l l b e l e s s t h a n t h a t o n t h e s h e l f (*x<.<*,). M e a s u r e m e n t s w e r e made t o d e t e r m i n e t h e v a r i a t i o n i n w a v e a m p l i t u d e a l o n g t h e l a t e r a l w a v e f r o n t ( a l o n g t h e l i n e B B ' i n f i g u r e 3 1 ) , T h e r e s u l t s o f t h e s e m e a s u r e m e n t s a r e s h o w n i n f i g u r e 3 5 , w h e r e t h e v a l u e o f t h e a m p l i t u d e s a r e n o r m a l i z e d t o t h e m e a s u r e d v a l u e a t L,= 0. 745m ( x , y = 1 . 1m,0.15m). , E q u a t i o n 2 . 1 8 3 F i g u r e 3 4 . L a t e r a l w a v e a m p l i t u d e v a r i a t i o n w i t h p a t h l e n g t h L ~ . F i g u r e 3 5 . V a r i a t i o n o f w a v e a m p l i t u d e a l o n g t h e l a t e r a l w a v e f r o n t . 145 p r e d i c t s t h a t the amplitude of the l a t e r a l wave decays as L a A l a t e r a l wave f o l l o w i n g the geo m e t r i c a l c p t i c s path shown i n f i g u r e 31 w i l l reach a point on the l a t e r a l wavefront (Be'} which i s dependent upon the p a t h l e n g t h . Of course, when L 2 changes, so does pathlength l , , . Consequently, p o i n t s of o b s e r v a t i o n along BB are d i r e c t l y r e l a t e d with pathlength L 2 and pathlength L,. The amplitude a t t e n u a t i o n w i l l be dependent upon both p a t h l e n g t h s , 1, and Lx, as w e l l as the f r i c t i o n a l a t t e n u a t i o n along each r e s p e c t i v e l e n g t h (dependent upon oc, i n depth h, and i n depth h^). A l l r a y s a r r i v i n g at BB ' t r a v e l the same pathlength L e and experience the same a t t e n u a t i o n . . Thus, the r e l a t i v e amplitude decay does not depend upon t h i s p o r t i o n of the g e o m e t r i c a l o p t i c s path. Assuming t h a t the l a t e r a l wave i s a plane wave, the wave amplitude w i l l decay along the pathlength L, because of f r i c t i o n , while along the pathlength L v i t w i l l decay p r o p o r t i o n a l l y with (L t) •& . The o v e r a l l v a r i a t i o n i n amplitude along the l a t e r a l wavefront may then be determined from *lCF> - L,. 3.13 where P i s an o b s e r v a t i o n point along the wavefront (BC', or i n the case of the experiment BB ). To obtain a l e a s t sguares f i t to the data i n f i g u r e 35, us i n g the expression given i n eguation 3.13, i s d i f f i c u l t . I n s t e a d , a curve was c a l c u l a t e d u s i n g the range of values of L ( and l ^ experienced i n these measurements - i -I and values ov,=0.71m and *Cx=0.32m obtained e a r l i e r f o r the r e f l e c t e d and l a t e r a l waves, r e s p e c t i v e l y . . T h i s curve i s shown 146 as a s o l i d l i n e i n f i g u r e 35, which f i t s the data very w e l l . ; 3.2.4. The E f f e c t s of D i s p e r s i o n In chapter 2 ( s e c t i o n 2.1.6), an examination of the e f f e c t of d i s p e r s i o n upon the mathematical model was made. I t was found t h a t , f o r s m a l l amplitude shallow water waves, eguation 2.172 must be s a t i s f i e d , and the a p p r o p r i a t e model equations t o be a p p l i e d are determined by 1) the D r s e l l parameter (eguation 2.173) and 2) the i n i t i a l c o n d i t i o n s under which the wave f i e l d i s generated. Hammack and Segur (1978) c o n s t r u c t a d i m e n s i o n l e s s U r s e l l parameter U e (eguation 2.174) and a dimensionless volume V to determine the a p p r o p r i a t e model equations which are a p p l i c a b l e to the problem. T h e i r r e s u l t s can be a p p l i e d to the experimental model used i n t h i s i n v e s t i q a t i o n . For waves generated i n the s h e l f r e gion (h v=1.28 cm), V i s found to be of the order of u n i t y ; t h i s a l s o a p p l i e s to waves i n the deep water (h=2. 19 cm). T h i s p l a c e s the waves observed i n the experiment i n the range i n which the a p p r o p r i a t e model i s d e f i n e d by the Korteweg-de V r i e s eguation. Consequently, the exact wave shapes ( c n c i d a l or s o l i t a r y ) w i l l not be p r e d i c t e d by small amplitude t h e o r y , but the wave speeds should agree with those found using shallow water theory. Looking to the experimental data, an examination i s made of the e l o n g a t i o n of the wave packet with propagation time or d i s t a n c e t r a v e r s e d , d i f f e r e n c e s i n the phase and group v e l o c i t i e s and changes i n the shape of the wave packet. As a f i r s t i n g u i r y , c o n s i d e r s m a l l amplitude, l i n e a r , shallow water waves propagating i n water of uniform depth. The d i s p e r s i o n r e l a t i o n f o r these waves i s 147 3. 14 Using the expansion T> ^ vs ~ " 3. 15 approximate e x p r e s s i o n s f o r the phase and group speeds are given by 3. 16 and V * i y : 0 - ^ v + a I k V - - J 3.17 i r e s p e c t i v e l y . R e t a i n i n g terms of order (kh) , vie f i n d <± -VkV 3.18 — - i _ Using the val u e s c ? = c ? = 0. 350±-0. C064 msec , cu. = 0.344 ±0. 0088 msec ~* (where the average energy speed i s used i n l i e u of the group speed) and yj~qh=0.354 msec"' , the l e f t - h a n d s i d e of equation 3.18 y i e l d s a value of 0.017*0.01-5. T h i s mean value of 1.7$ normalized d i f f e r e n c e between the phase and qroup speeds al l o w s us to c o n s i d e r the e f f e c t s of d i s p e r s i o n upon the wave speeds as n e q l i q i b l e . The r i g h t - h a n d s i d e of equation 3.18 can 148 be e v a l u a t e d by s e l e c t i n g a r e p r e s e n t a t i v e v a l u e c f k f o r t h e w a v e p a c k e t p r o p a g a t i n g i n w a t e r o f d e p t h h . F o r h = 0 . 0 1 2 8 m , t h e mean p e r i o d o f t h e l e a d i n g w a v e i n t h e p a c k e t m e a s u r e s a b o u t o n e s e c o n d . W i t h t h e s e v a l u e s , t h e r i g h t - h a n d s i d e o f 3.18 h a s a v a l u e o f 0 . 0 1 7 , w h i c h i s t h e same a s t h e mean v a l u e f o u n d f o r t h e l e f t - h a n d s i d e . H e n c e , t h e o b s e r v e d d i f f e r e n c e i n t h e p h a s e v e l o c i t y , c , a n d t h e g r o u p v e l o c i t y , c ^ , i s d u e t o p h a s e d i s p e r s i o n . An a l t e r n a t e m e t h o d o f a s s e s s i n g d i s p e r s i o n i s t o e x a m i n e t h e e l o n g a t i o n o f t h e w a v e p a c k e t , e i t h e r i n p a r t o r i n w h o l e , a s a f u n c t i o n o f p r o p a g a t i o n t i m e o r d i s t a n c e . C o n s i d e r a w a v e p a c k e t o f o v e r a l l l e n g t h L , f o r w h i c h a p o i n t o f c o n s t a n t p h a s e ( p o i n t A, f i g u r e 17) a n d t h e c e n t r o i d o f t h e p o t e n t i a l e n e r g y o f t h e w h o l e w ave p a c k e t a r e s e p a r a t e d b y a d i s t a n c e 1. I f e l o n g a t i o n i s u n i f o r m t h r o u g h o u t t h e p a c k e t , we h a v e t h e r e l a t i o n w h e r e L\ L a n d & 1 a r e t h e i n c r e m e n t a l c h a n g e s i n L a n d 1 , r e s p e c t i v e l y , i n a n i n t e r v a l o f t i m e C\t. L e t t i n g c ^ c . , we c a n w r i t e A L 1 L 3. 19 £ 1 3.20 a n d i t f o l l o w s t h a t 149 3. 21 Using dR=cdt, equations 3.20 and 3.21 become 3.22 arc. ' c and 3 d ' 1 V c J 3. 23 Using the values found e a r l i e r , c=c p and c =c^, the righ t - h a n d s i d e of equation 3.22 g i v e s a value o f 0.017*0,021* Equation 3.23 can be rearranged to o b t a i n the approximation where the values used to c a l c u l a t e the l e f t - h a n d s i d e are found frcm the experimental data. These mean values are given as f o l l o w s : ££=0.0175m, &R=0. 1m, 1=0.361m, 1=0.0349m. The r e s u l t i n g value of the l e f t - h a n d s i d e c f 3.24 i s 0.0169, which i s i n reasonable agreement with the value of the right-hand s i d e , again i n d i c a t i n g t h a t d i s p e r s i o n i s due to f i n i t e wavelength. A curve showing the v a r i a t i o n of 1 with R i s shown i n f i g u r e 36. The mean value, 1, l i e s at a value of R=1.1m. I t i s seen t h a t the changes i n wave speeds are very s m a l l and e l o n g a t i o n of the wave packet i s comparably s m a l l . These L ^ C - Co, 3. 24 I I ; 1 1— O 1.0 , 2 0 R (meters) F i g u r e 3 6 . V a r i a t i o n i n l e n g t h 1 a s a f u n c t i o n o f R. 151 changes are none-the-less measureable and a s t r e t c h i n g cf the waveform i s observed. In f i g u r e 37, p r o f i l e s are shown f o r d i r e c t waves recorded at d i s t a n c e s of 1.4m, 1.8m and 2.05m from the source. These waves were propagating i n water depth h 2cm. I t i s i n t e r e s t i n g t c note that at 1.8m and 2.05m, the second c r e s t (when the p r o f i l e i s m u l t i p l i e d by -1) i s h i g h e r than the c r e s t of the l e a d i n g wave. Murty (1977,p.43) d i s c u s s e s the work or C a r r i e r (1971) and others who have d e a l t with t h i s occurrence through the use of a n a l y t i c a l models. In C a r r i e r (1971,p.168) a wave form i s generated to simulate the Alask a earthquake o f 1964. The shape of t h i s wave i s s i m i l a r t c those produced e x p e r i m e n t a l l y i n c e r t a i n r e g i o n s of the wavetank during t h i s experiment. The f i n i t e f r o n t a l area of the wave source renders i t not t r u l y a point source and at c e r t a i n d i s t a n c e s E, the e f f e c t s of d i s p e r s i o n become evident. The shape of the l e a d i n g edge of the wave packet i s observed to resemble that of an A i r y f u n c t i o n . These phenomena are the same ones which cause the d i s p e r s i o n i n c and c ^ seen e a r l i e r . 3.2.5.' Summary of the Experimental R e s u l t s The wave f i e l d seen during the experiment r e s u l t e d i n wave amplitudes and wave speeds which measured wit h i n 235 of those p r e d i c t e d by the a n a l y t i c a l model. The decay of the wave amplitudes and wave e n e r g i e s were approximately the same as those p r e d i c t e d t h e o r e t i c a l l y , once c o r r e c t i o n s f o r the damping due to f r i c t i o n had been made. The f r i c t i o n a l decay observed i n the experiment agreed with the r e s u l t s of other experimenters. The wave amplitudes o f the d i r e c t and r e f l e c t e d waves were found to decay (once the e f f e c t s of f r i c t i o n had been removed) as fi F i g u r e 37. Change i n wave p a c k e t p r o f i l e due t o d i s p e r s i o n . 153 a n d (R ) , r e s p e c t i v e l y , a s i s e x p e c t e d f o r c y l i n d r i c a l w a v e s m e a s u r e d a t d i s t a n c e s R a n d R ' f r o m t h e s o u r c e S a n d i m a g e s o u r c e S ', r e s p e c t i v e l y . T h e l a t e r a l w a v e a m p l i t u d e w a s f o u n d t o d e c a y a s ( I •) a n d t h e w a v e f r o n t o f t h e wave w a s f o u n d t o b e p l a n e . D i s p e r s i o n , w h i l e n o t i c e a b l e a n d m e a s u r a b l e , p l a y s l i t t l e p a r t i n t h e w a v e s p e e d s a n d w a v e a m p l i t u d e a n d / o r e n e r g y d e c a y . T h e w a v e s h a p e was m o d i f i e d b y s o m e p h a s e d i s p e r s i o n w h i c h r e s u l t s m o s t l i k e l y f r o m t h e f i n i t e d i m e n s i o n s o f t h e s o u r c e . I n g e n e r a l , t h e e x p e r i m e n t a l wave f i e l d a g r e e s w e l l w i t h t h e f i e l d f o u n d a n a l y t i c a l l y a n d c a n r e a d i l y b e u s e d i n r e a l s i t u a t i o n s o f s i m p l e g e o m e t r y t o p r e d i c t t h e w a v e f i e l d ( a s i n t h e c a s e o f t s u n a m i w a v e s ) , 15 4 4. C O N C L U S I O N S T h i s s t u d y h a s e x a m i n e d t h e f i e l d o f l o n g s u r f a c e g r a v i t y w a v e s g e n e r a t e d by a l o c a l i z e d s o u r c e o n a s h e l f , s u c h a s m i g h t c c c u r i n t s u n a m i g e n e r a t i o n o r w h i c h m i g h t be p r o d u c e d b y l o c a l e x c i t a t i o n d u e t o w i n d s o r t i d e s . B o t h a n a l y t i c a l a n d e x p e r i m e n t a l a s p e c t s c f t h e p r o b l e m h a v e b e e n s t u d i e d . T h e a n a l y t i c a l m o d e l e x a m i n e s t h e f i e l d o f s m a l l a m p l i t u d e s h a l l o w w a t e r w a v e s p r o p a g a t i n g o n a s h e l f o f u n i f o r m d e p t h , w h e r e t h e s h e l f r e g i o n i s s e p a r a t e d f r o m a d e e p w a t e r r e g i o n o f u n i f o r m d e p t h b y a l i n e a r v e r t i c a l s t e p . T h e p r o b l e m i s s o l v e d o n a r o t a t i n g c o o r d i n a t e s y s t e m , a l t h o u g h m o r e p r e c i s e r e s u l t s a r e o b t a i n e d f o r r o t a t i o n r a t e s much l e s s t h a n t h e wave f r e q u e n c y c r f o r t h e c a s e o f z e r o - r o t a t i o n . E x a c t s o l u t i o n s f o r t h e wave f i e l d i s s u i n g f r o m a l o c a l i z e d s o u r c e a r e o b t a i n e d i n t h e f o r m o f p a r a b o l i c c y l i n d e r f u n c t i o n s . A s y m p t o t i c e x p r e s s i o n s a r e f o u n d f o r t h e w a v e s i n t h e f a r - f i e l d f r o m g e o m e t r i c a l o p t i c s c o n s i d e r a t i o n s , T h e p r o b l e m i s r e l a x e d t o t h e c a s e o f z e r o -r o t a t i o n a n d s o l u t i o n s a r e d e t e r m i n e d f o r b o t h t h e c a s e o f t i m e -h a r m o n i c a n d i m p u l s e e x c i t a t i o n ( g i v i n g t h e s t e a d y - s t a t e a n d t r a n s i e n t s o l u t i o n s , r e s p e c t i v e l y ) . T h e e x p e r i m e n t c o n d u c t e d i n t h e l a b o r a t o r y e x a m i n e d t h e f i e l d o f s h a l l o w w a t e r g r a v i t y w a v e s g e n e r a t e d u n d e r c o n d i t i o n s c h o s e n t o s i m u l a t e t h e a n a l y t i c a l m o d e l i n t h e c a s e o f z e r o -r o t a t i o n . T h e e x c i t a t i o n o f t h e f i e l d was made w i t h a s m a l l c y l i n d r i c a l s o u r c e w h i c h was p u l s e d t o s i m u l a t e a t e m p o r a l i m p u l s e f u n c t i o n . T h e r e s u l t i n g f i e l d o f w a v e s on t h e s h e l f was f o u n d t o b e c o m p o s e d o f c y l i n d r i c a l d i r e c t a n d r e f l e c t e d w a v e s a n d o f a l a t e r a l wave w h i c h w a s g e n e r a t e d when t o t a l r e f l e c t i o n 155 o c c u r r e d . T h e w a v e s p e e d s o f t h e d i r e c t a n d r e f l e c t e d w a v e s w e r e f o u n d t o be i n a g r e e m e n t w i t h t h o s e f o u n d f r o m l i n e a r s h a l l o w w a t e r t h e o r y . T h e l a t e r a l w a v e s p e e d w a s f o u n d t o b e a f u n c t i o n c f t h e r e l a t i v e p o s i t i o n s o f t h e s o u r c e a n d t h e p o i n t o f o b s e r v a t i o n t o t h e d e p t h d i s c o n t i n u i t y a s w e l l a s t c t h a t p o r t i o n o f t h e r a y p a t h w h i c h l a y i n t h e d e e p w a t e r . The a r r i v a l - t i m e o f t h e l a t e r a l w a v e was f o u n d t o a g r e e w i t h t h a t d e t e r m i n e d b y r a y t h e o r y , a n d t h e r e g i o n s c f t h e s h e l f i n w h i c h t h e l a t e r a l wave w a s t h e f i r s t d i s t u r b a n c e t o a r r i v e w e r e d e t e r m i n e d . T h e w a v e a m p l i t u d e s o f t h e d i r e c t a n d r e f l e c t e d w a v e s w e r e f o u n d t o d e c a y a s c y l i n d r i c a l w a v e s w i t h f r i c t i o n a l d a m p i n g . T h e l a t e r a l w a v e a m p l i t u d e was f o u n d t o d e c a y w i t h t h e - 1 . 5 p o w e r o f t h e r a y p a t h l e n g t h i n t h e d e e p w a t e r , w i t h a n a d d i t i o n a l d e c a y f a c t o r d u e t o f r i c t i o n . T h e l a t e r a l w a v e f r o n t w a s f o u n d t o b e p l a n e a n d t h e a m p l i t u d e a l o n g t h e w a v e f r o n t v a r i e s a c c o r d i n g t o t h e - 1 . 5 p o w e r o f t h e q e o m e t r i c a l o p t i c s p a t h l e n g t h i n t h e d e e p w a t e r . A g r e e m e n t o f t h e o r y w i t h o b s e r v a t i o n w a s o v e r a l l v e r y g o o d . T h e w a v e f i e l d f o u n d f r o m b o t h t h e a n a l y t i c a l a n d e x p e r i m e n t a l m o d e l s r e s u l t s i n d i r e c t a n d r e f l e c t e d w a v e s w h i c h d e c a y l e s s r a p i d l y w i t h d i s t a n c e f r o m t h e s o u r c e t h a n t h e l a t e r a l w a v e . U n d e r i m p u l s e e x c i t a t i o n , t h e a r r i v a l o f t h e l a t e r a l w a v e c a n p r e c e e d t h a t o f a n y o t h e r d i s t u r b a n c e . T h i s f a c t s u g g e s t s t h a t , i n t h e c a s e o f t s u n a m i s g e n e r a t e d u n d e r s i m p l e g e o m e t r i c a l c o n d i t i o n s , t h e l a t e r a l wave may a c t a s a p r e c u r s o r o f t h e l a r g e r d i r e c t a n d r e f l e c t e d w a v e s . K n o w i n g t h e g e o m e t r i c a l o p t i c s p a t h l e n g t h f o l l o w e d b y t h e l a t e r a l w a v e , i t i s p o s s i b l e t o d e t e r m i n e t h e w a v e s p e e d , w a v e e n e r g y a n d e n e r g y 156 d e c a y . F o r p o i n t s s u f f i c i e n t l y f a r f r o m t h e s o u r c e , t h i s may r e s u l t i n s u f f i c i e n t w a r n i n g t i m e t o a l l o w p r e p a r a t i o n s t o b e made f o r t h e o n c o m i n g l a r g e r w a v e s i n t h e t s u n a m i . 157 REFERENCES Abramowitz, M. and Stegun, I. A. 1964. Handbook of Mathematical lE£ctions x with Formulas^ Graphs, and Mathematical T a b l e s . Nat. Bur. Standards Appl. Math. S e r i e s , 55. Superintendent of Documemts, U.S. Government P r i n t i n g O f f i c e , W a s h i n g t o n , D.C. 1046p. Backus, G.E. 1962. The e f f e c t of the e a r t h ' s r o t a t i o n on the propagation of ocean waves over long d i s t a n c e s . Deep-Sea Res. 9 pp. 165-197 Brac e w e l l , E. 1965. The F o u r i e r Transform and I t s A ^ j l i c a t i o n s A McGraw-Hill, New York 381p. Erekhovskikh, L.M. 1960. Saves In Layered Media. Academic Press, Mew York 561p. Cagniard, L. 19 39. 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S o u r c e m e c h a n i s m o f t h e i r a j c r t s u n a m i i n T h e G r e a t A l a s k a E a r t h q u a k e o f 1964..; N a t i o n a l A c a d e m y o f S c i e n c e s , W a s h i n g t o n , D.C. pp.111-121.. W a t s o n , G.N. 1966. A T r e a t i s e o n t h e T h e o r y o f B e s s e l F u n e j t i o n S j s . C a m b r i d g e U n i v e r s i t y P r e s s , C a m b r i d g e , E n g l a n d 804p. W h i t h a m , G.B. 1974. L i n e a r a n d N o n l i n e a r W a y e s x J o h n W i l e y & S o n s , New Y o r k 636p. W h i t t a k e r , E. T. . a n d W a t s o n , G.N. 1952., A C o u r s e - o f M o d e r n lHi-2.2§iS3L C a m b r i d g e U n i v e r s i t y P r e s s , C a m b r i d g e , E n g l a n d 608p. 164 APPENDIX A. The Method of Steepest Descents Consider the i n t e g r a l (Copson, 1967,p.65} v A. 1 i n the complex © -plane, where and P(<s) are a r b i t r a r y a n a l y t i c f u n c t i o n s of O , r e g u l a r i n the domain c o n t a i n i n g the path of i n t e g r a t i o n p , and independent of oC , where o«- i s a l a r g e p o s i t i v e r e a l parameter. The b a s i c i d e a of t h i s technigue of asymptotic e v a l u a t i o n i s to deform the path, T,to a " s t e e p e s t descent" path such t h a t along the new path, P, the value of the i n t e g r a l i s determined mainly from the c o n t r i b u t i o n from a s h o r t p o r t i o n of the path. The value cf the o r i g i n a l i n t e g r a l i s not changed provided no s i n g u l a r i t i e s are c r o s s e d during deformation. I f poles or branch cuts are encountered, the r e s i d u e s and branch i n t e g r a l s must be evaluated to account f o r t h e i r c o n t r i b u t i o n s . Write i n terms of i t s r e a l and imaginary parts as A.2 The i n t e g r a n d of eguation A.I thus becomes A. 3 The s t e e p e s t descent path i s t h a t along which the f u n c t i c r tye) has a maximum a t some point on the path and decreases as r a p i d l y as p o s s i b l e away from t h a t p o i n t . By a n a l y t i c f u n c t i o n theory. 165 the l i n e of most r a p i d decrease of ^QCCJ^ c o i n c i d e s with the l i n e of constant value o f ^(a) , c a l l e d the l i n e of co n s t a n t phase, The poin t a t which n a s a maximum i s known as the saddle p o i n t . The d e r i v a t i v e fR(©") at t h i s p o i n t i s z e r o , as i s $ s i n c e $ t « s ) i s constant along t h i s path. T h e r e f o r e , the saddle p o i n t can be found by s e t t i n g Since and are r e a l and Fffc), which i s assumed to be a s l o w l y v a r y i n g f u n c t i o n of O , does not depend on , the i n t e g r a n d does not o s c i l l a t e r a p i d l y on the new path f o r l a r g e values of <*• . L e t eguation A.H be s o l v e d and the saddle p o i n t l o c a t e d at (}•-©<,. Define a new v a r i a b l e of i n t e g r a t i o n along the s t e e p e s t descent path by where $ i s the new v a r i a b l e and v a r i e s frcn; saddle p o i n t corresponds to 5-Q . Equating r e a l p a r t s of equations A. 2 and A.5 g i v e s -co to + <* . The and imaginary 166 A. 6 In t h i s , the imaginary p a r t s t a y s constant, while the r e a l part i s a maximum when ^ -o and decreases as \ i n c r e a s e s . The i n t e g r a l of eguation A.1 may now be w r i t t e n i n the new v a r i a b l e as 1 - e » - ^* fl#7 - A3 where Since «C i s l a r g e , i t i s convenient to expand <^ft) i n a power s e r i e s i n : S u b s t i t u t i o n i n t o e g u a t i o n A.7 g i v e s Using the values of the d e f i n i t e i n t e g r a l s I 67 - A3 ~ * ^ X III J ^ ^ = A.1 1 g i v e s A. 12 T h i s s e r i e s i s g e n e r a l l y not convergent but i s asymptotic. Assuming t h a t <*. i s s u f f i c i e n t l y l a r g e and t h a t fat") v a r i e s s u f f i c e n t l y slowly compared with e , i . e . , i f i t s d e r i v a t i v e s cfi (V) are s u f f i c e n t l y s m a l l , the f i r s t term i n eguation fl.10 i s a good approximation of the i n t e g r a l and i s c a l l e d the asymptotic' e x p r e s s i o n of I i n . The e x p r e s s i o n g i v e n i n eguation A.8, i n g e n e r a l , does not have a c l o s e d form. A method must be devised to o b t a i n <£($) i n a power s e r i e s i n $ . D i f f e r e n t i a t i n g eguation A.5 with r e s p e c t to © g i v e s r ' or T h e r e f o r e , I 6 8 Expanding -fte>) and F f e ) i n powers of €-©-<90 g i v e s '"r,^ 1 A. 14 Equation A.5 then g i v e s T h i s s e r i e s can be i n v e r t e d thus r e p r e s e n t i n q £- i n terms of a power s e r i e s i n $ . L e t ft. 16 S u b s t i t u t i o n i n t o equation A.15 q i v e s E q u a t i n g c o e f f i c i e n t s o f l i k e p o w e r s o f g i v e s _J J u A. 18 1 _ *h crV - 3 f e c^'V a , -S u b s t i t u t i o n o f e g u a t i o n A.14 i n t o A. 13 g i v e s I h e n e g u a t i o n A . 1 6 , w i t h t h e v a l u e s l i s t e d i n e q u a t i o n A . 1 8 , s u b s t i t u t e d i n t o e g u a t i o n A. 1 9 , t h e r e s u l t i s l7'o T"p HO jL 20 In equation fi.12, the e x p r e s s i o n s - z (TSW £l + ± £ ^ - * <3>V~ Fr A.21 are needed. Higher order terms can be found i n a l i k e manner. The asymptotic expansion, f o r ^G*>\, to the f i r s t order i s thus given as A. 22 As an example of the a p p l i c a t i o n of the method of s t e e p e s t descents, c o n s i d e r the i n t e g r a l r e p r e s e n t i n g the r e f l e c t e d wave, cM9- A. 23 T h i s i n t e g r a l i s of the form shown i n eguation A.1, where R^-i.R.(o), »Hft.' , -Pfo'UCcji.C.G-Oi). The time dependence ^"t<**t i s i m p l i e d and suppressed. The saddle p o i n t , found by s o l v i n g eguation A,4, i . e . , <*VA/\G&-0^I=O, l i e s a t £ = - 0 ^ The eguation f o r I 7 1 the s t e e p e s t descent path i s found by expanding i n terms of i t s r e a l and imaginary components, i . e . , ?CeW SvwC®'- <90 svwU ©" -f I Cv> (©- fl^ c*r> U Q-" 1,24 and using t h i s r e l a t i o n i n eguation A.5. T h i s r e s u l t s i n the eguation Eguating the r e a l and imaginary parts of t h i s eguation and using the second p a r t of equation A.6 r e s u l t s i n the eguation of the ste e p e s t descent path, which i s <L*vl&'-«0 CWJU©" =. [ A.26 The angle of i n t e r c e p t i o n of t h i s path with the r e a l a x i s (at ©--© v) i s r e a d i l y found. Expanding about i n a T a y l o r s e r i e s g i v e s Let fi'-^ + ^ e A. 28 By s u b s t i t u t i o n . I 7 ^ $C<sf> - {CcO "CtoOv'-e A. 29 R e t a i n i n g o n l y t h e f i r s t t w o t e r m s i n e g u a t i o n A.29 a n d e q u a t i n g i t w i t h e q u a t i o n A.5 q i v e s $ ^ - ^ C © , V <• A.30 E v a l u a t i n q 4-"c©0 , we f i n d ) - — A.31 F o r -oc-<^ <<*» # w e f i n d ^ l i s r e a l a n d p o s i t i v e . T h e e x p o n e n t i a l i n e q u a t i o n A. 31 i s t h u s s a t i s f i e d w h e n , A . 3 2 T h u s , t h e s t e e p e s t d e s c e n t p a t h m o v e s f r o m -"v*/x + C«<» t o t h e r e a l © - a x i s , i n t e r s e c t s i t a t <5-© x a t a n a n q l e o f w i t h r e s p e c t t o t h e p o s i t i v e < & ' - a x i s , l e a v e s t h e r e a l a x i s a t a n a n q l e o f a n d p r o c e e d s t o X - t * . T h u s , t h e i n t e r s e c t i o n o f t h e s t e e p e s t d e s c e n t p a t h w i t h t h e r e a l © - a x i s i s a t a n a n g l e o o f 45 ( s e e f i q u r e A - 1 ) . u r e A - l . P a t h o f s t e e p e s t d e s c e n t i n t h e c o m p l e x © - p l a n e . 174 APPENDIX B. The Have Tank The wave tank used i n these i n v e s t i g a t i o n s i s i n the h y d r a u l i c s l a b o r a t o r y belonging to the Department of C i v i l E ngineering. The tank, which was modified f o r t h i s experiment, i s c o n s t r u c t e d of plywood and p l e x i g l a s s , supported by a heavy wooden framework: the tank bottom and three s i d e s are made of plywood, while one of the l o n g i t u d i n a l s i d e s i s made o f p l e x i g l a s s . The l a t t e r s i d e a l l o w s a c l e a r l a t e r a l view of the water column frcm without. The top of the tank remains open. A diagram of the wave tank i s shown i n f i g u r e B-1. The i n t e r i o r dimensions cf the tank are as f o l l o w s : 5.0m i n l e n g t h , 2.2m i n width and 0,6m i n depth. The tank bottom, which i s f l a t and l e v e l throughout, i s 0.75m above the l a b o r a t o r y f l o o r . In order to e s t a b l i s h a l a b o r a t o r y model which was r e p r e s e n t a t i v e o f the mathematical model developed i n chapter 2, i t was necessary to c o n s t r u c t a s h e l f r e g i o n which l a y a d j a c e n t t o a deep water r e g i o n . To accomplish t h i s , the wave tank was d i v i d e d l o n g i t u d i n a l l y i n t o two r e g i o n s . Along one s i d e c f the tank a s h e l f r e g i o n , c o v e r i n g n e a r l y 55% of the tank bottom area, and measuring 1.2m i n width and 5.0m i n l e n g t h , was e s t a b l i s h e d . The remaining p o r t i o n o f the tank bottom, which amounts to about 45% of the t o t a l bottom area, and measuring 1.0m i n width and 5.0m i n l e n g t h , c o n s t i t u t e d the deep water re g i o n . The two r e g i o n s were separated by a 5.0m long r e c t i l i n e a r step which re p r e s e n t e d a v e r t i c a l c o n t i n e n t a l slope. Tc c o n s t r u c t a s h e l f r e g i o n , sheets of one inch (2.54 cm) t h i c k p l e x i g l a s s , measuring 1.2m i n width, were p o s i t i o n e d above and p a r a l l e l to the tank bottom. These sheets were held r i g i d by A • — i i ' 5.0m U — l _ J i I 1 0.6 m J 2.2m--1.2m-/ / / / / / / / / / / / / / 77 F i g u r e B - l . Wave t a n k g e o m e t r y . 176 a system of metal supports and f a s t e n e r s . The volume between these sheets and the bottom was f i l l e d with water d u r i n g the experiment. Thus, t h i s c o n f i g u r a t i o n represented a f l a t and l e v e l s u r f a c e r e p r e s e n t a t i v e of a s h e l f r e g i o n . In order t h a t the c o n t i n e n t a l slope appear as a r i g i d and impermeable v e r t i c a l boundary (so t h a t the r e g i o n l o c a t e d beneath the s h e l f i s i n a c c e s s i b l e t o wave motion), a 5.0m length of metal box-channel was placed as the support c f the p l e x i g l a s s sheet along i t s inner edge. Ey s e l e c t i n g the t h i c k n e s s of the support members, the d i f f e r e n c e between the water depth on the s h e l f and that i n the deep water was e s t a b l i s h e d . During the experimental procedure, i t was d e s i r a b l e to be able to generate and observe waves i n the absence of a s h e l f r e g i o n , i . e . , one wanted the water depth throughout the wave tank to be uniform. Since i t was a major undertaking to remove and r e p l a c e the s h e l f c o n f i g u r a t i o n , a means cf removing the deep water r e g i o n was i n t r o d u c e d . To accomplish t h i s , four sheets of one i n c h (2.54 cm) t h i c k p l e x i g l a s s , measuring 1.0m i n width and 1.25m i n l e n g t h , were l a i d f l a t upon the tank bottom during the experiment when the deep water r e g i o n was r e g u i r e d . In e f f e c t , t h i s r a i s e d the bottom of the deep water r e g i o n by 2.54 cm. With the use of s p a c e r s , these sheets could be r a i s e d to a p o s i t i o n where they were on the same l e v e l as the s h e l f . In t h i s c o n f i g u r a t i o n the water depth was uniform throughout the wave tank and egual to that on the s h e l f . T h i s c o n f i g u r a t i o n was used i n the experimental procedure d e s c r i b e d i n method A of chapter 3. In v a r y i n g the deep water depth i n t h i s manner, i t was not necessary to remove any of the s t r u c t u r a l m a t e r i a l from 177 w i t h i n the water column and thus, the f r e e s u r f a c e water l e v e l w i t h i n the tank remains unchanged. I t was a l s o p o s s i b l e to change the v e r t i c a l p o s i t i o n of o n l y some of the p l e x i g l a s s sheets i n the deep water r e g i o n . C o n s i d e r i n g the tank bottom, as shown i n f i g u r e B-1, t o be d i v i d e d i n t o f o u r quadrants, i t was thus p o s s i b l e t o r a i s e the sheets i n guadrant 3 wh i l e l e a v i n g those i n quadrant 2 unchanged. T h i s c o n f i g u r a t i o n r e s u l t e d i n three guadrants (guadrants 1,3 and 4) of the tank having a measured water depth equal to t h a t on the s h e l f , while the remaining quadrant (quadrant 2) had a depth equal to t h a t of the deep water. T h i s c o n f i q u r a t i o n was used durinq the experimental procedure d e s c r i b e d i n method B c f c h a p t e r 3, A " s i x i n c h " (15.2 cm) aluminum " I 11 beam, extendinq the lenqth of the wave tank, was mounted atop one of the l o n q i t u d i n a l s i d e w a l l s of the tank. T h i s beam served as a t r a c k along which two c a r r i a q e s , used t o hold and p o s i t i o n the wave d e t e c t o r s , t r a v e r s e d . Each c a r r i a q e (as shown i n f i q u r e B-2) was con t i n u o u s l y p c s i t i o n a b l e alonq the l e n q t h of the tank. Attached to each c a r r i a q e was an aluminum t r u s s measurinq 1.7m i n le n q t h . Attached to the h o r i z o n t a l member of the t r u s s was a s l i d e mechanism which was p o s i t i o n a b l e from one end of the t r u s s to the other. T h i s s l i d e h e l d a micrometer adjustment mechanism by which the attached wave d e t e c t o r could be p o s i t i o n e d v e r t i c a l l y to an accuracy of 0.1mm i n the wave tank. These three deqrees of freedom allowed continuous p c s i t i o n i n q of the wave sensor throuqhout the r e q i o n of the wave tank i n which measurements were to be made. 178 F i g u r e B - 2 . T h e d e t e c t o r c a r r i a g e m e c h a n i s m . 178a. 179 I n a d d i t i o n t o t h e t w o m o v a b l e d e t e c t o r s , a t h i r d d e t e c t o r was p o s i t i o n e d w i t h i n t h e w a v e t a n k . T h i s d e t e c t o r was h e l d r i g i d l y i n p o s i t i o n i n t h e t a n k b y m e a n s o f f i x e d a t t a c h m e n t . T h e p u r p o s e o f t h i s s e n s o r was t o m o n i t o r t h e wave p a c k e t g e n e r a t e d b y t h e s o u r c e m e c h a n i s m . T h e s i g n a l r e c e i v e d b y t h i s d e t e c t o r w a s u s e d a s a c o n t r o l i n t h e e x p e r i m e n t . A b a s e s t a n d u p o n w h i c h was m o u n t e d a m i c r o m e t e r c a l i b r a t e d v e r t i c a l p o i n t e r was u s e d t o m e a s u r e t h e p o s i t i o n o f t h e e g u i l i b r i u m f r e e s u r f a c e a s w e l l a s t o d e t e r m i n e t h e w a t e r d e p t h a t v a r i o u s p o s i t i o n s t h r o u g h o u t t h e w a v e t a n k . T h i s a p p a r a t u s i s s h o w n i n f i g u r e B-3. gure B-3. Apparatus f o r measuring water depth and f r e e s u r f a c e l e v e l . 180 OL 181 A P P E N D I X C. T h e S a v e G e n e r a t o r T h e d i r e c t w a v e f i e l d e x a m i n e d i n t h e t h e o r e t i c a l m o d e l i n c h a p t e r 2 was c o n s i d e r e d t o c o n s i s t o f c y l i n d r i c a l w a v e s a r i s i n g f r o m a p o i n t s o u r c e . I t w a s n e c e s s a r y t o r e p r o d u c e a wave s o u r c e i n ' t h e e x p e r i m e n t a l m o d e l w h i c h r e a s o n a b l y r e s e m b l e d t h a t d e s c r i b e d i n t h e t h e o r e t i c a l m o d e l . T h u s , t h e e x p e r i m e n t a l wave g e n e r a t o r h a d t o r e a s o n a b l y i s o t r o p i c a n d , a t l e a s t a t s o m e d i s t a n c e f r o m t h e s o u r c e , t h e w a v e s h a d t c a p p e a r t o a r i s e f r o m a p o i n t . A f t e r some d e l i b e r a t i o n a n d e x a m i n a t i o n o f s e v e r a l m e t h o d s o f g e n e r a t i o n , a r e l a t i v e l y s i m p l e d e v i c e was c o n s t r u c t e d w h i c h p r o v e d t o b e a d e q u a t e f o r t h e p u r p o s e s o f t h i s e x p e r i m e n t . T h e w a v e g e n e r a t o r c h o s e n c o n s i s t e d o f a n i n v e r t e d v e g e t a b l e s t e a m - b a s k e t , m o u n t e d r i g i d l y o n t o p o f a h e a v y b r a s s c y l i n d e r w h i c h a c t s a s a p e d e s t a l h o l d i n g a h o r i z o n t a l c i r u l a r r i n g f r c m w h i c h a c i r c u l a r c u r t a i n , made o f t h e l e a v e s , h a n g s . T h e b a s k e t c o n s i s t s c f 18 l e a v e s , t r a p e z o i d a l i n s h a p e , w h i c h a r e a r r a n g e d i n a n o v e r l a p p i n g f a s h i o n . T h e w a v e g e n e r a t o r i s s h o w n i n i t s u n a c t i v a t e d s t a t e i n t h e u p p e r p h o t o g r a p h i n f i q u r e C - 1 . T h e d i a m e t e r o f t h e s u p p o r t i n g r i n g m e a s u r e s 1 2 . 5 cm. T h e l e a v e s m e a s u r e 6 cm i n l e n g t h , 2,5 cm a c r o s s t h e t o p e d g e a n d 5 cm a c r o s s t h e b o t t o m e d g e . T h e o v e r l a p p i n g a r r a n g e m e n t c f t h e l e a v e s i s s u c h t h a t when a n y l e a f i s r o t a t e d o u t w a r d l y a b o u t i t s h o r i z o n t a l a x i s , t h e e n t i r e l e a f s y s t e m s i m u l t a n e o u s l y r o t a t e s o u t w a r d l y . T h e a c t i v a t e d s o u r c e i s s h o w n i n t h e l o w e r p h o t o g r a p h o f f i g u r e C - 1 . T h e c y l i n d r i c a l b r a s s p e d e s t a l w h i c h h c l d s t h e l e a f s y s t e m was p o s i t i o n e d o n t h e p l e x i g l a s s s h e l f w h e r e i t w a s h e l d i n p o s i t i o n b y a s m a l l b r a s s p i n . T h i s c y l i n d r i c a l p i n w a s g u r e C - l . T h e w a v e g e n e r a t o r . 183 i n s e r t e d i n t o t h e c e n t e r o f t h e p e d e s t a l a s w e l l a s i n t o a s m a l l h o l e i n t h e p l e x i g l a s s . T h i s a r r a n g e m e n t p o s i t i o n e d a n d f i x e d t h e c e n t e r c f t h e s o u r c e . I n o r d e r t o a c t i v a t e t h e l e a f s y s t e m , t w o n y l o n l i n e s o f e g u a l l e n g t h w e r e j o i n e d a t a common p o i n t d i r e c t l y a b o v e t h e c e n t e r o f t h e s o u r c e , w i t h t h e o t h e r e n d s a t t a c h i n g t o t w o l e a v e s , l o c a t e d d i a m e t r i c a l l y o p p o s i t e o n e a n o t h e r . When t h e common p o i n t o f t h e s e l i n e s w a s m o v e d v e r t i c a l l y u p w a r d t h e l e a f s y s t e m r e s p o n d e d , a s o n e , by r o t a t i n g o u t w a r d . A t t a c h m e n t o f t h e a c t i v a t i n g l i n e t o t w o l e a v e s was s u f f i c e n t t o a s s u r e t h a t t h e m o t i o n o f a l l t h e l e a v e s o c c u r r e d s i m u l t a n e o u s l y . A f o u r i n c h b o x - c h a n n e l was f i x e d i n p o s i t i o n l a t e r a l l y a c r o s s t h e t o p o f t h e t a n k . A s o l e n o i d was m o u n t e d o n t h e c h a n n e l w h i c h was u s e d t o a c t i v a t e t h e s o u r c e . T h e n y l o n l i n e a t t a c h i n g w i t h t h e l e a f s y s t e m was c o n n e c t e d w i t h t h e m o v a b l e p o r t i o n o f t h e s o l e n o i d , w h i c h w a s i n t h e f o r m o f a n a r m w h i c h m o v e d v e r t i c a l l y u p w a r d u p o n a c t i v a t i o n . T h e s o l e n o i d was e n e r g i z e d b y a P o w e r s t a t V a r i a b l e T r a n s f o r m e r , t y p e 116, w h i c h i n t u r n w a s t r i g g e r e d b y a n e l e c t r o n i c r e l a y s w i t c h . A s c h e m a t i c d i a g r a m o f t h e s o u r c e c o n t r o l s y s t e m i s s h o w n i n f i g u r e C - 2 . T h e r a p i d i t y o f m o v e m e n t o f t h e l e a v e s w a s c o n t r o l l e d b y v a r y i n g t h e o u t p u t v o l t a g e c f t h e P o w e r s t a t . T h e l e n g t h o f a r c t h r o u g h w h i c h t h e l e a v e s r o t a t e d was c o n t r o l l e d b y v a r y i n g t h e l e n g t h c f t h e s o l e n o i d a r m . A d j u s t m e n t o f t h e s e p a r a m e t e r s r e s u l t e d i n t h e d e s i r e d o u t w a r d s w e e p o f t h e l e a f s y s t e m t o p r o d u c e t h e e f f e c t o f a w i d e n i n g c i r c l e . T h e v e r t i c a l p l a c e m e n t o f t h e wave g e n e r a t o r c o u l d b e a d j u s t e d s o t h a t t h e d e p t h t o w h i c h t h e l e a v e s h u n g i n t h e w a t e r c o l u m n was a p p r o p r i a t e t o g e n e r a t e t h e P O W E R STAT T R A N S F O R M E R 1 1 0 V 3 * o A C T I V A T I N G S W I T C H -o o 6 v S O L E N O I D A C T I V A T I N G R E L A Y W A V E G E N E R A T O R Figure C-2. Wave generator a c t i v a t i n g c i r c u i t . 185 c y l i n d r i c a l w a v e s w i t h o u t s p l a s h . W h i l e t h i s m e c h a n i s m was q u i t e s i n p i e i t p r o v e d t o be v e r y s a t i s f a c t o r y i n q e n e r a t i n q t h e w a v e s d e s i r e d . T h e w a v e p a c k e t s q e n e r a t e d w i t h i n t h e p e r i o d o f a n e x p e r i m e n t s h o w e d r e m a r k a b l e c o n s t a n c y a n d v e r y l i t t l e a d j u s t m e n t o f e i t h e r t h e s o l e n o i d a r m l e n q t h o r v a r i a c v o l t a g e was r e q u i r e d . I n a c t i v a t i n q t h e s o l e n o i d t w o m e t h o d s w e r e t r i e d . F i r s t , t h e s o l e n o i d was e n e r q i z e d a n d t h e v o l t a q e k e p t a p p l i e d , t h u s h o l d i n q t h e l e a v e s i n t h e e x t e n d e d p o s i t i o n u n t i l a l l t h e w a v e s h a d b e e n r e c o r d e d . S e c o n d , t h e v o l t a q e w a s r e d u c e d t o z e r o i m m e d i a t e l y a f t e r t h e l e a v e s h a d r e a c h e d t h e i r maximum e x t e n d e d p o s i t i o n . , T h e l e a v e s t h e n r e t u r n e d t o t h e i r r e s t p o s i t i o n , a l l o f t h i s o c c u r r i n q w h i l e m e a s u r e m e n t s w e r e b e i n q t a k e n . T h e d i f f e r e n c e i n t h e s e t w o t e c h n i g u e s made v i r t u a l l y n o d i f f e r e n c e i n t h e w a v e f o r m s r e c o r d e d , s o t h e l a t t e r t e c h n i q u e was u s e d s i n c e i t r e m o v e d t h e u n d u e s t r e s s o n t h e s o l e n o i d c a u s e d b y a l l o w i n q i t t o r e m a i n e n e r q i z e d f o r p r o l o n q e d p e r i o d s o f t i m e . A l l o f t h e e l e c t r i c a l a n d m e c h a n i c a l e q u i p m e n t u s e d i n c o n n e c t i o n w i t h t h e w a v e q e n e r a t o r w e r e i s o l a t e d b o t h m e c h a n i c a l l y a n d e l e c t r i c a l l y f r o m t h e d e t e c t o r s a n d o t h e r e q u i p m e n t . T h e wave q e n e r a t o r a n d a s s o c i a t e d a p p a r a t u s was s h o c k - m o u n t e d s c a s t o r e m o v e a n y m e c h a n i c a l v i b r a t i o n w h i c h m i q h t - o t h e r w i s e b e t r a n s m i t t e d t o t h e w a v e d e t e c t o r s . C a r e was t a k e n t o a r r a n q e t h e e l e c t r i c a l w i r i n q a n d t o u s e s h i e l d i n q w h e n n e c e s s a r y t o m i n i m i z e e x t r a n e o u s e l e c t r o n i c p i c k u p . T h e s i q n a l f r o m t h e a c t i v a t i n q s w i t c h o f t h e w a v e q e n e r a t o r c o n t r o l s y s t e m was u s e d a s a t i m i n g r e f e r e n c e o n t h e c h a r t r e c o r d e r . 186 A P P E N D I X D. T h e H a v e A b s o r b e r s H h e n t h e wave g e n e r a t o r w a s a c t i v a t e d , t h e g e n e r a t e d w a v e s p r o p a g a t e d o u t w a r d l y u n t i l t h e y i n t e r a c t e d e i t h e r w i t h t h e e d g e o f t h e s h e l f o r t h e t a n k w a l l s . W a v e s r e f l e c t i n g f r o m t h e l a t t e r w e r e u n d e s i r a b l e s i n c e t h e i r p r e s e n c e t e n d e d t o o b s c u r e t h e c o n s t i t u e n t s a r i s i n g f r o m t h e s t e p . T h i s b e c a m e p a r t i c u l a r l y a p p a r e n t when t h e t a n k w a l l s w e r e n o t c o v e r e d w i t h seme a b s o r b i n g m a t e r i a l a n d when t h e p a t h l e n g t h o f t h e s e r e f l e c t e d w a v e s was n e a r l y t h e same a s f o r t h e o t h e r w a v e s . T h u s , w h e n t h e w a v e d e t e c t o r s w e r e f a r r e m o v e d f r o m t h e s o u r c e , i n t e r f e r e n c e was g r e a t e r ; t h e l e a d i n g e d g e o f t h e w a l l - r e f l e c t e d p a c k e t c o u l d a r r i v e c o n c u r r e n t l y w i t h t h e m i d d l e o r t r a i l i n g p o r t i o n o f t h e w a v e s f r o m t h e s h e l f e d g e . I t was d e s i r a b l e t o a b s o r b o r d a m p e n a s much a s p o s s i b l e a l l r e f l e c t i o n s a r i s i n g f r o m t h e t a n k w a l l s . T e s t s w e r e c o n d u c t e d u s i n g v a r i o u s a b s o r b e r m a t e r i a l s a n d c o n f i g u r a t i o n s . T h e m a t e r i a l w h i c h p r o v e d m o s t s a t i s f a c t o r y was r u b b e r i z e d h o r s e h a i r . T h i s c o u l d b e p u r c h a s e d i n l a r g e r o l l s o f t w o i n c h t h i c k n e s s a n d t h e m a t e r i a l w a s e a s i l y c u t a n d s h a p e d t o t h e d e s i r e d g e o m e t r y . S i n c e i t was o f p a r t i c u l a r i n t e r e s t t o r e m o v e t h e w a l l r e f l e c t i o n s f r o m t h e r e g i o n s i n w h i c h t h e w a v e s a r i s i n g f r o m t h e s t e p w e r e t o be m e a s u r e d , i t w a s i n t h e s e r e g i o n s t h a t t h e e f f e c t s o f a b s o r b i n g t h e w a l l - r e f l e c t e d w a v e s w e r e e x a m i n e d . T e s t s w e r e f i r s t c o n d u c t e d w i t h t h e t a n k w a l l s l e f t b a r e . T h e s h e l f w a s r e m o v e d i n o r d e r t h a t o n l y t h e d i r e c t w a ve c o m p o n e n t a n d t h e w a l l - r e f l e c t e d w a v e s w e r e i n e v i d e n c e . T e s t s w e r e c o n d u c t e d , u s i n g w a t e r d e p t h s w h i c h r a n g e d f r o m 0 . 0 1 5 m t c 0 . 0 2 0 m. T h e wave g e n e r a t o r was p o s i t i o n e d i n t h e c e n t e r o f t h e t a n k . 187 a n d t h e w a v e d e t e c t o r s w e r e l o c a t e d a t v a r i o u s p o i n t s a l o n g t h e l o n g i t u d i n a l c e n t e r l i n e o f t h e t a n k . The wave f i e l d a r r i v i n g a t o n e i n t e r m e d i a t e p o s i t i o n , P , i s s h o w n i n f i g u r e D - 1 ( A ) . A f t e r t h e a r r i v a l c f t h e i n c i d e n t w a v e , r e f l e c t i o n s f r o m t h e s i d e w a l l s a n d t h e e n d w a l l w e r e r e c o r d e d . T h e w a v e a m p l i t u d e o f t h e w a l l - r e f l e c t e d w a v e s c a n be s e e n t o b e s i g n i f i c a n t . I n o r d e r t c r e m o v e a n y c o m p o n e n t s c o m i n g f r o m t h e t a n k w a l l s , t h e a b s o r b i n g m a t e r i a l w as i n t r o d u c e d a s a l i n e r o n t h e t a n k w a l l s . T h e f i r s t g e o m e t r y a t t e m p t e d was a f l a t s u r f a c e , p a r a l l e l t o t h e t a n k w a l l , made o f r u b b e r i z e d h o r s e h a i r . S t r i p s m e a s u r i n g t w o i n c h e s ( 5 . 1 cm) i n v e r t i c a l h e i g h t a n d 2 5 cm i n h o r i z o n t a l d e p t h w e r e p l a c e d a r o u n d t h e i n s i d e o f t h e t a n k w a l l s . T h u s , t h e t a n k d i m e n s i o n s w e r e r e d u c e d b y 50 cm i n w i d t h a n d l e n g t h , a n d t h e w a l l s w e r e t h e n made o f h o r s e h a i r . W i t h t h i s c o n f i g u r a t i o n i n p l a c e , w a v e s w e r e g e n e r a t e d a n d t h e f i e l d a r i s i n g a t P was r e c o r d e d a s s h o w n i n f i g u r e D - 1 ( B ) . C l e a r l y , t h e p l a c e m e n t o f t h e h o r s e h a i r l i n e r i n t h e t a n k h a d l i t t l e e f f e c t u p o n t h e a m p l i t u d e o f t h e w a v e s r e f l e c t i n g f r o m t h e w a l l s . T h e w a v e l e n g t h s g e n e r a t e d a p p e a r t o h a v e b e e n s u f f i c i e n t l y l o n g t h a t t h e h o r s e h a i r s t r u c t u r e a p p e a r e d a s r e f l e c t i n g a s t h e b a r e w a l l . Some d i f f e r e n c e was e v i d e n c e d i n t h e a r r i v a l t i m e o f t h e w a v e s r e f l e c t i n g f r o m t h e b a r e a n d h o r s e h a i r s i d e s , b u t t h i s d i f f e r e n c e i s a c c o u n t e d f o r b y t h e d i f f e r e n c e i n t r a v e l d i s t a n c e o f t h e t w o w a v e s . S i n c e t h e h o r s e h a i r i t s e l f d o e s n o t a d e q u a t e l y a b s o r b w a ve e n e r g y , t h e q e o m e t r y o f t h e a b s o r b e r f r o n t w as m o d i f i e d . A s a w t o o t h p a t t e r n ( a s s e e n f r o m a b o v e ) w a s c u t i n t o t h e h o r s e h a i r . T h e s e t e e t h h a d m e a s u r e d b a s e s o f 10 cm a n d h e i q h t s 1*8 1 A) Bare wal Is 1 B) Plane-front horsehair 1 D) Sawtooth- f ront and extending fingers of horsehair F i g u r e D - l . R e f l e c t i o n s o f f v a r i o u s w a l l - a b s o r b e r c o n f i g u r a t i o n s , where 1 denotes the d i r e c t wave, 2 the s i d e - w a l l r e f l e c t i o n , 3 the e n d - w a l l r e f l e c t i o n , and 4 m u l t i p l e r e f l e c t i o n s . 189 o f 20 cm, w h e r e t h e b a s e l a y p a r a l l e l t o t h e a d j a c e n t t a n k w a l l . W a v e s r e f l e c t i n g f r o m t h i s s u r f a c e s h o w e d a l a r g e r e d u c t i o n i n w a v e a m p l i t u d e . T h e f i e l d a r i s i n g a t P, f o r t h i s c o n f i g u r a t i o n i s s h o w n i n f i g u r e D-1 ( C ) . T o f u r t h e r r e d u c e s i d e - w a l l r e f l e c t i o n s , s t r i p s o f h o r s e h a i r , m e a s u r i n g 5.1 cm i n v e r t i c a l t h i c k n e s s , 3 cm i n h o r i z o n t a l w i d t h a n d 37 cm i n l e n g t h w e r e p l a c e d b e t w e e n t h e s a w t e e t h . I n t h i s t e s t , t h e c e n t e r p o r t i o n o f e a c h s i d e w a l l h a d , i n a d d i t i o n t o a s a w t o o t h p r o f i l e , a n a r r a y o f p a r a l l e l s t r i p s w h i c h l a y p e r p e n d i c u l a r t o t h e s i d e w a l l s , o r i g i n a t e d a t t h e s a w t o o t h b a s e a n d p r o j e c t e d t o w a r d t h e t a n k c e n t e r l i n e . P i t e ( 1 9 7 3 , A p p e n d i x 7) u s e d t h i s c o n f i g u r a t i o n i n e x a m i n i n g f r i c t i o n a l l y d a m p e d w a v e s . W a v e s r e f l e c t e d f r o m t h i s g e o m e t r y a n d r e c o r d e d a t P , a r e s h o w n i n f i g u r e D - 1 { D ) . T h i s c o n f i g u r a t i o n a d e q u a t e l y d a m p e d t h e w a l l - r e f l e c t e d w a v e s a n d w a s u s e d a s t h e a b s o r b i n g g e o m e t r y i n t h e e x p e r i m e n t . I n t h e e v e n t t h a t t h e w a v e s o u r c e w a s p o s i t i o n e d c l c s e t o an e n d w a l l , a s i m i l a r a r r a y o f s t r i p s was p l a c e d i n t h e s a w -t o o t h p a t t e r n a g a i n s t t h e e n d w a l l n e a r e s t t h e s o u r c e . T h i s p r o v e d s a t i s f a c t o r y i n d a m p i n g w a v e s r e f l e c t e d f r o m t h e e n d w a l l . A d i a g r a m o f t h e t a n k w i t h a b s o r b i n g m a t e r i a l a l o n g t h e w a l l s i s s h o w n i n f i g u r e D-2. 191 A P P E N D I X E. T h e Wave D e t e c t o r T h e p u r p o s e o f t h e w a v e d e t e c t o r i s t o m e a s u r e t h e i n s t a n t a n e o u s f r e e s u r f a c e e l e v a t i o n a t a p o i n t a s a c o n t i n u o u s f u n c t i o n o f t i m e . I n v e s t i g a t i o n s h o w e d t h a t a c a p a c i t a n c e p r o b e , a c t i n g a s a t r a n s d u c e r , met t h i s n e e d m o s t s a t i s f a c t o r i l y . T h e p r o b e d e v e l o p e d c o n s i s t s o f a p a i r o f i n s u l a t e d p a r a l l e l c o p p e r p l a t e s m e a s u r i n g 10 cm i n l e n g t h , 3 cm i n w i d t h a n d 0.1 cm i n t h i c k n e s s , A d r a w i n g o f t h e d e t e c t o r i s s h o w n i n f i g u r e E-1 a n d p h o t o g r a p h s s h o w i n g f r o n t a n d s i d e v i e w s o f a d e t e c t o r a r e s h o w n i n f i g u r e s E-2 a n d 1 - 3 , r e s p e c t i v e l y . The p l a t e s a r e f i x e d a t t h e i r u p p e r e n d t o a p l e x i g l a s s b l o c k w h i c h h o l d s t h e m i n a p a r a l l e l p o s i t i o n a n d s e p a r a t e s t h e m b y a d i s t a n c e o f 0.5 cm. E a c h p l a t e i s c a r e f u l l y c o a t e d w i t h a t h i n u n i f o r m l a y e r o f i n s u l a t i n g v a r n i s h a n d t e f l o n . T h i s r e s u l t s i n t h e p r o b e h a v i n g t h e d e s i r e d d i e l e c t r i c p r o p e r t i e s a n d a i d s i n r e d u c i n g m e n i s c u s e f f e c t s . A n u m b e r o f c o a t i n g c o m b i n a t i o n s w e r e t e s t e d p r i o r t o s e l e c t i n g t h e o n e u s e d . T e s t s w e r e c o n d u c t e d i n w h i c h t h e p l a t e w i d t h was v a r i e d . I n a l l o f t h e s e t e s t s , v e r y l i t t l e d i f f e r e n c e was n o t e d i n t h e d e t e c t o r r e s p o n s e s i n c e , i n e a c h c a s e , t h e p l a t e w i d t h m e a s u r e d much l e s s t h a n t h e s h o r t e s t w a v e l e n g t h o f i n t e r e s t i n t h e e x p e r i m e n t . T h e f i n a l c h o i c e o f p l a t e w i d t h a n d s e p a r a t i o n u s e d i n c o n s t r u c t i n g t h e w a v e d e t e c t o r s was b a s e d u p o n e a s e o f h a n d l i n g a n d f a b r i c a t i o n o f t h e i n s t r u m e n t . I f t h e d e t e c t o r p l a t e s a r e u n i f o r m l y c o a t e d w i t h i n s u l a t i o n a n d i f e d g e e f f e c t s a r e n e g l e c t e d , t h e c a p a c i t a n c e c f t h e d e t e c t o r i s a p p r o x i m a t e l y g i v e n b y t h e f o r m u l a H Z 10 c m 3cm-0.1cm • 0.5cm F i g u r e E - l . Wave p r o b e . 193 Figure E-2. Wave detector, front view. 1 9 3 OL 194 F i g u r e E-3. Wave d e t e c t o r , s i d e v i e w . 194 a 195 w h e r e flw i s t h e p l a t e a r e a i m m e r s e d i n w a t e r , d i s t h e d i s t a n c e c f s e p a r a t i o n o f t h e p l a t e s a n d £ w i s t h e d i e l e c t r i c c o n s t a n t o f t h e w a t e r . T h e c a p a c i t a n c e d u e t o t h a t p a r t o f t h e p l a t e w h i c h i s e x p o s e d t o t h e a i r i s n e g l e c t e d s i n c e t h e d i e l e c t r i c c o n s t a n t i n a i r i s much l e s s t h a n t h a t i n w a t e r . , T h e e n d e f f e c t s o f t h e p l a t e s c a n a l s o b e c o n s i d e r e d n e g l i g i b l e ( M c G o l d r i c k , 1 9 6 9 ) . T h u s , o n c e t h e p l a t e s a r e i n i t i a l l y i m m e r s e d t o a p r e d e t e r m i n e d d e p t h i n t h e w a t e r c o l u m n , a n y s u b s e g u e n t c h a n g e i n t h e f r e e s u r f a c e e l e v a t i o n i d e a l l y r e s u l t s i n a l i n e a r c h a n g e i n c a p a c i t a n c e . T h e w a v e d e t e c t o r t h e n i s a l i n e a r t r a n s d u c e r a n d i t i s o n l y n e c e s s a r y t o c o n v e r t t h e c h a n g e i n c a p a c i t a n c e i n t o a p r o p o r t i o n a l c h a n g e i n v o l t a g e t o d e t e r m i n e t h e c h a n g e i n wave h e i g h t . T o make t h i s c o n v e r s i o n , t h e c i r c u i t r y s h o w n i n t h e b l o c k d i a g r a m o f f i g u r e E-4 i s u s e d . T h e c h a n g e i n c a p a c i t a n c e r e g i s t e r e d b y t h e wave d e t e c t o r i s f e d i n t o a p u l s e w i d t h m o d u l a t o r , w h i c h i n t u r n i s d r i v e n by a 2 5 k h z o s c i l l a t o r . T h e r e s u l t i n g c h a n g e i n p u l s e w i d t h i s i n t e g r a t e d a n d t h e o u t p u t o f t h e i n t e g r a t o r i n p u t t o a s u m m i n g a m p l i f i e r . T h e o u t p u t o f t h e s u m m i n g a m p l i f i e r i s a v o l t a g e p r o p o r t i o n a l t o t h e c h a n g e i n c a p a c i t a n c e . T h e g a i n c f t h e s u m m i n g a m p l i f i e r c a n b e c o n t r o l l e d a n d t h e d . c , v o l t a g e l e v e l a d j u s t e d by means o f a d . c , v o l t a g e o f f s e t . T h u s , t h e p l a t e s c a n be i m m e r s e d t o a n y p r a c t i c a l d e p t h i n t h e w a t e r c o l u m n a n d t h e d . c , v o l t a g e l e v e l , c o r r e s p o n d i n g t o t h e f r e e e g u i l i b r i u m s u r f a c e a t r e s t , s e t t c z e r o . 25 khz Pulse width Osc i l l a to r Modu la to r Transducer Summat ion Amp l i f i e r O u t p u t Vo l tage P robe ' F i g u r e E-4. S c h e m a t i c o f wave d e t e c t o r v o l t a g e r o u t i n g . 6-197 I n p r a c t i c e , t h e p r o b e d o e s n o t b e h a v e a s a p u r e c a p a c i t o r s i n c e t h e d i e l e c t r i c h a s a f i n i t e l e a k a g e r e s i s t a n c e . T h u s , c a r e m u s t b e t a k e n t c e n s u r e t h a t t h e i n s u l a t i o n i s g o o d a s w e l l a s u n i f o r m , s i n c e t h e i n s u l a t i o n w i l l e v e n t u a l l y a b s o r b w a t e r , t h u s c h a n g i n g t h e l e a k a g e r e s i s t a n c e , c a r e m u s t b e t a k e n t o c a l i b r a t e t h e d e v i c e w i t h s u f f i c e n t r e g u l a r i t y t o e n s u r e t h a t t h e i n s u l a t i o n i s b e i n g m a i n t a i n e d . T h e p r o b e s u s e d i n t h e e x p e r i m e n t w e r e l e f t i m m e r s e d o n l y d u r i n g t h e c o u r s e o f a n e x p e r i m e n t o r d u r i n g c a l i b r a t i o n . T e s t s w e r e c o n d u c t e d t o d e t e r m i n e c h a n g e s c a u s e d b y p r o l o n g e d i m m e r s i o n , a n d t h e r e s u l t s s h o w e d t h a t o n l y a f t e r h a v i n g b e e n l e f t i n w a t e r f o r a m a t t e r o f d a y s d i d a b s o r p t i o n b y t h e i n s u l a t i o n e f f e c t t h e p e r f o r m a n c e o f t h e d e t e c t o r . T e s t s l a s t i n g p e r i o d s c o m p a r a b l e w i t h t h e d u r a t i o n c f a n e x p e r i m e n t s h o w e d n e g l i g i b l e e f f e c t s . D e t e c t o r C a l i b r a t i o n F o r an i d e a l c a p a c i t a n c e - t y p e d e t e c t o r , t h e c a p a c i t a n c e v a r i e s l i n e a r l y w i t h f r e e s u r f a c e e l e v a t i o n , To v e r i f y t h a t t h e i n s t r u m e n t i s i n d e e d l i n e a r r e g u i r e s t h a t a c a l i b r a t i o n c u r v e f o r t h e i n s t r u m e n t b e d e t e r m i n e d . T h i s f u n c t i o n a l r e l a t i o n s h i p b e t w e e n t h e o u t p u t v o l t a g e o f t h e d e t e c t o r a n d t h e f r e e s u r f a c e e l e v a t i o n w i l l e n a b l e u s t o a c c u r a t e l y d e t e r m i n e t h e wave s h a p e s a n d a m p l i t u d e s o b s e r v e d i n t h e e x p e r i m e n t . I t i s a l s o n e c e s s a r y t o d e t e r m i n e w h a t e f f e c t s , i f a n y , t h e a d j u s t m e n t o f d . c . l e v e l s a n d g a i n s o f t h e i n s t r u m e n t e l e c t r o n i c s h a v e u p o n t h e r e s p o n s e . When t h e d e t e c t o r was i n i t i a l l y p o s i t i o n e d , t h e p l a t e s w e r e i m m e r s e d t o some p r e s c r i b e d d e p t h i n t h e w a t e r . I t was n e c e s s a r y 198 to determine what e f f e c t , i f any, the depth of immersion had on the measurements made. L i k e w i s e , the e f f e c t s of the meniscus were examined. Two methods of c a l i b r a t i o n were used t o make these d e t e r m i n a t i o n s . S t a t i c C a l i b r a t i o n During t h i s c a l i b r a t i o n , the wave d e t e c t o r was mounted on a rod which could be v e r t i c a l l y p o s i t i o n e d i n increments of as l i t t l e as 0,001 f t , (approximately 0.30 5mm). T h i s p o s i t i o n i n g was accomplished by means of a micrometer s c a l e mechanism. With the d e t e c t o r immersed i n the water t o a predetermined depth and, with the f r e e s u r f a c e a t r e s t , the output d.c. vo l t a g e from the d e t e c t o r was l e v e l l e d t o zer o v o l t s . The d e t e c t o r was then r a i s e d and lowered through a s e r i e s of st e p s , d u r i n g which the output v o l t a g e of the d e t e c t o r was recorded by both a d i g i t a l voltmeter and a Brush r e c o r d e r . The Brush recorder provided a continuous record of the output voltage of the instrument d u r i n g the time i t was r a i s e d or lowered o r duri n g the time i t was a t r e s t . The d i g i t a l v o l t m e t e r was used only while the d e t e c t o r was at r e s t , any v a r i a t i o n i n output v o l t a g e o c c u r r i n g while the instrument was at r e s t was recorded. C a l i b r a t i o n curves f o r two of the d e t e c t o r s are shown i n f i g u r e E-5. To obtain these curves, the d e t e c t o r was immersed to a depth whereby the lower 1.5 cm of the p l a t e s were covered with water. The d e t e c t o r was then lowered through 18 equal increments of 0.71 mm, which amounted to a t o t a l lowerinq of about 1.28 cm. Hhen the lowerinq was completed, the instrument was r a i s e d i n the same incremental steps u n t i l i t was returned to i t s i n i t i a l immersion depth of 1.5 cm. The d e t e c t o r output v o l t a g e was 20 0 r e c o r d e d a t e a c h s t e p o f r a i s i n g a n d l o w e r i n g . N e x t , t h e d e t e c t o r w a s r a i s e d i n 0.71 mm i n c r e m e n t s u n t i l t h e a m o u n t o f p l a t e s t i l l c o v e r e d by w a t e r m e a s u r e d a p p r o x i m a t e l y 0.22 cm. O p e n r e a c h i n g t h i s u p p e r m o s t l e v e l , t h e d e t e c t o r was a g a i n l o w e r e d i n t h e s ame i n c r e m e n t a l c h a n g e s u n t i l i t was r e t u r n e d t o t h e s t a r t i n g i m m e r s i o n d e p t h o f 1,5 cm. The v o l t a g e s r e c o r d e d t h r o u g h o u t t h i s r a i s i n g a n d l o w e r i n g r e s u l t e d i n t h e c a l i b r a t i o n c u r v e s s h o w n i n f i g u r e E - 5 , T h e d i f f e r e n c e i n s l o p e o f t h e s e t w o c u r v e s i s a r e s u l t o f t h e d i f f e r e n c e i n t h e g a i n s e t t i n g s o f t h e t w o d e t e c t o r s . B o t h d e t e c t o r s e x h i b i t e d a r e s p o n s e w h i c h w a s s u f f i c i e n t l y l i n e a r , e s p e c i a l l y w i t h i n t h e r a n g e o f s u r f a c e e l e v a t i o n s a n d v o l t a g e s r e c o r d e d d u r i n g t h e e x p e r i m e n t . I n o r d e r t o o b t a i n a c a l i b r a t i o n c u r v e o f f i n e r r e s o l u t i o n , a c a l i b r a t i o n was c o n d u c t e d w h e r e i n m e a s u r e m e n t s w e r e made a t i n t e r v a l s o f 0 , 001 ' ( 0.3 mm), T h e r e s u l t s f r o m t h i s c a l i b r a t i o n a r e s h o w n i n f i q u r e E - 6 . D i r e c t w a v e f i e l d s g e n e r a t e d d u r i n q t h i s c a l i b r a t i o n w e r e m e a s u r e d a t d i s t a n c e s f r o m 0.5 t o 1.5 m f r c m t h e s o u r c e . T h e s e m e a s u r e m e n t s , u s i n g t h e c a l i b r a t i o n g i v e n i n f i g u r e E - 6 , s h o w e d t h a t t h e s e w a v e s , p r c p a g a t i n q i n w a t e r o f 1,28 cm d e p t h , h a d w a ve h e i g h t s o f a m i l l i m e t e r o r l e s s . Seme v a r i a t i o n i n t h e d e t e c t o r o u t p u t v o l t a q e was o b s e r v e d , d e p e n d i n g u p o n w h e t h e r t h e i n s t r u m e n t w a s r a i s e d o r l o w e r e d i n t o p o s i t i o n . T h i s r e l a t i v e d i f f e r e n c e , w h i c h a m o u n t e d t c n o m o r e t h a n 3 o r was a t t r i b u t e d m a i n l y t o t h e m e n i s c u s e f f e c t . When t h e p l a t e s w e r e m o v i n g u p w a r d , t h e m e n i s c u s a p p e a r e d • s t r e t c h e d * , w h i l e i t a p p e a r e d • c o m p r e s s e d ' when t h e p l a t e s w e r e m o v i n g d o w n w a r d . Some t r a n s i e n t b e h a v i o u r i n t h e p o s i t i o n o f t h e m e n i s c u s o n t h e p l a t e s u r f a c e was s e e n o n c e t h e p l a t e s c a m e t o F i g u r e E - 6 . F i n e r e s o l u t i o n c a l i b r a t i o n c u r v e f o r d e t e c t o r T 2 0 2 r e s t . T h i s c h a n g e r e s u l t e d i n a c h a n g e i n t h e a m o u n t o f p l a t e a r e a c o v e r e d b y w a t e r , w i t h a c o n s e g u e n t c h a n g e i n d e t e c t o r c a p a c i t a n c e a n d o u t p u t v o l t a g e . T h e t i m e s p a n d u r i n g w h i c h t h e m e n i s c u s ' s o u g h t * t o r e a c h a n e q u i l i b r i u m p o s i t i o n was much g r e a t e r t h a n t h e p e r i o d t a k e n t o make a n e x p e r i m e n t a l m e a s u r e m e n t ( o f t h e o r d e r o f s e c o n d s ) . T h i s r e s u l t e d i n t h e c o n c l u s i o n t h a t t h e v a r i a t i o n o f t h e o u t p u t v o l t a q e d u e t o t h e m o t i o n o f t h e m e n i s c u s c o u l d be d i s r e g a r d e d . C a l i b r a t i o n c u r v e s d r a w n f r c m t h e d a t a r e c o r d e d d u r i n q t h e r a i s i n q o r l o w e r i n q o f t h e d e t e c t o r p r o v e d t o b e l i n e a r . C a r e w a s t a k e n t o c o n s i s t e n t l y r e c o r d t h e v o l t a q e r e a d i n g s a t t h e same t i m e f o l l o w i n g t h e t i m e w h e n t h e p l a t e s c a m e t o r e s t . I n a l l c a s e s t h e s l o p e s o f t h e c a l i b r a t i o n c u r v e s w e r e t h e s a m e a n d c n l y t h e p o s i t i o n o f t h e c u r v e s d i f f e r e d . T h e s e c u r v e s w e r e d i s p l a c e d w i t h r e s p e c t t o o n e a n o t h e r , b u t by no m o r e t h a n 2 o r 355, a t t h e o r i g i n . To d e t e r m i n e t h e e f f e c t o f d r i f t s i n t h e q a i n c f t h e d e t e c t o r e l e c t r o n i c s , t e s t s w e r e c o n d u c t e d o v e r a n a m p l i t u d e r a n q e f a r e x c e e d i n q t h o s e e x p e r i e n c e d d u r i n q t h e e x p e r i m e n t . C h a n q e s i n i n s t r u m e n t g a i n r e s u l t e d i n a c h a n g e s i n t h e s l o p e s o f t h e c a l i b r a t i o n c u r v e s b u t h a d n o e f f e c t o n t h e l i n e a r i t y o f t h e c u r v e s . O f c o u r s e , w h e n e v e r i t was n e c e s s a r y t o c h a n g e t h e i n t e r n a l g a i n o f an i n s t r u m e n t d u r i n g t h e e x p e r i m e n t , i t was l i k e w i s e n e c e s s a r y t o r e c a l i b r a t e t h e i n s t r u m e n t . T h i s g a v e a s s u r a n c e t h a t a g u a n t i t a t i v e m e a s u r e o f t h e f r e e s u r f a c e e l e v a t i o n was m a i n t a i n e d . C h a n g e s i n t h e d . c v o l t a g e l e v e l s s e r v e d t o s h i f t t h e c a l i b r a t i o n c u r v e a l o n g t h e v o l t a g e o u t p u t a x i s , b u t d i d n o t 203 a p p e a r t o a l t e r t h e l i n e a r i t y o r o v e r a l l r e s p o n s e o f t h e i n s t r u m e n t . S i m i l a r l y , c h a n g e s i n t h e a m o u n t o f p l a t e a r e a i n i t i a l l y s u b m e r g e d i n t h e w a t e r r e s u l t e d o n l y i n s h i f t i n g t h e c a l i b r a t i o n c u r v e a l o n g t h e d e t e c t o r v o l t a g e a x i s . B y a d j u s t i n g t h e d . c . v o l t a g e l e v e l o f t h e d e t e c t o r e l e c t r o n i c s , a n y s h i f t s r e s u l t i n g f r o m o v e r o r u n d e r i m m e r s i o n c o u l d b e c o m p e n s a t e d f o r ; t h e r e s u l t was t o g e t z e r o o u t p u t v o l t a g e f r o m t h e ' d e t e c t o r f o r t h e f r e e s u r f a c e a t r e s t . G r e a t c a r e was t a k e n t o a s s u r e t h a t t h e p l a t e s w e r e i m m e r s e d t o t h e s a m e d e p t h e a c h t i m e a m e a s u r e m e n t w a s made. S m a l l a d j u s t m e n t s o f t h e d . c . l e v e l w e r e made t o c o m p e n s a t e f o r e r r o r s i n p o s i t i o n i n g . T h e e f f e c t o f t e m p e r a t u r e c h a n g e o f t h e w a t e r u p o n t h e i n s t r u m e n t r e s p o n s e w a s a l s o d e t e r m i n e d . I t was f o u n d t h a t , w i t h i n t h e t e m p e r a t u r e r a n g e e x p e r i e n c e d d u r i n g a n y i n d i v i d u a l e x p e r i m e n t , t h e s e e f f e c t s w e r e n e g l i g i b l e , , C a r e w a s t a k e n t o a s s u r e t h a t t h e d a y t o d a y s t a r t i n g w a t e r t e m p e r a t u r e was t h e s a n e . D _ y j n a j i c C a l i b r a t i o n U n d e r e x p e r i m e n t a l c o n d i t i o n s , t h e f r e e s u r f a c e e l e v a t i o n v a r i e s c o n t i n u o u s l y d u e t o t h e w a v e m o t i o n g e n e r a t e d b y t h e s o u r c e . T o d e t e r m i n e t h e r e s p o n s e o f t h e d e t e c t o r t c t h e s e f l u c t u a t i o n s , a d y n a m i c c a l i b r a t i o n w a s p e r f o r m e d , d u r i n g w h i c h t h e i n s t r u m e n t p l a t e s w e r e m o v e d i n a n d o u t o f t h e w a t e r . T h e m o t i o n i m p a r t e d t o t h e d e t e c t o r was d e s i g n e d t o r e s e m b l e t h e m o t i o n t h a t t h e d e t e c t o r e x p e r i e n c e d d u e t o t h e w a v e m o t i o n i n t h e e x p e r i m e n t . T h e w a v e p e r i o d s a n d a m p l i t u d e s u s e d i n t h e c a l i b r a t i o n c o v e r e d t h e r a n g e o f t h o s e o b s e r v e d i n t h e e x p e r i m e n t . 204 To c o n d u c t t h i s c a l i b r a t i o n , a n a p p a r a t u s was c o n s t r u c t e d w h i c h r a i s e d a n d l o w e r e d t h e d e t e c t o r , w h i l e a l l o w i n g t h e p e r i o d o f o s c i l l a t i o n a n d t h e w a v e a m p l i t u d e t o be v a r i e d . T h e d e v i c e c o n s i s t e d o f a v a r i a b l e s p e e d e l e c t r i c m o t o r ( f i g u r e E-7) w h i c h h a d a c i r c u l a r d i s k m o u n t e d c o n c e n t r i c a l l y o n t h e m o t o r s h a f t . T h e d i s k h a d a n u m b e r o f h o l e s d r i l l e d i n t o i t a t v a r i o u s p o s i t i o n s a l o n g t h e d i s k r a d i u s . T h e s e h o l e s s e r v e d a s p o i n t s o f a t t a c h m e n t f o r an a r m w h i c h i n t u r n a t t a c h e d w i t h a g u i d e d v e r t i c a l r o d w h i c h h e l d t h e d e t e c t o r . T h e u p p e r e n d o f t h e arm was a t t a c h e d t o t h e d i s k a t t h e a p p r o p r i a t e p o s i t i o n , w h i l e t h e l o w e r e n d was a t t a c h e d t o t h e u p p e r e n d o f t h e g u i d e d r o d . T h e d e t e c t o r w a s a t t a c h e d a t t h e l o w e r e n d o f t h e r o d . T h u s , a s t h e m o t o r t u r n e d , t h e r o t a t i n g d i s k m o v e d t h e u p p e r e n d o f t h e a r m i n a c i r c l e i n t h e p l a n e o f t h e v e r t i c a l . T h i s m o t i o n was t r a n s m i t t e d t o t h e g u i d e d r o d i n t h e f o r m o f v e r t i c a l r e c t i l i n e a r t r a n s l a t i o n . T h e a m p l i t u d e o f t h e o s c i l l a t i o n was c o n t r o l l e d b y a t t a c h i n g t h e a r m t o p o s i t i o n s o f d i f f e r e n t r a d i a l d i s t a n c e f r o m t h e c e n t e r o f t h e d i s k , T h e p e r i o d o f o s c i l l a t i o n was c o n t r o l l e d b y c h a n g i n g t h e m o t o r s p e e d b y m e a n s o f a r h e o s t a t w h i c h r e g u l a t e d t h e d . c , v o l t a g e a p p l i e d t o t h e m o t o r . T h e a p p a r a t u s w a s p o s i t i o n e d o v e r a r e s e r v o i r o f w a t e r a n d t h e p o s i t i o n o f t h e r e s e r v o i r a d j u s t e d s o t h a t t h e p l a t e s w e r e i m m e r s e d t o t h e d e s i r e d l e v e l i n t h e w a t e r . T h e o u t p u t v o l t a g e o f t h e d e t e c t o r i s a m e a s u r e o f c a p a c i t a n c e , w h i c h d e p e n d s o n t h e f r e e s u r f a c e e l e v a t i o n . T h i s m e a s u r e m e n t , o f c o u r s e , i n c l u d e s s u c h f a c t o r s a s t h e m e n i s c u s e f f e c t . I t was n e c e s s a r y t h a t we a l s o d e t e r m i n e , c o n c u r r e n t l y D.C. M O T O R RHEOSTAT D E T E C T O R F R E E S U R F A C E F i g u r e E - 7 . D y n a m i c c a l i b r a t i o n a p p a r a t u s . 2 0 6 and independently, the absolute mechanical p o s i t i o n c f the d e t e c t o r i n order t h a t a comparison c o u l d be made between the t r u e p o s i t i o n of the water s u r f a c e and the p o s i t i o n as seen by the wave d e t e c t o r . To measure the p o s i t i o n of the d e t e c t o r with r e s p e c t to the water s u r f a c e , a 10 K Armaco potentiometer having a t r a n s l a t i o n a l type c o n t r o l was attached to the s l i d e mechanism guide. The c o n t r o l l e v e l was attached d i r e c t l y to the rod which held the d e t e c t o r . The potentiometer used was a C-1 Taper type f o r which the displacement of the c o n t r o l r e s u l t e d i n a d i r e c t l y p r o p o r t i o n a l change i n the output voltage.,As the d e t e c t o r was moved up and down, the same motion was imparted to the c o n t r o l l e v e r and the v o l t a g e out of the potentiometer was a d i r e c t measure of the d e t e c t o r p o s i t i o n . S i n c e we d e s i r e d t c r e c o r d t h i s s i g n a l on the Brush r e c o r d e r , i t was i n p u t i n t o a v o l t a g e f o l l o w e r i n order t o o b t a i n a high impedence l o o k i n g i n t o the r e c o r d e r . A schematic diagram o f the potentiometer c i r c u i t i s i n c l u d e d i n f i g u r e E-7. The d.c. l e v e l <pen p o s i t i o n ) c o n t r o l on the Brush r e c o r d e r was used to adjust the potentiometer v o l t a g e , corresponding to the e g u i l i b r i u m f r e e s u r f a c e , to zero v o l t s . Thus, the Brush r e c o r d e r recorded zero v o l t s from the potentiometer at the same time as i t recorded z e r o v o l t s frcm the wave d e t e c t o r . The gain adjustments on the Brush r e c o r d e r were used to e g u a l i z e the voltage maxima r e c e i v e d from the potentiometer and the wave de t e c t o r . Thus, a d i r e c t comparison between the s i g n a l s r e c e i v e d from the wave d e t e c t o r and the potentiometer were made and any d i f f e r e n c e s noted. In the t e s t s performed, wave p e r i o d s ranging from 0.25 seconds t o 2.0 seconds were examined. The wave amplitudes were 2 0 7 v a r i e d f r o m 1 mm t o 2 cm. T h i s c o v e r e d t h e r a n q e o f w a v e s o b s e r v e d i n t h e e x p e r i m e n t . T h e r e s u l t s o f o n e s u c h t e s t a r e s h e w n i n f i g u r e E - 8 . F o r t h i s c a l i b r a t i o n , t h e m o t o r was r u n a t o n e r e v o l u t i o n p e r s e c o n d a n d t h e a m p l i t u d e w a s s e t a t 6 mm. T h e c u r v e r e p r e s e n t i n g t h e a c t u a l d i s p l a c e m e n t o f t h e d e t e c t o r , w i t h r e s p e c t t o t h e e g u i l i b r i u m s u r f a c e , a l o n g w i t h t h e c u r v e s h o w i n g t h e c h a n g e i n v o l t a g e d u e t o t h e c h a n g e i n c a p a c i t a n c e c f t h e d e t e c t o r , a r e s h o w n . S u p e r i m p o s e d o n t h e s e c u r v e s a r e p o i n t s r e s u l t i n g f r o m n o r m a l i z e d d a t a o b t a i n e d b y s t a t i c c a l i b r a t i o n . T h e r a n g e o v e r w h i c h t h e p r o b e was r a i s e d o r l o w e r e d i s i n d i c a t e d i n f i g u r e E - 8 . T h e s e d a t a a r e i n r e a s o n a b l e a g r e e m e n t , w i t h s eme d i f f e r e n c e i n a m p l i t u d e a n d p h a s e i n e v i d e n c e . T h e s e d i f f e r e n c e s , w h i c h a r e a t t r i b u t a b l e m a i n l y t c t h e m e n i s c u s , a r e s m a l l ; t h e d i f f e r e n c e i n a m p l i t u d e s m e a s u r e a b o u t -i% a t m o s t , w h i l e t h e p h a s e s h i f t a t t h e o r i g i n i s a b o u t 5%, 108 x S t a t i c ca l i b ra t i on D e t e c t o r output Displacement P o t e n t i o m e t e r output F i g u r e E - 8 . D y n a m i c c a l i b r a t i o n d e t e c t o r r e s p o n s e . 209 A P P E N D I X F. E l e c t r o n i c C i r c u i t r y D u r i n g t h e e x p e r i m e n t , w a v e s w e r e m e a s u r e d s i m u l t a n e o u s l y a t t h r e e l o c a t i o n s i n t h e w a v e t a n k . T h e o u t p u t v o l t a g e s f r o m t i e i n d i v i d u a l d e t e c t o r s w e r e r o u t e d t h r o u g h t h e t w o n e t w o r k s s h o w n i n f i g u r e F - 1 , T h e s i g n a l f r o m t h e s t a t i o n a r y d e t e c t o r , w h i c h was u s e d a s a m o n i t o r o r c o n t r o l , was i n p u t t o t h e c i r c u i t s h e w n i n p a r t (a) o f f i g u r e F - 1 . T h e s i g n a l s f r o m t h e t w o m o v e a b l e d e t e c t o r s , u s e d t o m e a s u r e t h e wave f i e l d a t v a r i o u s p o s i t i o n s t h r o u g h o u t t h e t a n k , w e r e i n p u t t o t h e c i r c u i t s h o w n i n p a r t (b) o f f i g u r e F - 1 . A p h o t o g r a p h o f t h e a m p l i f i e r a n d c o n t r o l c i r c u i t s i s s e e n i n f i g u r e F - 2 . I n p a r t (a) o f f i g u r e F - 1 , t h e o u t p u t v o l t a g e o f d e t e c t o r T 3 , e 3 , i s i n p u t t o t h e o p e r a t i o n a l a m p l i f i e r A 3 , w h i c h h a s a g a i n o f o n e . T h e d . c . l e v e l o f t h i s a m p l i f i e r c a n be f i n e l y a d j u s t e d b y m e a n s o f t h e d . c . l e v e l l i n g r e s i s t o r , R 3 . T h u s , f o r t h e f r e e s u r f a c e a t r e s t , t h e o u t p u t v o l t a g e o f h % c a n be s e t t o z e r o v o l t s . T h e s i g n a l s e e n a t t h e p o i n t P ^ i n t h e c i r c u i t i s t h u s t h e i n v e r t e d o u t p u t s i g n a l o f d e t e c t o r T ^ , a d j u s t e d s o t h a t z e r o v o l t s i n d i c a t e s t h e f r e e s u r f a c e a t r e s t . I n p a r t ( b) o f f i g u r e F - 1 , t h e o u t p u t v o l t a g e f r o m d e t e c t o r T , , e v , i s i n p u t t o a m p l i f i e r A , . T h i s a m p l i f i e r h a s a n a d j u s t a b l e g a i n , K, t h u s , t h e o u t p u t v o l t a g e o f A , , when p r o p e r l y d . c . l e v e l l e d by m e a n s o f r e s i s t o r fi^ , i s - K e ( v o l t s . T h e o u t p u t v o l t a g e o f d e t e c t o r T 1 , e J _ , i s i n p u t t o a m p l i f i e r A t , w h i c h h a s a g a i n o f u n i t y a n d c a n be d . c . l e v e l l e d w i t h r e s i s t o r B ^ . T h e o u t p u t o f A i s - e 7 > v o l t s . I t w a s n e c e s s a r y a t o n e p o i n t i n t h e e x p e r i m e n t t o d i f f e r e n c e t h e s i g n a l s f r o m d e t e c t o r s T, a n d T^, s o t h e s i g n a l f r o m a m p l i f i e r A z i s i n v e r t e d u s i n q e 3 A o — V \ A R. C A A A A -R 1 1 -12 v — v v R K/v-+12v A, 3 o P. 3 R1 =10KJ7 R2 = 5 6 K A R 3 - - 3 . 3 K X I K =0-1 Kfl C =.22jJ a. .Detec to r c i rcu i t (contro l ) . C - 1 2 v — A / N K V ^ — +12v 2^ O A/ty-R1 -12v ^W>A—+12v R^ C 4 f A A A A b.Detector circuit (movable) F i g u r e F - l . D e t e c t o r a m p l i f i e r e l e c t r o n i c s a n d s i g n a l r o u t i n g . g u r e F-2. A m p l i f i e r n e t w o r k . 211a. 2 1 2 a m p l i f i e r A ^ , w h i c h h a s a g a i n o f o n e . T o a c c o m p l i s h t h i s d i f f e r e n c i n g , t h e o u t p u t s o f a m p l i f i e r s fl, a n d A^ a r e i n p u t t o a m p l i f i e r A, z , w h i c h a l s o h a s a g a i n o f o n e . T h u s , t h e o u t p u t o f a m p l i f i e r A l 2_, s e e n a t p c i n t P(^_ i n p a r t (b) o f f i g u r e F-1, i s K e j - e ^ v o l t s . T h e p u r p o s e o f t h i s d i f f e r e n c i n g i s t o e x t r a c t s i g n a l s r e p r e s e n t i n g c e r t a i n c o m p o n e n t s o f t h e wave f i e l d f r o m t h e s i g n a l r e p r e s e n t i n g t h e t o t a l f i e l d . B y p o s i t i o n i n g d e t e c t o r s T, a n d Tj_ s o t h a t t h e y w e r e e g u i d i s t a n t f r c m t h e s o u r c e , t h e d i r e c t f i e l d s e e n b y e a c h d e t e c t o r was t h e s a m e . T h e a d j u s t a b l e g a i n o n a m p l i f i e r A, a l l o w s f o r a n y n e c e s s a r y a d j u s t m e n t t c a s s u r e t h a t t h e s e s i g n a l s a r e i n d e e d e g u a l . S m a l l v a r i a t i o n s i n p o s i t i o n a s w e l l a s s l i g h t v a r i a t i o n s i n d e p t h c a n c a u s e t h e t w o s i g n a l s t o d i f f e r s l i g h t l y f r c m o n e a n o t h e r , e i t h e r i n a m p l i t u d e o r a r r i v a l t i m e . O nce t h e s e t w o s i g n a l s a r e s u f f i c i e n t l y a l i k e , t h e g e o m e t r i e s o f t h e s h e l f a n d n o r - s h e l f r e g i o n s c a n b e a d j u s t e d s o t h a t d e t e c t o r T> w i l l r e c o r d t h e t o t a l w a v e f i e l d , d e t e c t o r T ^ w i l l r e c o r d t h e i n c i d e n t wave c o m p o n e n t , a n d t h e d i f f e r e n c e w i l l r e p r e s e n t t h e r e f l e c t e d p l u s l a t e r a l w a v e c o m p o n e n t s w h i c h a r i s e f r o m t h e s h e l f e d g e . T h e s e t y p e c f m e a s u r e m e n t s c a n b e made a t a n y a n d a l l p o s i t i o n s w i t h i n t h e t a n k . E a c h d a y d u r i n g t h e e x p e r i m e n t a l p e r i o d , t h e v a r i o u s o p e r a t i o n a l a m p l i f i e r s u s e d i n t h e a m p l i f i e r n e t w o r k w e r e c h e c k e d a n d a d j u s t e d b y a p p l y i n g a c a l i b r a t i o n s i g n a l f r o m a s i g n a l g e n e r a t o r . T h i s c h e c k a n d a d j u s t m e n t , i f n e e d e d , a s s u r e d t h a t t h e g a i n o f e a c h o p e r a t i o n a l a m p l i f i e r r e m a i n e d o n e . T h e l i n e a r i t y o f t h e v a r i o u s a m p l i f i e r s w a s a l s o c h e c k e d . T h e o u t p u t v o l t a g e s f r o m v a r i o u s p o i n t s i n t h e c i r c u i t w e r e 2 1 3 i n p u t i n t o a s i x - c h a n n e l C l e v i t e - B r u s h H a r k 2 6 0 r e c o r d e r . T h u s , s i g n a l s f r o m a l l t h r e e wave d e t e c t o r s , Ty , T a n d T^ w e r e r e c o r d e d o n i n d i v i d u a l c h a n n e l s , w h i l e t h e d i f f e r e n c e b e t w e e n s i g n a l s s e e n b y d e t e c t o r s T, a n d T ^ was r e c o r d e d o n a n o t h e r c h a n n e l . T h i s r e c o r d p r o v i d e s a g r a p h i c a l r e c o r d o f t h e t o t a l w a v e f i e l d , t h e d i r e c t wave a n d t h e c o m b i n a t i o n o f t h e r e f l e c t e d a n d l a t e r a l w a v e s . Two s e t s o f s u c h r e c o r d i n g s a r e s h o w n i n f i g u r e F - 3 . T h e B r u s h r e c o r d e r was r e g u l a r l y c a l i b r a t e d w i t h a s i g n a l g e n e r a t o r t o d e t e r m i n e t h e l i n e a r i t y a n d g a i n o f t h e i n d i v i d u a l c h a n n e l s . T h e i n d i v i d u a l c h a n n e l g a i n s ( s e n s i t i v i t y s c a l e s ) w e r e a d j u s t e d t o u n i t y , a n d t h e d . c . l e v e l s ( p e n p o s i t i o n s ) s e t t o r e c o r d z e r o o u t p u t f o r a n u l i n p u t . A p h o t o g r a p h o f t h e r e c o r d e r , a l o n g w i t h t h e e l e c t r o n i c c o n t r o l c i r c u i t , i s s h o w n i n f i g u r e F - 4 . R = 1.1 m R = 1.0 m F i g u r e F - 3 . Wave t r a c e s o f t h e t o t a l and d i r e c t w a v e f i e l d s w i t h t h e d i f f e r e n c e b e t w e e n t h e t w o f o u n d e l e c t r o n i c a l l y . gure F-4. E l e c t r o n i c s and recorder table. 2/5"CL 216 APPENDIX G. Data Anaysis The experimental data was recorded i n g r a p h i c a l form on c h a r t r e c ords produced by using a C l e v i t e - B r u s h r e c o r d e r . The graph i s a r e c o r d of output v o l t a g e from the d e t e c t o r a m p l i f i e r network shown as a f u n c t i o n of the wave a r r i v a l - t i m e , or d i s t a n c e of the point of o b s e r v a t i o n from the source, s i n c e these two q u a n t i t i e s are l i n e a r l y r e l a t e d . A l l the experimental data was recorded i n t h i s form. C e r t a i n f i e l d p r o p e r t i e s , such as phase speed, wave amplitude and wave shape can be determined d i r e c t l y frcm the graph, but other p r o p e r t i e s , such as the speed of energy propagation, energy f l u x and energy d i s p e r s i o n cannot. To determine these, a p p r o p r i a t e numerical procedures must be a p p l i e d . I t i s d i f f i c u l t to determine the f i e l d a r i s i n g from the i n t e r a c t i o n of the d i r e c t wave with the s h e l f edge, i . e . , the d i f f e r e n c e between the t o t a l f i e l d on the s h e l f and the d i r e c t wave, without the a i d o f mechanical or e l e c t r i c a l means. In some cases, t h i s f i e l d was found e l e c t r o n i c a l l y by d i f f e r e n c i n g the s i g n a l s coming d i r e c t l y from the wave d e t e c t o r s , but when t h i s was not p o s s i b l e the data recorded f o r the t o t a l f i e l d and d i r e c t f i e l d had t o be d i f f e r e n c e d p o i n t by p o i n t . To perform a l l of these v a r i o u s c a l c u l a t i o n s , a s m a l l computer system was used. The system used i s composed of a Hewlett-Packard (HP) model 9820A c a l c u l a t o r used i n c o n j u n c t i o n with an HP model 9864A d i g i t i z e r and an HP model 9862A p l o t t e r . The graphs on the data c h a r t r e c o r d s are d i g i t i z e d and the data s t o r e d i n the c a l c u l a t o r memory. The d i g i t i z e r i s capable of sampling a t i n t e r v a l s of 0,01 i n c h . The storage c a p a c i t y of the c a l c u l a t o r 2 1 7 memory i s s u f f i c i e n t t h a t t h e d a t a f r o m s e v e r a l t r a c e s c a n b e s t o r e d c o n c u r r e n t l y . T h e c a l c u l a t o r , w h i c h i s p r o g r a m m a b l e , t h e n p e r f o r m s t h e p r e s c r i b e d n u m e r i c a l o p e r a t i o n s a n d o u t p u t s t h e r e s u l t s e i t h e r i n t h e f o r m o f n u m e r i c a l r e s u l t s o n p a p e r t a p e o r i n g r a p h i c a l f o r m b y m e a n s o f t h e p l o t t e r . S u c h r e s u l t s a s wave s p e e d a n d w a v e e n e r g y p e r u n i t a r e a a r e p r o d u c e d i n n u m e r i c a l f o r m . Wave f o r m s r e s u l t i n g f r o m t h e d i f f e r e n c i n g o f t w o g r a p h s a r e o u t p u t g r a p h i c a l l y . S t a n d a r d s m o o t h i n g t e c h n i g u e s a r e a p p l i e d t o t h e c a l c u l a t e d r e s u l t s p r i o r t o p l o t t i n g . A a e a s u r e c f r e p r o d u c i b i l i t y was o b t a i n e d b y c o m p a r i n g a c u r v e w h i c h h a d b e e n p r o c e s s e d ( d i g i t i z e d , s m o o t h e d a n d p l o t t e d ) w i t h t h e o r i g i n a l c u r v e . I n a l l c a s e s r e p r o d u c i b i l i t y was w i t h i n t h e l i m i t s o f e r r o r i m p o s e d b y t h e d i g i t i z a t i o n p r o c e s s ( a t m o s t a 3 t o 4% ) . A n e x a m p l e o f p l o t s o f t h e t o t a l f i e l d , t h e d i r e c t f i e l d a n d t h e d i f f e r e n c e o f t h e t w o , f o r an o b s e r v a t i o n p o i n t l o c a t e d o n t h e s h e l f , i s s h o w n i n f i g u r e G-1,, I n o r d e r t o m e a s u r e t h e v a r i a t i o n o f w a v e e n e r g y p e r u n i t a r e a w i t h d i s t a n c e f r o m t h e s o u r c e , r , t h e wave p o t e n t i a l e n e r g y p e r u n i t a r e a a t e a c h p o s i t i o n r i s c a l c u l a t e d f r c m t h e e x p r e s s i o n w h e r e 1|(r) i s t h e w a v e a m p l i t u d e a t d i s t a n c e r , a n d r 0 a n d r ^ a r e t h e p o s i t i o n s o f t h e l e a d i n g e d g e a n d t a i l o f t h e w a v e p a c k e t , r e s p e c t i v e l y ( s e e f i g u r e G - 3 ) . O u t s i d e t h e s e l i m i t s , t h e G. 1 o f0 1* T 1 1 r F i g u r e G - 1 . D i g i t i z e d f o r m s o f t h e t o t a l a n d d i r e c t w a v e f i e l d s w i t h t h e i r c a l c u l a t e d d i f f e r e n c e . 2.(9 wave amplitude i s e s s e n t i a l l y zero. The geometry used with equation G. 1 i s shown i n f i g u r e G -2 . I t i s a l s o cf i n t e r e s t to determine the f i r s t moment or c e n t r o i d of the wave packet p o t e n t i a l energy. T h i s i s found from the e x p r e s s i o n r0  y - _ G . 2 f \r)Ar tLG Using a c o n s i s t a n t means of determining a r e p r e s e n t a t i v e a r r i v a l - t i m e of the wave packet, t , the energy propagation speed i s given by In a d d i t i o n to f i n d i n g the wave p o t e n t i a l energy per u n i t area and the c e n t r o i d o f the p o t e n t i a l energy f o r the whole wave packet, these q u a n t i t i e s are determined f o r var i o u s p o r t i o n s o f the wave packet. The p o t e n t i a l energy per un i t area f o r any p o r t i o n of the wave packet i s found from E : * fj G . 4 cU <L<9-where t h e l i m i t s r ; and r • , can be s e l e c t e d from the range 220 Z X F i g u r e G - 2 . W a v e f r o n t geometry f o r c y l i n d r i c a l waves . F i g u r e G-3. P r o f i l e o f t he wave p a c k e t . 7 0 222 i=C,1,,,.,n as shown i n f i q u r e G-3, Likew i s e , the c e n t r o i d of the enerqy i s found f o r any part of the packet frcm G.5 Frcm these c a l c u l a t i o n s i s found what f r a c t i o n cf the enerqy of the wave packet i s i n the va r i o u s p o r t i o n s of the packet. T h i s i n f o r m a t i o n provides a aeasure of enerqy d i s p e r s i o n . 

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