T S U N A M I S - T H E PROPAGATION O F
LONG WAVES O N T O A S H E L F
b y
D e r e k G a r a r d G o r i n g
W. M. Keck Laboratory of Hydraulics and Water Resources
Division of Engineering and Applied Science
C A L I F O R N I A I N S T I T U T E O F T E C H N O L O G Y
Pasadena, California
by
L A N D B O U W H O G E S C H O O L Bibliotheek Cultuurtechniek, Hydraulica en Afvosrhydrologie, en Weg,- en Waterbouwkunde en Irrigatie
Nieuwe Kanaal 11, Wageningen
Derek Garard G o r i n g P r o j e c t S u p e r v i s o r : F r e d r i c R a i c h l e n P r o f e s s o r o f C i v i l E n g i n e e r i n g S u p p o r t e d by N a t i o n a l Science F o u n d a t i o n
Grant Numbers ENV72-03587 and ENV77-20499 The New Zealand M i n i s t r y o f Works and Development
M. Keck L a b o r a t o r y o f H y d r a u l i c s and Water Resources D i v i s i o n o f E n g i n e e r i n g and A p p l i e d S c i e n c e C a l i f o r n i a I n s t i t u t e o f Technology Pasadena, C a l i f o r n i a KH-R-38 ( V 0 B I B L I O T H E E K DKK LANDBOUWHOGKSCHOOL WAGENINGEN November 1978 C E N T R A L E L A N D S O U W C A T A L O Q U S OOOO 0814 7635
ACKNOWLEDGMENTS
S e v e r a l p e o p l e a s s i s t e d i n t h e e x e c u t i o n o f t h i s i n v e s t i g a t i o n and i t i s w i t h s i n c e r e g r a t i t u d e t h a t t h e w r i t e r acknowledges t h e i r h e l p h e r e .
P r o f e s s o r F r e d r i c R a i c h l e n , my t h e s i s a d v i s o r , g e n e r o u s l y p r o -v i d e d g u i d a n c e , encouragement and a s s i s t a n c e i n a l l aspects o f t h e p r o j e c t .
P r o f e s s o r Thomas J . R. Hughes gave a d v i c e and encouragement i n t h e development o f t h e f i n i t e element program. D i s c u s s i o n s w i t h Dr. R o b e r t C. Y. Koh were o f g r e a t h e l p i n t h e development o f many o f t h e n u m e r i c a l t e c h n i q u e s used i n t h e a n a l y s i s and d a t a r e d u c t i o n . H i s program, MAGIC, was used e x t e n s i v e l y , e s p e c i a l l y f o r p l o t t i n g many o f t h e f i g u r e s . F e l l o w s t u d e n t T h i e r r y L e p e l l e t i e r o f t e n a c t e d as a s o u n d i n g board and h i s a d v i c e was h e l p f u l .
Mr. E l t o n F. D a l y , s u p e r v i s o r o f t h e shop and l a b o r a t o r y , gave i n v a l u a b l e a s s i s t a n c e i n a l l a s p e c t s o f t h e d e s i g n , c o n s t r u c t i o n and maintenance o f l a b o r a t o r y equipment w h i c h made t h e e x p e r i m e n t a l phase o f t h i s p r o j e c t a p l e a s u r e . Mr. Joseph J . Fontana and Mr. R i c h a r d E a s t v e d t c o n s t r u c t e d t h e l a b o r a t o r y equipment; Mr. D a v i d Byrum a s s i s t e d w i t h t h e e x p e r i m e n t s and d r a f t e d t h e f i g u r e s ; Mr. P e t e r Chang and Miss E l l a Wong a s s i s t e d w i t h t h e e x p e r i m e n t s and w i t h d a t a r e d u c t i o n ; Mrs. A d e l a i d e R. Massengale t y p e d t h e m a n u s c r i p t .
My w i f e , T r i s h , and c h i l d r e n , Sonia and Todd, s u p p o r t e d and h e l p e d me w i t h t h e i r p a t i e n c e and l o v e .
The r e s e a r c h was s u p p o r t e d by NSF Grant Nos. ENV72-03587 and ENV77-20499. The New Zealand M i n i s t r y o f Works and Development g e n e r o u s l y g r a n t e d t h e w r i t e r l e a v e on f u l l pay w i t h a l l o w a n c e s f o r t h e e n t i r e p e r i o d o f s t u d y . E x p e r i m e n t s were conducted a t t h e W. M. Keck L a b o r a t o r y o f H y d r a u l i c s and Water Resources.
T h i s r e p o r t i s e s s e n t i a l l y t h e t h e s i s o f t h e same t i t l e sub-m i t t e d by t h e w r i t e r on 17 Novesub-mber 1978 t o t h e C a l i f o r n i a I n s t i t u t e
o f Technology i n p a r t i a l f u l f i l 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 t h e degree o f D o c t o r o f P h i l o s o p h y i n C i v i l E n g i n e e r i n g .
iv
ABSTRACT
The various aspects of the propagation of long waves onto a shelf (i.e., reflection, .transmission and propagation on the shelf)
~
are examined experimentally and theoretically. The results are applied to tsunamis propagating onto the continental shelf.
A numerical method of solving the one-dimensional Boussinesq equations for constant depth using finite element techniques is presented. The method is extended to the case of an arbitrary variation in depth (i.e., gradually to abruptly varying depth) in the direction of wave propagation. The scheme is applied to the propagation of solitary waves over a slope onto a shelf and is confirmed by experiments.
A theory is developed for the generation in the laboratory of long waves of permanent form, i.e., solitary and cnoidal waves. The theory, which incorporates the nonlinear aspects of the problem, applies to wave generators which consist of a vertical plate which moves horizontally. Experiments have been conducted and the results agree well with the generation theory, In addition, these results are used to compare the shape, celerity and damping characteristics of the generated waves with the long wave theories.
The solution of the linear nondispersive theory for harmonic waves of a single frequency propagating over a slope onto a shelf is extended to the case of solitary waves. Comparisons of this analysis with the nonlinear dispersive theory and experiments are presented.
Comparisons of experiments with solitary and cnoidal waves with the predictions of the various theories indicate that, apart from propagation, the reflection of waves from a change in depth is a linear process except in extreme cases. However, the transmission and the propagation of both the transmitted and the reflected waves in general are nonlinear processes. Exceptions are waves with heights which are very small compared to the depth. For these waves, the
entire process of propagation onto a shelf in the vicinity of the shelf is linear. Tsunamis propagating from the deep ocean onto the continental shelf probably fall in this class.
v i Chapter 1 2 3 TABLE OF CONTENTS INTRODUCTION 1.1 O b j e c t i v e s and Scope LITERATURE SURVEY THEORETICAL ANALYSIS 3.1 O u t l i n e D e r i v a t i o n o f t h e Long Wave E q u a t i o n s and Exact S o l u t i o n s 3.1.1 The S o l i t a r y Wave 3.1.2 C n o i d a l Waves 3.2 Wave G e n e r a t i o n Page 1 2 4 9 1 1 25 28 32 3.2.1 The D e r i v a t i o n o f a G e n e r a t i o n E q u a t i o n f o r Long Waves 3.2.1.1 The G e n e r a t i o n o f S o l i t a r y Waves 3.2.1.2 The G e n e r a t i o n o f C n o i d a l Waves 3.3 The P r o p a g a t i o n o f Long Waves o n t o a S h e l f by t h e
N o n l i n e a r D i s p e r s i v e Theory 32 37 40 43 3.3.1 The N u m e r i c a l S o l u t i o n o f t h e Boussinesq J E q u a t i o n s f o r Constant Depth by a F i n i t e Element Method i. 3.3.1.1 A n a l y t i c a l F o r m u l a t i o n o f t h e 44 Problem 44 3.3.1.2^ F i n i t e Element F o r m u l a t i o n 47 3.3.1.Ï The Time I n t e g r a t i o n A l g o r i t h m 3.3.1.4 The I t e r a t i v e Scheme 52 3.3.1.5 Convergence and A c c u r a c y 53 3.3.2 E x t e n s i o n t o t h e Case o f V a r i a b l e D e p t h 65 The P r o p a g a t i o n o f Long Waves Onto a S h e l f by
69 t h e L i n e a r N o n d l s p e r s l v e Theory 69 The P r o p a g a t i o n o f Long Waves t o I n f i n i t y by t h e
80 N o n l i n e a r D i s p e r s i v e Theory 80 3.5.1 Summary o f t h e I n v e r s e S c a t t e r i n g Theory 8 1
Chapter 3.5.2 The A n a l y t i c S o l u t i o n f o r a Wave w i t h sech^ Shape 3.5.3 N u m e r i c a l S o l u t i o n s f o r Waves w i t h A r b i t r a r y Shape 3.5.3.1 Scheme 1 ; A Sum o f F u n c t i o n s 3.5.3.2 Scheme 2: A S i n g l e F u n c t i o n EXPERIMENTAL EQUIPMENT AND PROCEDURES
4.1 The Wave Tank 4.2 The Wave G e n e r a t o r 82 85 87 89 92 92 96 4.2.1 The H y d r a u l i c System 4.2.2 The Servo-System
4.2.3 The C a r r i a g e and Wave P l a t e 4.4 The Measurement o f Wave A m p l i t u d e s RESULTS AND DISCUSSION OF RESULTS
5.1 Wave G e n e r a t i o n and P r o p a g a t i o n i n a Constant Depth 114 121 125 5 1 1 The G e n e r a t i o n o f S o l i t a r y Waves 125 5*.1*.2 The P r o p a g a t i o n o f S o l i t a r y Waves m a C o n s t a n t Depth , 5 1 3 The G e n e r a t i o n o f C n o i d a l Waves 5^1.4 The P r o p a g a t i o n o f C n o i d a l Waves i n a Constant Depth
5.2 The R e f l e c t i o n o f Long Waves f r o m a Change i n Depth 5.2.1 The R e f l e c t i o n o f S o l i t a r y Waves f r o m a Step 5.2.2 The R e f l e c t i o n o f C n o i d a l Waves f r o m a Step . 5.2.3 The R e f l e c t i o n o f S o l i t a r y Waves f r o m a Slope
5.3 The T r a n s m i s s i o n o f Long Waves over a Change i n D e p t h 5.3.1 The T r a n s m i s s i o n o f S o l i t a r y Waves o v e r a Step 176 192 205 226 226
v i i i
Chapter Page
5.3.2 The T r a n s m i s s i o n o f C n o i d a l Waves over
a Step 232 5.3.3 The T r a n s m i s s i o n o f S o l i t a r y Waves over
a Slope 234 5.4 The P r o p a g a t i o n o f Long Waves on t h e S h e l f 258
5.4.1 The P r o p a g a t i o n o f S o l i t a r y Waves on t h e S h e l f 258 5.4.2 The P r o p a g a t i o n o f C n o i d a l Waves on t h e S h e l f 285 5.5 Waves P r o p a g a t i n g o f f t h e S h e l f 292 5.6 A p p l i c a t i o n o f t h e R e s u l t s t o t h e Tsunami Problem 294 6 CONCLUSIONS 301 LIST OF REFERENCES 306 LIST OF SYMBOLS 311 APPENDIX A C n o i d a l Wave R e l a t i o n s h i p s and N u m e r i c a l Methods
o f E v a l u a t i o n 315 APPENDIX B The E q u a t i o n f r o m Boussinesq (1872) , t h e Boussinesq
E q u a t i o n s and S o l i t a r y Waves 320 APPENDIX C The L i n e a r N o n d l s p e r s l v e Theory f o r a S i n g l e
Harmonic Wave 322 APPENDIX D T e s t s o f t h e I n v e r s e S c a t t e r i n g N u m e r i c a l Schemes 327
APPENDIX E The N o n l i n e a r N o n d i s p e r s i v e Theory f o r t h e
P r o p a g a t i o n o f sech^ Waves 322
LIST OF FIGURES
(a) I n c i d e n t wave p r o p a g a t i n g towards t h e s h e l f ,
(b) Wave t r a n s f o r m i n g on t h e s l o p e and ( c ) R e f l e c t e d and t r a n s m i t t e d waves
D e f i n i t i o n s k e t c h o f f l o w s i t u a t i o n
(a) and ( b ) The p r o p a g a t i o n o f a sech^ wave by v a r i o u s t h e o r i e s
( c ) and ( d ) The p r o p a g a t i o n o f a sech^ wave by v a r i o u s t h e o r i e s
V a r i o u s c n o i d a l waves
Comparison o f t h e f i r s t t h r e e harmonic components o f c n o i d a l waves ( ) and Stokes waves ( )
Page 10 11 20 21 28 31 33 Wave g e n e r a t i o n phase p l a n e
Phase p l a n e showing t y p i c a l wave p l a t e t r a j e c t o r y f o r a s o l i t a r y wave T y p i c a l wave p l a t e t r a j e c t o r y f o r c n o i d a l wave g e n e r a t i o n D e f i n i t i o n s k e t c h f o r n u m e r i c a l scheme F i n i t e element mesh Wave p r o f i l e s c a l c u l a t e d u s i n g t h e n u m e r i c a l scheme f o r ( a ) H/h=0.7 and ( b ) H / h = 0 . 1
Shapes o f t h e waves a f t e r t h e y have t r a v e l l e d t e n wave l e n g t h s f o r ( a ) H/h=0.7 and ( b ) H/h = 0.1
Comparison o f wave p r o p a g a t i o n u s i n g t h e scheme developed f o r t h i s p r o j e c t ( — ) v i t h t h e scheme o f Hughes, L i u and Zimmermann (1978) ( ;
Wave p r o p a g a t i o n u s i n g t h e scheme o f Hughes L i u and Zimmermann (1978) w i t h two elements i n t h e d e p t h D e f i n i t i o n s k e t c h f o r e x t e n s i o n t o v a r i a b l e d e p t h The v a r y i n g b o t t o m c o n s i d e r e d as a s e r i e s o f s t e p s 45 48 58 59 63 64 65 68
X Page D e f i n i t i o n s k e t c h f o r 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 71 Schematic d r a w i n g o f a t y p i c a l t a n k module ( a f t e r F r e n c h ( 1 9 6 9 ) ) 93 (a) C r o s s - s e c t i o n o f t h e s h e l f , ( b ) E l e v a t i o n o f t h e s l o p e s (L = 150 cm, 300 cm and 450 cm) and ( c ) E l e v a -t i o n o f -t h e h a l f - s i n e -t r a n s i -t i o n 95 (a) Schematic d r a w i n g o f t h e wave g e n e r a t o r w i t h t h e
" l o n g " c y l i n d e r and ( b ) Schematic d r a w i n g o f t h e wave
g e n e r a t o r w i t h t h e " s h o r t " c y l i n d e r 97 98 99 O v e r a l l v i e w o f t h e wave g e n e r a t o r View o f t h e h y d r a u l i c s u p p l y system B l o c k c i r c u i t d i a g r a m o f t h e f u n c t i o n g e n e r a t o r 105 View o f t h e f r o n t f a c e o f t h e e l e c t r o n i c s 106 B l o c k c i r c u i t d i a g r a m o f t h e s e r v o - c o n t r o l l e r 111 Examples o f t h e a c t u a l and programmed wave p l a t e
d i s p l a c e m e n t s f o r ( a ) s o l i t a r y wave g e n e r a t i o n and
(b) c n o i d a l wave g e n e r a t i o n 113 Drawing o f t h e b l a d e h o l d e r 115 Drawing o f a t y p i c a l wave gauge ( a f t e r R a i c h l e n ( 1 9 6 5 ) ) 116
C i r c u i t diagram f o r wave gauges ( a f t e r Okoye ( 1 9 7 0 ) ) 116 (a) View o f t h e c a l i b r a t i o n d e v i c e and ( b ) View o f
t h e m a s t e r c o n t r o l 118 C a l i b r a t i o n c u r v e s f o r ( a ) manual c a l i b r a t i o n and (b) c a l i b r a t i o n u s i n g t h e A/D c o n v e r t e r 120 The l a y o u t and t h e v a r i o u s a s p e c t s o f a t y p i c a l e x p e r i m e n t 122 O s c i l l o g r a p h r e c o r d f r o m a t y p i c a l e x p e r i m e n t o f a s o l i t a r y wave p r o p a g a t i n g o v e r a s t e p o n t o a s h e l f 123 O s c i l l o g r a p h r e c o r d o f t h e waves g e n e r a t e d by a
ramp t r a j e c t o r y (S = 10.33 cm, T = 0.80 sec and
F i g u r e 5.4 S o l i t a r y wave g e n e r a t i o n t r a j e c t o r i e s , H/h = 0.1 5.12 5.14 t o 0.7 129 130 132 135 138 141 5.5 O s c i l l o g r a p h r e c o r d o f t h e waves g e n e r a t e d by t h e s o l i t a r y wave t r a j e c t o r y w i t h H/h= 0.2 (S = 10.33 cm, T = 2.044 sec and h = 10 cm)
5.6 Comparison o f t h e shape o f s o l i t a r y waves w i t h r e l a t i v e h e i g h t s ( a ) H/h=0.15 and ( b ) H/h = 0.61 w i t h t h e t h e o r i e s o f B o u s s i n e s q and McCowan 5.7 V a r i a t i o n o f H/S w i t h t h e r e l a t i v e wave h e i g h t , H/h, f o r s o l i t a r y wave g e n e r a t i o n 5.8 Comparison o f t h e " f r e q u e n c y " o f e x p e r i m e n t a l s o l i t a r y waves w i t h t h a t o f t h e B o u s s i n e s q t h e o r y 5.9 V a r i a t i o n o f t h e I n v e r s e s c a t t e r e d t o measured wave h e i g h t r a t i o , H^^/H, w i t h r e l a t i v e wave h e i g h t , H/h 140 5 10 Comparison o f t h e volume under e x p e r i m e n t a l s o l i t a r y
waves w i t h t h a t o f t h e B o u s s i n e s q and McCowan t h e o r i e s 5.11 V a r i a t i o n o f t h e damping e x p o n e n t , ƒ, w i t h r e l a t i v e wave h e i g h t H/h f o r s o l i t a r y waves V a r i a t i o n o f t h e I n v e r s e s c a t t e r e d t o measured h e i g h t r a t i o , HiNv/H, w i t h r e l a t i v e wave h e i g h t H/h o f ' s o l i t a r y waves as t h e y p r o p a g a t e 5.13 V a r i a t i o n o f t h e " f r e q u e n c y , " , w i t h r e l a t i v e wave h e i g h t , H/h, o f s o l i t a r y waves as t h e y p r o p a g a t e 147 V a r i a t i o n o f c e l e r i t y , c / / i h . w i t h _ ^ r e l a t i v e wave h e i g h t , H/h, f r o m ( a ) t h i s s t u d y and ( b ) Naheer (1977) 149 5.15 T r a j e c t o r y shapes, waves and a s s o c i a t e d d a t a f o r
c n o i d a l wave t r a j e c t o r i e s CNl t o CN6 5 16 ( a ) Comparison o f t h e shape o f e x p e r i m e n t a l c n o i d a l waves w i t h t h e o r y ( T r a j e c t o r i e s C N l , CN2 and CN3) 154 S 16 ( b ) Comparison o f t h e shape o f e x p e r i m e n t a l c n o i d a l waves w i t h t h e o r y ( T r a j e c t o r i e s CN4, CN5 and CN6) 155 5 17 O s c i l l o g r a p h r e c o r d showing t h e waves g e n e r a t e d by t r a j e c t o r y CN4 w i t h h = 2 0 cm, S = 11.18 cm and T = 2.90 sec -"-^^ 152
x i i Ii£H£e Page 5.18 O s c i l l o g r a p h r e c o r d showing t h e waves g e n e r a t e d by t r a j e c t o r y CN4 w i t h h = 20 cm, S = 11.18 and T = 4.28 sec I57 5.19 V a r i a t i o n o f t h e r e l a t i v e t i m e , t /T, w i t h n o n -d i m e n s i o n a l p e r i o -d , T/g7h, f o r c n o i -d a l wave g e n e r a t i o n t r a j e c t o r i e s 150 5.20 T h e o r e t i c a l v a r i a t i o n o f H/S w i t h t h e i n v e r s e n o n -d i m e n s i o n a l p e r i o -d , l/Tl/gph, f o r c n o i -d a l waves 162 5.21 V a r i a t i o n o f H/S w i t h t h e i n v e r s e n o n d i m e n s i o n a l p e r i o d 1/T/g/h f o r c n o i d a l wave g e n e r a t i o n 164 5.22 O s c i l l o g r a p h r e c o r d o f t h e waves g e n e r a t e d by t r a j e c t o r y CN6 w i t h h = 5 , S = 6 . 0 7 cm and T = 3 . 4 0 sec 167 5.23 V a r i a t i o n o f t h e damping exponent, ƒ, w i t h r e l a t i v e wave h e i g h t , H^/h, f o r c n o i d a l waves 170 5.24 Comparison o f t h e a m p l i t u d e s o f t h e f i r s t t h r e e F o u r i e r components o f e x p e r i m e n t a l c n o i d a l waves w i t h t h e t h e o r y 172 5.25 V a r i a t i o n o f t h e c e l e r i t y p a r a m e t e r , a, w i t h U r s e l l Number, HL^/h^, f o r c n o i d a l waves 173 5.26 V a r i a t i o n o f c e l e r i t y o f s o l i t a r y and c n o i d a l waves w i t h r e l a t i v e wave h e i g h t , H/h 175 5.27 V a r i a t i o n o f t h e wave h e i g h t r a t i o , H„/Hj, w i t h t h e r e l a t i v e i n c i d e n t wave h e i g h t , \l\ 179 5.28 V a r i a t i o n o f t h e i n v e r s e s c a t t e r e d wave h e i g h t r a t i o , \ w i t h t h e r e l a t i v e i n c i d e n t wave h e i g h t , 'Q.Jh. 180 INV ^ 5.29 V a r i a t i o n o f t h e volume r a t i o , Vr,/Vj., w i t h t h e r e l a t i v e i n c i d e n t wave h e i g h t , Hj/h^^ 181 5.30 V a r i a t i o n o f t h e wave h e i g h t r a t i o , YLjE^, w i t h d e p t h r a t i o , h^lh^ ^ ^ 188 5.31 V a r i a t i o n o f t h e i n v e r s e s c a t t e r e d wave h e i g h t r a t i o , H^ l\, w i t h d e p t h r a t i o , h./h,. 189 INV ^ ^ 5.32 V a r i a t i o n o f t h e volume r a t i o , V^/V,, w i t h d e p t h r a t i o , h^/h2 190
F i g u r e — ^ 5.33 I n c i d e n t c n o i d a l waves (.nj/h-^ = 0.1, T / i 7 h = 2 7 . 2 ) and t h e waves r e f l e c t e d f r o m t h e s t e p f o r v a r i o u s d e p t h s 5.34 V a r i a t i o n o f t h e wave h e i g h t r a t i o , w i t h r e l a t i v e i n c i d e n t wave h e i g h t , H-^/hj^, f o r c n o i d a l waves 5.35 V a r i a t i o n o f t h e wave h e i g h t r a t i o , Hj^/H^, w i t h d e p t h r a t i o , h-^/h2, f o r c n o i d a l waves 5.36 Comparison o f t h e e x p e r i m e n t a l r e f l e c t e d c n o i d a l waves w i t h t h o s e c a l c u l a t e d f r o m t h e l i n e a r d i s p e r s i v e t h e o r y 5.37 Comparison o f t h e e x p e r i m e n t a l r e f l e c t e d c n o i d a l waves w i t h t h o s e c a l c u l a t e d f r o m t h e n o n l i n e a r d i s p e r s i v e t h e o r y 5.38 T h e o r e t i c a l v a r i a t i o n o f ( a ) t h e F o u r i e r t r a n s f o r m o f a s o l i t a r y wave and ( b ) 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 f r e q u e n c y
5.39 The waves r e f l e c t e d when a s o l i t a r y wave p r o p a g a t e s up v a r i o u s s l o p e s as p r e d i c t e d b y t h e 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 5 40 V a r i a t i o n o f t h e r e f l e c t e d wave h e i g h t r a t i o , % / % , w i t h l e n g t h r a t i o , L/i, as p r e d i c t e d by t h e l i n e a r n o n d l s p e r s l v e t h e o r y 5 4 1 V a r i a t i o n o f t h e r e l a t i v e r e f l e c t e d wave h e i g h t , H R(L/ « , ) / % ( 0 ) , w i t h l e n g t h r a t i o , L/l, as p r e d i c t e d by t h e 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 5 42 ( a ) V a r i a t i o n o f t h e r e f l e c t e d wave h e i g h t r a t i o , H R / H I, w i t h l e n g t h r a t i o , L/l, f o r a d e p t h r a t i o o t V ^ 2 " ^ 5 42 ( b ) V a r i a t i o n o f t h e i n v e r s e s c a t t e r e d r e f l e c t e d wave h e i g h t r a t i o , w i t h l e n g t h r a t i o , L/l, f o r a d e p t h r a t i o o f h i / h 2 = 3 5 43 ( a ) V a r i a t i o n o f t h e r e f l e c t e d wave h e i g h t r a t i o , H R / H I, w i t h l e n g t h r a t i o , L/l, f o r a d e p t h r a t i o o f h^/h2 = 4 194 196 198 201 202 207 210 212 216 217 218
XV F i g u r e 5.54 Page 5.55 5.56 Comparison o f p h y s i c a l and n u m e r i c a l e x p e r i m e n t s f o r a s o l i t a r y wave w i t h h e i g h t Hx = 3.1 cm P ^ o p a g a t i n g f r o m a d e p t h h i = 31.08 cm o v e r a «lope w i h l e n g t h L = 450 cm o n t o a s h e l f w i t h d e p t h h.^ = 15.54 cm Comparison o f t r a n s m i t t e d waves f o r i n c i d e n t waves
o f v a r i o u s h e i g h t s f o r a l e n g t h r a t i o o f L / ) l - 2.00 24b (a) Comparison o f t r a n s m i t t e d waves f o r i n c i d e n t
waves o f v a r i o u s h e i g h t s and f o r v a r i o u s l e n g t h r a t i o s f o r a d e p t h r a t i o o f " 5 56 ( b ) Comparison o f t r a n s m i t t e d waves f o r i n c i d e n t waves o f v a r i o u s h e i g h t s and f o r v a r i o u s l e n g t h r a t i o s f o r a d e p t h r a t i o o f h.j^/li2 = ^ 5.57 V a r i a t i o n w i t h l e n g t h r a t i o o f ( a ) t h e ^ t i v e d i f f e r e n c e f o r s l o p e s , {S - S^^^)ISj^^ and ( b ) t h e r e l a t i v e d i f f e r e n c e f o r wave ^ ^ ^ ^ . f ^ ^ ' / ^ " V , / 5 5 and t h e U r s e l l Number, U ( f o r h^/h2= 3) l m c o m p a r i s o n o f t h e a c t u a l and t h e p r e d i c t e d q u a n t i t i e s g i v e n by ( a ) Eq. (5.24) and ( b ) Eq. (5.25)
E x p e r i m e n t a l wave r e c o r d s showing f f ' a p p r o x i m a t e r e l a t i v e wave h e i g h t o f % / h 2 - 0 . 1 p r o p a g a t i n g on t h e s h e l f E x p e r i m e n t a l wave r e c o r d s showing f f ' a p p r o x i m a t e r e l a t i v e wave h e i g h t o f H^/h2 - 0.3 p r o p a g a t i n g on t h e s h e l f
E x p e r i m e n t a l wave r e c o r d s showing sech^ waves w i t h a p p r o x i m a t e r e l a t i v e wave h e i g h t o f H^/h2-u.D p r o p a g a t i n g on t h e s h e l f 5.58 5.59 5.60 5.61 5.62 5.63 5.64
Comparison o f waves measured ^^P^^^^^^^^^^J^^^^
t h o s e c a l c u l a t e d by v a r i o u s t h e o r i e s a t l o c a t i o n s g i v e n by x/h2 = 0, 8.88 and 13.31
Comparison o f waves measured f P ^ ^ i ^ ^ ^ ^ ^ ^ ^ i ^ ^ ^ ^ ' J l ^ t h o s e c a l c u l a t e d by v a r i o u s t h e o r i e s a t l o c a t i o n s g i v e n by x / h 2 = 1 7 . 7 5 , 22.19 and 26.63
Comparison o f waves measured ^^P^^^^^^^^^f ^ " j ^ J ; , t h o s e c a l c u l a t e d by v a r i o u s t h e o r i e s a t l o c a t i o n s g i v e n by x / h 2 = 3 1 . 0 7 , 35.50 and 39.94
266
267
5.65 Views o f t h e s e p a r a t i o n caused by a s o l i t a r y wave p r o p a g a t i n g o v e r ( a ) t h e s t e p and ( b ) t h e h a l f - s i n e t r a n s i t i o n ( h ^ = 20.5 cm, h2 = 4.96 cm, H j = 2 . 0 cm) 270 5.66 V a r i a t i o n o f t h e r e l a t i v e h e i g h t , H/h2, o f a wave as i t p r o p a g a t e s on t h e s h e l f . Comparison o f e x p e r i m e n t s i n w h i c h t h e s t e p was used w i t h t h o s e i n w h i c h t h e h a l f - s i n e t r a n s i t i o n was used 272 5.67 T h e o r e t i c a l v a r i a t i o n o f t h e d i s t a n c e a sech^ wave p r o p a g a t e s t o b r e a k i n g , x^//gh7, w i t h r e l a t i v e wave h e i g h t , H^/h2 i o ^ 275 2 5.68 T h e o r e t i c a l v a r i a t i o n o f t h e U r s e l l Number, mjhjg, w i t h p r o p a g a t i o n d i s t a n c e , x/x, , f o r an i n i t i a l r e l a t i v e wave h e i g h t o f H^/h2 = 0.1; n o n l i n e a r d i s p e r s i v e and n o n d l s p e r s l v e t h e o r i e s 277 5.69 T h e o r e t i c a l v a r i a t i o n o f t h e U r s e l l Number, Vn^h^/g, w i t h p r o p a g a t i o n d i s t a n c e , x/x, , f o r an i n i t i a l r e l a t i v e wave h e i g h t o f H^/h2 = 0.3; n o n l i n e a r d i s p e r s i v e and n o n d i s p e r s i v e t h e o r i e s 278 5.70 T h e o r e t i c a l v a r i a t i o n o f t h e U r s e l l Number, Vü^h^/g, w i t h p r o p a g a t i o n d i s t a n c e , x / x j ^ , f o r an i n i t i a l r e l a t i v e wave h e i g h t o f H^/h2 = 0.5; n o n l i n e a r d i s p e r s i v e and n o n d i s p e r s i v e t h e o r i e s 279 5.71 T h e o r e t i c a l v a r i a t i o n o f t h e d i s t a n c e f o r d i s p e r s i v e e f f e c t s t o become i m p o r t a n t , x,/h„, w i t h r e l a t i v e wave h e i g h t , H„/h„ ^ ^ 282 5.72 T h e o r e t i c a l v a r i a t i o n o f t h e d i s t a n c e f o r n o n l i n e a r e f f e c t s t o become i m p o r t a n t , fix //gh„, w i t h r e l a t i v e wave h e i g h t , H^/h2 ^ 284 5.73 O s c i l l o g r a p h r e c o r d s o f e x p e r i m e n t s i n w h i c h c n o i d a l waves o f t h e same h e i g h t , H /h_ = 0.28, p r o p a g a t e on t h e s h e l f ^ 287 5.74 O s c i l l o g r a p h r e c o r d s o f e x p e r i m e n t s i n w h i c h c n o i d a l waves o f t h e same p e r i o d , T/g/hj = 5 7 . 1 , p r o p a g a t e on t h e s h e l f 289 5.75 Comparison f o r t h e p r o p a g a t i o n o f c n o i d a l waves on t h e s h e l f between e x p e r i m e n t and t h e n o n l i n e a r d i s p e r s i v e t h e o r y 291
x v i i F i g u r e 5.76 Comparison f o r t h e p r o p a g a t i o n o f s o l i t a r y waves o f f t h e s h e l f i n t o deep w a t e r between e x p e r i m e n t and t h e l i n e a r d i s p e r s i v e t h e o r y 5.77 Schematic d r a w i n g o f t h e c o n t i n e n t a l s l o p e o f f t h e c o a s t o f C a l i f o r n i a
E . l The x - t p l a n e f o r a sech^ wave p r o p a g a t i n g i n t o s t i l l w a t e r by t h e n o n 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
T a b l e Page 3.1 Number o f i t e r a t i o n s f o r convergence f o r v a r i o u s
n o d a l s p a c i n g numbers, N^, and t i m e s t e p numbers, N^ 55 3.2 Comparison o f i n i t i a l and f i n a l conserved q u a n t i t i e s
f o r t h e n u m e r i c a l scheme 56 3.3 Maximum U r s e l l Numbers f o r a p a r t i c u l a r number o f
s o l i t a r y waves t o emerge f r o m a sech^ wave 85 5.1 S o l u t i o n s o f t h e s o l i t a r y wave due t o B o u s s i n e s q ,
McCowan and L a i t o n e (Naheer (1977)) 133 5.2 Comparison o f g e n e r a t i o n t r a j e c t o r i e s f o r t Q / T = 0 . 2 0 0 .
5/?max f o r a h a l f p e r i o d f o r v a r i o u s wave h e i g h t and
p e r i o d c o m b i n a t i o n s 1 6 1 5.3 Maximum r e l a t i v e i n c i d e n t wave h e i g h t s f o r n o n - b r e a k i n g waves on t h e s h e l f as p r e d i c t e d by t h e 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 185 5.4 Wave h e i g h t r a t i o s f o r e x p e r i m e n t s , (H„/H-j.) , 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 , (H^^/Hj.) , Expt l i n e a r d i s p e r s i v e t h e o r y , (Hp/Hj) * and n o n l i n e a r d i s p e r s i v e t h e o r y , (H P / H T) 204 ^ ^ N.D. 5.5 R e f l e c t e d wave h e i g h t r a t i o s , ( a ) And ( b ) /Ej, f o r v a r i o u s l e n g t h r a t i o s and r e l a t i v e ^INV ^ i n c i d e n t wave h e i g h t s f o r d e p t h r a t i o h-,/h2 = 3 ( n o n l i n e a r d i s p e r s i v e t h e o r y ) 221 5.6 R e f l e c t e d wave h e i g h t r a t i o s , ( a ) H R / H I and ( b ) Ht j ^ „ / H T, f o r v a r i o u s l e n g t h r a t i o s and r e l a t i v e i n c i d e n t wave h e i g h t s f o r d e p t h r a t i o h-|^/h2 = 1.5 ( n o n l i n e a r d i s p e r s i v e t h e o r y ) 223 5.7 T r a n s m i t t e d waves c a l c u l a t e d u s i n g t h e n o n l i n e a r d i s p e r s i v e t h e o r y f o r h^/h2 = 3 228 5.8 D e t a i l s o f t h e e x p e r i m e n t s f o r p r o p a g a t i o n o f s o l i t a r y waves on a s h e l f , shown i n F i g s . 5.58 t o 5.60 264 5.9 Comparison o f t r a n s m i t t e d wave d a t a f o r t h e e x p e r i m e n t s p r e s e n t e d i n F i g . 5.66 273 5.10 R e l a t i v e d i f f e r e n c e s between t h e t h e o r i e s i n ( a ) t h e s l o p e o f t h e f r o n t f a c e and ( b ) t h e t r a n s m i t t e d wave h e i g h t f o r t s u n a m i s w h i c h a r e s o l i t a r y waves 298
1
CHAPTER 1 INTRODUCTION
Long waves a r e waves w i t h l e n g t h s w h i c h a r e l a r g e compared t o the d e p t h o f w a t e r i n w h i c h t h e y a r e p r o p a g a t i n g . Among t h e waves w h i c h f a l l i n t h i s c l a s s a r e " t s u n a m i s " o r , as t h e y a r e sometimes c a l l e d , " t i d a l waves." The word " t s u n a m i " i s a Japanese word w h i c h means " h a r b o r wave." I t has been adopted by t h e s c i e n t i f i c community i n p r e f e r e n c e t o " t i d a l wave" t o mean an e a r t h q u a k e - g e n e r a t e d sea wave.
The e a r t h q u a k e s w h i c h g e n e r a t e tsunamis u s u a l l y i n v o l v e v e r t i c a l movements o f t h e sea bed. Such an e a r t h q u a k e o c c u r r e d i n A l a s k a i n 1964; i t g e n e r a t e d a t s u n a m i w h i c h p r o p a g a t e d t h r o u g h o u t t h e P a c i f i c c a u s i n g damage a t v a r i o u s l o c a t i o n s a l o n g t h e West Coast o f t h e U n i t e d S t a t e s , p a r t i c u l a r l y i n C r e s c e n t C i t y , C a l i f o r n i a . An I m p o r t a n t a s p e c t i n t r y i n g t o e i t h e r a v o i d o r p r e p a r e f o r such a d i s a s t e r i s t o under-s t a n d how a t under-s u n a m i p r o p a g a t e under-s .
I n t h e deep ocean where t h e d e p t h may be 3500 m a t s u n a m i m i g h t t y p i c a l l y have a l e n g t h o f about 300 km and a h e i g h t o f 1 m and t r a v e l a t a speed o f 700 km/hr. The p r o p a g a t i o n o f t h e t s u n a m i would proceed e s s e n t i a l l y i n c o n s t a n t d e p t h t h r o u g h t h e deep ocean u n t i l i t reached t h e r e g i o n o f s h a l l o w e r d e p t h w h i c h s u r r o u n d s most l a n d m a s s e s — t h e c o n t i n e n t a l s h e l f . Here t h e d e p t h decreases c o n s i d e r a b l y ; o f i n t e r e s t i n t h i s i n v e s t i g a t i o n was t o d e t e r m i n e
the c o a s t , the i n v e s t i g a t i o n was c a r r i e d ont hy -eans ot p h y s i c a l and a n a l y t i c a l models.
1,1 Ob-] e c t i v e s and Scope
The o b j e c t i v e of t h i s i n v e s t i g a t i o n was to examine, both e x p e r i m e n t a l l y and t h e o r e t i c a l l y , the v a r i o u s a s p e c t s of the propa-g a t i o n of lonpropa-g waves onto a s h e l f , i . e . , the r e f l e c t i o n , t r a n s m i s s i o n and propagation of the waves on the s h e l f , f o r both abrupt and g r a d u a l
Changes i n depth. Of e , u a l importance was to determine i f the l i n e a r mathematical models which commonly a r e used i n the a n a l y s i s of
4f -ii- -fp n e c e s s a r y t o use more c o m p l i c a t e d t s u n a m i s a r e s u f f i c i e n t o r i f x t xs n e c e s s a r y
n o n l i n e a r models.
The waves used i n t h i s study were p r i m a r i l y s o l i t a r y waves. These were chosen because i t can be shown t h e o r e t i c a l l y that waves which
have net p o s i t i v e volume e v e n t u a l l y , i f the propagation d i s t a n c e i s s u f f i c i e n t , w i l l b r e a . up i n t o a a e r i e s of s o l i t a r y waves. For a n a l y s i s , s o l i t a r y waves have the advantage t h a t , although n o n l i n e a r ,
they can be d e s c r i b e d w i t h J u s t two parameters: the wave h e i g h t and the depth. A d d i t i o n a l b e n e f i t s a r e : they propagate w i t h c o n s t a n t form i n c o n s t a n t depth and g e n e r a l l y they can be separated from
r e f l e c t e d waves. P e r i o d i c waves i n the form of c n o i d a l waves a l s o were c o n s i d e r e d f o r propagation over abrupt changes i n depth.
TO f a c i l i t a t e the experimental i n v e s t i g a t i o n , a t h e o r y was developed f o r the g e n e r a t i o n I n the l a b o r a t o r y of long waves of
permanent f o r m , i . e . , s o l i t a r y and c n o i d a l waves. The t h e o r e t i c a l i n v e s t i g a t i o n i n c l u d e d t h e development o f a f i n i t e element t e c h n i q u e of s o l v i n g t h e o n e - d i m e n s i o n a l Boussinesq e q u a t i o n s . T h i s was a p p l i e d t o t h e f u l l problem o f s o l i t a r y waves p r o p a g a t i n g o v e r a s l o p e o n t o a s h e l f and was c o n f i r m e d by p h y s i c a l e x p e r i m e n t s . A r e v i e w o f p r e v i o u s s t u d i e s o f t h e p r o p a g a t i o n o f l o n g waves onto a s h e l f i s p r e s e n t e d i n Chapter 2. The t h e o r e t i c a l a n a l y s i s w h i c h i n c l u d e s a r e v i e w o f t h e c l a s s i c a l l o n g wave t h e o r i e s and t h e i r a p p l i c a t i o n t o t h i s p r o b l e m , wave g e n e r a t i o n t h e o r y and t h e development o f t h e f i n i t e element n u m e r i c a l method a r e p r e s e n t e d i n Chapter 3. The e x p e r i m e n t a l equipment and p r o c e d u r e s a r e d e s c r i b e d i n Chapter 4. The r e s u l t s o f t h e I n v e s t i g a t i o n a r e p r e s e n t e d and d i s c u s s e d i n Chapter 5, and c o n c l u s i o n s based upon t h e s e a r e d e s c r i b e d i n Chapter 6.
CHAPTER 2
LITERATURE SURVEY
The n o n l i n e a r p a r t i a l d i f f e r e n t i a l agnations which govern t h e
However, u n t i l r e c e n t l y , only the equation a r i a l n g f r o . a l i n e a r approximation t o theae e,uatlon3 has heen used f o r p r e d i c t i n g the
propagation of long waves onto a s h e l f .
The theory a r i s i n g from t h i s equation i s termed the l i n e a r ^ ^ ^ . . ^ theory. The s o l u t i o n s of t h e theory f o r long waves
„f a r b i t r a r y shape propagating over abrupt and gradual s l o p e s a r e presented i n .amb ( 1 « . ) • ^ „ , . b e w o r . . I t was o r i g i n a l l y p u b l i s h e d i n 18...) . a m b a , 3 . . . n e ) Shows, f o r a s t e p , t h e 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 a r e g i v e n b y :
( i
- / h 7 r 2 ) (2
.1)
I L , = — i z r » ^ C l + / h ^ ) and (2.2) 1 + v / h j / h ^r e s p e c t i v e l y , where h , i s the upstream depth and h^ i s t h e depth on
t h e s h e l f .
,„r a " g r a d u a l " s l o p e , i . e . , a s l o p e on which the depth changes by only a s m a l l f r a c t i o n of i t s e l f w i t h i n the l i m i t s of a wavelength.
5
Lamb ( 1 9 3 2 , § 1 8 5 ) shows t h e 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 a r e g i v e n by Green's Law: K R = 0 , ( 2 . 3 ) and = ( h ^ / h 2 ) ^ , ( 2 . 4 ) r e s p e c t i v e l y . S o l u t i o n s o f t h e 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 f o r t h e s l o p e s between an a b r u p t s l o p e ( i . e . , a s t e p ) and a g r a d u a l s l o p e have been p r e s e n t e d by K a j i u r a ( 1 9 6 1 ) , Wong et al. ( 1 9 6 3 ) and Dean ( 1 9 6 4 ) . For a l l o f t h e s e s t u d i e s t h e s o l u t i o n was o b t a i n e d f o r an harmonic wave w i t h a s i n g l e f r e q u e n c y i n t h e s t e a d y s t a t e .
K a j i u r a ( 1 9 6 1 ) p r o p o s e d a method o f s o l u t i o n f o r s l o p e s o f g e n e r a l shape and p r e s e n t e d t h e s o l u t i o n s f o r two cases:
i ) A s l o p e on w h i c h t h e d e p t h v a r i e s as t h e square o f t h e d i s t a n c e a l o n g i t . The s o l u t i o n f o r t h e wave on t h e s l o p e i s a f u n c t i o n o f x^. i i ) A c o n t i n u o u s s l o p e d e t e r m i n e d such t h a t t h e b a s i c e q u a t i o n i s t r a n s f o r m e d i n t o an e q u a t i o n w h i c h g i v e s s i m p l e e x p r e s s i o n s f o r t h e 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 .
Wong et al. ( 1 9 6 3 ) and Dean ( 1 9 6 4 ) o b t a i n e d t h e s o l u t i o n f o r a s l o p e on w h i c h t h e d e p t h v a r i e s l i n e a r l y as a f u n c t i o n o f t h e d i s t a n c e a l o n g i t . The s o l u t i o n o f t h e wave on t h e s l o p e i s a f u n c t i o n o f B e s s e l f u n c t i o n s .
. . e a u . the a o l u U o n a f o . the » o a . . e . a a of abrupt and gra.uaX aiopos »ore f o r W »a,es „f a r b i t r a r y shaped t b a r e f o r a . i f i t . s . a i i . to ao s o . t b s s o l u t i o n s oan ba a p p i i a . d i r e o t i y to s o l i t a r y or
„ o i a a i waves. However, f o r s l o p e s between tbe two extremes, tbe s o l u t i o n s a r e f o r barmonio waves w i t b a s i n g l e frequency o n l y .
. e r e f o r e tbe s o l u t i o n s , even i f v a l i d , cannot be a p p l i e d d i r e c t l y to
s o l i t a r y o r c n o i d a l waves.
f i r s t s o l v e d f o r t h e p r o b l e m The f u l l n o n l i n e a r e q u a t i o n s were f i r s t s o l v
n*-n a s h e l f by Madsen and M e i (1969) . o f l o n g waves p r o p a g a t i n g o n t o a s h e i t oy
„Sing tbe equations developed by Mel and M.baut. ( l . « , , wbicb
C i « « developed a n u m e r i c a l metbod of s o l u t i o n based on tbe metbod-^ o f Long (1964). The s l o w l y v a r y i n g d e p t h o f - c h a r a c t e r i s t i c s scheme o f Long ^
A M p i (1969) i s e q u i v a l e n t t o t h e g r a d u a l a s s u m p t i o n used by Madsen and M e i (1969;
s l o p e m e n t i o n e d e a r l i e r .
„adsen and «ei (196,) found t b e o r e t i c a l l y and e x p e r i m e n t a l l y tba as a s o l i t a r y wave propagates up a gradual s l o p e i t s abape changes. With the f r o n t f a c e of tbe wave steepening and secondary waves emerging
, „ m tbe b a c . f a c e of tbe wave. E v e n t u a l l y , e i t h e r on the s l o p e or > a s e r i e s of s o l i t a r y waves on the s h e l f , the waves s e p a r a t e i n t o a
by a t r a i n of o s c i l l a t o r y waves. B a r l i e r , S t r e e t . • a x p e r i m e n t a l l y had observed s i m i l a r behavior but over a p r o p a g a t i
d i s t a n c e which was i n s u f f i c i e n t f o r the s o l i t a r y waves to emerge f u l l y f r o m t h e m a i n t r a i n .
7
A n a l y t i c a l s o l u t i o n s o f t h e p r o b l e m o f s o l i t a r y waves p r o p a g a t i n g over a g r a d u a l s l o p e were f o u n d i n d e p e n d e n t l y by T a p p e r t and Zabusky
(1971) and Johnson ( 1 9 7 3 ) . By assuming zero r e f l e c t i o n and s l o w l y v a r y i n g d e p t h , a v a r i a b l e d e p t h f o r m o f t h e KdV e q u a t i o n can be
d e r i v e d and, u s i n g t h e same t e c h n i q u e s as were used by Gardner et al. (1967) t o s o l v e t h e KdV i n c o n s t a n t d e p t h , as3rmptotic s o l u t i o n s f o r the s o l i t a r y waves w h i c h emerge on t h e s h e l f can be o b t a i n e d . The number o f s o l i t a r y waves w h i c h w i l l emerge on t h e s h e l f i s a f u n c t i o n of o n l y t h e d e p t h r a t i o , h^/h2, as g i v e n by:
1 + h + 8 — - [
I
1
^2/
i
(2.5)
N < P
where t h e number o f waves, N, i s s t r i c t l y l e s s t h a n P. The h e i g h t of t h e s o l i t a r y waves w h i c h emerge i s g i v e n by:
H / h / - 2 n ] ^o ^ \^2, n = 1 , 2, , N where i s t h e h e i g h t o f t h e i n c i d e n t s o l i t a r y wave. To summarize, p r e v i o u s i n v e s t i g a t i o n s i n t h e f i e l d o f l o n g waves p r o p a g a t i n g o n t o a s h e l f have d e a l t w i t h one o f t h e f o l l o w i n g a s p e c t s o f t h e problem:
( 1 ) L i n e a r waves o f a r b i t r a r y shape p r o p a g a t i n g o v e r an extreme s l o p e ( i . e . , e i t h e r g r a d u a l o r a b r u p t ) ;
( i l ) L i n e a r harmonic „avea w i t h a a i n g l e £re,nency propagating
over a s l o p e ; or
( U l ) S o l i t a r y waves propagating over a gradual s l o p e .
The , u e s t l o n o£ which of the t h e o r i e s to use f o r the propagation of long waves I n v a r i o u s s i t u a t i o n s I s addressed hy HammacU and Segur
(1978). They show, u s i n g a s y m p t o t i c arguments and a r e c t a n g u l a r
of the i n i t i a l wave and an U r s e l l Numher hased on the amplitude and l e n g t h of the I n i t i a l wave. Applying t h e i r c r i t e r i a to tsunamis, they Show the U s H E - É - B " ^ theory i s the r e l e v a n t theory f o r the propagation of t h e l e a d i n g wave of a tsunami i n a c o n s t a n t depth from the g e n e r a t i o n r e g i o n to the beach.
9 CHAPTER 3 THEORETICAL ANALYSIS The t h e o r e t i c a l a s p e c t s o f t h e p r o b l e m can be d e s c r i b e d r e f e r r i n g t o F i g . 3,1 w h i c h shows t h e s e r i e s o f e v e n t s w h i c h t a k e s p l a c e as a l o n g wave p r o p a g a t e s o n t o a s h e l f .
F i g , 3.1(a) shows t h e i n c i d e n t wave p r o p a g a t i n g towards t h e s h e l f i n a r e g i o n o f c o n s t a n t d e p t h . The v a r i o u s t h e o r i e s f o r l o n g waves p r o p a g a t i n g i n a c o n s t a n t d e p t h a r e r e v i e w e d i n S e c t i o n 3.1 and e x a c t s o l u t i o n s a r e d e s c r i b e d .
As w i t h o t h e r I n v e s t i g a t o r s ( e . g . Madsen and M e l ( 1 9 6 9 ) ) , f o r t h e a n a l y s i s t h e i n c i d e n t wave was assumed t o be a s o l i t a r y wave ( a l t h o u g h , as m e n t i o n e d p r e v i o u s l y , r e c e n t work by Hammack and Segur (1978) has c a s t some doubt on t h e p r a c t i c a l v a l i d i t y o f t h i s ) . A t h e o r y f o r t h e g e n e r a t i o n , i n t h e l a b o r a t o r y , o f s o l i t a r y waves and a l s o o f c n o i d a l waves i s p r e s e n t e d i n S e c t i o n 3,2.
As t h e wave p r o p a g a t e s t h r o u g h a r e g i o n o f v a r i a b l e d e p t h i t s shape changes as shown i n F i g . 3 . 1 ( b ) , and e v e n t u a l l y t h e wave s p l i t s up i n t o two waves: a r e f l e c t e d wave t r a v e l i n g t o t h e l e f t i n t h e deep w a t e r and a t r a n s m i t t e d wave t r a v e l i n g t o t h e r i g h t on t h e s h e l f , see F i g . 3 . 1 ( c ) . Two t h e o r i e s a r e p r e s e n t e d which s o l v e t h e p r o b l e m . I n S e c t i o n 3.3 a f i n i t e element method o f s o l u t i o n o f t h e B o u s s i n e s q e q u a t i o n s f o r t h e case o f waves p r o p a g a t i n g i n a c o n s t a n t d e p t h i s p r e s e n t e d , t h e n t h e method i s extended t o t h e case o f waves p r o p a g a t i n g
INCIDENT WAVE
F i g . 3.1(a) I n c i d e n t wave p r o p a g a t i n g t o w a r d s t h e s h e l f , F i g . 3.1(b) Wave t r a n s f o r m i n g on t h e s l o p e .REFLECTED WAVE
TRANSMITTED WAVE
F i g . 3 . 1 ( c ) R e f l e c t e d and t r a n s m i t t e d waves.11
i n a r e g i o n w i t h v a r i a b l e d e p t h . T h i s s o l u t i o n i s t h e more a c c u r a t e of those c o n s i d e r e d because i t i n c o r p o r a t e s , up t o second o r d e r , t h e e f f e c t s o f d i s p e r s i o n and n o n l i n e a r i t y . A f i r s t o r d e r s o l u t i o n i n xjhich t h e s e e f f e c t s a r e n e g l e c t e d i s p r e s e n t e d i n S e c t i o n 3.4 where the t h e o r y developed by o t h e r s f o r t h e s o l u t i o n f o r i n c i d e n t waves which a r e harmonic i s r e v i e w e d and a p p l i e d t o t h e case o f an
i n c i d e n t wave w h i c h i s a s o l i t a r y wave.
F i n a l l y i n t h i s c h a p t e r t h e t e c h n i q u e o f i n v e r s e s c a t t e r i n g i s d e s c r i b e d and n u m e r i c a l schemes f o r i t s s o l u t i o n a r e p r e s e n t e d . I n v e r s e s c a t t e r i n g a l l o w s one t o d e t e r m i n e t h e f i n a l s t a t e o f a l o n g wave i f i t p r o p a g a t e s t o i n f i n i t y i n c o n s t a n t d e p t h i n t h e absence o f f r i c t i o n . I t was used i n t h i s s t u d y t o a n a l y z e t h e r e f l e c t e d wave. ( T h i s w i l l be d i s c u s s e d i n d e t a i l i n S e c t i o n 5.2.)
3.1 O u t l i n e D e r i v a t i o n o f t h e Long Wave E q u a t i o n s and Exact S o l u t i o n s The l o n g wave e q u a t i o n s can be d e r i v e d i n numerous ways; t h e
approach w h i c h i s o u t l i n e d h e r e f o l l o w s t h a t o f Whitham ( 1 9 7 4 ) . C o n s i d e r t h e f l o w s i t u a t i o n shown i n F i g . 3.2 w h i c h shows a 00 h CD
y / / / / / y-j-y-y-y-y
F i g . 3,2 D e f i n i t i o n S k e t c h o f t h e Flow S i t u a t i o nwave 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 i n a r e g i o n o f i n f i n i t e h o r i z o n t a l e x t e n t . The v e r t i c a l y a x i s has i t s o r i g i n a t t h e s t i l l w a t e r l e v e l . The d i s p l a c e m e n t o f t h e f r e e s u r f a c e f r o m t h e s t i l l w a t e r l e v e l i s . ( x , t ) . Assuming I n v l s c l d , I r r o t a t i o n a l . i n c o m p r e s s i b l e f l o w , t h e r e e x i s t s a v e l o c i t y p o t e n t i a l . ( x , y , t ) w h i c h s a t i s f i e s t h e L a p l a c e e q u a t i o n : V2$ = 0 - h < y < r ) (3.1) The boundary c o n d i t i o n s a r e : i ) No f l o w t h r o u g h t h e b o t t o m boundary: $ = 0 y = -^ y 11) K i n e m a t i c boundary c o n d i t i o n a t t h e s u r f a c e : ( 3 . 2 ) ( 3 . 3 ) (3.4) i i i ) Dynamic boundary c o n d i t i o n a t t h e s u r f a c e : $ + 1 (* 2 + $ 2) + g n = o y = n t 2 ^ x y
The waves under c o n s i d e r a t i o n a r e l o n g waves w h i c h a r e d e f i n e d as waves Whose c h a r a c t e r i s t i c h o r l s o n t a l l e n g t h . i s l a r g e compared t o
t h e d e p t h h, i . e . . > h . For l o n g waves t h e h o r i z o n t a l v e l o c i t y IS a p p r o x i m a t e l y c o n s t a n t o v e r t h e d e p t h so t h e v e l o c i t y p o t e n t i a l can he expanded i n terms o f t h e p a r a m e t e r y - h + y w h i c h i s s m a l l
i ^ l ^ l
WÊm r13
H^,y,t) = J 2 f ( x , t ) . ( 3 . 5 )
n=0
By s u b s t i t u t i n g Eq. ( 3 . 5 ) i n t o Eq. ( 3 . 1 ) , e q u a t i n g l i k e powers o f Y and a p p l y i n g t h e boundary c o n d i t i o n '^y = 0 a t Y - 0 , t h e e x p a n s i o n i s s i m p l i f i e d t o : ,2n 3^"f Each v a r i a b l e i s now n o r m a l i z e d by s c a l i n g by a c h a r a c t e r i s t i c q u a n t i t y : X* y* n = ^ il JIH > g where 5, i s t h e c h a r a c t e r i s t i c h o r i z o n t a l l e n g t h and H i s t h e c h a r a c t e r i s t i c h e i g h t o f t h e wave and s t a r r e d symbols denote t h e o r i g i n a l d i m e n s i o n a l v a r i a b l e s . ( H e n c e f o r t h a l l e q u a t i o n s w i l l be d i m e n s i o n l e s s u n l e s s s p e c i f i c a l l y s t a t e d o t h e r w i s e . ) When t h e s e v a r i a b l e s a r e s u b s t i t u t e d i n t o t h e e x p a n s i o n , Eq. ( 3 . 6 ) , and t h e r e m a i n i n g boundary c o n d i t i o n s , Eqs. ( 3 . 3 ) and ( 3 . 4 ) , two d i m e n s i o n l e s s numbers emerge: a = H/h and & = h^/Z^. I n w r i t i n g t h e e x p a n s i o n , Eq. ( 3 . 5 ) , i t was assumed t h a t B < 1 ( i . e . t h e l e n g t h o f t h e wave i s l a r g e compared t o t h e d e p t h ) . I t i s a l s o n e c e s s a r y t o assume t h a t a < l ( i . e . t h e wave h e i g h t i s s m a l l compared t o t h e d e p t h ) .
-
-
-
-
-
-
-
:
r
7
:
;
:
:
»
'
and r e t a i n i n g t e r m s t o o r d e r a , 3 n Ril'i and a r e as f o l l o w s , e q u a t i o n s a f t e r B o u s s i n e s q (1872) | ( l + a r i) u | ^ - | p ^ x x x " ° 1 Ril = 0 u ^ + a u u^ + n ^ - ^ P' ^ x x t ( 3 . 7 ) ( 3 . 8 ) . (3 7) and ( 3 . 8 ) t h e d i m e n s i o n l e s s numbers, a N o t i c e i n Eqs. (3.7) and1 . . The number a appears b e f o r e
and 3, have d i f f e r e n t r o l e s . ^^^^^ i n d i c a t i n g t h e i r i m p o r t a n c e r e l a t x v
n o n l i n e a r t e r m s i n d i c a t i n g ^^^^^
K i - h l r d d e r i v a t i v e terms w h i c h a r e
e m o d i f i e s t h e t h i r d ^^^^
(or, a c u i v a l e n t l y , tha preaaure d x s t r x h
h y d r o s t a t i c ) . , ,. i s f u r t h e r r e q u i r e d t h a t
n r, and (3.8) to apply, i t ^
" , , (.0 i l l u s t r a t e the r e a s o n f o r t h i s . „ and e he of the same o r d e r . ^ ^ ^ ^
1, v<= ft i s so much g r e a t e t
c o n s i d e r t h e case where 3 i s . r e f e r e n c e t o terms of o r d e r s h o u l d be i n c l u d e d m p r e f e r e n t h e n terms o f o r d e r p ^„„r-iate.) T h i s d (3 8) a r e n o t a p p r o p r i a t e . ; of o r d e r a, and Eqs. ( 3 . 7 ) an . ^^^^^^ ^^^^^^
- ' ^ — : ; : : r p : p i
' e . u c i t i y
expounded - ,ne B o u s s i n e s , equationst-ViP r a t i o a/p. nt;u«_>^ t h e i m p o r t a n c e o f t h e r a t
15
to be a p p l i c a b l e , t h e U r s e l l Number must be o f o r d e r u n i t y . S i n c e a r e p r e s e n t s t h e m a g n i t u d e o f n o n l i n e a r e f f e c t s and 3 r e p r e s e n t s t h e magnitude o f d i s p e r s i v e e f f e c t s , t h e U r s e l l Number o f o r d e r u n i t y i m p l i e s a b a l a n c e o f n o n l i n e a r and d i s p e r s i v e e f f e c t s .
The v e l o c i t y u a p p e a r i n g i n Eqs. ( 3 . 7 ) and ( 3 . 8 ) i s t h e v e l o c i t y a t t h e b o t t o m y = - l . I t i s o f t e n more c o n v e n i e n t t o use the d e p t h averaged v e l o c i t y : n Ü =
J
$^dy . ( 3 . 9 ) The B o u s s i n e s q e q u a t i o n s t h e n t a k e t h e f o r m : j( l + a n ) ü | ^ = 0 , (3.10) ü , + aG5^ + n, 4 3 G ^ ^ , = 0 . (3.11) The B o u s s i n e s q e q u a t i o n s c a n n o t , i n g e n e r a l , be s o l v e d i n c l o s e d f o r m so i t i s n e c e s s a r y t o r e s o r t t o a n u m e r i c a l scheme such as t h a t w h i c h w i l l be d e s c r i b e d i n S e c t i o n 3 . 3 . The B o u s s i n e s q e q u a t i o n s a r e t h e most g e n e r a l f o r m o f t h e l o n g wave e q u a t i o n s s i n c e t h e o t h e r w e l l known e q u a t i o n s can be deduced f r o m them. These w i l l now be l i s t e d a l o n g w i t h t h e i r g e n e r a l s o l u t i o n s :i ) F o r s m a l l a m p l i t u d e , v e r y l o n g waves (a « 1, B « 1) Eqs.
and ïit-'^x^^ (3.12) n - a = 0 o r ^ t t XX a = f ( k x - a ) t ) + g C l ^ + ' ^ t ^ . ,2 =,21,2 and c ^ = > ^ • w h e r e o Waves p r o p a g a t e e o . . a « s p e e d end w U . i „ a n d -X d i r e c t i o n s . ,3 „„t a s g r e a t a s i « ^ « - - - - - ¬ t h o s e c o n s i d e r e d a b o v e ( a« l , and (3.11) r e d u c e t o : \ + ^ x ' ° and 1-^ t t - 1-^ x x - S 1-^ 1-^ x t t r e the U n e a r . i i f f l a £ ^ These a r e t h e i i i i S — —
1„ dimensional ter»s o i the form:
(3.14)
w h i c h h a v e s o l u t i o n s
17
c 2 k 2
T h i s I m p l i e s t h a t waves p r o p a g a t e w i t h speeds w h i c h a r e a f u n c t i o n o f t h e l e n g t h o f t h e wave and t h e waves do n o t have a permanent shape.
For f i n i t e a m p l i t u d e , v e r y l o n g waves ( a < l , 3 « 1 and U » l ) , Eqs. (3.10) and (3.11) reduce t o :
{ ( l + an)ü(^= 0
( 3 . 1 6 ) + aüü^ + " ° '
w h i c h a r e t h e n o n l i n e a r n o n d l s p e r s l v e e q u a t i o n s
(sometimes c a l l e d t h e A i r y e q u a t i o n s ) . By r e v e r t i n g back t o d i m e n s i o n a l q u a n t i t i e s , Eqs. (3.16) can be expressed more s i m p l y i n c h a r a c t e r i s t i c f o r m :
^ ( I I ± 2 c ) = 0 o n ^ = G± c , (3.17)
where ^ = / g ( h + n )
For waves p r o p a g a t i n g t o t h e r i g h t i n t o s t i l l w a t e r , Eqs. (3.17) p r e d i c t t h a t t h e wave a m p l i t u d e and t h e v e l o c i t y a r e c o n s t a n t a l o n g t h e c h a r a c t e r i s t i c c u r v e s d x / d t = ü + c, w h i c h a r e s t r a i g h t l i n e s . Thus, each p o r t i o n o f t h e wave
t r a v e l s a t i t s own speed, ü + c. T h i s p r o c e s s was termed a m p l i t u d e d i s p e r s i o n by L i g h t h l l l and Whitham ( 1 9 5 5 ) . A t
t h e l e a d i n g edge t r a v e l s a t speed / i h ; under a c r e s t t h e v e l o c i t y and t h e a m p l i t u d e a r e each g r e a t e r t h a n z e r o ,
hence t h e c r e s t moves f a s t e r t h a n t h e l e a d i n g edge. E v e n t u a l l y t h e r e f o r e t h e c r e s t w i l l o v e r t a k e t h e l e a d i n g
edge and t h e wave w i l l b r e a k . B r e a k i n g may a c t u a l l y occur b e f o r e t h i s depending on t h e shape o f t h e wave,
i v ) For waves t r a v e l i n g t o t h e r i g h t o n l y , t h e v e l o c i t y can be expressed i n terms o f t h e a m p l i t u d e :
and t h e Boussinesq e q u a t i o n s t h e n r e d u c e t o t h e KdV e q u a t i o n ( a f t e r Korteweg and de V r i e s ( 1 8 9 6 ) ) :
S i n c e ,
^ =_Ti + 0 ( a , B )
Eq. (3.19) can be C K p r e s s e d t o t h e same o r d e r a s :
w h i c h i s more amenable to n u m e r i c a l s o l u t i o n ( s e e , f o r
example. P e r e g r i n e ( 1 9 6 6 ) ) .
The KdV e q u a t i o n has e x a c t a n a l y t i c a l s o l u t i o n s i n t h e form
of waves of permanent s h a p e - s o l i t a r y waves and c n o i d a l waves. B e f o r e d i s c u s s i n g these waves i n d e t a i l , an example i s p r e s e n t e d
( i l ) L i n e a r D i s p e r s i v e ( i i i ) N o n l i n e a r N o n d i s p e r s i v e ( i v ) N o n l i n e a r D i s p e r s i v e R e f e r r i n g t o F i g . 3 . 3 ( a ) , t h e p r o b l e m i s posed where a t t = 0 t h e r e e x i s t s , i n w a t e r o f c o n s t a n t d e p t h and I n f i n i t e e x t e n t , a wave w i t h p r o f i l e g i v e n i n d i m e n s i o n a l t e r m s by: n( x , 0 ) = H sech^KX , (3.21)
For t h e example shown, t h e f o l l o w i n g c o n d i t i o n s a p p l y :
^ = 0.05 and k= \-^ h > 4 h3
and f o r t > 0 t h e wave i s assumed t o p r o p a g a t e t o t h e r i g h t i n t o s t i l l w a t e r . F i g s . 3 . 3 ( a ) , ( b ) , ( c ) and ( d ) show t h e wave p r o f i l e s
c a l c u l a t e d u s i n g t h e v a r i o u s t h e o r i e s l i s t e d above a t I n t e r v a l s o f n o n d i m e n s i o n a l t i m e , t / g / h , o f 25. The a b s c i s s a s a r e t w h i c h means t h a t t h e f i g u r e s a r e t h e s e r i e s o f e v e n t s an o b s e r v e r w o u l d see i f he were t r a v e l i n g a t speed Vgïï.
I n F i g . 3 . 3 ( a ) t h e p r o f i l e s f r o m a l l f o u r t h e o r i e s a r e p l o t t e d t o g e t h e r . I n F i g s . 3 . 3 ( b ) , ( c ) and ( d ) t h e 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 compared r e s p e c t i v e l y w i t h t h e l i n e a r d i s p e r s i v e t h e o r y , t h e n o n 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 and t h e n o n l i n e a r d i s p e r s i v e
t h e o r y . Under t h e 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 , t h e wave w o u l d r e m a i n s t a t i o n a r y and r e t a i n i t s o r i g i n a l shape. Under t h e l i n e a r d i s p e r s i v e t h e o r y t h e wave w o u l d p r o p a g a t e as i f i t c o n s i s t e d o f a l i n e a r
V<( (a) /7-;<0<r\ NONLINEAR / / ' / ' \ \ > NONDISPERSIVE I W — ^ L I N E A R ^"^'''^NJoTNONDISPERSIVE -30 -20 -10 F i g . 0 10 20 x / h - t y g 7 h " A / I I ' -V -3' • •
x / h - t T g T h x/h-tygTïï
t h e o v e r a l l shape w o u l d change. _ . l e t s t h a t t h a w a v e w o u l d t e t a l u i t a i n t e g t l t y h o t t h a t t h e c o o r d i n a t e s behxnd t h e c r e b 4 . 0 ^ + - T h e wave w o u l d 3,eepeu w h i l e t h e c t e a t h e i g h t t e m a l u e d c o n s t a n t . h e . , l n t o h t e a . When t h e . o n t . a c e hecame v e r t i c a l c a t t / i T K - , 0 . 9 i n t h i s c a s e ) . /o o n none X. f a c t f o r t h e wave c h o s e n a n d d e s c t l h e d hy K,. C3..1) s o l u t i o n o f t h e KdV e q u a t i o n . Hence,t . a p e a s Shown l n . l g . 3 . 3 C d , . ^ ^ ^J^^: „onld u n d e r t h e U , , . . ^ - . "
^ ' t T r .
t h e wave t a . e s a s I t p r o p a g a t e s i n a . r t l c u l a r c a s e . .he r e l a t i v e M a g n i t u d e s o£ t h e d i s p e r s i v e t e r . ^ ^ o " ^ k . depends on t h e r e i a c i v e & 3 !o t h e d i m e n s i o n a l f o r m o f t h e KdV and t h e n o n l i n e a r t e r m ^ ^ ^^x e q u a t i o n : ^3^22) n + c _ ( l + U b . + l ^ o ^ ^ K x x - ° c = where23
V
3 Hr e p r e s e n t s t h e case where t h e n o n l i n e a r t e r m b a l a n c e s t h e d i s p e r s i v e r 3 H
t e r m and t h e wave shape r e m a i n s c o n s t a n t . I f K < < ^ — ^ t h e n o n l i n e a r t e r m i s l a r g e r t h a n t h e d i s p e r s i v e term (U >> 1) and a m p l i t u d e d i s p e r s i o n as shown i n F i g . 3 . 3 ( c ) t a k e s p l a c e . I f
K » ( i . e . t h e wave i s more peaked t h a n a s o l i t a r y wave o f t h e same h e i g h t ) t h e d i s p e r s i v e t e r m i s l a r g e r t h a n t h e n o n l i n e a r t e r m
( U < < 1 ) and f r e q u e n c y d i s p e r s i o n as shown i n F i g . 3 . 3 ( b ) t a k e s p l a c e . Since t h e KdV o r Boussinesq e q u a t i o n s can be s o l v e d i n t h e
near f i e l d o n l y by a p p r o x i m a t e n u m e r i c a l t e c h n i q u e s , i t i s d e s i r a b l e to use t h e o t h e r e q u a t i o n s wherever p o s s i b l e s i n c e t h e y can be s o l v e d e x a c t l y i n many cases. The p r o b l e m o f w h i c h o f t h e e q u a t i o n s t o
use i n v a r i o u s c i r c u m s t a n c e s i s addressed by Hammack and Segur ( 1 9 7 8 ) . They show t h a t f o r i n i t i a l c o n d i t i o n s o f a r e c t a n g u l a r wave, t h e
a p p l i c a b l e e q u a t i o n depends on t h e I n i t i a l volume and i n i t i a l U r s e l l Number, b u t t h a t e v e n t u a l l y , a f t e r a p r o p a g a t i o n t i m e w h i c h i s a f u n c t i o n o f t h e i n i t i a l c o n d i t i o n s , o n l y t h e KdV e q u a t i o n w i l l a p p l y . T h i s I n t r o d u c e s a n o t h e r i m p o r t a n t p a r a m e t e r i n l o n g wave p r o p a g a t i o n : the p r o p a g a t i o n t i m e .
I t i s e v i d e n t f r o m F i g . 3.3 t h a t i f t h e t i m e o f I n t e r e s t i s 0 < t / g / h < 25 t h e n any o f t h e f o u r t h e o r i e s can be used s i n c e t h e y a l l p r o v i d e e s s e n t i a l l y t h e same r e s u l t s . However f o r t / g / h > 25 the s o l u t i o n s become q u i t e d i f f e r e n t . The I n t e r p r e t a t i o n o f t h i s i s t h a t b o t h d i s p e r s i v e and n o n l i n e a r e f f e c t s t a k e some p r o p a g a t i o n t i m e
..eo.. a o h a . a c t e . i e . i c p . o p a s a t i o . ti»e i a t h e t i m e . nondis£ersive t h e o r y a . K V Ea (3.21) i s a . i ^ i t i a i e o . . i t i o n s^ven B,. „ p „ . i . a t e i , , . . i e t . e o t . a n . t o t t . i s ^vpe o i wave, t . e p r o p a g a t i o n t i m e I h n a f o r t h i e t h e o r y p e r c e n t a g e of . f f e c t a to become i m p o r t a n t i s some p d i s p e r s i v e t h e o r y . t h e . . . • • : . . —
: : : : r •••-•"*•'•
„f t h e o t h e r t h e o r i e s , a p p r o a c h i s to a s s . e t h . - - ^ ^ ^ - e s s t h e J „ of t h i s o f f p c t s t o become i m p o r t a n i : , , , . , e r s i v e e f f e c t r : I f „ . t h e c o n : : ! : : : : : : ^ - - t h e o r i e s e same r e s n i t s . The s o r t Of c o n c i s i o n W h i c h c a n he ' i s t h a t i f , f o r e x a m p i e , t h e 2 0 , U n H ^ i - 2 - H . . i s p e r a i v e e f f e c t s to become i m p o r t a n t .25
3.1.1 The S o l i t a r y Wave
The s o l i t a r y wave was observed f i r s t by S c o t t R u s s e l l ( 1 8 4 4 ) . I t c o n s i s t s o f a s i n g l e hump o f w a t e r e n t i r e l y above s t i l l w a t e r l e v e l and extends f r o m x = -oo t o x = oo. Three t h e o r i e s a r e a v a i l a b l e w h i c h d e s c r i b e t h e wave p r o f i l e ; t h o s e o b t a i n e d by: Boussinesq ( 1 8 7 2 ) , McCowan (1891) and L a i t o n e ( 1 9 6 3 ) . The most i m p o r t a n t o f t h e s e i s t h a t due t o Boussinesq (1872) s i n c e i t i s t h i s f o r m w h i c h i s an e x a c t s o l u t i o n o f t h e KdV e q u a t i o n . I n d i m e n s i o n a l q u a n t i t i e s t h e B o u s s i n e s q s o l i t a r y wave i s : n ( x , t ) = H sech2 ( x - c t ) (3.25) where c = / g ( h + H)
The McCowan and L a i t o n e s o l i t a r y waves r e s u l t f r o m h i g h e r o r d e r t h e o r i e s b u t do n o t f i t e x p e r i m e n t a l d a t a any b e t t e r t h a n does Eq.
(3.25) (see f o r example Naheer ( 1 9 7 7 ) , F r e n c h ( 1 9 6 9 ) ) .
The s o l i t a r y wave has t h e u n i q u e p r o p e r t y t h a t i n a d e p t h h i t i s c o m p l e t e l y d e f i n e d by t h e wave h e i g h t , H. T h i s s i m p l i c i t y o f shape a l o n g w i t h i t s ease o f g e n e r a t i o n i n t h e l a b o r a t o r y and i t s p r o p a g a t i o n w i t h c o n s t a n t shape make t h e s o l i t a r y wave a p a r t i c u l a r l y
s u i t a b l e model wave t o s t u d y e x p e r i m e n t a l l y . For t h i s s t u d y i t had t h e added advantage t h a t when c o n s i d e r i n g r e f l e c t i o n s f r o m a s l o p e o r a s t e p t h e r e f l e c t e d wave was c o m p l e t e l y s e p a r a t e f r o m t h e i n c i d e n t wave.
3.1.2 C n o i d a l Waves
i n d i m e n s i o n a l f o r m t h e y a r e d e f i n e d ( e . g . Svendsen ( 1 9 7 4 ) ) a s :
n( x , t ) = y , - h +
Hcn2l2K
(M)M
, (3.26) where m=a/B i s t h e e l l i p t i c p a r a m e t e r (sometimes c a l l e d k^) , K = K(m)i s t h e f i r s t c o m p l e t e e l l i p t i c i n t e g r a l , c n i s one o f t h e J a c o b i a n e l l i p t i c f u n c t i o n s (hence t h e name c n o i d a l ) , y , i s t h e h e i g h t o f t h e t r o u g h above t h e b o t t o m , L i s t h e wave l e n g t h and T i s t h e p e r i o d .
I t i s n o t e d t h a t f o r g i v e n d e p t h h , c n o i d a l waves a r e d e f i n e d by any two o f t h e f o l l o w i n g :
i ) t h e wave l e n g t h L ( o r t h e p e r i o d T ) , i i ) t h e wave h e i g h t H,
i i i ) t h e e l l i p t i c parameter m ( o r t h e e l l i p t i c i n t e g r a l K ) . The r e l a t i o n s h i p s between t h e s e and t h e o t h e r p a r a m e t e r s were
d e s c r i b e d by W i e g e l (1960) and Svendsen ( 1 9 7 4 ) . They a r e p r e s e n t e d i n Appendix A a l o n g w i t h t h e n u m e r i c a l t e c h n i q u e s w h i c h were d e v e l o p e d d u r i n g t h i s s t u d y f o r t h e i r e v a l u a t i o n .
The e l l i p t i c parameter m, by d e f i n i t i o n , i s t h e U r s e l l Number, i . e . . U = a/3. A n o t h e r t y p e o f U r s e l l Number w h i c h can be d e f i n e d i n t e r m l o f p h y s i c a l p a r a m e t e r s i s HL^/hB. xhe d i f f e r e n c e i n t h e s e two d e f i n i t i o n s i s i n t h e use o f t h e c h a r a c t e r i s t i c l e n g t h . f o r U = a/B and t h e use o f t h e wave l e n g t h L f o r HL^/hS. ^he two numbers a r e
27
hence t h e l e n g t h s a r e r e l a t e d by:
L = -;= K£ (3.28)
Since t h e e l l i p t i c I n t e g r a l K I s a f u n c t i o n o f o n l y t h e parameter m, HL^/h^ i s a l s o a f u n c t i o n o n l y o f m; hence e i t h e r o f t h e U r s e l l
Numbers can be used t o d e f i n e t h e shape o f t h e c n o i d a l wave. The parameter m can t a k e v a l u e s between 0 and 1 . A t t h e two e x t r e m e s :
1) As m->0 ( a n d , c o n s e q u e n t l y , HL^/h^ -^0) , t h e J a c o b i a n
e l l i p t i c f u n c t i o n , c n , becomes t h e t r i g o n o m e t r i c f u n c t i o n , c o s , and K^^' Hence t h e e q u a t i o n f o r c n o i d a l waves, Eq. (3.26) becomes:
i . e . , a harmonic wave,
i i ) As m-»-l ( a n d , c o n s e q u e n t l y , HL^/h^->-"), t h e J a c o b i a n e l l i p t i c f u n c t i o n , c n , becomes t h e h y p e r b o l i c f u n c t i o n , sech, and K, L and T-^". Hence t h e e q u a t i o n f o r c n o i d a l waves, Eq. ( 3 . 2 6 ) , becomes:
The range o f c n o i d a l wave shapes f r o m m = 0 and HL^/h^ = 0 t o xa^l and HL^/h^^-oo i s shown i n F i g . 3.4. F o r H L ^ / h ^ l l O , t h e shape
appears s i n u s o i d a l t o t h e eye b u t i n f a c t a t HL^/h^ = 10 t h e c r e s t a m p l i t u d e i s about 20% g r e a t e r t h a n t h e t r o u g h a m p l i t u d e , i . e . ,
(3.29)
(3.30)
