ON IMPACT PRESSURES CAUSED BY BREAKING WAVES
hy
CONSTANTINOSMOUTZOURIS Research Fellow
Delft University of Technology Department of Civil Engineering
Delft - The Netherlands
PART I : INTRODUCTION page
1. INTRODUCTION 1 2. EXISTING STUDIES ON THE PRESSURES DUE TO THE WAVE BREAKING 3
2 . 1 . I n t r o d u c t i o n 3 2.2. S t u d i e s i g n o r i n g t h e e n t r a p p e d a i r 3
2.3. S t u d i e s c o n s i d e r i n g t h e e n t r a p p e d a i r 7
3. EXPERI1«1ENTAL SET-UP 10
PART I I ; WAVE PROPAGATION AND BREAKING ON A SLOPING STRUCTURE
4. PHENOMENOLOGICAL DESCRIPTION OF THE WAVE PROPAGATION AND 13 BREAKING
5. PRESSURES ON THE STRUCTURE DUE TO THE WAVE BREAKING 16
PART I I I : DETERMINISTIC MODELING OF THE PRESSURES ON THE STRUCTURE
6. FACTORS LOADING THE STRUCTURE DURING THE WAVE BREAKING 19 6.1. Water l a y e r o v e r t h e b r e a k i n g zone o f t h e s t r u c t u r e 19
6.1.1. I n t r o d u c t i o n 19 6.1.2. E v o l u t i o n i n t i m e o f t h e w a t e r l a y e r h e i g h t 19
6.1.3. Maximum and minimum h e i g h t s o f t h e w a t e r l a y e r 25 6.2. Shock p r e s s u r e due t o t h e i m p a c t o f t h e w a t e r j e t 26 6.2.1. I n t r o d u c t i o n 26 6.2.2. Shock p r e s s u r e o n t h e dot«i-rushing w a t e r l a y e r . 28 s u r f a c e 6.2.3. T r a n s m i s s i o n o f t h e shock p r e s s u r e t h r o u g h t h e d o t m - 30 r u s h i n g w a t e r l a y e r 6.2.4. On t h e p r e s s u r e on t h e s t r u c t u r e due t o t h e shock 37 p r e s s u r e o n t h e do\<m-rushing w a t e r l a y e r 6.3. A i r p o c k e t 6.3.1. I n t r o d u c t i o n 39 6.3.2, Movements o f t h e a i r p o c k e t 41 6.3.2.1. D e c r e a s e o f t h e s e c t i o n 41 6.3.2.2. H o r i z o n t a l movement 43 6.3.2.3. V e r t i c a l movement 45
6.3.2.4. R o t a t i o n 48 6.3*-2.5. R a d i a l o s c i l l a t i o n s 48 6.3.3. M o d e l i n g o f t h e a i r p o c k e t 50 6.3.3.1. A i r p o c k e t r a d i u s 50 6.3.3.2. F l o w p a t t e r n 52 6.3.3.3. P r e s s u r e 54 7. TOTAL PRESSURE ON THE STRUCTURE 64
PART I V : PROBABILISTIC ANALYSIS OF THE PRESSURES ON THE STRUCTURE
8 . 1 . I n t r o d u c t i o n 68 8.2. Maximum r e c o r d e d p r e s s u r e i n one wave p e r i o d 69
8.3. Minimum r e c o r d e d p r e s s u r e i n one wave p e r i o d 73
8.4. F i r s t peak o f t h e p r e s s u r e 74
PART V; CONCLUSIONS
9, CONCLUSIONS - FURTHER WORK 85
10. ACKNOWLEDGEMENTS 86
PART V I : APPENDICES
A l . PHOTOGRAPHICAL DESCRIPTION OF THE FLOW I N THE BREAKING ZONE 88
A2, PRESSURE TIME HISTORIES gg
A3. L I S T OF SYMBOLS ] Q 5 A4. L I S T OF REFERENCES i n o
F i g . 3.1 : D e f i n i t i o n s k e t c h o f t h e e x p e r i m e n t s F i g . 4.1 : Zones o f a s l o p i n g m a r i t i m e s t r u c t u r e F i g . 5 . 1 . : T y p i c a l p r e s s u r e - t i m e h i s t o r i e s r e c o r d e d b y t h e t r a n s d u c e r s on t h e s t r u c t u r e F i g . 5.2 : D e c o m p o s i t i o n o f a p r e s s u r e - t i m e h i s t o r y F i g . 6.1 : E v o l u t i o n o f t h e w a t e r s u r f a c e as f u n c t i o n o f t i m e i n t h e b r e a k i n g zone o f t h e s t r u c t u r e F i g . 6.2 : E v o l u t i o n o f t h e w a t e r h e i g h t o v e r t h e t r a n s d u c e r s as f u n c t i o n o f t i m e i n one wave p e r i o d . F i g . 6.3 : Water h e i g h t o v e r t h e t r a n s d u c e r s F i g . 6.4 : H o r i z o n t a l movement o f t h e w a t e r j e t f r o m t h e wave b r e a k i n g
F i g , 6.5 : E v o l u t i o n o f h and h . i n t h e a f t e r - b r e a k i n g zone o f a s l o p i n g " max m m t - o s t r u c t u r e ( f r o m 24 ) , F i g . .6,6 : D e f i n i t i o n s k e t c h o f t h e w a t e r j e t movement F i g . 6.7 : P r e s s u r e - h e a d due t o t h e w a t e r j e t i m p a c t F i g . 6,8 : Some shock p r e s s u r e s as t h e y w e r e r e c o r d e d by t h e t h r e e t r a n s d u c e r s F i g . 6.9 : C o r r e l a t i o n b e t w e e n t h e damping c o e f f i c i e n t and t h e s h o c k p r e s s u r e F i g . 6.10: Damping c o e f f i c i e n t 3 v e r s u s peak p r e s s u r e P^^^ F i g . 6.11: D e c o m p o s i t i o n o f t h e p r e s s u r e I F i g . 6,12: F o r m a t i o n o f t h e a i r p o c k e t d u r i n g a wave b r e a k i n g F i g , 6.13: D e c r e a s e o f t h e c r o s s - s e c t i o n o f t h e a i r p o c k e t ( e x p e r , ) F i g , 6,14: D e c r e a s e o f t h e d i a m e t e r o f t h e a i r p o c k e t ( e x p e r . ) F i g , 6,15: H o r i z o n t a l movement o f t h e a i r p o c k e t ( e x p e r , ) F i g , 6.16: E v o l u t i o n o f t h e h e i g h t o f t h e w a t e r l a y e r belov; t h e a i r p o c k e t ( e x p e r . ) F i g . 6.17: R o t a t i o n o f t h e a i r p o c k e t F i g . 6.18: P r e s s u r e o s c i l l a t i o n s due t o t h e a i r p o c k e t v i b r a t i o n s ( e x p e r . ) F i g . 6.19: A m p l i t u d e o f t h e p r e s s u r e o s c i l l a t i o n s ( e x p e r . ) F i g , 6.20: F r e q u e n c y o f t h e p r e s s u r e o s c i l l a t i o n s ( e x p e r . ) F i g . 6.21: F r e q u e n c y o f t h e a i r p o c k e t v i b r a t i o n s ( t h e o r . ) F i g . 6,22: I n i t i a l f r e q u e n c y o f t h e a i r p o c k e t v i b r a t i o n s ( t h e o r , : ) F i g , 6,23: Minimum speed o f p r o p a g a t i o n o f t h e sound i n t h e d o T O - r u s h i n g
l a y e r ( t h e o r . )
F i g , 6.24: E v o l u t i o n o f t h e h e i g h t o f t h e w a t e r l a y e r above t h e a i r p o c k e t ( e x p e r . )
F i g . 7 . 1 : P r e s s u r e - h e a d s on t h e s t r u c t u r e f r o m t h e t h r e e f a c t o r s ( t h e o r . ) F i g . 8 . 1 : C h a r a c t e r i s t i c s t a t i s t i c a l v a l u e s o f P and P . max mm F i g . 8.2: S i g n i f i c a n t v a l u e s o f P and P . max m m h F i g . 8.3: C u m u l a t i v e d i s t r i b u t i o n s o f P and P . ( v ^ = 0.08) max m m L h F i g . 8.4: C u m u l a t i v e d i s t r i b u t i o n s o f P and P . ° (-2- = 0.14) max m m h L F i g . 8.5: C u m u l a t i v e d i s t r i b u t i o n s o f P and P . (^^ = °0.19) max m m L F i g . 8.6: C u m u l a t i v e d i s t r i b u t i o n s o f P and P . u n 3 e r a s h i f t o f t h e max m m t i m e s c a l e F i g . 8.7: C u m u l a t i v e d i s t r i b u t i o n s o f P, L I S T OF TABLES T a b l e 3 . 1 : Examined waves T a b l e 8 . 1 : C h a r a c t e r i s t i c T a b l e 8.2: C h a r a c t e r i s t i c T a b l e 8.3: C h a r a c t e r i s t i c and a m b i e n t p r e s s u r e s s t a t i s t i c a l v a l u e s o f P max s t a t i s t i c a l v a l u e s o f P .
mm
s t a t i s t i c a l v a l u e s o f P,1. INTRODUCTION
E n g i n e e r s a r e i n t e r e s t e d i n t h e dynamics o f t h e i n t e r f a c e w a t e r s t r u c t u r e . I n case o f b r e a k i n g o f w a t e r waves on a s t r u c t u r e h i g h p o s i t i v e and sometimes n e g a t i v e p r e s s u r e s o f v e r y s h o r t d u r a t i o n o c c u r .
N o t o n l y t h e maxima and m i n i m a o f t h e p r e s s u r e s on t h e s t r u c t u r e a r e i m p o r t a n t t o a d e s i g n i n g e n g i n e e r , b u t a l s o t h e p r e s s u r e - t i m e h i s t o r y : f a i l u r e o f t h e s t r u c t u r e may o c c u r when f o r t h e f i r s t t i m e t h e p r e s s u r e r e a c h e s a c e r t a i n l e v e l o r when t h e a c c u m u l a t i o n o f s m a l l damages due t o l o w e r p r e s s u r e s r e a c h e s a c e r t a i n l e v e l .
The b r e a k i n g o f w a t e r waves on t h e s t r u c t u r e i s a p r o c e s s i n a t h r e e -p h a s e s - s y s t e m , because a l m o s t a l w a y s a i r i s e n t r a -p -p e d i n t h e mass o f t h e b r e a k i n g wave. The l a r g e number o f p a r a m e t e r s w h i c h c o n t r o l t h e d e v e l o p e d p r e s s u r e s can be c l a s s i f i e d i n t h r e e g r o u p s , a c c o r d i n g t o t h e t h r e e phases w a t e r , s o l i d a i r :
- w a t e r and wave p a r a m e t e r s such as w a t e r d e p t h , wave p e r i o d and h e i g h t . - s t r u c t u r e p a r a m e t e r s such as g e o m e t r y o f t h e i n t e r f a c e , m a t e r i a l
c o n s t a n t s
- a i r p a r a m e t e r s s u c h as a m b i e n t p r e s s u r e .
Many o f t h e p a r a m e t e r s a r e n o t c o n t r o l l a b l e and change c o n t i n u o u s l y . F o r t h i s r e a s o n t h e r e i s no p r e s s u r e h i s t o r y i d e n t i c a l t o a p r e v i o u s one. O n l y g e n e r a l t r e n d s r e m a i n t h e same. As most o f t h e l o a d s on a s t r u c t u r e , p r e s s u r e s due t o t h e wave b r e a k i n g a r e s t o c h a s t i c .
U n t i l r e c e n t l y i t was c o n s i d e r e d t h a t t h e wave p a r a m e t e r s w e r e t h e most i m p o r t a n t : t h e b r e a k i n g mechanism depends m a i n l y o n them. Recent e x p e r i m e n t a l i n d i c a t i o n s show t h a t t h e a i r p a r a m e t e r s have a l s o an i m p o r t a n t r o l e i n t h e wave b r e a k i n g l o a d s . A i r p o c k e t s and b u b b l e s e n t r a p p e d i n t h e w a t e r mass ^ d u r i n g t h e b r e a k i n g i n f l u e n c e t h e p r e s s u r e . The d y n a m i c s o f t h e i n t e r f a c e
a i r - w a t e r a r e i m p o r t a n t : some o s c i l l a t i n g p r e s s u r e s on t h e s t r u c t u r e a r e due t o t h e v i b r a t i o n s o f a m a j o r a i r p o c k e t .
I n s c a l e model i n v e s t i g a t i o n s o f i m p a c t p r e s s u r e s caused by waves b r e a k i n g on o r a g a i n s t a s t r u c t u r e , t h e c o m p r e s s i b i l i t y o f t h e a i r i s n o r -m a l l y n o t s c a l e d down. Thus, s c a l e e f f e c t s a r e t o be e x p e c t e d when t h e -m o d e l r e s u l t s a r e c o n v e r t e d i n t o l a r g e r - s c a l e p r o t o t y p e v a l u e s a c c o r d i n g t o F r o u d e ' s
2
model l a w . One way o f i n v e s t i g a t i n g t h e n a t u r e and m a g n i t u d e o f such e f f e c t s i s t o p e r f o r m p r e s s u r e measurements f o r t h e same s t r u c t u r e and i n c i d e n t waves b u t w i t h d i f f e r e n t a m b i e n t a i r p r e s s u r e s , s i n c e t h e c o m p r e s s i b i l i t y o f t h e a i r v a r i e s w i t h t h e p r e s s u r e .
I t i s w i t h t h i s o b j e c t i v e t h a t t h e Board o f M a r i t i m e Works i n t h e N e t h e r l a n d s c o m m i s s i o n e d t h e D e l f t H y d r a u l i c s L a b o r a t o r y t o c a r r y o u t an i n v e s t i g a -t i o n on -t h e p r e s s u r e s o c c u r r i n g d u r i n g -t h e b r e a k i n g o f w a -t e r waves on a s l o p i n g s t r u c t u r e w i t h d i f f e r e n t i n p u t waves and two a m b i e n t p r e s s u r e s : a t m o s -p h e r i c and n e a r vacuum. E x -p e r i m e n t s w e r e -p e r f o r m e d by t h e D e l f t H y d r a u l i c s L a b o r a t o r y . i n a s m a l l wave f l u m e o f t h e L a b o r a t o r y o f F l u i d M e c h a n i c s o f t h e C i v i l E n g i n e e r i n g D e p a r t m e n t , D e l f t U n i v e r s i t y o f T e c h n o l o g y , i n x<rhich t h e a m b i e n t a i r p r e s s u r e c o u l d be v a r i e d . A r e p o r t on t h e e x p e r i m e n t s , t h e r e s u l t s o b t a i n e d , a n d i n t e r p r e t a t i o n s t h e r e o f , i s i n p r e p a r a t i o n w i t h i n t h e D e l f t H y d r a u l i c s L a b o r a t o r y . I n t h e m e a n t i m e , t h e e x p e r i m e n t a l d a t a o b t a i n e d w e r e made a v a i l a b l e t o t h e a u t h o r , who was t h e n a R e s e a r c h F e l l o w a t 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 p u r p o s e s o f a n a l y s i s . Such a n a l y s i s was t o be c a r r i e d o u t i n d e p e n d e n t l y , and was i n no way a p a r t i a l f u l f i l l m e n t o f t h e c o n t r a c t u a l o b l i -g a t i o n s o f t h e D e l f t H y d r a u l i c s L a b o r a t o r y . Thus, a l t h o u -g h t h e d a t a u s e d i n t h e p r e s e n t s t u d y w e r e c o l l e c t e d b y t h e D e l f t H y d r a u l i c s L a b o r a t o r y , t h e i n t e r p r e t a t i o n s t h e r e o f and c o n c l u s i o n s d e r i v e d f r o m them as s t a t e d i n t h e p r e s e n t r e p o r t a r e t h e a u t h o r ' s own.
The p r e s e n t r e p o r t c o n t a i n s t h e r e s u l t s o f a t h e o r e t i c a l a n a l y s i s o f t h e p r e s s u r e s on a s l o p i n g s t r u c t u r e due t o t h e wave b r e a k i n g and an a n a l y s i s o f d a t a f r o m t h e above m e n t i o n e d e x p e r i m e n t s . S p e c i a l a t t e n t i o n i s p a i d t o t h e r o l e o f t h e e n t r a p p e d a i r p o c k e t and o f t h e a m b i e n t p r e s s u r e . A s t r i c t o v e r a l l m o d e l l i n g o f t h e p r e s s u r e s i s i m p o s s i b l e . F o r t h i s r e a s o n t h e mecha-n i s m amecha-nd g e mecha-n e r a l t r e mecha-n d s o f t h e p r e s s u r e h i s t o r y a r e a mecha-n a l y s e d amecha-nd d e s c r i b e d b y means o f p a r t i a l d e t e r m i n i s t i c m o d e l s , w h i c h seems t o be a q u i t e r e a l i s t i c a p p r o a c h . On t h e o t h e r h a n d , t h e v a l u e s o f t h e p r e s s u r e s a r e d e s c r i b e d s t a t i s -t i c a l l y , w h i c h i s a more r e a s o n a b l e a p p r o a c h because o f -t h e i r s -t o c h a s -t i c c h a r a c t e r . The p l a n o f t h e r e p o r t i s as f o l l o w s : - A b r i e f a n a l y s i s o f e x i s t i n g a p p r o a c h e s t o t h e p r o b l e m i s g i v e n f i r s t . - A p h e n o m e n o l o g i c a l d e s c r i p t i o n o f t h e wave p r o p a g a t i o n and b r e a k i n g on a s l o p i n g s t r u c t u r e i s t h e n made.
T y p i c a l p r e s s u r e t i m e s i g n a l s a r e p r e s e n t e d , as t h e y have been r e c o r -ded. Each one o f them i s t h e r e s u l t o f a s u p e r p o s i t i o n o f p a r t i a l p r e s s u r e - t i m e h i s t o r i e s due t o a number o f l o a d i n g f a c t o r s .
- The m a i n l o a d i n g f a c t o r s a r e a n a l y s e d and d e t e r m i n i s t i c m o d e l i n g i s a t t e m p t e d .
- A s t a t i s t i c a l a n a l y s i s o f some c h a r a c t e r i s t i c v a l u e s o f t h e p r e s s u r e i s f i n a l l y made.
2. EXISTING STUDIES ON THE PRESSURES DUE TO THE WAVE BREAKING
2.1 I n t r o d u c t i o n
A q u i t e l a r g e number o f s t u d i e s has been made, b o t h e x p e r i m e n t a l and t h e o r e t i c a l , on t h e p r e s s u r e s o c c u r r i n g on a s t r u c t u r e due t o t h e wave b r e a k i n g on i t . The i n f l u e n c e o f t h r e e g r o u p s o f p a r a m e t e r s has been more o r l e s s c h e c k e d i n t h e s e s t u d i e s :
- The wave p a r a m e t e r s :
They a r e t h e most e x t e n s i v e l y e x a m i n e d , a l t h o u g h t h e r e a r e n o t enough r e s u l t s r e l a t i n g t h e p r e s s u r e w i t h t h e wave p a r a m e t e r s . - The s t r u c t u r e p a r a m e t e r s : I n most o f t h e s t u d i e s t h e s t r u c t u r e - w a t e r i n t e r f a c e i s p l a n e and v e r t i c a l . - The a i r p a r a m e t e r s : Some t h e o r e t i c a l s t u d i e s e x i s t i n w h i c h i t i s t r i e d t o m o d e l t h e i n f l u e n c e o f t h e e n t r a p p e d a i r . I n some e x p e r i m e n t a l s t u d i e s t h e i n f l u e n c e o f t h e a m b i e n t p r e s s u r e and o f t h e t h i c k n e s s o f t h e e n -t r a p p e d a i r l a y e r on -t h e shock p r e s s u r e i s c h e c k e d .
Some o f t h e most p i o n e e r i n g s t u d i e s a r e now b r i e f l y r e v i e w e d . They a r e d i v i d e d i n t o txro c a t e g o r i e s a c c o r d i n g t o t h e scope o f t h i s i n v e s t i g a t i o n : t h o s e w h i c h i g n o r e t h e e n t r a p p e d a i r and t h o s e w h i c h t a k e i t i n t o a c c o u n t . 2.2. S t u d i e s i n g o r i n g t h e e n t r a p p e d a i r E a r l y e x p e r i m e n t a l s t u d i e s on t h e ' p r e s s u r e s d u r i n g t h e wave b r e a k i n g w e r e c a r r i e d o u t w i t h o u t s e n s i t i v e t r a n s d u c e r s . F o r t h a t r e a s o n many d e t a i l s o f t h e p r e s s u r e r e s p o n s e , s u c h as t h e i n f l u e n c e o f t h e e n t r a p p e d a i r , w e r e n o t d e t e c t e d . Examples o f s u c h e a r l y s t u d i e s a r e t h e w o r k s o f G a i l l a r d 12 , H i r o i 15 and de R o u v i l l B e s s o n P e t r y 31 on c o n s t r u c t e d v / a l l s and b r e a k
-4 w a t e r s , L a r r a s 18 was t h e f i r s t t o c o l l e c t e x p e r i m e n t a l d a t a i n a l a b o r a t o r y f l u m e . A l l t h e s e e x p e r i m e n t a l w o r k s d i d n o t p r e s e n t any r e l a t i o n b e t w e e n t h e p r e s s u r e and t h e wave p a r a m e t e r s . Some o t h e r i n v e s t i g a t o r s c o l l e c t e d e x p e r i m e n t a l d a t a and a t t e m p t e d t o d e v e l o p e m p i r i c a l r e l a t i o n s based on t h e s e d a t a : Rundgren 32 r e p o r t s t h e r e s u l t s o f an e x p e r i m e n t a l s t u d y on t h e b r e a k i n g wave p r e s s u r e s on a w a l l . He c o n c l u d e s t h a t t h e l a r g e s t peak P ^ ^ max of t h e p r e s s u r e - t i m e h i s t o r y , due t o t h e s h o c k , i s r e l a t e d t o t h e
deep-w a t e r deep-wave h e i g h t H^ and l e n g t h L^. -Based on h i s r e s u l t s and t h e r e s u l t s o f o t h e r i n v e s t i g a t o r s t h e a u t h o r p r o p o s e s t h e f o l l o w i n g e x p r e s s i o n : w o o w h e r e e i s t h e u n i t w e i g h t o f w a t e r , C, and C„ two c o n s t a n t s , w 1 z M i n i k i n 20 p r o p o s e s an e q u a t i o n f o r P caused b y b r e a k i n g waves on _ max v e r t i c a l b r e a k w a t e r s p l a c e d on t h e t o p o f a s l o p e . T h i s e q u a t i o n i s d e r i v e d f r o m t h e e x p e r i m e n t a l d a t a o f B a g n o l d and h i s o\m: h. H P = 102.4 h ^ ( l + r ^ ) — i n ton/m max t h L o o w h e r e h ^ i s t h e w a t e r d e p t h a t t h e t o e o f t h e w a l l and h ^ t h e w a t e r d e p t h a t t h e t o e o f t h e s l o p i n g b o t t o m . A c c o r d i n g t o M i n i k i n , P o c c u r s a l w a y s a t t h e s t i l l w a t e r l e v e l . A t max •' o t h e r p o i n t s on t h e w a t e r - s t r u c t u r e s u r f a c e t h e p r e s s u r e P i s g i v e n by: P = P (1 - 1 ^ ) ^ max H o w h e r e X i s .the d i s t a n c e o f t h e c o n s i d e r e d p o i n t f r o m t h e s t i l l w a t e r l e v e l . M i n i k i n ' s e q u a t i o n i s v e r y w i d e l y used i n c o a s t a l e n g i n e e r i n g . N a g a i 26 p e r f o r m e d e x t e n s i v e e x p e r i m e n t s on t h e b r e a k i n g v;ave p r e s s u r e s on a v e r t i c a l w a l l p l a c e d a l s o on t h e t o p o f a s l o p e . He assumes t h e s h o c k p r e s s u r e t o be due t o t h e i n s t a n t a n e o u s momentum change d u r i n g t h e i m p a c t and p r e s e n t s a t h e o r e t i c a l e x p r e s s i o n f o r P ,.A c o n s t a n t i n c l u d e d i n t h i s
max
e x p r e s s i o n i s e v a l u a t e d f r o m h i s e x p e r i m e n t a l d a t a w h i c h he d i v i d e s i n t o two c a t e g o r i e s : p r e s s u r e s o f o r d i n a r y b r e a k i n g waves and p r e s s u r e s o f e x t r a -o r d i n a r y b r e a k i n g waves. E x t r a -o r d i n a r y b r e a k i n g waves a r e d e f i n e d as waves
w h i c h l o a d t h e s t r u c t u r e w i t h e x t r a o r d i n a r y h i g h shock p r e s s u r e s . N a g a i does n o t s p e c i f y t h e l i m i t b e t w e e n t h e two c a t e g o r i e s . F i n a l l y he p r o p o s e s : 2
K
H 9 P = 20 + 500 £ r ^ - i n gr/cm max w h L o o f o r o r d i n a r y b r e a k i n g waves and h^ H O O and hl E . P = 280 (0.04 + — - — ) ' i n gr/cm max h L ^ o of o r e x t r a o r d i n a r y b r e a k i n g waves, where h ^ and h ^ a r e t h e same as i n M i n i k i n ' s a p p r o a c h . I t i s n o t e d t h a t t h e n u m e r i c a l c o e f f i c i e n t s i n N a g a i ' s e x p r e s s i o n s have a l l k i n d o f d i m e n s i o n s . C o n c e r n i n g t h e v e r t i c a l d i s t r i b u t i o n o f t h e p r e s s u r e o v e r t h e w a l l , he d i s t i n g u i s h e s two t y p e s o f d i s t r i b u t i o n s . I n t h e f i r s t t y p e , P o c c u r s a t III 3.x the s t i l l w a t e r l e v e l and t h e p r e s s u r e i s d i s t r i b u t e d as:
P = P (1 - r ^ ) ^ max H o where .X i s t h e d i s t a n c e f r o m t h e s t i l l w a t e r l e v e l measured i n b o t h d i r e c t i o n s on t h e s t r u c t u r e - w a t e r i n t e r f a c e . I n t h e second t y p e P o c c u r s a t t h e t o e max of t h e w a l l and t h e p r e s s u r e i s d i s t r i b u t e d as: ^ = ^max('
- 2 I - ) '
owhere X i s measured i n one d i r e c t i o n .
N a g a i w r i t e s t h a t t h e v e r t i c a l d i s t r i b u t i o n o f t h e p r e s s u r e o v e r t h e w a l l i s a f f e c t e d b y t h e shape o f t h e b r e a k w a t e r and t h e b e h a v i o r o f t h e b r e a k i n g wave. He does n o t s p e c i f y u n d e r w h i c h c o n d i t i o n s t h e two p r o p o s e d d i s t r i b u t i o n s a p p e a r .
G a r c i a 13 makes an a t t e m p t t o e s t a b l i s h an e m p i r i c a l r e l a t i o n b e t w e e n P and t h e wave l e n g t h and h e i g h t :
max ^ L
P = e H f ( - ^ ) max w o H
6 L where f ( - r ^ ) i s a f u n c t i o n t o be d e f i n e d f r o m e x p e r i m e n t a l d a t a . F i n a l l y H he p r o p o s e s : P = 5 0 ^ 2 / 3 ^ 1 / 3 max w o w h e r e i s t h e d e e p - w a t e r wave e n e r g y p e r u n i t c r e s t l e n g t h : E = i e H^L o 8 w o o
The l o c a t i o n o f t h e maximum s h o c k p r e s s u r e i s f o u n d t o depend on t h e s l o p e i n f r g n t o f t h e w a l l , on t h e w a t e r d e p t h h ^ a t t h e ;toe o f t h e w a l l and f i n a l l y on - — . XJ The v e r t i c a l d i s t r i b u t i o n o f P i s as f o l l o w s , a c c o r d i n g t o G a r c i a : max ' 2X 2 P = P ( 1 - r — ) above t h e l o c a t i o n o f P max h ^ max 1 S X 9 P = P ( 1 - -Ll2±.)^ b e l o w t h e l o c a t i o n o f P max h ^ max The Waterways E x p e r i m e n t S t a t i o n F o r m u l a 16 i s b a s e d on t h e s h o c k f r o n t s w h i c h a p p e a r d u r i n g t h e b r e a k i n g . They a r e due t o t h e n o n l i n e a r c h a -r a c t e -r o f t h e i m p a c t on s o l i d s and -r e p -r e s e n t s u -r f a c e s o f d i s c o n t i n u i t y c -r o s s e d by t h e f l o w .
D u r i n g t h e b r e a k i n g two shock f r o n t s a r e c r e a t e d : t h e one p r o p a g a t e s i n t h e w a t e r w i t h a speed C and t h e o t h e r i n t h e s t r u c t u r e w i t h a speed C .
^ w s Ahead o f t h e f i r s t f r o n t t h e w a t e r i s m o v i n g w i t h a c o n s t a n t v e l o c i t y V , and
s n ahead o f t h e second one t h e s t r u c t u r e i s a t r e s t w h i l e b e h i n d i t t h e s t r u c t u r e moves w i t h a c o n s t a n t v e l o c i t y V . The p r e s s u r e b e h i n d b o t h shock f r o n t s
s t ^ r e p r e s e n t s t h e shock p r e s s u r e P .
max
C o n s e r v a t i o n o f mass and momentum a t t h e two f r o n t s g i v e s f i n a l l y : p C p C s s s s ^max " p ( V ^ + C ) + p C ^w^-^sh sh ~ p C + p C ^wS^^sh w sh w "^s s "^w w "^s s w h e r e p ( p ) i s t h e mass d e n s i t y o f w a t e r ( s t r u c t u r e ) , C (C ) i s t h e speed w s w s o f sound i n w a t e r ( s t r u c t u r e ) , and V^^ i s t h e v e l o c i t y o f i m p a c t . When t h e s h o c k f r o n t i s t r a v e l l i n g f r o m t h e p o i n t o f i m p a c t t o t h e n e a r e s t f r e e s u r f a c e t h e wave i s a c o m p r e s s i o n one. A f t e r i t s r e f l e c t i o n on
t h e f r e e s u r f a c e i t becomes a t e n s i o n one. The d u r a t i o n o f t h e s h o c k p r e s s u r e i s e q u a l t o t h e t i m e n e c e s s a r y f o r a wave t o t r a v e l w i t h t h e sound speed C^
i n w a t e r f r o m t h e i m p a c t t o t h e f r e e s u r f a c e and t h e n b a c k .
Kamel 16 compared t h e shock p r e s s u r e s as t h e y a r e p r e d i c t e d by t h e above f o r m u l a w i t h e x p e r i m e n t a l d a t a o b t a i n e d b y d r o p p i n g a p l a t e i n t o a w a t e r mass. The i m p a c t s u r f a c e was p a r a l l e l t o t h e w a t e r s u r f a c e . A c c o r d i n g
t o t h e a u t h o r , W.E.S. f o r m u l a p r e d i c t s v a l u e s much l a r g e r t h a n t h e e x p e r i m e n -t a l ones. The d i s c r e p a n c y i s due -t o -t h e p r e s e n c e o f e n -t r a p p e d a i r b e -t w e e n -t h e a c c e l e r a t e d p l a t e and t h e w a t e r s u r f a c e .
Von Karman [35] g i v e s a s i m p l e e x p r e s s i o n f o r t h e maximum shock p r e s s u r e •^max ^^-^"^^ c a n , o c c u r d u r i n g a w a t e r hammer i m p a c t : P = p V max w s h w where V , i s t h e v e l o c i t y o f i m p a c t , s rl 2.3. S t u d i e s c o n c e r n i n g t h e e n t r a p p e d a i r Some i n v e s t i g a t i o n s h a v e b e e n c a r r i e d o u t i n t h e d i r e c t i o n o f t h e i n f l u -ence o f t h e e n t r a p p e d a i r d u r i n g t h e b r e a k i n g on t h e p r e s s u r e s l o a d i n g t h e s t r u c t u r e . B a g n o l d 2 was t h e f i r s t t o p r o p o s e a m a t h e m a t i c a l m o d e l . I t i s a w a t e r -hammer model a c c o r d i n g t o w h i c h a t h i n l a y e r o f a i r o f t h i c k n e s s d i s e n t r a p p e d and compressed b e t w e e n t h e f a c e o f t h e b r e a k i n g wave and t h e s t r u c t u r e . The c o m p r e s s i n g w a t e r mass has a u n i t c r o s s - s e c t i o n a r e a and a l e n g t h K. I t moves w i t h t h e same v e l o c i t y V^^ as t h e s t r i k i n g wave f r o n t . The c o m p r e s s i o n i s a d i a b a t i c a l and t h e a m b i e n t p r e s s u r e p^ i s e q u a l t o t h e a t m o s p h e r i c one. The shock p r e s s u r e P i s due t o t h e sudden r e d u c t i o n o f t h e w a t e r mass momentum,
max The e q u a t i o n o f m o t i o n o f t h e w a t e r mass o v e r t h e a i r l a y e r i s : p K ^ - p + p = 0 w ^^2 o y w h e r e y i s t h e a d i a b a t i c c o n s t a n t o f t h e a i r and y t h e d i s t a n c e i n t h e d i r e c t i o n o f c o m p r e s s i o n . T h i s e q u a t i o n i s i n t e g r a t e d g r a p h i c a l l y and g i v e s p r e s s u r e -t i m e c u r v e s f r o m w h i c h P i s -t a k e n : max 2.7 p v \ p = ^ L ^ K max d K i s a p p r o x i m a t e l y e q u a l t o H /5. o
B a g n o l d ' s model i s t h e f i r s t t o t a k e i n t o a c c o u n t t h e a i r p o c k e t e n -t r a p p e d i n -t h e w a -t e r mass d u r i n g -t h e b r e a k i n g . I -t c o n s i d e r a b l y s i m p l i f i e s t h e r e a l mechanism o f b r e a k i n g and p r e d i c t s v a l u e s P much h i g h e r t h a n
max ^ t h e o b s e r v e d ones,
Weggel and M a x w e l l 36] p r o p o s e a d i f f e r e n t model f o r t h e t e m p o r a l and s p a t i a l d i s t r i b u t i o n o f t h e shock p r e s s u r e . They assume t h a t t h e a i r i s u n i f o r m l y m i x e d w i t h t h e w a t e r and n o t t h a t t h e two phases a r e s e p a r a t e d , as i t was i n t h e model o f B a g n o l d , They u s e two e q u a t i o n s f o r t h e t w o d i m e n -s i o n a l c o n -s e r v a t i o n o f momentum, t h e e q u a t i o n o f c o n t i n u i t y and an e q u a t i o n o f s t a t e f o r t h e c o m p r e s s i b l e m i x t u r e . They assume t h a t b o t h t h e b o t t o m and f r e e s u r f a c e a r e h o r i z o n t a l and t h a t t h e f r e e s u r f a c e i s i n d e p e n d e n t o f t i m e . A d i s t u r b a n c e r e p r e s e n t i n g t h e shock i s i n t r o d u c e d i n t h e i n t e r f a c e s t r u c t u r e -w a t e r . The s p a t i a l and t e m p o r a l v a r i a t i o n s o f t h e d i s t u r b a n c e a r e f o l l o -w e d w i t h a n u m e r i c a l s o l u t i o n o f t h e e q u a t i o n s . The n u m b e r i c a l r e s u l t s compare f a v o r a b l y x ^ i t h e x p e r i m e n t a l d a t a c o l l e c t e d i n a l a b o r a t o r y f l u m e . S e l l a r s 33 p r o p o s e s an e x p r e s s i o n f o r t h e maximum shock p r e s s u r e P max d u r i n g t h e i m p a c t o f a l i q u i d - a i r m i x t u r e on an e l a s t i c s t r u c t u r e : max I P^^ 6 Pgg 6 w h e r e P^^ i s t h e a m b i e n t p r e s s u r e , V^^ t h e v e l o c i t y a t t h e i m p a c t , t h e mass d e n s i t y o f p u r e l i q u i d , t h e sound speed i n p u r e l i q u i d , c t h e s t r u c -t u r e impedance r a -t i o and 6 -t h e l i q u i d - a i r m i x -t u r e v o l u m e -t r i c impedance r a -t i o . C o e f f i c i e n t c i s e q u a l t o 9.1 f o r a r i g i d s t r u c t u r e and t o 0.1 f o r a f l e x i b l e s t r u c t u r e , Führböter 11 a r r i v e s a t a s e m i - e m p i r i c a l f o r m u l a f o r t h e s h o c k p r e s s u r e P due t o t h e wave a t t a c k on a s t r u c t u r e , max P = P .V ^.C ( - ^ A ) ' / ^ max w s h w V , sh
where p i s t h e w a t e r d e n s i t y , V . t h e i m p a c t v e l o c i t y , C t h e sound speed w sb w
i n w a t e r and A t h e i m p a c t - n u m b e r d e f i n e d a s :
where E and E a r e t h e e l a s t i c i t y o r Young's modulus o f w a t e r and a i r , R. the h y d r a u l i c r a d i u s o f t h e i m p a c t zone and d t h e t h i c k n e s s o f t h e a i r
c u s h i o n . The t i m e o f p r e s s u r e r i s e t ^ ^ ^ i s g i v e n b y : R. t sh w T h e r e a r e n o t many e x p e r i m e n t a l r e s u l t s c o n c e r n i n g t h e i n f l u e n c e o f t h e a i r c u s h i o n and o f t h e a m b i e n t p r e s s u r e on t h e shock p r e s s u r e due t o t h e wave b r e a k i n g . C o n c l u s i o n s have been dratTO f r o m e x p e r i m e n t s where t h e s h o c k was p r o d u c e d by t h e i m p a c t o f a r i g i d s u r f a c e on a w a t e r l a y e r . The r e s u l t s o f t h e s e e x p e r i m e n t s were e x t r a p o l a t e d t o t h e wave b r e a k i n g , w h i c h i s an i m p a c t b e t w e e n two w a t e r masses. Here a r e some o f t h e s e e x p e r i m e n t a l r e s u l t s :
Kamel 16] c a r r i e d o u t e x p e r i m e n t s i n a t a n k f i l l e d w i t h s t i l l w a t e r . A p l a t e was r e l e a s e d f r o m a c e r t a i n p o s i t i o n and t h e p r e s s u r e was r e c o r d e d . The same e x p e r i m e n t s w e r e c o n d u c t e d w i t h a d i s t u r b e d w a t e r s u r f a c e . He c o n -c l u d e s t h a t t h e sho-ck p r e s s u r e i n -c r e a s e s when t h e r e i s no a i r and d e -c r e a s e s w i t h i n c r e a s i n g a i r l a y e r t h i c k n e s s .
B a g n o l d 2 w r i t e s t h a t a c c o r d i n g t o h i s e x p e r i m e n t s t h e p r e s s u r e due t o t h e shock i s l a r g e r when t h e e n t r a p p e d a i r c u s h i o n i s t h i n n e r . The s h o c k p r e s s u r e due t o i d e n t i c a l waves d i f f e r s f r o m one e x p e r i m e n t t o t h e o t h e r b e -cause o f s m a l l i r r e g u l a r i t i e s o n t h e wave f r o n t b u t t h e i m p u l s e i s r a t h e r c o n s t a n t .
R i c h e r t [ 2 9 w r i t e s t h a t t h e l a r g e s t p r e s s u r e on t h e s t r u c t u r e a l w a y s o c c u r s where t h e e n t r a p p e d a i r c u s h i o n was i n i t i a l l y s i t u a t e d . I f t h e e n t r a p p e d a i r c u s h i o n i s t h i n n e r , t h e maximum shock p r e s s u r e w i l l be h i g h e r and t h e
d u r a t i o n s h o r t e r .
A c c o r d i n g t o Ross [ 3 0 ] , t h e a i r , w h i c h i s a l w a y s e n t r a p p e d b y t h e i r r e g u l a r i t i e s o f t h e wave f r o n t and as b u b b l e s i n t h e w a t e r mass, i s com-p r e s s e d . I t s c u s h i o n i n g e f f e c t l o w e r s t h e t o com-p com-p r e s s u r e and i n c r e a s e s t h e d u r a t i o n o f t h e p r e s s u r e .
N e g a t i v e p r e s s u r e s show t h a t t h e a i r had been compressed so much t h a t i n r e - e x p a n d i n g i t t h r e w t h e w a t e r b a c k t o cause t h e p r e s s u r e o f t h e t r a p p e d
10
a i r t o d r o p n e g a t i v e . A r e g u l a r v i b r a t i o n o f t h e p r e s s u r e i n d i c a t e s r e p e a t e d c o n t r a c t i o n s and e x p a n s i o n s o f a b u b b l e .
Ackerman and Chen 1 c o n d u c t e d e x p e r i m e n t s i n a vacuum t a n k i n o r d e r t o i n v e s t i g a t e t h e e f f e c t o f t h e a i r on i m p a c t l o a d s o f b r e a k i n g waves. The i m p a c t was p r o d u c e d on a s t i l l w a t e r s u r f a c e b y a f l a t p l a t e w i t h r i n g s . D i f f e -r e n t v o l u m e s o f a i -r w e -r e e n t -r a p p e d b e t w e e n t h e f a l l i n g p l a t e and t h e w a t e -r .
The e x p e r i m e n t s show t h a t a p a r t o f t h e p r e s s u r e i s due t o t h e e n t r a p p e d a i r and t h a t t h e shock p r e s s u r e d e c r e a s e s w i t h r e d u c t i o n i n t h e v o l u m e o f t h e e n t r a p p e d a i r and i n t h e a m b i e n t p r e s s u r e . Even when a i r was t o t a l l y r e m o v e d , w a t e r hammer c o n d i t i o n s w e r e n e v e r f o u n d .
I t i s n o t e d t h a t t h e c o n c l u s i o n o f Ackerman and Chen a c c o r d i n g t o w h i c h t h e s h o c k p r e s s u r e due t o t h e i m p a c t d e c r e a s e s w i t h t h e v o l u m e o f t h e e n t r a p p e d a i r i s i n c o n t r a d i c t i o n w i t h a l l t h e p r e v i o u s l y r e v i e w e d r e s u l t s . I n c o n c l u s i o n , i f t h e r e s u l t s c o n c e r n i n g t h e i n f l u e n c e o f t h e e n t r a p p e d a i r and o f t h e a m b i e n t p r e s s u r e on t h e shock p r e s s u r e c o u l d be e x t r a p o l a t e d t o t h e wave b r e a k i n g , i t s h o u l d be c o n c l u d e d t h a t : - t h e a i r p a r a m e t e r s have an i m p o r t a n t r o l e i n t h e p r e s s u r e s on t h e s t r u c t u r e due t o t h e wave b r e a k i n g - t h e e n t r a p p e d a i r h a s a c u s h i o n e f f e c t on t h e shock p r e s s u r e w h i c h i n c r e a s e s w i t h t h e q u a n t i t y o f t h e a i r and w i t h t h e a m b i e n t p r e s s u r e . 3. EXPERIMENTAL SET-UP E x p e r i m e n t s w e r e c o n d u c t e d b y t h e D e l f t H y d r a u l i c s L a b o r a t o r y i n a s h o r t wave l a b o r a t o r y f l u m e w i t h h o r i z o n t a l b o t t o m o f t h e D e l f t U n i v e r s i t y o f Techn o l o g y . The t r a Techn s v e r s a l s e c t i o Techn o f t h e f l u m e i s composed o f aTechn e x t e r i o r m e t a l -l i c s e c t i o n and an i n t e r i o r s e c t i o n o f P e r s p e x m a t e r i a -l . The i n t e r i o r s e c t i o n has a w i d t h o f 50 cm and a h e i g h t o f 60 cm. S h o r t waves o f u n i f o r m p e r i o d T w e r e c r e a t e d b y t h e t r a n s l a t i o n movement o f a p l a n e g e n e r a t o r s i t u a t e d a t one end o f t h e f l u m e . A s t r a i g h t s l o p i n g s t r u c t u r e o f c o n c r e t e was p l a c e d a t t h e o t h e r end ( s e e f i g . 3 . 1 ) ,
The s l o p e was 1 : 6 ( a = 9.46°), The d i s t a n c e b e t w e e n t h e mean p o s i t i o n o f t h e m o v i n g p a r t o f t h e g e n e r a t o r and t h e t o e o f t h e s t r u c t u r e was 556 cm. A f i l t e r composed o f w i r e s c r e e n s o f 160 cm l e n g t h was p l a c e d a t a d i s t a n c e o f 70 cm f r o m t h e g e n e r a t o r .
F i g . 3 . 1 : D e f i n i t i o n s k e t c l i o f t t i e ex{3eriments
Tlie maximum d i s p l a c e m e n t o f t h e g e n e r a t o r f r o m i t s mean p o s i t i o n was k e p t c o n s t a n t d u r i n g t h e e x p e r i m e n t s and e q u a l t o 30.1 cm. T h r e e wave gauges measured t h e wave h e i g h t a l o n g t h e h o r i z o n t a l p a r t o f t h e f l u m e . I r r e g u l a r i
-t i e s due -t o -t h e wave g e n e r a -t o r and -t o -t h e r e l a -t i v e l y s m a l l d i s -t a n c e b e -t w e e n t h e g e n e r a t o r and t h e s t r u c t u r e caused t h e wave h e i g h t o f t h e g e n e r a t e d waves t o be n o t v e r y u n i f o r m a l o n g t h e f l u m e . F o r t h e d i f f e r e n t v a l u e s o f T u s e d i n t h i s s t u d y t h e waves had a h e i g h t b e t w e e n 5 and 8 cm. Because o f t h e i m p r e c i s i o n i n t h e v a l u e s o f t h e wave h e i g h t t h e s t e e p n e s s Y ( = H /L )
o o o o f t h e waves w i l l n o t be m e n t i o n e d i n t h e r e s t o f t h e r e p o r t .
The h e i g h t o f t h e s t i l l w a t e r l e v | l S\IL o v e r t h e h o r i z o n t a l b o t t o m i s c a l l e d h and t h e wave l e n g t h L ( = ^ — t h - ^ 1 ^ ^ ) . The r a t i o h /L i s c a l l e d
O O ZTI L Q O O
i n i t i a l r e l a t i v e d e p t h .
T h r e e d i f f e r e n t i a l p r e s s u r e t r a n s d u c e r s , model PDCR 20, D r u c k L t d . r e -c o r d e d t h e p r e s s u r e s on t h e s t r u -c t u r e d u r i n g t h e wave b r e a k i n g . One s i d e o f t h e membrane was u n d e r t h e p r e s s u r e due t o t h e wave b r e a k i n g and t h e o t h e r s i d e u n d e r t h e p r e s s u r e due t o t h e a m b i e n t a i r .
The t h r e e t r a n s d u c e r s w e r e p l a c e d on an a l u m i n i u m p l a t e o f d i m e n s i o n s 150 X 119 X 18 mm, w h i c h was f i x e d i n t h e b r e a k i n g zone o f t h e s t r u c t u r e :
- The f i r s t t r a n s d u c e r , c a l l e d t r a n s . 1 , was a t a h e i g h t h j = 32.5 cm above t h e h o r i z o n t a l b o t t o m
- The second t r a n s d u c e r , c a l l e d t r a n s . 2, was p l a c e d a t a h e i g h t = 33.5 cm a t a d i s t a n c e o f 6 cm f r o m t r a n s . 1 - The t h i r d t r a n s d u c e r , c a l l e d t r a n s . 3, was a t a h e i g h t h ^ = 34.0 cm above t h e h o r i z o n t a l b o t t o m and a t a d i s t a n c e o f 3 cm f r o m t r a n s . 2. The p r e s s u r e r a n g e p o f t h e t r a n s d u c e r s was + 10 p s i . T h e i r n a t u r a l f r e ¬ quency was 2500 p ^ ^ 7900 HZ. The f r e q u e n c y o f t h e s y s t e m a l u m i n u m p l a t e - t r a n s d u c e r ( w i t h o u t w a t e r )
12
a p p r o x i m a t e l y 2700 Hz. The n a t u r a l f r e q u e n c y o f t h e t r a n s d u c e r f i x e d o n t h e p l a t e u n d e r t h e w a t e r was q u i t e h i g h and d i d n o t i n f l u e n c e t h e r i s i n g t i m e s o f t h e r e c o r d e d p r e s s u r e .
The s i g n a l s f r o m t h e t h r e e t r a n s d u c e r s were r e c o r d e d on m a g n e t i c t a p e s and some o f them on p h o t o s e n s i t i v e Kodak p a p e r .
N o r m a l speed f i l m s ( 6 0 f r a m e s / s e c ) and h i g h speed f i l m s ( 4 0 0 f r a m e s / s e c ) were made o f t h e b r e a k i n g p r o c e s s .
Water and a i r p a r a m e t e r s w e r e c h e c k e d d u r i n g t h e e x p e r i m e n t s : waves w e r e g e n e r a t e d w i t h d i f f e r e n t v a l u e s o f h ^ and u n d e r a t m o s p h e r i c p r e s s u r e , c a l l e d a t m , p r e s s . , and u n d e r n e a r vacuum c o n d i t i o n s , c a l l e d vacuum. The s o - c a l l e d vacuum c o r r e s p o n d e d i n r e a l i t y w i t h an a m b i e n t p r e s s u r e e q u a l t o 'V 2% o f t h e a t m o s p h e r i c p r e s s u r e . Of t h e v a r i o u s c o m b i n a t i o n s o f h and L ' o o used i n t h e e x p e r i m e n t a l s t u d y , o n l y t h r e e a r e examined i n t h i s r e p o r t , u n d e r b o t h a m b i e n t p r e s s u r e s . They a r e l i s t e d i n t a b l e 3 . 1 : h o T L 0 h /L 0 0 a m b i e n t p r e s s u r e 41.0 cm 1.30 sec 218 cm 0.19 atm. p r e s . 41.0 cm 1.30 sec 218 cm 0.19 vacuum 41 .8 1 .62 293 0.14 atm. p r e s . 41 .8 1 .62 293 0.14 vacuum 42.0 2.75 537 0.08 atm. p r e s . 42.0 2.75 537 0.08 vacuum
P a r t I I : WAVE PROPAGATION M D BREAKING ON A SLOPING STRUCTURE
4. PHENOMENOLOGICAL DESCRIPTION OF THE WAVE PROPAGATION AND BREAKING
A q u a n t i t a t i v e d e s c r i p t i o n o f t h e b r e a k i n g o f w a t e r waves and o f t h e a s s o c i a t e d phenomena does n o t e x i s t . Knowledge o f t h e w a t e r p a r t i c l e m o t i o n and v e l o c i t y as f u n c t i o n o f t i m e i s n e c e s s a r y f o r such a m o d e l . The o l d c r i t e r i o n a c c o r d i n g t o w h i c h b r e a k i n g s t a r t s when t h e maximum h o r i z o n t a l v e l o c i t y o f t h e w a t e r p a r t i c l e s i n t h e wave c r e s t becomes l a r g e r t h a n t h e wave c e l e r i t y has been q u i t e w e l l v e r i f i e d .
A q u a l i t a t i v e d e s c r i p t i o n o f t h e wave p r o p a g a t i o n and b r e a k i n g o n a s l o p i n g s t r u c t u r e i s g i v e n i n t h e f o l l o w i n g :
The c h a r a c t e r i s t i c s o f s h o r t w a t e r waves a r r i v i n g f r o m t h e open sea and p r o p a g a t i n g s h o r e w a r d s o n a s l o p i n g s t r u c t u r e change g r a d u a l l y . V e r t i c a l and h o r i z o n t a l d e f o r m a t i o n s appear and i n c r e a s e as t h e w a t e r d e p t h d e c r e a s e s The waves become u n s t a b l e and f i n a l l y b r e a k when i n s t a b i l i t y becomes i r r e v e r s i b l e . Two r a d i c a l t r a n s f o r m a t i o n s o f t h e f l o w p a t t e r n o c c u r d u r i n g t h e p r o p a -g a t i o n : t h e s t a b l e o s c i l l a t o r y f l o w o v e r t h e l o w e r p a r t o f t h e s t r u c t u r e i s t r a n s f o r m e d i n t o an u n s t a b l e o s c i l l a t o r y f l o w w h i c h f i n a l l y changes i n t o a r a p i d l y c h a n g i n g up and d o \ m - r u s h i n g f l o w . The t o t a l p r o p a g a t i o n zone o n t h e s t r u c t u r e c a n be d i v i d e d i n t o t h r e e m a i n z o n e s , c o r r e s p o n d i n g t o t h e t h r e e t y p e s o f f l o w : s t a b i l i t y zone, i n s t a -b i l i t y zone and f i n a l zone ( s e e f i g . 4 . 1 ) .
^ ' t o t a l p r o p a g a t i o n zone '
Xp,inax
F i g . 4 . 1 : Zones o f a s l o p i n g m a r i t i m e s t r u c t u r e14 T h e i r l e n g t h s have been f o u n d c o r r e l a t e d w i t h t h e c h a r a c t e r i s t i c s o f r e g u l a r waves and t h e s t r u c t u r e s l o p e [22] : - I n t h e s t a b i l i t y zone t h e waves r e m a i n s t a b l e , t h e i r h e i g h t d e c r e a s e s , r e f l e c t i o n and d e f o r m a t i o n s a r e s m a l l . - I n t h e i n s t a b i l i t y zone c o u l d be d i v i d e d i n t o two s m a l l e r z o n e s : t h e p r e - b r e a k i n g zone and t h e b r e a k i n g zone. I n t h e p r e - b r e a k i n g zone 23 t h e waves become u n s t a b l e , d e f o r m a t i o n s and r e f l e c t i o n i n c r e a s e . Wave h e i g h t i n c r e a s e s a l s o and shows a maximum v a l u e , c a l l e d b r e a k i n g h e i g h t H^, a t t h e b r e a k i n g l i n e x ^ a f t e r w h i c h i t d e c r e a s e s r a p i d l y . The l o c a t i o n where t h e b r e a k i n g f i r s t a p p e a r s on t h e wave p r o f i l e depends on t h e i n i t i a l wave s t e e p n e s s , r e l a t i v e d e p t h and s t r u c t u r e s l o p e s : as i n i t i a l wave s t e e p n e s s d e c r e a s e s o r / a n d i n i t i a l r e l a t i v e d e p t h i n -c r e a s e s i t moves f r o m t h e t o p o f t h e p r o f i l e t o w a r d s i t s l o w e r p a r t and d i s a p p e a r s ( t o t a l r e f l e c t i o n ) . B r e a k i n g s l o c a t e d m a i n l y on t h e t o p or i n t h e u p p e r p a r t o f t h e wave p r o f i l e a r e c o n s i d e r e d i n t h e r e s t of t h i s r e p o r t . The l e n g t h o f t h e p r e - b r e a k i n g zone i n c r e a s e s w i t h d e c r e a s i n g s t r u c t u r e s l o p e . I t shows a maximum v a l u e f o r a c e r t a i n wave s t e e p n e s s .
The b r e a k i n g l i n e i s t h e l i m i t b e t w e e n t h e p r e - b r e a k i n g and b r e a k i n g zones.
I n case o f a b r e a k i n g s t a r t i n g on t h e t o p o f t h e wave p r o f i l e , a w a t e r j e t i s c r e a t e d a t t h e b r e a k i n g l i n e . I t moves s h o r e w a r d s i n t h e b r e a k i n g zone 24 and s t r i k e s t h e w a t e r l a y e r a r r i v i n g f r o m t h e f i n a l zone a t x , The p o i n t x i s t a k e n t o d e f i n e t h e end o f t h e b r e a k i n zone,
s ^ s
Between t h e w a t e r j e t and t h e w a t e r l a y e r a m a j o r a i r p o c k e t i s e n -t r a p p e d . The w a -t e r j e -t o v e r -t h e a i r p o c k e -t has a s h o r e w a r d h o r i z o n -t a l v e l o c i t y and a v e r t i c a l one due t o t h e g r a v i t y a c c e l e r a t i o n . The w a t e r l a y e r b e l o w t h e a i r p o c k e t moves s e a w a r d s . A c i r c u l a t i o n a p p e a r s a r o u n d t h e a i r p o c k e t . A t t h e same t i m e , i t i s e n t r a i n e d s h o r e w a r d s b y t h e u p - r u s h i n g w a t e r mass. The a i r p o c k e t soon t a k e s t h e f o r m o f a c i r c u l a r v o r t e x m o v i n g s h o r e w a r d s . The c r o s s s e c t i o n d e c r e a s e s w i t h t i m e , b e c a u s e o f a mass exchange w i t h t h e a m b i e n t w a t e r : s m a l l a i r b u b b l e s a r e c o n -t i n u o u s l y c r e a -t e d and d i s p e r s e d i n -t h e w a -t e r mass. They f o l l o w a c i r c u l a r f l o w p a t h due t o t h e a i r p o c k e t c i r c u l a t i o n .
A t t h e end o f t h e a i r p o c k e t l i f e t h e w a t e r h e i g h t o v e r t h e s t r u c t u r e i n c r e a s e s as i f an e x p l o s i o n h a d t a k e n p l a c e i n t h e w a t e r mass. A t t h e same t i m e s m a l l a i r b u b b l e s a r e t h r o m b e h i n d l i k e a j e t .
Many s m a l l a i r b u b b l e s a r e a l s o c r e a t e d and d i s p e r s e d i n t h e w a t e r , d u r i n g t h e i m p a c t . They r i s e t o t h e f r e e s u r f a c e , f o r m i n g foam.
- I n t h e f i n a l zone, t h e f a l l e n w a t e r mass i s r e f l e c t e d upwards b y t h e w a t e r l a y e r and t h e n f a l l s a g a i n . A second a i r p o c k e t i s f o r m e d
sometimes. S m a l l v o r t i c e s a r e g e n e r a t e d i n t h e w a t e r mass. The doxm-r u s h i n g movement o f t h e w a t e doxm-r l a y e doxm-r changes i n t o u p doxm-r u s h i n g . A k i n d o f s o l i t a r y wave i s c r e a t e d b y t h e f a l l e n w a t e r mass. I t p o r p a g a t e s shore-wards s u p e r i m p o s e d on t h e u p - r u s h i n g l a y e r . When k i n e t i c e n e r g y has b e e n t o t a l l y t r a n s f o r m e d i n t o p o t e n t i a l , t h e s y s t e m u p - r u s h i n g w a t e r l a y e r s o l i t a r y wave s t a r t s m o v i n g seawards. The w a t e r l i n e o s c i l l a t e s b e t w e e n x and x . . The w a t e r l i n e o f t h e s t i l l w a t e r i s a t
p, max p, m m X = X .
o
The d e s c r i b e d f l o w i n t h e b r e a k i n g zone and a t t h e b e g i n n i n g o f t h e f i n a l zone i s i m p o r t a n t f o r t h e r e s t o f t h i s s t u d y . The p h o t o g r a p h s o f A p p e n d i x A l show t h e s u c c e s s i v e phases o f t h i s f l o w . They p r e s e n t t h e e v o -l u t i o n o f t h e f -l o w as f u n c t i o n o f t i m e f o r t h e t h r e e chosen v a -l u e s o f i n i t i a -l r e l a t i v e d e p t h u n d e r b o t h a m b i e n t p r e s s u r e . C o n c e r n i n g t h e t i m e s c a l e , t = 0 c o r r e s p o n d s w i t h t h e moment a t w h i c h t h e w a t e r j e t s t r i k e s t h e w a t e r l a y e r . Some g e n e r a l c o n c l u s i o n s f r o m t h e p h o t o g r a p h s and t h e f i l m s a r e t h e f o l l o w i n g : a. I n f l u e n c e o f t h e i n i t a l r e l a t i v e d e p t h h /L . . . C Q O - The a i r p o c k e t l i f e i s s h o r t e r i n case o f h /L = 0.14 t h a n i n c a s e o o o f 0.18. I n t h e f i r s t case t h e a i r p o c k e t was e a s i l y d i s t i n g u i s h e d up t o t - 0.20 sec. and i n t h e second case up t o t ~ 0.25 s e c . I n case o f h^/L^ = 0.08, t h e r e i s h a r d l y any a i r p o c k e t . The l i f e o f t h e a i r p o c k e t i s p r o p o r t i o n a l t o i t s i n i t i a l s i z e .
- The b r e a k i n g zone becomes l a r g e r when h /L i n c r e a s e s . I n case o f o o
h /L = 0 . 0 8 t h e r e i s h a r d l y any w a t e r j e t : t h e c r e s t o f t h e wave
o O J J J
f a l l s on t h e s h o r e w a r d wave f r o n t .
- The s o l i t a r y wave i n t h e f i n a l zone becomes h i g h e r w i t h i n c r e a s i n g h /L .
o o
- The f i r s t p a r t o f t h e f i n a l zone i s much more t u r b u l e n t i n c a s e o f h /L = 0 . 0 8 .
o o
b. I n f l u e n c e o f t h e a m b i e n t p r e s s u r e
The a i r p o c k e t d i s a p p e a r s much sooner u n d e r vacuum t h a n u n d e r a t m o s p h e r i c c o n d i t i o n s . T h e r e a r e l e s s b u b b l e s c r e a t e d . The c i r c u
16
- The e v o l u t i o n o f t h e w a t e r s u r f a c e w i t h t i m e does n o t seem t o be m o d i f i e d b y t h e a m b i e n t p r e s s u r e .
5. PRESSURES ON THE STRUCTURE DUE TO THE WAVE BREAKING
P r e s s u r e - t i m e s i g n a l s , as t h e y w e r e r e c o r d e d b y t h e t h r e e t r a n s d u c e r s i n t h e b r e a k i n g zone d u r i n g one wave p e r i o d u n d e r a t m o s p h e r i c and vacuum c o n d i t i o n s , a r e s h o r n i n A p p e n d i x A2.
A l l o f them show a v e r y s h o r t and s t e e p r i s i n g t i m e a f t e r t h e b e g i n n i n g a t t = 0. I t i s f o l l o w e d b y a decay t i m e . D u r i n g t h a t t i m e ( i n case o f atmos-p h e r i c c o n d i t i o n s ) o s c i l l a t i o n s a atmos-p atmos-p e a r . The atmos-p r e s s u r e atmos-p a s s e s t o n e g a t i v e v a l u e s and t h e n a g a i n t o t h e p o s i t i v e ones. F i n a l l y , i t d e c r e a s e s v e r y r e g u l a r l y .
A t y p i c a l p r e s s u r e h i s t o r y i s shotm i n F i g . 5 . 1 . I t was r e c o r d e d d u r i n g b r e a k i n g u n d e r a t m o s p h e r i c c o n d i t i o n s i n o r d e r t o be a b l e t o r e c o r d t h e i n -f l u e n c e o -f t h e a i r p o c k e t .
A q u a l i t a t i v e a n a l y s i s o f a t y p i c a l p r e s s u r e h i s t o r y will now be made. The g o a l i s t o e s t a b l i s h r e l a t i o n s b e t w e e n some c h a r a c t e r i s t i c p o i n t s o f t h e d i a g r a m and t h e phases o f a wave b r e a k i n g as t h e y have been d e s c r i b e d e a r l i e r .
The p r e s s u r e d i a g r a m s t a r t s a t t = 0, w h i c h i s t h e moment o f t h e s t r i k i n g b e t w e e n t h e w a t e r mass f r o m t h e b r e a k i n g and t h e d o \ m - r u s h i n g w a t e r l a y e r . The p r e s s u r e r i s e s a f t e r t = 0 because o f t h e . i n c r e a s i n g w a t e r h e i g h t o v e r t h e t r a n s d u c e r s . A s h o c k wave due t o t h e i m p a c t a r r i v e s t o t h e t r a n s d u c e r s v e r y soon a f t e r t = 0. The p r e s s u r e on t h e s t r u c t u r e due t o t h e s h o c k wave i s t h e n s u p e r i m p o s e d o n t h e w a t e r h e i g h t r e s u l t i n g i n a s t e e p p r e s s u r e d i a g r a m .
The f i r s t peak P^ o f t h e p r e s s u r e a p p e a r s a t t = t ^ . A t t^^ ends t h e r i s i n g t i m e o f t h e p r e s s u r e f r o m t h e s h o c k wave. I t s b e g i n n i n g i s n o t a l w a y s easy t o d i s t i n g u i s h . I r r e g u l a r i t i e s and s m a l l a i r b u b b l e s i n t h e f a l l i n g w a t e r mass sometimes g i v e a n i r r e g u l a r f o r m t o t h e d i a g r a m d u r i n g t h e r i s i n g t i m e . I n g e n e r a l , t ^ ^ i s e q u a l t o t h e r i s i n g t i m e o f t h e shock p r e s s u r e o n t h e s t r u c -t u r e p l u s -t h e -t i m e n e c e s s a r y f o r -t h e s h o c k -t o r e a c h -t h e s -t r u c -t u r e . I n some r e c o r d s i t i s p o s s i b l e t o d i s t i n g u i s h t h e moment a t w h i c h t h e s h o c k p r e s s u r e a r r i v e s a t t h e t r a n s d u c e r : t h e p r e s s u r e shows s u d d e n l y a much s t e e p e r s l o p e
t h a n t h e i n i t i a l one, w h i c h was due t o i n c r e a s i n g w a t e r h e i g h t o n l y . Between t = 0 and t ^ ^ an a i r p o c k e t i s f o r m e d i n t h e w a t e r mass. The a i r p o c k e t i s compressed by t h e w a t e r . C o m p r e s s i o n s t a r t s a t t = t ^ . The
b p r e s s u r e i n c r e a s e s and shows a second peak. V i b r a t i o n s o f t h e a i r p o c k e t f o l l o w t h e i n i t i a l c o m p r e s s i o n g i v i n g a n o s c i l l a t o r y f o r m t o t h e p r e s s u r e d i a g r a m . The o s c i l l a t i o n s a r e s u p e r i m p o s e d on a d e c r e a s i n g p r e s s u r e due t o
t h e d e c r e a s i n g w a t e r h e i g h t , and t o t h e decay o f t h e shock p r e s s u r e on t h e s t r u c t u r e . A n o t h e r f a c t o r c o n t r i b u t i n g t o t h e d e c r e a s i n g p r e s s u r e i s t h e a i r p o c k e t : when i t p a s s e s o v e r t h e t r a n s d u c e r s t h e w a t e r h e i g h t i s r e d u c e d t o t h e h e i g h t s above and b e l o w t h e a i r mass.
O s c i l l a t i o n s a r e c l e a r e r a t t h e t r a n s d u c e r 1 b e c a u s e t h e a i r p o c k e t was f o r m e d o v e r i t . F r e q u e n c y and a m p l i t u d e d e c r e a s e w i t h t i m e .
From a c e r t a i n moment o s c i l l a t i o n s become l e s s c l e a r because t h e a i r p o c k e t s e c t i o n has d e c r e a s e d much. The end o f o s c i l l a t i o n s t ^ i s n o t a l w a y s easy t o d i s t i n g u i s h on t h e r e c o r d s .
I n case o f vacuum c o n d i t i o n s , t h e p r e s s u r e d i a g r a m does n o t show any o s c i l l a t i o n s due t o t h e a i r p o c k e t . I t m i g h t be due e i t h e r t o a n o n - e x i s t e n c e o f v i b r a t i o n s o f t h e a i r p o c k e t u n d e r vacuum o r t o a v e r y s m a l l f r e q u e n c y o f v i b r a t i o n s w h i c h does n o t p e r m i t t o d i s t i n g u i s h t h e p r e s s u r e o s c i l l a t i o n s on t h e r e c o r d .
A f t e r t h e end o f t h e decay t i m e o f t h e shock p r e s s u r e and o f t h e a i r p o c k e t o s c i l l a t i o n s , t h e p r e s s u r e d i a g r a m f o l l o w s t h e e v o l u t i o n o f t h e w a t e r h e i g h t o f t h e t r a n s d u c e r s . I t passes t o p o s i t i v e v a l u e s and t h e n de-c r e a s e s r e g u l a r l y t o a minimum a t t = T, w h i de-c h i s e q u a l t o t h e minimum h e i g h t o f t h e w a t e r . I t i s o b v i o u s f r o m t h e a n a l y s i s g i v e n above t h a t t h e p r e s s u r e s on t h e s t r u c t u r e a r e m a i n l y due t o t h r e e l o a d i n g f a c t o r s : - t h e w a t e r l a y e r o v e r t h e s t r u c t u r e - t h e shock p r e s s u r e due t o t h e i m p a c t - t h e a i r p o c k e t o s c i l l a t i o n s .
Each one o f them l o a d s t h e s t r u c t u r e . I f i t i s assumed t h a t t h e s u p e r p o s i t i o n i s l i n e a r , t h e n f o r t h e t i m e b e i n g i t i s p o s s i b l e t o decompose a t y p i c a l p r e s s u r e h i s t o r y i n t o two p a r t i a l p r e s s u r e h i s t o r i e s (see f i g . 5 . 2 ) : - The f i r s t one, c a l l e d p r e s s u r e h i s t o r y I , shows a r i s i n g t i m e t o a p e a k
and t h e n a decay t i m e d u r i n g w h i c h i t r e a c h e s n e g a t i v e v a l u e s . A f t e r a
c e r t a i n t i m e i t passes i n t o p o s i t i v e v a l u e s and t h e n i t d e c r e a s e s r e g u l a r l y . I t i s due t o t h e s h o c k p r e s s u r e and t o t h e w a t e r l a y e r .
- The second one, c a l l e d p r e s s u r e h i s t o r y I I , has an o s c i l l a t o r y f o r m and i s damped. I t i s due t o t h e a i r p o c k e t v i b r a t i o n s .
I n t h e f o l l o w i n g s e c t i o n s i t w i l l be t r i e d t o decompose f u r t h e r t h e p r e s s u r e I and t o make a m a t h e m a t i c a l model f o r t h e p r e s s u r e I I ,
J8 F i g . 5 . 1 : T y p i c a l p r e s s u r e - t i m e h i s t o r i e s r e c o r d e d by t h e t r a n s d u c e r s on t h e s t r u c t u r e
p r e s s u r e I
SWL
t=0
trans.1
SWL e
t = T
t = T
t r a n s . 3
p r e s s u r e I I
trans.1
F i g . 5.2: D e c o m p o s i t i o n o f a p r e s s u r o - t i m e h i s t o r yP a r t I I I : DETERMINISTIC MODELING OF THE PRESSURES ON THE STRUCTURES
6. FACTORS LOADING THE STRUCTURE DURING THE WAVE BREAKING
6.1. W a t e r l a y e r o v e r t h e b r e a k i n g zone o f t h e s t r u c t u r e
6.1.1. I n t r o d u c t i o n
The h e i g h t h o f t h e w a t e r l a y e r o v e r t h e b r e a k i n g zone o f t h e s t r u c t u r e c o n t r i b u t e s a c t i v e l y t o t h e f o r m a t i o n o f t h e p r e s s u r e s r e c o r d e d b y t h e t r a n s -d u c e r s -d u r i n g t h e b r e a k i n g , as i t i s a n a l y s e -d i n p a r . 5. Because o f t h e wave m o t i o n , h changes i n t i m e and i n space.
I n o r d e r t o d e t e m i n e t h e c o n t r i b u t i o n o f t h e w a t e r h e i g h t t o t h e t o t a l p r e s s u r e , i t i s n e c e s s a r y t o i n v e s t i g a t e t h e e v o l u t i o n o f t h e w a t e r s u r f a c e as f u n c t i o n o f t i m e i n one wave p e r i o d i n t h e b r e a k i n g zone o f t h e s t r u c t u r e . T h i s e v o l u t i o n w i l l be i n v e s t i g a t e d i n t h e f o l l o w i n g p a r a g r a p h s and an e m p i r i c a l r e l a t i o n r e p r e s e n t i n g t h e e v o l u t i o n o f h w i l l be p r o p o s e d .
6.1.2. E v o l u t i o n i n t i m e o f t h e w a t e r l a y e r h e i g h t
The e v o l u t i o n o f t h e w a t e r s u r f a c e as f u n c t i o n o f t i m e i n one wave p e r i o d i n t h e b r e a k i n g zone o f t h e s t r u c t u r e was s t u d i e d f r o m t h e f i l m s . F i g . 6 . 1 . shows such an e v o l u t i o n . I t c a n be seen t h a t :
On t h e d o x m - r u s h i n g w a t e r l a y e r ( 1 ) a mass o f w a t e r i s added ( 2 ) . T h i s i s due t o t h e w a t e r j e t f o r m e d a t t h e b r e a k i n g and r e s u l t s i n a m o d i f i -c a t i o n o f t h e f l o w p a t t e r n : t h e w a t e r mass s t a r t s u p - r u s h i n g . A p a r t of t h e f a l l e n w a t e r mass i s r e f l e c t e d upward: b e c a u s e o f t h i s r e f l e c t i o n a p a r t o f t h e b r e a k i n g zone shows minimum w a t e r h e i g h t o v e r i t ( a r o u n d t h e p o i n t s i n ( 3 ) , ( 4 ) and ( 5 ) ) . The w a t e r h e i g h t d e c r e a s e s ' s e a w a r d o f s and i n c r e a s e s s h o r e w a r d o f i t . A t t h e same t i m e s moves s h o r e w a r d . A f t e r t h e r e f l e c t e d w a t e r mass h a s r e a c h e d a maximum h e i g h t ( 5 ) , i t s t a r t s f a l l i n g a g a i n ( 6 ) . A s o l i t a r y wave i s c r e a t e d b y t h e p e r t u r b a t i o n due t o t h e f a l l e n w a t e r mass. I t p r o p a g a t e s s h o r e w a r d o n t h e u p - r u s h i n g w a t e r mass w h i l e i t s h e i g h t and c e l e r i t y d e c r e a s e . A f t e r t h e end o f t h e u p -r u s h i n g movement, t h e f l o w d i -r e c t i o n changes a g a i n : t h e w a t e -r mass s t a -r t s do\TO-ruwhing and t h e s o l i t a r y wave p r o p a g a t i n g s e a w a r d s , a l t h o u g h i t s h e i g h t i s much r e d u c e d ( 7 ) , ( 8 ) . A f t e r t h e passage o f t h e s o l i t a r y wave t h e w a t e r h e i g h t o v e r t h e b r e a k i n g zone d e c r e a s e s g r a d u a l l y b e f o r e t h e
. , w a t e r mass f r o m t h e new b r e a k i n g a r r i v e s ( 9 ) , ( 1 0 ) , ( 1 1 ) .
The e v o l u t i o n o f t h e w a t e r h e i g h t h o v e r t h e t r a n s d u c e r s as f u n c t i o n o f t i m e i n one wave p e r i o d was f o l l o w e d on t h e f i l m s made d u r i n g t h e e x p e r i -ments (see f i g . 6 . 2 . ) :
h shows a minimum v a l u e h . a t t = 0: t h e d o w n - r u s h i n g w a t e r mass t a k e s
mm
a minimum h e i g h t t h e moment b e f o r e t h e s t r i k i n g w i t h t h e w a t e r j e t . A f t e r t = 0, h i n c r e a s e s v e r y r a p i d l y , because o f t h e added w a t e r mass. I t _ r e a c h e s a maximum h a t t = t , and t h e n s t a r t s d e c r e a s i n g b e c a u s e
max mwl , ^ o f g r a v i t y a c t i o n .
A q u a n t i t y o f w a t e r mass i s r e f l e c t e d upward and t h e w a t e r h e i g h t shows a second peak. T h i s second peak a p p e a r s on t h e p r e s s u r e r e c o r d e d by a t r a n s d u c e r o n l y i n case t h i s t r a n s d u c e r i s s i t u a t e d s h o r e w a r d o f s. I t m i g h t be l a r g e r t h a n t h e f i r s t peak: i t depends on t h e r e f l e c t i o n c o n d i t i o n s .
A f t e r t h e second peak, h d e c r e a s e s because o f t h e u p - r u s h i n g movement. A t h i r d peak o f h i s shown when t h e s o l i t a r y wave passes o v e r t h e
t r a n s d u c e r a f t e r w h i c h i t d e c r e a s e s g r a d u a l l y t o t h e minimum v a l u e h . .
mm
Many s m a l l i r r e g u l a r i t i e s a r e p r e s e n t i n t h e d i a g r a m o f h ( t ) .They a r e due t o a l l k i n d s o f s m a l l p e r t u r b a t i o n s i n t h e b r e a k i n g zone, s u c h as a i r b u b b l e s and v i b r a t i o n s o f t h e m a j o r a i r p o c k e t .
F o r a c e r t a i n s l o p e o f t h e s t r u c t u r e and a c e r t a i n wave on i t , ' t h e e v o l u t i o n i n t i m e o f h depends v e r y much on t h e b r e a k i n g l i n e and b r e a k i n g h e i g h t - . I n case o f m o d i f i c a t i o n s a t t h e b r e a k i n g l i n e and h e i g h t f r o m one wave t o t h e o t h e r , t h e w a t e r h e i g h t - t i m e h i s t o r y i s m o d i f i e d , m a i n l y b e t w e e n t = 0 and t h e t h i r d peak. B u t t h e g e n e r a l t e n d e n c y b e t w e e n h and h .
max m m r e m a i n s t h e same.
The minimum v a l u e h ^ ^ ^ a t a c e r t a i n p o i n t o f t h e b r e a k i n g zone depends on t h e l o c a t i o n o f t h e SWL, t h e wave c h a r a c t e r i s t i c s and t h e s l o p e o f t h e s t r u c t u r e . I t i s s l i g h t l y a f f e c t e d by t h e m o d i f i c a t i o n s i n the. b r e a k i n g l i n e due t o t h e i r r e g u l a r i t i e s o f t h e i n p u t waves, h depends on t h e same para-.\
max ^ ^ m e t e r s as h^^^^ and f u r t h e r m o r e on t h e f a l l e n w a t e r mass and t h e t h i r d p e a k
depends on t h e s o l i t a r y wave: b o t h o f them depend on a l a r g e r number o f p a r a m e t e r s t h a n h and h . .
max m m
The f i l m s showed t h a t f o r a c e r t a i n g e n e r a t e d wave t h e v a l u e s o f h max and h^_j^^ a r e l e s s s c a t t e r e d t h a n t h e v a l u e s o f t h e second and t h i r d p e a k s : t h e y a r e much more d e t e r m i n i s t i c .
l a y e r h e i g h t . I n o r d e r t o a p p r o a c h t h e w a t e r h e i g h t t i m e h i s t o r y by means o f a d e t e r -m i n i s t i c l a w , t h e -most d e t e r -m i n i s t i c v a l u e s o f t h e h i s t o r y -must be u s e d . A c c o r d i n g t o t h e above g i v e n a n a l y s i s , t h e s e v a l u e s a r e h and h . . I n max m m s u c h a case t h e h i s t o r y can be c o n s i d e r e d as a s u p e r p o s i t i o n o f a c u r v e w h i c h i n c r e a s e s a b r u p t l y f r o m h . t o h and t h e n d e c r e a s e s r e g u l a r l y m m max " •' f r o m h t o h . , o f two peaks due t o t h e w a t e r mass r e f l e c t i o n and t o t h e
max m m
s o l i t a r y V7ave passage and o f many s m a l l i r r e g u l a r i t i e s .
The m a i n and most d e t e r m i n i s t i c c o n t r i b u t i o n t o t h e w a t e r h e i g h t - t i m e h i s t o r y f o r m a t i o n comes f r o m t h e c u r v e o f a r e g u l a r l y d e c r e a s i n g s l o p e (see f i g . 6 . 3 ) . I t can be modeled as f o l l o w s : Between t = 0 and t , t h e w a t e r h e i g h t h i n c r e a s e s n e a r l y l i n e a r l y : mwl ° J J ( 6 . 1 ) h = h . + ( h - h . ) . — ^ f o r 0 < t < t , m m max mm tmwl — — mwl
Between t = t , and T t h e w a t e r h e i g h t h d e c r e a s e s and t h e s l o p e o f t h e
mwl ^ c u r v e i decrea'ses - f a s t e r when h i s l a r g e r . I t c o u l d be r e p r e s e n t e d b y : ( 6 . 2 )
41
= - Bh f o r t < t < T d t mwl — — w h e r e B i s a c o n s t a n t i n d e p e n d e n t o f t . Equ. ( 6 . 2 ) y i e l d s : ( 6 . 2 ) l o g h = - B t + C where C i s a c o n s t a n t t h a t can be e v a l u a t e d e i t h e r f r o m t h e i n i t i a l c o n d i t i o n ( t = t ,, h = h ) o r f r o m t h e f i n a l one ( t = T, h = h . ) . mwl max m m I f t h e i n i t i a l c o n d i t i o n i s used equ. ( 6 . 2 ) g i v e s : h _ ,„B(t , - t ) — = 10 mwl h max o r : ( 6 . 3 ) = a^^ ^mwl^^^^ ~ V w P f o r t , < t < T \ a x -hSome waves o f — = 0.14 u n d e r b o t h a t m o s p h e r i c and vacuum c o n d i t i o n s . w e r e s t u d i e d a g a i n f r o m t h e f i l m s . I t t u r n e d o u t t h a t equ. ( 6 . 3 ) w i t h a = 0.04 i s q u i t e c l o s e t o 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 h r e e t r a n s -d u c e r s an-d t h e two a m b i e n t p r e s s u r e s (see f i g . 6 . 3 ) . I t i s b e l i e v e -d t h a t a depends on t h e wave c h a r a c t e r i s t i c s . The e x p e r i m e n t a l d a t a o f t h i s s t u d y d i d n o t p e r m i t t o e s t a b l i s h any r e l a t i o n c o n c e r n i n g a.
