Energy dissipation in breaking solitary and periodic waves

1.

INTRODUCTION

1.

It is well known (see Divoky et al.,

1970, for a review) that

on gentle slopes (slope S

<

1:30, say) the wave height after

breaking does not decay in proportion to the mean depth.

The

curve

vs h/h

b

is concave upwards (for plane bottom).

The

concavity increases with decreasing S and with increasing Ho/L

_o

(Nakamura et al., 1966).

1.2

The often-used hypothesis H(x)

yh(x), Y

const. for given

(S, Ho/L

_o

)' does not incorporate the effects mentioned above.

Its success in the prediction of set-up can perhaps be ascribed

Co the fact that it he.s been tested mainly on plane and

relati-vely steep slopes, where indeed it is in reasonable agreement

with the data.

1.3

The hypothesis H

=

yh is not applicable in regions where the

depth is constant or increasing in the propagation direction,

such as in a bar-trough profile.

Visual observations of the

latter situation cannot fail but to give the impression that

the immediate post-breaking behavior is governed primarily by

the characteristics at breaking, with its own imposed length

scale.

1.4

The bottom slope is believed to affect this behavior only if it

is sufficiently steep, so that the rate of change of wave height

due to changes in depth ("shoaling" with either increasing or

decreasing depth) is comparable to that due to breaking.

5. DISCUSSION

A c o m p a r i s o n has b e e n g i v e n o f c a l c u l a t e d and m e a s u r e d wave

h e i g h t d e c a y , f o r s o l i t a r y waves on a p l a n e , 1:100 s l o p e , and f o r p e r i o d i c waves on a h o r i z o n t a l b o t t o m and o n p l a n e s l o p e s o f w i d e l y v a r y i n g i n c l i n a t i o n . F o r a l l t h e s e c a s e s a f a i r o v e r a l l a g r e e m e n t i s o b t a i n e d f o r B = 2, i n w h i c h B i s t h e c o e f f i -c e n t i n t h e e s t i m a t e o f t h e d i s s i p a t i o n r a t e , eq. ( 5 ) . T h i s l e n d s s u p p o r t t o ( 5 ) and t o t h e h y p o t h e s i s t h a t B s h o u l d be a c o n s t a n t , i n d e p e n d e n t o f b o t t o m s l o p e , and t h a t i t s h o u l d be o f o r d e r one. 5.2 W h i l e t h e o v e r a l l a g r e e m e n t has been c a l l e d f a i r , i t s h o u l d be p o i n t e d o u t t h a t i n m o s t c a s e s t h e i n i t i a l d e c a y r a t e i s o v e r -e s t i m a t -e d b y t h -e p r -e s -e n t m o d -e l ( f o r B = 2 ) . I n o r d -e r t o o b t a i n

a more r e a l i s t i c m o d e l one may h a v e t o use d i f f e r e n t l a w s f o r

t h e d i s s i p a t i o n i m m e d i a t e l y a f t e r t h e i n c e p t i o n o f b r e a k i n g and f o r t h e d i s s i p a t i o n f a r t h e r f r o m t h a t p o i n t , as has b e e n s u g -g e s t e d by B u h r - H a n s e n e t a l . ( 1 9 7 8 ) . 5.3 The i n i t i a l d e c a y r a t e i n t h e p r e s e n t m o d e l , as g i v e n by ( 2 0 ) f o r p e r i o d i c w a v e s , c a n be w r i t t e n i n a s u i t a b l y n o r m a l i z e d f o r m as The f i r s t t e r m i n t h e r h s r e p r e s e n t s t h e ( l i n e a r ) s h o a l i n g e f -f e c t , t h e s e c o n d t h a t o -f t h e d i s s i p a t i o n . On s u -f -f i c i e n t l y g e n t l e s l o p e s , t h e s e c o n d t e r m p r e d o m i n a t e s , i n w h i c h c a s e t h e i n i t i a l d e c a y r a t e i s f o r a l l p r a c t i c a l p u r p o s e s g o v e r n e d by t h e wave p a r a m e t e r s a t t h e b r e a k p o i n t . ( I t may be n o t e d , f r o m a c o m p a r i s o n w i t h ( 2 2 ) , t h a t K i s p r o p o r t i o n a l t o t h e r a t i o o f t h e i n i t i a l wave h e i g h t v a r i a t i o n i n d u c e d b y d i s s i p a t i o n t o