Wave transmission by overtopping

Po:..u-s I , ; - j u L i - r

Ralph M. Parsons L a b o r a t o r y

For Water Resources and Hydrodynamics

Department o f C i v i l E n g i n e e r i n g M a s s a c h u s e t t s I n s t i t u t e o f T e c h n o l o g y WAVE TRANSMISSION by R a l p h H. C h a r l e s K. BY OVERTOPPING Cross S o l l i t t T e c h n i c a l Note No. 15 December 1970 P r e p a r e d Under C o n t r a c t No. DACW-72-68-C-0032 C o a s t a l E n g i n e e r i n g Research C e n t e r

U.S. Army Corps o f E n g i n e e r s W a s h i n g t o n , D.C. DSR P r o j e c t 71135

ABSTRACT

This report presents a theory for ocean wave transmission past breakwaters by overtopping, based on an evaluation of the energy content of the overtopping water. While several co-efficients are subject to further investigation, the data shows

that the general form of the equations developed is correct. Comparison with large-scale model tests reinforces this belief, and comparison of an intermediate theoretical result predicting the volume of overtopping water with published data again shows reasonable agreement. An envelope curve for the transmission coefficient, based on all available data, gives a simple tool for preliminary design estimates of the transmission coefficient.

ACKNOWLEDGEMENTS

T h i s s t u d y o f wave t r a n s m i s s i o n p a s t b r e a k w a t e r s was

sponsored by t h e C o a s t a l E n g i n e e r i n g Research Center o f t h e U. S.

Army Corps o f Engineers under C o n t r a c t No. DACW-72-68-C-0032.

C o n t r a c t a d m i n i s t r a t i o n was p r o v i d e d by t h e M.I.T. D i v i s i o n o f

Sponsored Research under DSR 71135.

The work was c a r r i e d o u t i n t h e Ralph M. Parsons L a b o r a t o r y

f o r Water Resources and Hydrodynamics o f t h e Department o f C i v i l

E n g i n e e r i n g a t M.I.T. Mr. Ken W i l s o n , a Research A s s i s t a n t i n

the L a b o r a t o r y , a s s i s t e d w i t h t h e d a t a r e d u c t i o n and numerous o t h e r

INTRODUCTION The d e s i g n f o r a h a r b o r i n an exposed l o c a t i o n g e n e r a l l y i n c l u d e s a b r e a k w a t e r t o p r o v i d e an a r e a s h e l t e r e d f r o m t h e waves. As b r e a k w a t e r s a r e u s u a l l y d e s i g n e d t o p e r m i t some o v e r -t o p p i n g b y . -t h e waves d u r i n g s e v e r e s -t o r m s , i -t becomes n e c e s s a r y t o p r e d i c t t h e c h a r a c t e r i s t i c s o f t h e waves so t r a n s m i t t e d i n t o the h a r b o r , t o a s s u r e t h a t t h e wave a c t i o n i n t h e s h e l t e r e d a r e a i s w i t h i n a c c e p t a b l e l i m i t s . T h i s r e p o r t p r e s e n t s a t h e o r y f o r wave t r a n s m i s s i o n by o v e r t o p p i n g , based on an e v a l u a t i o n o f t h e energy c o n t e n t o f t h e o v e r t o p p i n g w a t e r . W h i l e t h e c o e f f i c i e n t s f o r r e f l e c t i o n and r e g e n e r a t i o n a r e open t o q u e s t i o n , t h e l a b o r a t o r y d a t a shows t h a t t h e g e n e r a l f o r m o f t h e e q u a t i o n s

developed i s c o r r e c t . Comparison w i t h l a r g e - s c a l e model t e s t s

r e i n f o r c e s t h i s b e l i e f , and c o m p a r i s o n o f a t h e o r e t i c a l p r e d i c -t i o n o f -t h e volume o f o v e r -t o p p i n g w a -t e r w i -t h a p o r -t i o n o f -t h e p u b l i s h e d d a t a a g a i n shows r e a s o n a b l e agreement. An e n v e l o p e c u r v e f o r t h e t r a n s m i s s i o n c o e f f i c i e n t , based on d a t a f r o m s e v e r a l s o u r c e s , g i v e s a s i m p l e t o o l f o r p r e l i m i n a r y e s t i m a t e s o f t h e t r a n s m i s s i o n c o e f f i c i e n t . THEORY

When a wave o v e r t o p s a b r e a k w a t e r , some o f t h e i n c i d e n t wave

energy i s r e f l e c t e d , some i s d i s s i p a t e d , and some i s t r a n s m i t t e d

when t h e wave runup on t h e seaward f a c e o f t h e b r e a k w a t e r exceeds

the l e v e l o f t h e b r e a k w a t e r c r e s t . A s l u g o f w a t e r f r o m each

i n c i d e n t wave o v e r t o p p i n g t h e s t r u c t u r e f l o w s a c r o s s t h e t o p

and down t h e l e e f a c e o f t h e b r e a k w a t e r ; t h e t r a n s m i t t e d waves

are g e n e r a t e d i m p u l s i v e l y b y t h i s w a t e r mass, and t h u s t h e energy

c o n t e n t o f t h e s e t r a n s m i t t e d waves must be d e r i v e d f r o m t h e o v e r

-t o p p i n g w a -t e r .

T h i s energy c o n t e n t can be c o n v e n i e n t l y e v a l u a t e d i n s t e p s :

f i r s t , t h e energy c o n t e n t o f t h e w a t e r as i t c r o s s e s t h e b r e a k w a t e r

c r e s t must be e s t i m a t e d ; n e x t , t h e f r i c t i o n (and p e r c o l a t i o n ) l o s s e s

must be d e t e r m i n e d ; and f i n a l l y , t h e r e g e n e r a t i o n p r o c e s s must be

s t u d i e d .

The energy c o n t e n t o f t h e o v e r t o p p i n g w a t e r mass can be e s t i m a t e d

by assuming t h a t t h e t o t a l energy c o n t e n t o f t h e o v e r t o p p i n g w a t e r i s

the same as t h e energy o f t h a t p o r t i o n o f t h e wave runup t h a t w o u l d l i e

above t h e b r e a k w a t e r c r e s t were t h e b r e a k w a t e r f a c e e x t e n d e d t o - a h i g h e r

e l e v a t i o n as shown i n F i g . 1 . A t maximum r u n u p , f l o w v e l o c i t i e s a r e

e s s e n t i a l l y z e r o , and a l l t h e energy o f t h i s w a t e r i s i n t h e f o r m o f

p o t e n t i a l e n e r g y . By knowing t h e shape and p o s i t i o n o f t h i s h y p o t h e t

-i c a l runup wedge, t h -i s p o t e n t -i a l energy can be c a l c u l a t e d .

W i t h t h e above assumptions s t a t e d , t h e s o l u t i o n t o t h e p r o b l e m can

be o u t l i n e d as f o l l o w s :

I . The shape o f t h e runup wedge i s s p e c i f i e d as an n - t h

degree p a r a b o l a f r o m t h e f i r s t seaward wave t r o u g h t o t h e

p o i n t o f maximum runup on t h e extended b r e a k w a t e r f a c e .

-I -I , Mass c o n s e r v a t i o n r e q u i r e s t h a t t h e volume o f runup

above t h e s t i l l w a t e r l e v e l (SWL) equals t h a t "removed"

i n t h e t r o u g h , below t h e s t i l l w a t e r l e v e l , o v e r t h e

r e g i o n f r o m t h e f i r s t seaward wave t r o u g h ( a t maximum

runup) t o t h e b r e a k w a t e r .

I I I . The energy c o n t a i n e d i n t h e runup wedge i s e v a l u a t e d

f r o m t h e n e t energy f l u x i n t o a c o n t r o l volume e n c l o s i n g

t h e runup wedge and t h e p a r t i a l s t a n d i n g wave system j u s t

seaward o f t h e b r e a k w a t e r .

IV. The o v e r t o p p i n g energy i s e v a l u a t e d as t h e p o t e n t i a l

energy o f t h a t p o r t i o n o f t h e runup wedge e x t e n d i n g above

t h e a c t u a l b r e a k w a t e r c r e s t e l e v a t i o n a t maximum runup

on t h e e x t e n d e d f a c e .

V, The o v e r t o p p i n g w a t e r volume i s s i m i l a r l y e v a l u a t e d as

t h a t p o r t i o n o f t h e runup wedge l y i n g above t h e b r e a k w a t e r

c r e s t a t maximum r u n u p .

V I . The t r a n s m i t t e d wave energy i s e v a l u a t e d by a c c o u n t i n g

f o r t h e n e t energy f l u x i n t o a c o n t r o l volume w h i c h

en-c l o s e s t h e o v e r t o p p i n g f l o w and t h e t r a n s m i t t e d wave t r a i n .

The above g r o u p i n g p r o v i d e s a c o n v e n i e n t means o f s u m m a r i z i n g

the a n a l y t i c a l d e t a i l s o f t h e t h e o r y w h i c h f o l l o w s .

W h i l e t h e a n a l y s i s i s f o r a t w o - d i m e n s i o n a l s e c t i o n o f t h e

s t r u c t u r e , w i t h waves a r r i v i n g a t n o r m a l i n c i d e n c e t o t h e s t r u c t u r e ,

i t s h o u l d a p p l y a l s o t o waves a r r i v i n g n e a r l y p e r p e n d i c u l a r t o t h e

I . P a r a b o l i c Runup Wedge

At maximum r u n u p , t h e shape o f t h e runup wedge i s assumed t o

be a p a r a b o l a w i t h i t s v e r t e x a t t h e b o t t o m o f t h e f i r s t wave t r o u g h . The c o r r e s p o n d i n g e q u a t i o n i s Y = MX'^ - A where Y i s t h e w a t e r s u r f a c e e l e v a t i o n above t h e SWL, X i s d i s t a n c e f r o m t h e t r o u g h s h o r e w a r d , and A i s t h e a m p l i t u d e a t t h e t r o u g h ( F i g . 1 ) . T h i s e q u a t i o n s a t i s f i e s t h e c o n d i t i o n s o f s u r f a c e c o n t i n u i t y

and s l o p e a t i t s v e r t e x and a p p r o x i m a t e s t h e shape o f runup wedges

observed i n t h e l a b o r a t o r y . I t w i l l be assumed t h a t maximum runup

o c c u r s i n phase w i t h t h e extremum i n t h e p a r t i a l s t a n d i n g wave p r o f i l e .

Using l i n e a r wave t h e o r y t o d e s c r i b e t h e wave m o t i o n seaward o f t h e

f i r s t t r o u g h , A becomes A = A. + A 1 r = A^ ( 1 + k^) where = i n c i d e n t wave a m p l i t u d e A^ = r e f l e c t e d wave a m p l i t u d e and k = r e f l e c t i o n c o e f f i c i e n t * r

The runup e q u a t i o n may be w r i t t e n i n d i m e n s i o n l e s s f o r m by d i v i d i n g

t h r o u g h by A, i . e . .

where L i s t h e v a l u e o f X where Y = R, t h e runup h e i g h t ; t h u s ,

* The symbol c o n v e n t i o n used t h r o u g h o u t t h i s paper i s l o w e r case l e t t e r s r e p r e s e n t d i m e n s i o n l e s s q u a n t i t i e s and c a p i t a l l e t t e r s r e p r e s e n t

ML„ = R + A R and Y ^ R + A / i L \ _ The f o l l o w i n g d i m e n s i o n l e s s q u a n t i t i e s a r e d e f i n e d t o s i m p l i f y subsequent a l g e b r a : (H^^, H, X^, X^, and S a r e d e f i n e d i n F i g . 2 ) . u u 1 Y R ^ H A y , r , h ^ , h , l = , — , -1 = 2 L ^ 1 ^2 AS ^ X,X ,X , S , l , , , , ^ ^ ^R ^R ^R ^R^ ^R As a r e s u l t , t h e d i m e n s i o n l e s s runup s u r f a c e becomes y = ( r + l ) x - 1 ₍₁₎ I I . C o n s e r v a t i o n o f Mass R e f e r r i n g t o F i g . 2, mass c o n s e r v a t i o n r e q u i r e s t h a t t h e volume

o f w a t e r c o n t a i n e d i n t h e runup wedge between x = x^ and x = 1 e q u a l

volume m i s s i n g f r o m t h e v o i d between x = 0 and x = x.^. That i s .

1 2 ydx - ^ = 0 0 E v a l u a t i n g t h e i n t e g r a l y i e l d s , r + 1 - 1 s r _{+ 1} or 2_ r - n 2 n + 1 ( 2 )

The p h y s i c a l l i m i t s on n a r e n - 1,0 f o r t h e runup s u r f a c e t o be

E q u a t i o n (2) a p p l i e s s t r i c t l y t o impermeable s l o p e s , and

w i l l y i e l d c o n s e r v a t i v e l y h i g h e s t i m a t e s f o r o v e r t o p p i n g volumes

on permeable b r e a k w a t e r s .

I I I . Energy A n a l y s i s - Seaward Face

As t h e t r a n s m i t t e d wave energy comes f r o m t h e o v e r t o p p i n g

w a t e r , t h i s o v e r t o p p i n g e n e r g y , E^ i s u n a v a i l a b l e t o f o r m t h e

r e f l e c t e d wave. For t h e c o n t r o l volume o f F i g . 3, t h e energy f l u x

over a wave p e r i o d T can be w r i t t e n ,

The power i n i s t h a t o f t h e i n c i d e n t wave. The power o u t i n c l u d e s

t h a t o f t h e r e f l e c t e d wave and t h e o v e r t o p p i n g w a t e r . The l o s s e s

i n c l u d e f r i c t i o n l o s s e s on t h e s l o p e and l o s s e s i n t h e r e g e n e r a t i o n

of t h e r e f l e c t e d wave.

I f PE i s d e f i n e d as t h e p o t e n t i a l energy o f t h e e n t i r e runup

wedge above SWL a t maximum r u n u p , t h e n PE - E^ i s t h a t p o r t i o n o f t h e

runup energy w h i c h r e t u r n s seaward v i a rundown.

P a r t o f t h i s r e t u r n i n g energy i s l o s t t o s u r f a c e f r i c t i o n and

e n t r a n c e l o s s e s i n t h e rundown p r o c e s s . I n d e p e n d e n t s t u d i e s a t

M.I.T. have enumerated t h e l o s s e s f o r a s l u g o f f l u i d r e l e a s e d down

a smooth s l o p e , (Sy, 1969, K i n g , 1 9 7 0 ) . These s t u d i e s i n d i c a t e d t h a t

r e c o n v e r s i o n o f rundown energy t o r e f l e c t e d ( o r t r a n s m i t t e d ) wave

energy i s v e r y i n e f f i c i e n t : i f E i s t h e p o t e n t i a l energy o f t h e s l u g

a t t h e onset o f rundown, t h e n t h e r e c o n v e r s i o n t o wave energy r e s u l t s

i n energy l o s s e s e q u a l t o k. E where k i s a l o s s c o e f f i c i e n t and

f o r a smooth s l o p e o f 1:1.0.

R e c a l l i n g t h e energy f l u x r e l a t i o n s h i p s f r o m l i n e a r wave t h e o r y . Power i n - Power o u t Power l o s s

E q u a t i o n ( 3 ) becomes E^ (PE - E ^ ) k ^

h%

"^r^g " = - ( 4 ) where C = energy p r o p a g a t i o n r a t e ( g r o u p v e l o c i t y ) E. = i n c i d e n t wave energy d e n s i t y \ 1 A 2 2 ^ i Y = s p e c i f i c w e i g h t o f f l u i d E^ = r e f l e c t e d wave energy d e n s i t y 2 ^ i "^r L e t T = ^ = Hgve l e n g t h C H a v e c e l e r i t y PE p e = E o e

where pe and e^ a r e d i m e n s i o n l e s s p o t e n t i a l e n e r g i e s , and A =

A ^ ( l + k^) as p r e v i o u s l y d e f i n e d . S u b s t i t u t i o n o f t h e above i n t o E q u a t i o n ( 4 ) y i e l d s G T 1 ( 1 - k ^) ^ \ ( 1 + k ) ^ r

I-where k^ i s r e f l e c t i o n c o e f f i c i e n t o f t h e o v e r t o p p e d s t r u c t u r e . T h i s

w i l l be t a k e n as a l i n e a r i n t e r p o l a t i o n between Miches r e f l e c t i o n

co-e f f i c i co-e n t , k^, f o r a s t r u c t u r co-e whosco-e h co-e i g h t co-excco-eco-eds maximum r u n u p , and

zero r e f l e c t i o n f o r no s t r u c t u r e . T h a t i s ,

( h + h, )

k = k

—

( 6 )

E q u a t i o n ( 6 ) may be I n t e r p r e t e d as a r e d u c t i o n i n t h e r e f l e c t i o n

coe f f i c i coe n t by a f a c t o r w h i c h i s p r o p o r t i o n a l t o t h coe f i c t i t i o u s b r coe a k

-w a t e r e x t e n s i o n . P r e s e n t l y a v a i l a b l e i n f o r m a t i o n about p a r t i a l

r e f l e c t i o n f r o m o v e r t o p p e d s t r u c t u r e s does n o t j u s t i f y a more

e l a b o r a t e r e l a t i o n s h i p .

The p o t e n t i a l energy o f t h e runup wedge must be e v a l u a t e d

r e l a t i v e t o i t s a b i l i t y t o r e t u r n w o r k t o t h e r e f l e c t e d wave system.

The r e c o n v e r s i o n o f runup p o t e n t i a l energy t o wave energy o c c u r s

i n two s t e p s : ( 1 ) The p o t e n t i a l energy o f t h e s l u g i s c o n v e r t e d t o

k i n e t i c energy as t h e s l u g f a l l s t o t h e w a t e r s u r f a c e , and ( 2 ) t h i s k i n e t i c energy i s c o n v e r t e d t o f l u i d m o t i o n i n t h e r e - e n t r a n c e v i c i n i t y . Much o f t h e i n d u c e d f l u i d m o t i o n i s t u r b u l e n t and i s e v e n t u a l l y l o s t t o v i s c o u s d i s s i p a t i o n (as a c c o u n t e d f o r by t h e l o s s c o e f f i c i e n t , k ^ ) . The r e m a i n i n g m o t i o n c o n t r i b u t e s t o a component o f t h e r e f l e c t e d wave. I t i s i m p o r t a n t t o n o t e , h o w e v e r , t h a t once the c e n t e r o f g r a v i t y o f t h e s l u g f l o w has e n t e r e d t h e w a t e r , t h e s l u g i s n e u t r a l l y b o u y a n t and t h e r e f o r e has no f u r t h e r p o t e n t i a l t o

a c c e l e r a t e . The r e c o n v e r s i o n energy i s l i m i t e d t o t h e k i n e t i c energy

g a i n e d by t h e s l u g i n f a l l i n g t o t h e w a t e r s u r f a c e . C o n s e q u e n t l y , a t

maximum runup t h e p o t e n t i a l energy s h o u l d be measured r e l a t i v e t o t h e

w a t e r s u r f a c e . Due t o t h e u n c e r t a i n t y o f p o s i t i o n o f t h e w a t e r s u r

-f a c e d u r i n g r e - e n t r a n c e , t h e p o t e n t i a l energy w i l l be e v a l u a t e d w i t h

r e s p e c t t o t h e SWL.

I t f o l l o w s d i r e c t l y t h a t t h e d i m e n s i o n l e s s p o t e n t i a l e n e r g y o f t h e

runup wedge a t maximum runup i s s i m p l y (see F i g . 2 ) ,

-pe Z_ 2 2 dx - s r At X = x ^ , y = 0 1/n thus = ( ^TT ) and c o m p l e t i n g t h e above i n t e g r a t i o n y i e l d s pe = _{2n + 1} ( r + 1)-1 1 -( r + 1) 2 + 1/n 2 ( r + 1) n + 1. 1 -( r + 1) 1 + 1/n 1

-(r

+

D^^M

s r (7) IV. O v e r t o p p i n g Energy As d e s c r i b e d e a r l i e r , t h e o v e r t o p p i n g energy i s e q u a l t o

the p o t e n t i a l energy o f t h a t p o r t i o n o f t h e runup wedge e x t e n d i n g

above t h e b r e a k w a t e r c r e s t . The p o t e n t i a l energy i s e v a l u a t e d w i t h

r e s p e c t t o t h e SWL, t h u s e = o 1 2 • (y - h^) ( h ^ + dx ^ h^ + At X = X y = h t h u s x = h, + 1 b 1/n 2 [ r + 1 Completing t h e above i n t e g r a t i o n y i e l d s

1

j

( r + 1 ) 2 'o 2 2 ( r + 1) n + 1 ( 1 - h^^) 2n + 1

i J V ^ )

^ r + 1 / 2 + 1/n + 1 \ 1 + 1/n r + 1 + 1 h, + 1 1 - b ! , r + 1 /' ^ 1/n ^

I

( r - h ^ ) ^ ( r + 2h^) ( 8 ) ) S u b s t i t u t i o n o f E q u a t i o n s ( 7 ) and ( 8 ) i n t o E q u a t i o n ( 5 ) completes t h e energy c o n s e r v a t i o n a n a l y s i s on t h e seaward s i d e o f t h e b r e a k -w a t e r . V. O v e r t o p p i n g Volume The o v e r t o p p i n g volume i s e v a l u a t e d f r o m F i g . 2 as t h a t

volimie i n t h e runup wedge w h i c h extends above t h e b r e a k w a t e r c r e s t .

That i s , V = ^ s ( r - h, ) 2 (y - h^) dx E v a l u a t i n g t h e i n t e g r a l y i e l d s V = r + 1 + 1 ( 1 + h^) 1 1 -h, + 1 b . r + 1 1 + 1/n •) + 1 1 / n - l r + 1s ( r h ^ ) -( 9 )

V I . T r a n s m i t t e d Wave Energy

Energy c o n s e r v a t i o n r e q u i r e s t h a t d u r i n g each wave p e r i o d

t h e n e t energy accumulated i n t h e c o n t r o l volume s k e t c h e d i n F i g . 4

sums t o z e r o , i . e . ,

power i n - power o u t = power l o s s ( 1 0 )

The energy f l u x i n t o t h e c o n t r o l volume i s s i m p l y t h e o v e r t o p p i n g

energy d i v i d e d by t h e wave p e r i o d . The power l o s s e q u a l s t h e o v e r t o p p i n g

energy f l u x m u l t i p l i e d t i m e s t h e rundown l o s s c o e f f i c i e n t , k^. The

energy f l u x p a s s i n g o u t o f t h e c o n t r o l volume i s t h a t a s s o c i a t e d w i t h

t h e t r a n s m i t t e d wave. A l t h o u g h t h e r e g e n e r a t i o n p r o c e s s i s q u i t e

com-p l e x , i t r e com-p e a t s com-p e r i o d i c a l l y a t t h e f r e q u e n c y o f t h e i n c i d e n t wave,

and t h u s t h e f u n d a m e n t a l mode o f t h e t r a n s m i t t e d wave has t h e same f r e

-quency as t h e i n c i d e n t wave. E x p e r i m e n t s a t M.I.T. i n d i c a t e t h a t t h i s

mode c o n t a i n s most o f t h e t r a n s m i t t e d wave energy. As a f i r s t a p p r o x i

-m a t i o n , t h e n , h i g h e r h a r -m o n i c s -may be n e g l e c t e d and t h e power l e a v i n g

t h e c o n t r o l volume d u r i n g one wave p e r i o d i s s i m p l y E^C^ where

YA. 2 ' ^ t = ^ \ % A. and k = -7- = t r a n s m i s s i o n c o e f f i c i e n t . S u b s t i t u t i n g t h e above i n t o t A, 1 E q u a t i o n (10) y i e l d s E YA.2 , E 1 k C = — k T - 2 • t • ^g T 2 2 ^ t

But E = A. ( 1 + k ) L„. e and T = — , where = t r a n s m i t t e d wave

0 1 r R o C t

l e n g t h .

-I f t h e w a t e r d e p t h on t h e l e e s i d e o f t h e b r e a k w a t e r i s t h e same on

the seaward s i d e , t h e n = L and

\' = '%hr

+

( 1 - V

( 1 1 )

The problem as d e f i n e d i n c l u d e s f i v e d i m e n s i o n l e s s unknowns:

n , r , k^, k^ and s ( o r L ^ ) . However, o n l y f o u r e q u a t i o n s have been

p r e s e n t e d t o f a c i l i t a t e t h e s o l u t i o n t o t h e above. The p e r t i n e n t

e q u a t i o n s a r e ( 2 ) , ( 5 ) , ( 6 ) , and ( 1 1 ) . A l l o t h e r q u a n t i t i e s o f i n t e r

-e s t a r -e f u n c t i o n s o f t h -e f i v -e f u n d a m -e n t a l d-ep-end-ent v a r i a b l -e s l i s t -e d .

A f i f t h r e l a t i o n s h i p i s needed t o c o m p l e t e l y s p e c i f y t h e p r o b l e m . Two

a l t e r n a t i v e r e q u i r e m e n t s have been e x p l o r e d t o s a t i s f y t h e need f o r a

f i f t h e q u a t i o n . The f i r s t a t t e m p t r e q u i r e d t h a t t h e r u n u p wedge be t a n g e n t t o t h e extended b r e a k w a t e r s l o p e a t t h e h e i g h t o f maximum r u n u p . T h i s i s a c o n d i t i o n w h i c h has been o b s e r v e d f o r g e n t l e s l o p e s . T h i s r e s t r a i n t , however, y i e l d s runup h e i g h t s e x c e e d i n g t h o s e o b s e r v e d by a f a c t o r o f 1.5 and g r e a t e r . The second a t t e m p t s p e c i f i e d t h a t t h e d i s t a n c e f r o m t h e b r e a k -w a t e r t o t h e f i r s t t r o u g h be e q u a l t o h a l f a m o d i f i e d -wave l e n g t h . The

m o d i f i e d wave l e n g t h i s computed f r o m l i n e a r wave t h e o r y as a f u n c t i o n

o f w a t e r d e p t h on t h e b r e a k w a t e r s l o p e , and i s d e f i n e d as t h e i n t e g r a t e d

average wave l e n g t h i n t h e i n t e r v a l between t h e f i r s t t r o u g h and t h e

i n t e r s e c t i o n o f t h e SWL w i t h t h e b r e a k w a t e r s l o p e . T h i s r e q u i r e m e n t

reduces t o known r e s u l t s f o r t h e two extreme c o n d i t i o n s o f v e r t i c a l w a l l s

and h o r i z o n t a l s l o p e s . I m p o s i n g t h i s r e s t r a i n t , however, y i e l d s runup

o f 0.7 and l e s s . C o r r e s p o n d i n g 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

s i m i l a r l y low.

To s a t i s f y t h e immediate need f o r a f i f t h e q u a t i o n , t h e

a u t h o r s have r e l i e d on S a v i l l e ' s e x p e r i m e n t a l r e s u l t s (B.E.B. T.M. #64)

as an i n p u t f o r runup h e i g h t s . As a s p e c i f i c example, f o r a smooth

impermeable s l o p e o f 1:1.5, wave h e i g h t t o w a t e r d e p t h r a t i o s i n

excess o f 3.0, and wave cambers near 0.05, t h e a p p r o p r i a t e runup

h e i g h t i s t w i c e t h e i n c i d e n t wave h e i g h t . Use o f S a v i l l e ' s runup

r a t i o s f o r h i g h b r e a k w a t e r c r e s t s (^^/^ ^ 0.5) y i e l d s good c o r r e l a -t i o n be-tween e x p e r i m e n -t a l and -t h e o r e -t i c a l -t r a n s m i s s i o n c o e f f i c i e n -t s . However, f o r r e l a t i v e l y low b r e a k w a t e r c r e s t s (h, / r < 0.05) S a v i l l e ' s b d a t a u n d e r e s t i m a t e s t h e e q u i v a l e n t runup h e i g h t , and p r e d i c t e d t r a n s -m i s s i o n c o e f f i c i e n t s a r e so-mewhat low. I t s h o u l d be p o i n t e d o u t t h a t t h i s l a t t e r c a t e g o r y i s o f l i t t l e i n t e r e s t f o r p r a c t i c a l b r e a k w a t e r d e s i g n . Summarizing, t h e f i v e d i m e n s i o n l e s s unknowns, n, r , k^, k^,

and s may be s o l v e d u s i n g S a v i l l e ' s e x p e r i m e n t a l runup h e i g h t s and

E q u a t i o n s ( 2 ) , ( 5 ) , ( 6 ) , and ( 1 1 ) . E q u a t i o n s ( 7 ) and ( 8 ) must be u t i l i z e d t o f i n d pe and e i n terms o f t h e f i v e unknowns. An i t e r a

-^ o

t i v e p r o c e d u r e i s employed i n s e e k i n g a s o l u t i o n w h i c h s a t i s f i e s a l l

f i v e c o n d i t i o n s s i m u l t a n e o u s l y . The a u t h o r s have found t h a t t h i s i s

most q u i c k l y a c c o m p l i s h e d by i n c r e m e n t i n g n up f r o m a minimum v a l u e o f

u n i t y u n t i l E q u a t i o n s ( 2 ) , ( 5 ) and ( 6 ) a r e s a t i s f i e d and t h e n s o l v i n g

d i r e c t l y f o r t h e t r a n s m i s s i o n c o e f f i c i e n t and t h e o v e r t o p p i n g volume.

A s i m p l e d i g i t a l computer program has been w r i t t e n i n FORTRAN IV G t o

EXPERIMENTAL EQUIPMENT

The e x p e r i m e n t s were p e r f o r m e d a t M.I.T, by Lamarre

( 1 9 6 7 ) , u s i n g a g l a s s - w a l l e d wave f l u m e 2,5 f e e t w i d e and 105 f e e t

l o n g , u s i n g a c o n s t a n t w a t e r d e p t h o f 1.5 f e e t . A t t h e f a r end o f

t h e f l u m e was an i m p e r v i o u s beach a t a 5% s l o p e .

The wave g e n e r a t o r was o f t h e f l a p t y p e , 2.5 f e e t h i g h , and

h i n g e d a t t h e b o t t o m ; t h e t o p was moved back and f o r t h by a r o d

a t t a c h e d t o a crank arm h a v i n g v a r i a b l e speed and e c c e n t r i c i t y .

The smooth, impermeable b r e a k w a t e r was l o c a t e d 51.5 f e e t

f r o m t h e wave g e n e r a t o r and was 1.3 f e e t h i g h . I t c o u l d be r a i s e d by

s m a l l i n c r e m e n t s t o a maximum h e i g h t o f 1.77 f e e t by a d d i n g b l o c k i n g

u n d e r n e a t h . These b l o c k i n g s were made o f wood, shaped and i n s t a l l e d

i n such a way as t o m a i n t a i n c o n s t a n t f r o n t and r e a r s l o p e s o f 1 v e r t i c a l

t o 1.5 h o r i z o n t a l . The h o r i z o n t a l c r e s t o f t h e b r e a k w a t e r was 0.33 f e e t

w i d e . Roughness was o b t a i n e d when d e s i r e d by a d d i n g f l a t t e n e d expanded

m e t a l l a t h s h e e t s on t o p o f t h e smooth s u r f a c e s .

The i n s t r u m e n t s used i n t h e e x p e r i m e n t were p a r a l l e l - w i r e

r e s i s t a n c e t y p e wave gages connected by a w h e a t s t o n e b r i d g e t o a t w o

c h a n n e l Sanborn r e c o r d e r . The wave gages c o n s i s t e d o f two v e r t i c a l s t a i n

-l e s s s t e e -l w i r e s 1/8" i n d i a m e t e r and 1 f o o t -l o n g , mounted f r o m above

3/4" a p a r t , and p a r t i a l l y immersed i n t h e w a t e r .

EXPERIMENTAL PROCEDURE

S i x d i f f e r e n t wave p e r i o d s were i n v e s t i g a t e d . Once t h e

f r e q u e n c y o f t h e wave g e n e r a t o r was a d j u s t e d t o t h e p r o p e r v a l u e , and

b e f o r e i n s e r t i n g t h e b r e a k w a t e r i n t o t h e f l u m e , t h e e c c e n t r i c i t y o f

" i n c i d e n t " waves g e n e r a t e d a t each s e t t i n g were r e c o r d e d .

A f t e r t h e r e c o r d i n g o f t h e s e i n c i d e n t waves was c o m p l e t e d ,

t h e b r e a k w a t e r was i n s t a l l e d , and two wave gages were mounted a t

d i s t a n c e s o f one and two w a v e l e n g t h s r e s p e c t i v e l y beyond t h e c e n t e r

-l i n e o f t h e s t r u c t u r e . The f o u r d i f f e r e n t i n c i d e n t waves a -l r e a d y

measured were t h e n r e p r o d u c e d and t h e t r a n s m i t t e d waves r e c o r d e d .

A l l t h e e x p e r i m e n t s were r e p e a t e d a f t e r a d d i n g t h e expanded m e t a l

sheet f o r roughness.

The c o m p l e t e s e r i e s o f t e s t s was made f o r each i n c r e a s e i n

t h e h e i g h t o f t h e b r e a k w a t e r . A f t e r t h e b r e a k w a t e r was a t i t s

maximum h e i g h t , i . e . , when t h e r e was no more o v e r t o p p i n g , t h e s t r u c

-t u r e was p u l l e d o u -t o f -t h e w a -t e r , -t h e f r e q u e n c y o f -t h e wave-maker changed,

and t h e p r o c e s s r e p e a t e d f o r f i v e more p e r i o d s .

EXPERIMENTAL RESULTS AND DISCUSSION

The t r a n s m i t t e d wave h e i g h t s measured by gages 1 and 2 were

averaged t o o b t a i n an e s t i m a t e o f " t h e " t r a n s m i t t e d wave h e i g h t . Due

t o t h e presence o f t h e h i g h e r harmonics i n t h e t r a n s m i t t e d wave s y s t e m ,

t h e r e was g e n e r a l l y some v a r i a t i o n between t h e wave h e i g h t s measured

a t t h e two l o c a t i o n s ; t h i s v a r i a t i o n t y p i c a l l y amounted t o 5 t o 15 p e r

-c e n t . The i n -c i d e n t wave h e i g h t i s s i m i l a r l y t a k e n as t h e a v e r a g e o f

t h e two wave gages, b u t t h e d i f f e r e n c e o n l y amounts t o a p e r c e n t o r two.

S i m i l a r e f f e c t s were n o t e d by t h e U. S. Army Corps o f E n g i n e e r s (1965)

i n t h e Dana P o i n t model t e s t s .

e x p e r i m e n t a l r u n s . The curves p l o t t e d a r e s o l u t i o n s t o Eq. 11

and d e p i c t t h e o r e t i c a l bounds f o r smooth and rough s u r f a c e s . The

smooth s u r f a c e c u r v e corresponds t o a runup r a t i o R/H. = 1.8, a

l o s s c o e f f i c i e n t k„ ^ ,

£ 0,6, and an ' i n t r i n s i c " c o e f f i c i e n t o f r e f l e c

t i o n (as d e s c r i b e d by M i c h e ) p = 0.8. The rough s u r f a c e c u r v e c p r r e s

-ponds t o R/H^ = 1.6, = 0.8 and p = 0.7.

The runup r a t i o s used a r e t h o s e i n d i c a t e d by o u r own s t u d i e s .

They a r e somex^hat l o w e r t h a n t h o s e suggested by S a v i l l e (R/H. - 2.0)

and p r o b a b l y i n c l u d e some s c a l e e f f e c t s as w e l l as p e c u l i a r i t i e s o f

t h e e x p e r i m e n t a l a p p a r a t u s .

The l o s s c o e f f i c i e n t v a l u e s f o l l o w d i r e c t l y f r o m Sy's e x p e r i

-ments. He f o u n d f o r a smooth 1:1 s l o p e an average k ==0.7. I t i s

f e l t t h a t t h e r e g e n e r a t i o n p r o c e s s i s more e f f i c i e n t f o r h o r i z o n t a l

momentum t r a n s f e r (as i n a wave g e n e r a t o r f l a p ) t h a n f o r v e r t i c a l

momentum t r a n s f e r . Since t h e h o r i z o n t a l component o f rundown momentum

i n c r e a s e s f o r d e c r e a s i n g s l o p e s one m i g h t expect s m a l l e r l o s s

co-e f f i c i co-e n t s f o r morco-e g r a d u a l s l o p co-e s . Thco-erco-e i s , o f c o u r s co-e , a t r a d co-e o f f

t o s u r f a c e f r i c t i o n on v e r y g r a d u a l s l o p e s b u t f o r smooth 1:1.5 s l o p e s

a l o s s c o e f f i c i e n t e q u a l t o 0.6 seems a p p r o p r i a t e . T h i s i s i n c r e a s e d

t o 0.8 f o r t h e e q u i v a l e n t roughened s l o p e .

The i n t r i n s i c c o e f f i c i e n t o f r e f l e c t i o n i s a f u n c t i o n o f s u r

-f a c e roughness and p e r m e a b i l i t y . Miche suggests a v a l u e o -f p = 0.8

f o r smooth impermeable s l o p e s , and p ^ 0.33 f o r r u b b l e s l o p e s .

Con-s e q u e n t l y a v a l u e o f 0.8 waCon-s choCon-sen f o r t h e Con-smooth Con-s l o p e and 0.7 f o r

s l o p e s , Miche's t h e o r y y i e l d s = p,

For l o w b r e a k w a t e r s H^/R < 0.3± t h e t h e o r y u n d e r e s t i m a t e s

t h e t r a n s m i s s i o n c o e f f i c i e n t ; i t appears r e a s o n a b l e t h a t t h e

assumptions u n d e r l y i n g t h e t h e o r y a r e l e a s t v a l i d i n t h i s r e g i o n , and

moreover, t h e v a l u e s o f K and k used may n o t be a p p l i c a b l e . How¬

e v e r , t h i s range o f H^^/R i s o f l i t t l e p r a c t i c a l i n t e r e s t , as t h e

t r a n s m i s s i o n c o e f f i c i e n t s a r e t y p i c a l l y 0.3 t o 0.6.

For h i g h e r b r e a k w a t e r s (H^/R >0.5±), t h e t h e o r y g e n e r a l l y

o v e r e s t i m a t e s t h e t r a n s m i s s i o n c o e f f i c i e n t , p a r t i c u l a r l y f o r t h e

s m a l l e r v a l u e s o f r e l a t i v e d e p t h , H/L. F o r t h e most o f t h e range o f

H/L however, t h e t h e o r y p r o v i d e s an "upper e n v e l o p e " f o r k^, and t h u s

i s u s e f u l f o r p r e l i m i n a r y d e s i g n e s t i m a t e s .

I t i s i n t e r e s t i n g t o n o t e on F i g s . 5 t h r o u g h 10 t h a t t h e

i n c i d e n t wave s t e e p n e s s , H^/L has l i t t l e e f f e c t on t h e t r a n s m i s s i o n

c o e f f i c i e n t e x c e p t a t t h e l o w e s t v a l u e s o f E^/L.

As one m i g h t e x p e c t , a d d i n g roughness t o t h e s l o p e s reduces t h e

t r a n s m i s s i o n c o e f f i c i e n t f o r t h e l a b o r a t o r y d a t a , p r o b a b l y by r e d u c i n g

t h e runup and i n c r e a s i n g k .

There i s a s i g n i f i c a n t amount o f s c a t t e r i n t h e d a t a ; t h i s

can be a t t r i b u t e d t o s e v e r a l s o u r c e s .

1. Wave gage i n a c c u r a c i e s ; t h e s e gages t y p i c a l l y a r e o n l y

good t o 5 t o 10 p e r c e n t .

2. L a t e r a l r e s o n a n c e e f f e c t s can b i a s r e a d i n g s t a k e n a l o n g

t h e c h a n n e l c e n t e r l i n e . F o r s e v e r a l r u n s , i t was n o t i c e d

t h a t t h e wave c r e s t s were n o t u n i f o r m a c r o s s t h e c h a n n e l .

q u a r t e r - w a v e - l e n g t h and f o r t h e 1.0 second waves, a h a l f - w a v e - l e n g t h . For t h e o t h e r p e r i o d s , however, e s p e c i a l l y w i t h s t e e p e r waves, t h e p a r t i a l b r e a k i n g o c c a s i o n a l l y o b s e r v e d on t h e s t r u c t u r e can g e n e r a t e h i g h e r harmonics c a p a b l e o f l a t e r a l resonance. 3,- B r e a k i n g o f t h e waves on t h e s t r u c t u r e , observed f o r

t h e s t e e p e r waves, causes an a d d i t i o n a l energy l o s s n o t

accounted f o r by t h e t h e o r y . Because o f t h e s t r o n g

dependence o f b r e a k e r c h a r a c t e r i s t i c s on t h e backwash

f r o m t h e p r e c e d i n g wave, no v e r y u n i f o r m e f f e c t o f b r e a k

-i n g can be e x p e c t e d .

Besides t h e s c a t t e r , a n o t h e r s h o r t c o m i n g o f t h e d a t a i s t h a t t h e

ranges o f r e l a t i v e d e p t h , H/L, and wave s t e e p n e s s , EjL, do n o t f u l l y

cover t h o s e f o u n d i n t h e p r o t o t y p e . B r e a k w a t e r s a r e t y p i c a l l y b u i l t

i n 15 t o 40 f e e t o f w a t e r , and t h e i n c i d e n t waves t y p i c a l l y have wave

l e n g t h s f r o m 200 t o 400 f t ; thus H/L ranges f r o m a p p r o x i m a t e l y 0.04 t o

0.2 i n t h e p r o t o t y p e , w h i l e H/L i n t h e e x p e r i m e n t s ranges f r o m 0.16 t o

0.45. S i m i l a r l y t h e wave steepness H^/L, i n t h e p r o t o t y p e under s t o r m

c o n d i t i o n s w i l l be a p p r o x i m a t e l y 0.04 t o 0.10, w h i l e t h e maximum s t e e p

-ness a v a i l a b l e i n t h e e x p e r i m e n t s was 0.064. These d i f f i c u l t i e s stem

f r o m t h e d i f f i c u l t y o f g e n e r a t i n g s t e e p waves i n s h a l l o w w a t e r w i t h a

h i n g e d - f l a p wave g e n e r a t o r . The Dana P o i n t d a t a ( F i g . 1 1 ) , however,

r e p r e s e n t s p r o t o t y p e c o n d i t i o n s , and agrees w e l l w i t h t h e l a b o r a t o r y

d a t a , s u g g e s t i n g t h a t d e p t h i s n o t a v e r y i m p o r t a n t f a c t o r , e x c e p t

pos-s i b l y f o r wavepos-s w h i c h b r e a k b e f o r e r e a c h i n g t h e pos-s t r u c t u r e .

a more r e a l i s t i c b r e a k w a t e r f o r m . The e x p e r i m e n t a l d a t a

are from t h e U. S. Army Corps o f E n g i n e e r s Report d e s c r i b i n g a

s e r i e s o f s c a l e model t e s t s f o r t h e d e s i g n o f a h a r b o r a t Dana

P o i n t , C a l i f o r n i a . As t h e Dana P o i n t b r e a k w a t e r was a permeable

s t r u c t u r e , some wave energy was t r a n s m i t t e d even w i t h no o v e r

-t o p p i n g ( k ^ &l-t; 0 . 1 ) , and -t h u s , -t h e -t r a n s m i -t -t e d wave h e i g h -t s g i v e n

f o r s m a l l amounts o f o v e r t o p p i n g were n o t i n c l u d e d . The runup

r a t i o s used f o r t h e Dana P o i n t d a t a were R/H. = 1.0 and 1.1 f o r

p r o t o t y p e wave p e r i o d s o f 12 and 18 seconds r e s p e c t i v e l y . These

I

r a t i o s were e s t i m a t e d f r o m t h e d a t a f o r non o v e r t o p p i n g waves b u t

agree w e l l w i t h S a v i l l e ' s r e s u l t s . The t h e o r e t i c a l p o i n t s c o r r e s p o n d

t o t h e runup r a t i o s s t a t e d above, l o s s c o e f f i c i e n t k = 0.8, and

i n t r i n s i c c o e f f i c i e n t o f r e f l e c t i o n p= 0.4.

R e f e r r i n g t o F i g , 11 i t i s e v i d e n t t h a t a s c a l e e f f e c t

e x i s t s . The 1:50 s c a l e model y i e l d s l a r g e r t r a n s m i s s i o n c o e f f i c i e n t s

t h a n t h e 1:5 s c a l e model. T h i s i s p r o b a b l y a Reynolds e f f e c t w h e r e i n

t h e r e - e n t r a n c e l o s s e s f o r t h e l a r g e r , more t u r b u l e n t model (and

t h e r e f o r e p r o t o t y p e ) a r e h i g h e r t h a n i n t h e s m a l l e r model. The t h e o r e t i c a l s o l u t i o n l i e s between t h e two e x p e r i m e n t a l r e s u l t s . E x t r a p o l a t i n g t h e s e r e s u l t s t o p r o t o t y p e s c a l e , one expects t h e t h e o r y t o g i v e a c o n s e r v a t i v e l y h i g h e s t i m a t e f o r t h e t r a n s m i s s i o n c o e f f i c i e n t . For e n g i n e e r i n g p u r p o s e s , however, t h i s i s a d e s i r a b l e c o n d i t i o n . The t h e o r y does n o t a c c o u n t f o r d i r e c t t r a n s m i s s i o n t h r o u g h

a permeable b r e a k w a t e r . The c o n t i n u i t y e q u a t i o n , Eq. ( 2 ) , n e g l e c t s

f l o w i n t o t h e pores o f t h e b r e a k w a t e r and t h e r e b y o v e r e s t i m a t e s t h e

o v e r t o p p i n g volume. C o n s e q u e n t l y , t h e t r a n s m i s s i o n c o e f f i c i e n t due

-t o p u r e o v e r -t o p p i n g i s a l s o o v e r e s -t i m a -t e d . The -two e r r o r s a r e

compensating. However, t h e o v e r t o p p i n g r e g e n e r a t i o n p r o c e s s appears

t o be more e f f i c i e n t t h a n d i r e c t t r a n s m i s s i o n t h r o u g h b r e a k w a t e r s

o f low p e r m e a b i l i t y . The n e t r e s u l t i s t h a t t h e t r a n s m i s s i o n

co-e f f i c i co-e n t i s s l i g h t l y o v co-e r co-e s t i m a t co-e d whco-en t h co-e o v co-e r t o p p i n g t h co-e o r y i s a p p l i e d t o r u b b l e mound b r e a k w a t e r s . A g a i n , t h i s i s a d e s i r a b l e c o n d i t i o n f o r e n g i n e e r i n g e s t i m a t e s . A l l o f t h e e x p e r i m e n t a l d a t a , i n c l u d i n g Dana P o i n t , a r e p r e s e n t e d i n F i g . 12. I t d e m o n s t r a t e s t h e v a l i d i t y o f u s i n g H^/R as t h e d i m e n s i o n l e s s p a r a m e t e r f o r p l o t t i n g o v e r t o p p i n g t r a n s m i s s i o n d a t a . The e n v e l o p e c u r v e i s o f c o n s i d e r a b l e i n t e r e s t as i t appears t o be a f a i r l y c o n s i s t e n t upper bound f o r t h e t r a n s m i s s i o n c o e f f i c i e n t , o f t h e f o r m k^ = 0.65 (1.10 - Hj^/R), f o r H^/R < 1.0 ( 1 2 ) S a v i l l e (1955) has p u b l i s h e d d a t a on f l o w r a t e s a s s o c i a t e d w i t h t h e o v e r t o p p i n g o f v a r i o u s c o a s t a l s t r u c t u r e s . By m u l t i p l y i n g t h e g i v e n d i s c h a r g e s ( c f s / f t o f c r e s t w i d t h ) by t h e wave p e r i o d , an

o v e r t o p p i n g volume per wave i s o b t a i n e d . These r e s u l t s can be compared

w i t h Eq. ( 9 ) . F i g . 13 shows Eq. ( 9 ) p l o t t e d d i m e n s i o n l e s s l y as 2

V/(R - Hj^) (where V = A v , t h e d i m e n s i o n a l o v e r t o p p i n g volume per

f o o t o f b r e a k w a t e r c r e s t ) vs S a v i l l e ' s r e s u l t s . The d a t a shown a r e

f o r a s t r u c t u r e w i t h a smooth f a c e on a 1:1.5 s l o p e , f o r t h e d e p t h s

o f 4.5 and 9.0 f e e t a t t h e t o e , wave p e r i o d s o f 2.96 t o 6.4 seconds,

and f o r runup r a t i o s , based on n o n - o v e r t o p p i n g wave d a t a , r a n g i n g f r o m

been n e g l e c t e d . F o r t h e l a t t e r t h e h e i g h t s and runups a r e

r e p o r t e d o n l y t o t h e n e a r e s t f o o t , and t h e s m a l l v a l u e s o f R - H^^

computed a r e n o t r e l i a b l e . The t h e o r e t i c a l p o i n t s s o l v e d f o r used

the g i v e n runup r a t i o s , = 0,7 and p = 0.8. The c o r r e l a t i o n

between t h e o r y and e x p e r i m e n t ( p e r f e c t c o r r e l a t i o n i s t h e 45° l i n e )

f u r t h e r s u p p o r t s t h e a n a l y t i c a l a s s u i n p t i o n s . A g a i n t h e t h e o r y

con-s e r v a t i v e l y o v e r e con-s t i m a t e con-s t h e v o l u m e con-s , a p r o b a b l y concon-sequence o f

the h i g h runup r a t i o s used,

CONCLUSIONS

The t h e o r y p r e s e n t e d d e s c r i b e s t h e e s s e n t i a l f e a t u r e s o f

the p r o c e s s o f wave t r a n s m i s s i o n b y o v e r t o p p i n g , f o r waves a r r i v i n g

a t t h e b r e a k w a t e r w i t h o u t b r e a k i n g and a t n o r m a l i n c i d e n c e . The

r e l a t i o n s h i p d e r i v e d i s dependent on t h e c o e f f i c i e n t s o f r e f l e c t i o n

and l o s s , and t h e runup r a t i o . F u r t h e r i n v e s t i g a t i o n o f t h e s e

quan-t i quan-t i e s w o u l d p e r m i quan-t a r e f i n e m e n quan-t o f quan-t h e quan-t h e o r y .

The e n v e l o p e c u r v e may be used f o r p r e l i m i n a r y e s t i m a t e s

of t h e t r a n s m i s s i o n c o e f f i c i e n t . For r u b b l e mound b r e a k w a t e r s t h i s

e s t i m a t e can be i m p r o v e d by u s i n g t h e t h e o r y a l o n g w i t h a runup r a t i o

R/H_j^ = 1.0, a l o s s c o e f f i c i e n t k^ = 0.8, and an i n t r i n s i c r e f l e c t i o n c o e f f i c i e n t p = 0.4.

REFERENCES

U. S. Army C o r p s o f E n g i n e e r s ( 1 9 6 6 ) , " S h o r e P r o t e c t i o n , P l a n n i n g and D e s i g n " , C o a s t a l E n g i n e e r i n g R e s e a r c h C e n t e r , T e c h n i c a l R e p o r t No. 4, T h i r d E d i t i o n , U. S. Government P r i n t i n g O f f i c e . M i c h e , R., ( 1 9 5 3 ) , "The R e f l e c t i n g Power o f M a r i t i m e Works E x p o s e d

to A c t i o n o f t h e Waves", B u l l . B e a c h E r o s i o n B o a r d , U. S. Army C o r p s o f E n g i n e e r s , V o l . 7, No. 2, A p r i l 1 9 5 3 , p. l - , 7 . Sy, T., ( 1 9 6 9 ) , P e r s o n a l C o m m u n i c a t i o n K i n g , N. , ( 1 9 7 0 ) , "Wave R e g e n e r a t i o n i n B r e a k w a t e r O v e r t o p p i n g " , S. M. and Ocean E n g i n e e r T h e s i s s u b m i t t e d t o t h e D e p a r t m e n t o f N a v a l A r c h i t e c t u r e and M a r i n e E n g i n e e r i n g , M.I.T., C a m b r i d g e , M a s s . , A u g u s t 1 9 7 0 .

U. S, Army Corps of E n g i n e e r s ( 1 9 6 5 ) , " G e n e r a l D e s i g n f o r Dana P o i n t H a r b o r , Dana P o i n t , C a l i f o r n i a " , S e p t e m b e r 1 9 6 5 .

S a v i l l e , T., J r . , ( 1 9 5 5 ) , " L a b o r a t o r y D a t a on Wave Runup and O v e r -t o p p i n g on S h o r e S -t r u c -t u r e s " , U. S. Army C o r p s o f E n g i n e e r s , B e a c h E r o s i o n B o a r d T e c h n i c a l Memorandum No. 64. L a m a r r e , P., ( 1 9 6 7 ) , "Water Wave T r a n s m i s s i o n by O v e r t o p p i n g o f an I m p e r m e a b l e B r e a k w a t e r " , S. M. T h e s i s s u b m i t t e d t o t h e D e p a r t -ment o f C i v i l E n g i n e e r i n g a t M.I.T., C a m b r i d g e , M a s s . , S e p t e m b e r 1 9 6 7 .

C o n t r o l Volume

T 1 T — ^ 1 - T 1 — r T = 0.8 sec

Slopes

•

1 _{1 1 • i 1} 0.6 Dana P o i n t Model Data O 1:50 Model 0.5 A 1:5 Model Theory 0.4 -0.3 » A. A • ^ • ^ ^ A

•

A A

•

a

0

•

0.2 - i « % • ' 0 A -0.1 1 • • • 1 I 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 . H^/R

II 0.6 O* O O O 0.4 0.3 0.2 0.1 • A T i ï

Lab, Smooth Slope

Lab, Rough Slope

Dana P t . 1:50 S c a l e Dana P t . 1:5 S c a l e 0.65 (1.10 - H^/R) 0.4 0.8 0.1 0.2 0.3 0.4 0.5 ^076 0 ~ o.8„ ,^ 0.9 1.0

0.4

E x p e r i m e n t a l

g u r e 13 Comparison o f T h e o r e t i c a l O v e r t o p p i n g Volumes w i t h S a v i l l e ' s Data

LIST OF SYMBOLS A _{p a r t i a l s t a n d i n g wave a m p l i t u d e} A. X i n c i d e n t wave a m p l i t u d e A r r e f l e c t e d wave a m p l i t u d e

\

t r a n s m i t t e d wave a m p l i t u d e C wave c e l e r i t y c g energy p r o p a g a t i o n r a t e ( g r o u p v e l o c i t y ) E _{r u n down energy} E. X i n c i d e n t wave energy d e n s i t y E o o v e r t o p p i n g energy e o d i m e n s i o n l e s s o v e r t o p p i n g energy E r r e f l e c t e d wave energy d e n s i t y

\

t r a n s m i t t e d wave e n e r g y d e n s i t y H _{s t i l l w a t e r d e p t h} h _{d i m e n s i o n l e s s s t i l l w a t e r d e p t h}

\

b r e a k w a t e r c r e s t e l e v a t i o n above SWL

\

d i m e n s i o n l e s s b r e a k w a t e r c r e s t e l e v a t i o n H. X i n c i d e n t wave h e i g h t

h

rundown l o s s c o e f f i c i e n t k m Miche's r e f l e c t i o n c o e f f i c i e n t k r b r e a k w a t e r r e f l e c t i o n c o e f f i c i e n t

\

t r a n s m i s s i o n c o e f f i c i e n t L _{i n c i d e n t , r e f l e c t e d wave l e n g t h} h o r i z o n t a l d i s t a n c e f r o m f i r s t t r o u g h t o p o i n t o f maximum runup 36

LIST OF SYMBOLS ( c o n t i n u e d )

t r a n s m i t t e d wave l e n g t h

c o n s t a n t c o e f f i c i e n t i n p a r a b o l a e q u a t i o n

exponent i n p a r a b o l a e q u a t i o n

p o t e n t i a l energy o f runup wedge

d i m e n s i o n l e s s p o t e n t i a l energy o f runup wedge

runup h e i g h t d i m e n s i o n l e s s runup h e i g h t b r e a k w a t e r s l o p e , seaward f a c e " d i m e n s i o n l e s s " b r e a k w a t e r s l o p e , SA/L„ R wave p e r i o d o v e r t o p p i n g volume d i m e n s i o n l e s s o v e r t o p p i n g volume h o r i z o n t a l c o o r d i n a t e h o r i z o n t a l d i s t a n c e t o Y = 0, H^, r e s p e c t i v e l y d i m e n s i o n l e s s , X, X^, X2 v e r t i c a l c o o r d i n a t e d i m e n s i o n l e s s v e r t i c a l c o o r d i n a t e s p e c i f i c w e i g h t Miche's i n t r i n s i c r e f l e c t i o n c o e f f i c i e n t