semiemperical wave transformation

8/22/2019 Semiemperical Wave Transformation

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C o a s t a l E n g i n e e r i n g , 16 ( 1 9 9 2 ) 3 1 3 - 3 4 5 3 1 3

E l s e v i e r S c i e n c e P u b l i s h e rs B . V ., A m s t e r d a m

S e m i -e m p i r i c a l tr e a t m e n t o f w a v e t r a n s f o rm a t io n

o u t s i d e a n d i n s i d e t h e b r ea k e r l in e

K .P . B l a c k a n d M . A . R o s e n b e r g

Vic tor ian In s t i tute o f M arin e Sc iences , 14 Parl iam ent Place, Melbou rne 3002, Au stral ia

( R ece iv ed 1 4 Sep tem b er 1 9 9 0 ; accep ted a f t e r r ev i s io n 2 1 Mar ch 1 9 9 1 )

A B S T R A C T

B l a ck , K . P . a n d R o s e n b e r g , M . A . , 1 9 92 . S e m i - e m p i r i c a l t r e a t m e n t o f w a v e t ra n s f o r m a t i o n o u t s i d e

an d in s id e th e b r eak e r - l i n e . Coastal Eng. , 1 6 : 3 1 3 - 3 4 5 .

A s e m i - e m p i r i c a l m e t h o d i s d e v e l o p e d w h i c h p r e d i c t s t h e w a v e h e i g h t s h o a l i n g u p t o t h e b r e a k

p o i n t o n l a b o r a t o r y a n d n a t u r a l b e a c h e s . L i n e a r s h o a l i n g i s e n h a n c e d w i t h e m p i r i c a l c u r v e s f o r t h e

w a v e sh a p e , o b t a i n e d f r o m f i e ld m e a s u r e m e n t s o f w a v e h e ig h t t h r o u g h t h e b r e a k p o i n t a n d t h e s u r f

z o n e m a d e o n a n o c e a n b e a c h i n s o u t h e r n A u s t r a l ia f r o m 1 98 7 t o 1 98 9. W a v e h e ig h t s h o al i ng , b r e a k -

i n g a n d a t t e n u a t i o n a r e u n i f i e d in a m i x e d L a g r a n g i a n / E u l e r i a n n u m e r i c a l m o d e l o f r a n d o m w a v e

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

o f t h e h e i g h t t r a n s f o r m a t i o n o f i n d i v i d u a l w a v e s t h r o u g h t h e s u r f z o n e . T h i s i s d e m o n s t r a t e d f o r a

w i d e v a r i e t y o f c a se s f r o m t h e f i e l d a n d l a b o r a to r y .

I N T R O D U C T I O N

W h i le n u m e r i c a l b e a c h m o d e l s h a v e b e e n u n d e r d e v e l o p m e n t fo r se v er al

y e a r s ( e .g . E b e r s o l e a n d D a l r y m p l e , 1 9 7 9 , 1 9 8 0 ) , t h e r e i s s t il l a n e e d f o r f u r -

t h e r r e f i n e m e n t , o r at l ea s t, a n e e d f o r m o r e d a t a f o r c o n f i r m a t i o n o f t h e u n -

d e r ly i n g a s s u m p t i o n s . E x i s ti n g b e a c h n u m e r i c a l m o d e l s c o m m o n l y m a k e a

n u m b e r o f s im p l i f y in g a s s u m p t i o n s , d u e e i th e r t o t h e n e e d f o r s p e ed y s i m u -

l a t io n o r a l a c k o f d a t a t o j u s t i f y f u r t h e r c o m p l e x i t y .

T h e a v a il ab l e d a t a s e ts c o n t a i n i n g s i m u l t a n e o u s m e a s u r e m e n t s o f a n u m -

b e r o f s u r f z o n e p a r a m e t e r s a r e v e r y l im i t e d , s o a fi el d m e a s u r e m e n t p r o g r a m

w a s e s t a b l i s h e d in 1 98 7 , w h i c h c o n t i n u e d o v e r t h r e e s u c ce s s iv e s u m m e r s , t o

m e a s u r e s u r f z o n e c h a r a c t e ri s t ic s s p e c if ic a ll y f o r a p p l i c a t i o n t o n u m e r i c a l

m o d e l s i m u l a t io n s . T h i s c o m p r e h e n s i v e 3 -y ea r p r o g r a m i n c l u d e d n u m e r i c a l

m e a s u r e m e n t s o f w a v e c h a r a c t e r is t i c s a n d s u r f z o n e v e l o ci ti e s, a s w e l l a s s e d -

i m e n t t r a n s p o r t , o f f s h o r e a n d t h r o u g h t h e b r e a k p o i n t . I n t h is p a p e r , w e ex -

a m i n e t h e w a v e h e i g h t p r o p a g a t i o n : s h o a l in g , b r e a k i n g a n d a t t e n u a t i o n .

T h e p r e d i c t i o n o f w a v e s h o a li n g h a s b e e n c o n s i d e r e d b y a n u m b e r o f a u -t h o r s a n d i t is w e ll r e c o g n i z e d t h a t t h e a p p l i c a t i o n o f l in e a r t h e o r y r e su l ts i n

0 3 7 8 - 3 8 3 9 / 9 2 / $ 0 5 . 0 0 © 1 9 92 E l s e v i e r S c i e n c e P u b l i s h e r s B .V . A l l r ig h t s r e s e rv e d .

314 K.P. BLACK AND M.A. ROSENBERG

a n u n d e r - e s t i m a t e o f w a v e h e i g h t a t t h e b r e a k p o i n t ( e.g . B r i n k - K j a e r a n d

J o n s s o n , 1 97 3 ). T h i s l e d t o t h e e x a m i n a t i o n o f h i g h e r - o r d e r th e o r ie s , b u t d if -

f ic u lt ie s w e re s t il l e n c o u n t e r e d w i t h t h e i r a p p l i c a t i o n . F o r e x a m p l e , H a r d y

a n d K r a u s ( 1 98 8 ) t e s t e d c n o i d a l t h e o r y i n a s h o a l i n g m o d e l o n l y to f i n d th a t

s o m e s i g n i f i c a n t d e v i a t i o n s b e t w e e n t h e p r e d i c t i o n s a n d m e a s u r e m e n t s s t i l lo c c u r r e d o v e r a t w o - d i m e n s i o n a l e ll ip t ic a l s h oa l . M o r e o v e r , f ul ly n o n l i n e a r

s o l u t i o n s o f t h e d i f f e r e n t i a l w a v e e q u a t i o n s ( e. g. E lg a r a n d G u z a , 1 9 85 ; L i u

e t al ., 1 9 85 ) a r e v e r y d i f f i c u l t t o a p p l y i n n o n s t e a d y , w a v e h e i g h t p r o p a g a -

t i o n , b e a c h m o d e l s .

I n t h i s p a p e r , w e p r e s e n t a n e w s e m i - e m p i r i c a l m e t h o d t o p r e d i c t w a v e

h e i g ht sh o a l i n g w h i c h d r a w s o n t h e w o r k o f S v e n d s e n ( 1 9 8 4 ) . S v e n d s e n i n-

t r o d u c e d t h e w a v e s h a p e f a c t o r B 0 t o q u a n t i f y t h e r a t i o o f t i m e s e ri es v a r i a n c e

o v e r t r o u g h - t o -c r e s t w a v e h e i g h t. B o d i m i n i s h e s s h a r p ly a s t h e b r e a k p o i n t i s

a p p r o a c h e d , a s t h e s t e e p l y s h o a li n g w a v e s h a v e a t e n d e n c y t o a c c e n t th e i rh e i g h t fa r m o r e t h a n t h e i r v a r i a n c e. W e t a k e t h i s c o n c e p t a n d u t il is e it o n a

w a v e - b y - w a v e b a s is t o s p e c if y t h e s h o a l i n g o n n a t u r a l a n d l a b o r a t o r y b e a c h e s

r i g h t u p t o t h e b r e a k p o i n t . T h e m e t h o d i s t e s t e d a g a i n s t a w i d e v a r i e t y o f

m e a s u r e m e n t s , i n c l u d i n g s te e p c n o i d a l w a v e s , a n d is f o u n d t o s a t is f a ct o ri ly

p r e d i c t t h e n o n l i n e a r s h o a l i n g u p t o t h e b r e a k p o i n t i n a l l c as es .

T h e s e c o n d f u n d a m e n t a l f a c to r w e c o n s i d e r is t h e w a v e b r e a k i n g c r i te r i o n

a n d , a l t h o u g h c o n s i d e r a b l e r e s e a r c h h a s b e e n u n d e r t a k e n , t h e r e s til l a p p e a r s

t o b e w i d e v a r i a b i l i ty i n t h e d i f f e r e n t a u t h o r s ' s e l e c t i o n o f b r e a k i n g c r i te r i a.

A l t h o u g h m a n y a re v e r y s i m i l a r w h e n r e d u c e d t o t h e i r s h al lo w w a t e r f o r ma n d a s i m i l ar i ty la w w i t h t h e w a v e h e i g h t to w a t e r d e p t h r a t io a p p r o x i m a t e l y

e q u a l to 0 .7 8 is o ft e n i m p o s e d ( N o d a , 1 97 2; Y o u n g a n d L i n , 1 9 8 8 ) , t h e d if -

f e re n c e s i n t h e f o r m u l a e c a n b e h i g h l y i m p o r t a n t w i t h r e s p e c t to t h e l o c a t i o n

o f w a v e b r e a k i n g o n a n a t u r a l b e a c h . B e c a u s e t h e n a t u r e o f b r o k e n a n d u n b r o -

k e n w a v e s a r e c h a r a c t e r is t ic a l l y d if f e r e n t, a n a c c u r a t e p r e d i c t i o n o f b r e a k p o i n t

l o c a t i o n i s c r i ti c a l t o a w a v e - b y - w a v e n u m e r i c a l s o l u t i o n .

T h e t h i r d f a c t o r w e c o n s i d e r i s w a v e h e i g h t a t t e n u a t i o n i n s i d e t h e b r e a k

p o i n t . T h e D a l l y e t a l. ( 1 9 8 4 ) f o r m u l a t i o n , w h i c h a s s u m e s t h a t t h e b r o k e n

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

h e i g h t t o w a t e r d e p t h r a ti o , is e x a m i n e d a n d c o m p a r e d t o t h e p r e d i c t i o n s o f

b o r e t h e o r y ( e . g . B a t t j e s a n d J a n s s e n , 1 9 7 8 ; S t i v e , 1 9 8 4 ) . A f u l l k - e t u r b u -

l e nc e m o d e l , w h i c h h a s b e e n e x a m i n e d b y t h e a u t h o r s , w a s c o n s i d e r e d t o b e

b e y o n d t h e s c o p e o f t h i s p a p e r .

W e a r e p r i m a r i l y c o n c e r n e d w i t h a w a v e - b y - w a v e a n a ly s is f o r a p p l i c a t i o n

t o t h e g e n e r a t i o n o f lo w - f r e q u e n c y w a v e s a n d s u r f b e a t. H o w e v e r , t h e r e s u lt s

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

t e c h n i q u e s s u c h as t h o s e s u g g e s te d b y D a l l y a n d D e a n ( 1 9 8 6 ) f o r t h e a t te n -

u a t i o n , a n d t h e m e t h o d s p r e s e n t e d i n t h i s p a p e r f o r t h e s h o a li n g .

SEMI-EMPIRICALTREATMENT OF WAVETRANSFORMATION

W A V E H E I G H T D E T E R M I N A T I O N

T w o m a i n c a te g o ri e s o f n u m e r i c a l w a v e h e i g h t m o d e l a r e u t i l is e d o n

b e a c h e s . T h e s e a r e ( i ) d i r e c t w a v e h e i g h t s i m u l a t i o n s ( e. g. N o d a , 1 9 72 ) a n d( i i ) s e a l e v el s i m u l a t i o n s . I n t h e s e c o n d c a te g o r y a r e s o l u t i o n s o f t h e m i l d -

s l o p e e q u a t i o n s ( W a t a n a b e a n d D i b a j n i a , 1 9 8 8 ) a n d t h e B o u s s i n e s q e q u a -

t i o n s ( A b b o t t e t a l., 1 9 83 ) a n d h e i g h t s ar e f o u n d f r o m t h e s i m u l a t e d s e a l e v-

e ls . M a n y o f th e w a v e h e i g h t m o d e l s p r e s e n t e d i n t h e l i t e ra t u r e a r e s t e a d y-

s t a te s o l u t i o n s fo r m o n o c h r o m a t i c w a v e s , o r m o d e l s w h i c h u ti l is e t h e p r o b a -

b i l i ty d e n s i t y f u n c t i o n f o r w a v e h e i g h t s t o c r e a t e a v a r i a b l e - h e i g h t s o l u t i o n .

O n e o f t h e m o t i v a t i o n s f o r o u r s t u d y w a s t o s p e c if y t h e l o w - f r e q u e n c y o sc il -

l a t io n s i n t h e s u r f z o n e , a n d s o a f u ll y u n s t e a d y m o d e l w h i c h a c c o u n t e d f o r

t h e o r d e r o f a r r i v a l o f t h e w a v e s a s w e l l a s th e h e i g h t d i s t r i b u t i o n w a s r e-

q u i r ed . T h u s , w e e x t e n d e d t h e w a v e h e i g h t s i m u l a t i o n o f B l a ck a n d H e a l y

( 1 98 8 ) w h i c h c o n t a i n s t w o d i s t in c t , b u t c o u p l e d m o d e l s . T h e f ir st m o d e l s i m -

u l a t es t h e p r o p a g a t i o n o f w a v e h e i g h t a n d a n g le . T h e r a d i a t i o n s tr es s es a r e

d i r e c t l y o b t a i n e d f r o m t h e s e . I n t h e s e c o n d s t a g e , t h e r a d i a t i o n s t r e s s e s a r e

a p p l i e d a s th e d r i v i n g f o rc e s in a h y d r o d y n a m i c m o d e l t o c a lc u l a te t h e l ow -

f r e q u e n c y s e a le v e ls a n d c u r r e n ts . T h e m o d e l u s e s a t i m e s e ri es o f h e ig h t s a s

t h e i n p u t , a n d t h e r e f o r e a c c o m m o d a t e s t h e v a r i a b il i ty i n w a v e h e i g h t s f o u n d

o n n a t u r a l b e a c h es . A m i x e d L a g r a n g i a n / E u l e r i a n s o l u t io n is p r e s e n t e d w h i c h

h a s a n u m b e r o f a d v a n ta g e s o v e r t h e s t a n d a r d E u l e r i a n s c h e m e s , a l th o u g h a

E u l e r i a n s o l u t i o n w a s u t i li s e d i n a t w o - d i m e n s i o n a l t es t.

T h e m e t h o d s a n d m o d e l w e r e v a l i d a te d u s i n g a w i d e v ar ie t y o f d a ta . W e

c h o s e m e a s u r e m e n t s f r o m t h e l a b o ra t o r y a n d t h e f ie ld , st e ad y a n d u n s t e a d y

c o n d i t i o n s , t y p i c a l a n d s t e e p ly s h o a li n g w a v e s , a n d o n e - a n d t w o - d i m e n s i o n a l

c a se s ( T a b l e 1 ) . E x a m p l e s w i t h a n d w i t h o u t w a v e / c u r r e n t i n t e r a c t i o n w e r e

a ls o e x a m i n e d i n o n e - d i m e n s i o n a l u n s t e a d y c o n d i t i o n s . T o e n c o m p a s s t h es e

c o m b i n a t i o n s o f ca se s, t h e f ul l t w o - d i m e n s i o n a l t i m e - d e p e n d e n t h e i g h t e q ua -

t i o n i s g i v e n i n c l u d i n g t h e w a v e / c u r r e n t i n t e r a c t io n , a l t h o u g h o n l y th e t e r m s

w h i c h w e r e n e c e s s ar y w e r e i n c l u d e d i n e a ch o f t h e v a l i d a t i o n s i m u l a t i o n s

T A B L E 1

V a l i d a ti o n d a ta a n d m o d e l t y pe . T h e d a t a a r e c a te g o ri se d a s L a b o r a t o r y /F i e l d , P e r i o d i c / R a n d o m a n d

S t e a d y / U n s t e a d y . T h e m o d e l l i n g i s c a t e g o r is e d a s L a g r a n g i a n / E u l e r i a n , o n e - d i m e n s i o n a l / t w o - d i -

m e n s i o n a l a n d W a v e / C u r r e n t i n t e r a c t i o n in c l u d e d o r n o t i n c l u d e d

D a ta L / F P / R S / U L / E 1 /2 W / C

W a t a n a b e L P S L 1 N o

C n o i d a l L P S L 1 N o

A p o l l o B a y F R U L 1 Y e s

E l l i p s o i d L P S E 2 N o

3 16 K . P . B L A C K A N D M . A . R O S E N B E R G

( T a b le 1 ) . T h e t i m e - d e p e n d e n t h e i g h t e q u a t i o n i n tw o d i m e n s i o n s ( N o d a e t

a l. , 1 9 7 4 ) , o b t a i n e d b y d i f f e r e n t i a t i n g t h e e n e r g y c o n s e r v a t i o n e q u a t i o n , is :

O H O H o ) O H _ - H a0 t + ( U + c gc os 0 ) ~ x + ( V + c ~s in F o ( l a )

0 0 0 0 0 ~ - ~ + s i n _Oc~ [OU OV'~Q= - c , s in O ~ + c , co s O ~ + c o s O ~ + ~ + ~ ) + X ( l b )

a n d :

~ U ~ U O V O VX = [S xx~ + y x~ + x y~ + y y ~ /E ( l c )

w h e r e t is t im e , H is w a v e h e i g h t, U a n d V a r e x ( o n s h o r e ) a n d y ( l o n g s h o r e )

c o m p o n e n t s o f t h e c u r r e n t v e l o c i t i e s ( F i g . 1 ) , c~ t h e g r o u p s p e e d , 0 t h e w a v ea n g le ( p o s i t iv e a n t i- c l o c k w i s e o f t h e o n s h o r e - d i r e c t e d x a x i s ) , t h e r a d i a t i o n

s t r e s s e s ( S~ x , S x y, S y x, S y y ) a r e d e f i n e d b e l o w , a n d E i s t h e w a v e e n e r g y w h i c h ,

f o r l i n e a r w a v e s , e q u a l s O.125pgHz. Fo i s a c o m b i n e d d i s s ip a t io n t e ~ a c-

c o u n t i n g fo r b e d f r i c ti o n ( F r ) , a n d w a v e b r e a k i n g ( F b ) .

T h e d i s p e r s i o n r e l a t io n f o r l in e a r w a v e s i n s te a d y f lo w c o n d i t i o n s ( N o d a

e t a l . , 1 9 7 4 ) i s :

a + U k c o s 0 + V k s in 0 - ~ = 0 ( 2 a )

w h e r e :

a = ( g k t an h ( kd ) ) 1 /2 ( 2 b )

s h o r e l i n e

F i g . 1 . D e f i n i t i o n f o r w a v e m o d e l l i n g .

SEMI-EMPIRICAL TREATME NT OF WAVE TRANSFORMATIO N 317

w h e r e k is t h e w a v e n u m b e r ( k = 2 n / L , L t h e w a v e l e n g t h ) , co t h e a b s o l u t e

r a d i a n f r e q u e n c y (co = 2 z r /T , T t h e w a v e p e r i o d ) , tr is th e r e l a ti v e r a d i a n f re -

q u e n c y , g th e g r a v i t a t io n a l a c c e l e r a t i o n a n d d t h e t o ta l w a t e r d e p t h . A n i t e r-

a t iv e N e w t o n - R a p h s o n t e c h n i q u e i s e m p l o y e d to f i n d k , k n o w i n g t h e r a d i a n

f r e q u e n c y a n d U a n d 1~. T h e f u ll e q u a t i o n s f o r w a v e n u m b e r c o m p o n e n t s i na n u n s t e a d y f lo w a r e p r e s e n t e d b y s e v e r a l a u t h o r s ( e. g . Y a m a g u c h i , 1 98 8 ) .

T h e a p p l i e d b e d f r ic t io n a l r es i st a n ce t e r m i n th e w a v e h e i g h t m o d e l h a s

b e e n u t i li s ed in a n u m b e r o f n u m e r i c a l s i m u l a t io n s ( e.g . T h o r n t o n a n d G u z a ,

1 9 83 ; D a l l y a n d D e a n , 1 9 85 ; B l a c k a n d H e a l y , 1 9 8 8 ) a n d is ( a f t e r s o m e al-

g e b r a ic m a n i p u l a t i o n ) :

3 ~ ~ / s i n l ~ k d ) J ( 3 )

T h e w a v e h e i g h t lo ss d u e to b e d f r ic t io n d e p e n d s o n t h e m a x i m u m w a v e o r -

b i t a l v e l o c i ty in t h i s f o r m u l a t i o n . T h e c o e f f i c ie n t C r ( d e f i n e d i n t h e r e l a t io n -

sh ip zip = CfU~, w h e r e z is th e b e d s h e a r s tr es s a n d U b is th e m a x i m u m b e d

o r b i ta l v e l o c it y ) h a s th e s a m e d e f i n i t io n a s t h a t u s e d b y T h o r n t o n a n d G u z a

( 1 9 8 3 ) b u t is h a l f t h e m a g n i t u d e o f t h e c o e f f ic i e n t f u t i l i s e d b y D a l l y a n d

D e a n ( 19 85 ) . W a v e / c u r r e n t i n t e r a c t i o n m a y b e p a r a m e t e r i s e d b y a n a p p r o -

p r i a t e s e l e c t io n o f t h e c o e f f i c i e n t Cf, i f w a v e o r b i t a l m o t i o n is m u c h l a rg e r

t h a n t h e l o w - f r e q u e n c y c u r r e n t s ( e .g . B l a c k a n d M c S h a n e , 1 9 9 0 ) .

T h e r a d i a t io n s tr es s c o n c e p t w a s i n t r o d u c e d b y L o n g u e t - H i g g in s a n d S te w -

a r t ( 1 9 6 4 ) , w h o s h o w e d t h a t p r o g r e ss iv e w a v e s e n t e ri n g s h al lo w w a t e r in -d u c e a f lu x o f m o m e n t u m s h o r e w a r d s . F o r p u r e l y p r o g r es s iv e w a v e s ap -

p r o a c h i n g a b e a c h a t a n a n g l e 0 t h e r a d i a t i o n s tr e ss e s a re :

Sxx = E ( 1 . 5 n - 0 . 5 ) + 0 . 5 E - n c o s 2 0 ( 4 )

Syy = E ( 1 . 5 n - 0 . 5 ) - 0 . 5 E . n c os 20

Sxy =S~,~ = E . n s i n 0 c os 0 ( 5 )

n = c g / c = 0 . 5 ( 1 + 2 k d / s in h ( 2 k d ) )

a n d c is t h e p h a s e s p e ed . T h e h e i g h t m o d e l w a s c o u p l e d w i t h a h y d r o d y n a m i c

m o d e l ( B l a c k a n d H e a l y , 1 9 88 ) w i t h t h e r a d i a t i o n s tr e ss b e i n g t h e p r i n c i p l e

d r i v i n g f o rc e .

MIXED LAGRANGIAN AND EULERIAN MODEL

E v e n w i t h t h i r d - o r d e r a c c u r a t e d e r i v a t i v e s , h e i g h t d i s s i p a t i o n e r r o r s w e r e

s i g n i f ic a n t w h e n s o l v in g t h e h e i g h t p r o p a g a t i o n E q . 1 o n a n E u l e r i a n f in i t e

d i f f e r e nc e g r i d , pa r t i c u l a r l y on a c oa r s e g r i d o f m a ny c e ll s. M or e o ve r , t he s ha r ph e i g h t d i s c o n t i n u i t y at t h e b r e a k e r z o n e r e q u i r e d s p e c ia l t r e a t m e n t - - a t l ea s t

318 K.P. BLACKAND M.A. ROSENBERG

a t h ir d - o r d e r a c c u r a t e d e r iv a t i v e a p p r o x i m a t i o n w a s n e e d e d t o m i n i m i s e t h e

z i g -z a gg i ng o r i n s t a b i l i t y i n t he s o l u t i on ( B l a c k a n d H e a l y , 1988 ) .

L a g r a n g i a n m e t h o d s e l i m i n a t e th e s e p r o b l e m s , a n d a r e a n a t u r a l s e l ec ti o n

f o r s h al lo w b e a c h m o d e l li n g , a s w e r e t h e r a y t r a c k in g p r o c e d u r e s o f M u n k a n d

A r t h u r ( 1 9 52 ) . In p a r t i c u l a r , i n d i v i d u a l w a v e s n e e d t o b e t r a c k e d i n s p a c e tok n o w i f t h e y h a v e b r o k e n f u r t h e r o f fs h o re . T h e s im p l e s t m e t h o d o f a c h i e v i n g

t h i s is t o a s s i gn a t r ue o r f a l s e f la g t o i n d i c a t o r " pa r t i c l e s " r e p r e s e n t i n g e a c h

w a v e e n t e r i n g t h e o f f s h o r e b o u n d a r y . T h e f l a g c a r r i e d b y t h e p a r t i c l e t h e n

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

c o n d i t i o n ( b r o k e n o r n o t ) is u p d a t e d a s it p r o g r es s es . A n a t u r a l e x t e n s i o n o f

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

m o v e s s h o r e w a r d s a t p h a s e v e l o c i ti e s.

B y i n c o r p o r a t i n g th e s e p r in c i p l es , a m i x e d L a g r a n g i a n / E u l e r i a n s c h e m e w a s

d e v e l o p e d . T h e h e i g h t t r a n s f o r m a t i o n w a s t r e a te d u s i n g a L a g r a n g ia n s c h e m e ,w h i le t h e m o r e s m o o t h l y v a r y i n g v e l o c i ti e s a n d a n g le s w e r e t r e a t e d o n a E u -

l e r ia n f i n it e d i f f e r e n c e g r id . T h i s s c h e m e a v e r t e d t h e n e e d f o r a n y f i n i te d i f-

f e r e n c e a p p r o x i m a t i o n s f o r t h e h e i g h t . T h e s o l u t i o n w a s e x a c t i n t h e s e n s e

t h a t t h e i n d i c a t o r p a r t i c l e s h a d k n o w n p o s i t i o n s , h e i g h t s a n d h i s t o r i e s , a n d

n u m e r i c a l d i ss i p a ti o n o f h e ig h t o n t h e E u l e r i an g r i d w a s e l i m i n a t e d .

L a g r a n g ia n s c h e m e

I n o n e d i m e n s i o n , t h e p a r t ic l e p o s i t io n w a s u p d a t e d a t e a c h t im e s te p :

X 2 = X 1 --~ ( U - ~ C gC O S O)At ( 6 )

w he r e X 1 i s t he i n i t i a l pos i t i on , X 2 i s t he f i na l pos i t i on , a f t e r t i m e s t e p z l t .

A d v e c t i o n e r r o r s i n t r o d u c e d b y t h is f ir s t- o r d e r s c h e m e c a n b e r e d u c e d w i t h

h i g h e r o r d e r a c c u r a te a d v e c t i o n m e t h o d s ( e.g . B l a c k a n d G a y , 1 9 9 0 ) . H o w -

e v e r , c o n s i d e r a t i o n o f e rr o r s r e l a t e d to u n c e r t a i n t i e s i n t h e g r o u p s p e e d i n t h e

s u r f z o n e ( T h o r n t o n a n d G u z a , 1 9 83 ) s u g g e s t ed t h a t h i g h e r - o r d e r s o l u t io n s

m a y n o t b e w a r r a n t ed .

J o n s s o n ( 1 9 9 0 , e q . 6 0 ) p r e s e n t e d t h e e q u a t i o n f o r w a v e h e i g h t t r a n s fo r -m a t i o n ( f o r in i t i a l h e i g h t H i ) i n a c u r r e n t a l o n g a w a v e r a y w h i c h is:

H / H i = K c K s K ra K f ( 7 a )

w he r e K c is t he D op p l e r c oe f f i c i e n t , K s i s t he s h oa l i ng c oe f f i c i e n t , K ra i s t he

r e f r a c t i o n c o e f f i c i e n t a n d K f is t h e f r i c ti o n c o e f f i c ie n t . I n o n e d i m e n s i o n

K ra = 1 , a n d w e t r e a t f r i c t i on s e pa r a t e l y , s o t he e qu a t i o n r e du c e s to :

H 2 = H I" ( a 2/ a~ )1 /2 . ( Cgal Cga2 1/2 (7b)

w he r e a bs o l u t e g r o up s pe e ds C gal a n d Cga2 a n d t he r e l a t i ve f r e que nc i e s a~ a nd(72 a t t he i n i t i a l a nd f i na l pa r t i c l e pos i t i ons a t e a c h t i m e s t e p w e r e f oun d b y

SEMI-EMPIRICAL TREATMENT OF WAVE TRANSFORMATION 319

i n t e r p o l a t i o n o f v a lu e s p r e v i o u s l y s t o r e d a t th e c e l l m i d - p o i n t s o n t h e E u l e r-

i a n g r i d .

( T h e r e i s a n a l t e r n a t i v e t o t h is f o r m o r e c o m p l e x s i t u a ti o n s . S t ri c tl y , t h e

p a r t ic l e s o n l y i n d i c a t e t h e r a y p a t h a n d c a r r y a h e i g h t . E n o u g h i n f o r m a t i o n is

s t o r e d o n t h e t w o - d i m e n s i o n a l E u l e r i a n g r i d t o m a k e i t p o s si b le t o s o l v e f o rt h e r e f r a c t i o n c o e f f i c ie n t b y g r o u p i n g a ll t e r m s n o t d e p e n d e n t o n H ( i. e . n o n -

a d v e c t i o n t e r m s ) i n e q . 1 a n d i n t e r p o l a t i n g t h e r e su l t at t h e p a r ti c l e p o s i t io n .

T h i s i n t e r p o l a t e d v a l u e c a n t h e n b e u s e d t o u p d a t e t h e h e i g h t . W h i l e t h i s p r i n -

c ip l e w a s a p p l i e d i n t h e t w o - d i m e n s i o n a l L a g r a n g i a n / E u l e r i a n m o d e l , th e r e -

s u i ts a r e n o t d i s c us s e d i n t h i s pa pe r , s e e be l ow . )

T h e h e i g h t f o u n d w i t h e q . 7 w a s s u b s e q u e n t l y a d j u s t e d f o r b e d f r i c ti o n a n d

b r e a k i n g . F r o m e q s. 1 a n d 3 , t h e h e i g h t a d j u s t m e n t d u e to f r i c ti o n i n a m o d e l

t i m e s t e p A t w a s A H = FfAt. T h e a d j u s t m e n t d u e t o b r e a k i n g i s d e s c r i b e d la t er

( e q . 2 5 ) . T o i n t r o d u c e t h e r a d i a t i o n s tr es se s i n to t h e c o u p l e d h y d r o d y n a m i cm o d e l , t h e h e i g h t w i t h i n e a c h g r i d c e ll w a s t a k e n a s th e a v e r a g e h e i g h t o f th e

p a r t ic l e s c o n t a i n e d t h e r e i n .

I n d i c a t o r p a r ti c le s w e r e e l im i n a t e d ( i ) o n c e t h e y p a ss e d o u t o f t h e g r id o r

i n t o a d r y c e ll o n t h e b e a c h f a c e; o r ( i i ) i f t h e y m o v e d i n t o a c e ll w h e r e t h e

g r o u p s p e e d o f t h e w a v e s w a s le s s t h a n a n o p p o s i n g c u r r e n t s p e e d . I n th e l a t te r

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

w o u l d b e d i s si p a te d b y b r e a k i n g t u r b u l e n c e .

E U L E R I A N M O D E L S

W ave angle

T h e w a v e a n g l e w a s o b t a i n e d f r o m t h e z e r o v o r t i c i t y e q u a t i o n f o r w a v e

n u m b e r :

0 _ _ ( I k l s in 0) - ~ ( Ik l co s 0 ) = 0 ( 8 )0x y

T h e w a v e / c u r r e n t i n t e r a c t i o n is i m p l i ci t l y i n c l u d e d i n t h e v a l u e o f k w h i c hd e p e n d s o n c u r r e n t s tr e n g t h a n d t o t a l d e p t h . T o s o l v e f o r t h e a n g l e , E q . 8 w a s

w r i t t e n i n t h e f o r m :

0A 0B- - - - - = 0 ( 9 )0x 0y

a n d s o l v e d u s i n g t h e s p a c e - c en t r e d m i x e d i m p l ic i t s c h e m e r e c o m m e n d e d b yH a r d y a n d K r a u s ( 1 9 88 ) . T h a t i s:

A xAi , j=Ai_ l , j+-~ , . [ (1 - -a) (Bi_ l , j+l -Bi_ l , j_ l )+a(Bg , j+l -B i ,~_ l )] ( 1 0 )

w h e r e i i s t h e o n / o f f s h o r e g r i d ce ll s u b s c r ip t ( i n c r e a s in g s h o r e w a r d s ) a n d j is

d i r e c t e d lo n g s h o r e . T h e c o e f f i c i e n t " a " w a s s e t t o 0 .5 w h i c h c e n t r e s t h e y d e -

r i v a ti v e s in t h e x d i r e c t i o n . W e a d d e d a s m o o t h i n g fa c t o r ( w h i c h m i m i c s di f-

f r a c t io n ) i n t h e f o r m o f a n e d d y v i s c o si ty t e r m :

A'c~ =A i , j + N A [A i_ ~,j +A ~,j_ 1 +A~,~+ 1 - 3 .Ai,~] ( 1 1 )

w h e r e N~, w a s a s e l e c t e d c o e f f ic i e n t ( s e e la t e r ) . T h i s a d j u s t m e n t o c c u r r e d a f t e r

e a c h i t e r a t i o n a t e a c h c e ll . T h e s o l u t i o n m a r c h e s f o r w a r d f r o m o f f s h o r e , i te r -

a t i n g e a c h r o w u n t i l c o n v e r g e n c e , t y p i c a l l y a f t e r 3 - 4 i t e r a t io n s . T h e n :

0~,~ = s in - ~ Ai,~ /ki , j ) ( 12 )

T w o - d i m e n s i o n a l h e ig h t s i m u l a t i o n s

W h i le a L a g r a n g i a n / E u l e r i a n s c h e m e w a s d e v e l o p e d in t w o d i m e n s i o n s , a

E u l e r i a n s c h e m e w a s u s e d i n t h e t w o - d i m e n s i o n a l s i m u l a t i o n in t h is p a p e r

( T a b l e 1 ) b e c a u s e th e d a t a d i d n o t i n c l u d e w a v e b r e a k i n g o r w a v e / c u r r e n t

i n t e r a c ti o n . N o c u r r e n t m e a s u r e m e n t s w e r e a v a i la b l e f or v a l i d a t io n s o t h e

w a v e / c u r r e n t i n t e r a c t i o n w a s n e g l e c t e d f o r t h is c a se . T h u s , t o f i n d t h e w a v e

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

t h e f o r m e m p l o y e d b y H a r d y a n d K r a u s ( 1 98 8 ):

~ x ( F C os O ) + ~ ( F s in O ) = - F b ( 1 3 )

w h e r e F = E . Cg. H e i g h t w a s f o u n d o n a E u l e r i a n g r i d u s i n g t h e s a m e n u m e r i c a l

t e c h n i q u e s a p p l i e d t o f i n d t h e a n gl es . T h e d i s s i p a t i o n d u e t o f r ic t i o n w a s

F b = P 6 @ [H ~ o / s in h ( k d ) ]3 ( 1 4 )

a n d , i n f i n i t e d i f f e r e n c e f o r m :

F ~ i , j = ( F b i , j + F ~ i _ , ,~ ) / 2 ( 15 )T o i n c l u d e t h e w a v e / c u r r e n t i n t e r a c t i o n o n a E u l e r i a n gr id , t h e " w a v e a c -

t i o n " ( D a l r y m p l e , 1 9 88 ; J o n s s o n , 1 9 9 0 ) w a s s u b s t i t u t e d f o r F i n E q . 1 3.

H o w e v e r , n o r e s u l t s a r e p r e s e n t e d f o r th i s c a se .

H y d r o d y n a m i c m o d e l

T h e E u l e r ia n h y d r o d y n a m i c m o d e l is d e s c r ib e d b y B l a ck a n d H e a l y ( 1 98 8 ).

T h e m o d e l s o lv e s t h e tw o - d i m e n s i o n a l e q u a t i o n s f or m o m e n t u m a n d c o n ti -

n u i t y t o f i n d f lo w v e l o c it ie s a n d s e a le v e ls o n t h e s a m e g r i d e m p l o y e d f o r t h ew a v e a n g l e s.

SEMI-EMPIRICAL TREATM ENT OF WAVE TRANSF ORMATION 321

FIELD MEASUREMENTS

During a 3-year f ield program, b oth the circulat ion and sedim ent transport

on beaches were measured to facil i tate accurate calibrat ion of coupled beachdynamics numerical models (Black and Rosenberg, 1991 ). Relevant to this

paper are current and sea level time series of 17- or 34-minute duration re-

Fig. 2. Measurement platform at Apollo Bay. Instruments suspended from the front of he frame(from right to left) are capacitance wave probe, suspended sediment sampler, and current me-ter. The operators are measuring instrument elevations above the bed.

K . P . B L A C K A N D M . A . R O S E N B E R G

V I D E O

I - W A V E P R O B E I ]

~ (3,5

E~ 0 . 3. d

I JJ 0 . 1>~__.1 -0 .1

~ -o.atl J~ - 0 . 5

0 . 0 0 . 5 1 . 0 1 . 5 2 . 0

T I M E ( r a i n )

Fig. 3. Verification of wave probe accuracy comparing sea levels recorded with the capacitance

probe and digiti sed video observations.

0 . ~ A P R I L I O , 1 9 8 7 R I L 1 3 , 1 9 8 7

-2 ~ 1-4-5

0 4 0 8 0 1 2 0 1 6 0 0 4 0 8 0 1 20 16 1

M~ ~ ~ I A R C H 2 1 , 1 9 8 8

0 4 0 8 0 1 2 0 1 60

Fig. 4. Bathymet ry cross sections and ins tru ment positions for three wave height shoaling and

attenua tio n experiments. N.B. The wave height experiments were conducted on April 10 and

12, 1987 and March 22, 1988.

corded on a natural beach at Apollo Bay in southern Australia from 1987 to

1989. The beach was selected because of its nearly shore-parallel contours.

Instruments were deployed from alumi nium scaffolding frames, erected on

the beach and carr ied into the surf (Fig. 2). The frames were 1.83 m square

with hollow legs some 3 m tall, providing a work platform up to 6 m above

bed level, so that operations inside and outside the surf zone were possible in

most conditions.

Sea levels were record ed with capacitance-type wave probes which have a

thin teflon coated electrical wire along which the capacitance varies as the

immersio n level changes. One mat ter of concern was the selection of capaci-

tance wire thickness (Ti mpy an d Ludwick, 1983). If the wire is too thin,

stretching becomes a problem, causing zero shift and gain errors. Alterna-

tively, water drains o ff the wire more slowly as it thickens, causing the re-

sponse time o f the instrum ent to the red uced when the water level drops after

a wave has passed.

The instru ment beha viour was checked in the field by filming the mo tion

of water on the wave probe while simultaneously recording the instrume nt

output. Both the ins trument accuracy and the response time were found to be

of a high quality (Fig. 3 ). The response times were better in the field than instatic tank tests. The constan t vibration o f the wire by the waves apparently

SEMI-EMPIRICAL TREATM ENT OF WAVE TRANSFORMATION 323

c a u s e d th e i m p r o v e m e n t i n t h e f ie ld , a n d m a y e x p la i n t h e a p p a r e n t d i sc r ep -

a n c y b e t w e e n o u r r e su l ts u s i n g w ir e o f 2 .6 3 m m d i a m e t e r a n d t h e w i re t h ic k -

n e ss r e c o m m e n d a t i o n o f T i m p y a n d L u d w i c k ( 1 98 3 ) .

A l l i n s t r u m e n t a t i o n w a s d e p l o y e d i n t h e m o r n i n g a n d r e t ri e v e d la t er i n t h e

d a y , w h i c h a l l o w e d t h e c a l i b r a t i o n t o b e c h e c k e d tw i c e d a i ly . B a t h y m e t r y ( F ig .4 ) w a s m e a s u r e d w i t h a l e ve l a n d h a n d - h e l d s ta f f e i t h e r d u r i n g t h e s a m e d a y

o r o n t h e d a y s p r i o r t o o r f o l lo w i n g t h e e x p e r i m e n t s . R e p e t i t i o n o f m e a s u r e -

m e n t s i n d i c a t e d a v e r t i c a l a c c u r a c y o f +_ 4 c m a t t h e o f f s h o re l o c a t io n s w h e r e

w a v e a c t i o n m a d e i t d i ff i c u lt t o h o l d t h e s t a f f v e r ti c al .

An alysis techniques

P r i o r t o a z e ro d o w n - c r o s s i n g a n a l y s i s , F o u r i e r b a n d - p a s s f i l t e r in g w a s a p -

p l i e d t o t h e m e a s u r e d s e a le v e l t i m e s e ri es t o e l i m i n a t e t h e l o w - f r e q u e n c y s u r fb e a t b e l o w 0 . 0 3 H z . F o r t h e w a v e b r e a k i n g c r i t e r i o n a s s e s s m e n t , n o h i g h -f re -

q u e n c y f i lt e ri n g w a s a p p l i e d . F o r t h e w a v e s h o a l i n g a s s e s s m e n t , w a v e s w i t h

f r eq u e n c ie s a b o v e 0 . 33 H z w e r e r e m o v e d b e c a u s e t h e v a r ia n c e o f w a v e s o f

s h o r t e r p e r i o d s c o u l d n o t b e a d e q u a t e l y s p e c i f i e d a t t h e 2 H z s a m p l i n g in t e r v a l.

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

t r a n s f o r m a n d z e r o d o w n - c r o s s i n g s w e r e o b t a i n e d f r o m t h e f i l te r e d se ri es . T o

e l i m i n a t e s p e c tr a l c l i p p i n g o f t h e h e i g h t , t h e z e r o d o w n - c r o s s i n g h e i g h t s w e r e

o b t a i n e d f r o m t h e o r i g i n a l u n f i l t e r e d t i m e s e r i e s , b e t w e e n t h e z e r o d o w n -

c r o s s i n g t i m e s o b t a i n e d f r o m t h e f i l t e r e d r e c o r d s ( e .g . E b e r s o l e , 1 9 8 7 ) . T of u r t h e r r e d u c e t h e h e i g h t c l ip p i n g , a p a r a b o l a w a s f i t te d t h r o u g h t h e c r e s t a n d

t ro u g h o f e a ch w a v e , a n d t h e m a x i m u m a n d m i n i m u m s ea le ve l w a s o b ta i n e d

f r o m t h e s e c u r v e s . H o w e v e r , s o m e e r ro r , a s a f u n c t i o n o f s a m p l i n g in t e r v a l,

s ti ll o c c u r s i n v e r y s h a r p c r e s t e d w a v e s o r o n c o n c a v e w a v e f ac e s.

SHOALING

S v e n d s e n ( 1 9 8 4 ) i n t r o d u c e d t h e p a r a m e t e r B o g i v en b y:

B o = ( a z c /H z c ) 2 ( 1 6 )

w h e r e azc is t h e s t a n d a r d d e v i a t i o n o f t h e s e a le v e l r e c o r d ( w i t h z e r o m e a n )

b e t w e e n s u c c e s s i v e z e r o d o w n - c r o s s i n g s a n d H z c i s t h e c r e s t t o t r o u g h z e r o

d o w n - c r o s s i n g h e i g h t . B o i s a w a v e s h a p e f a c t o r w h i c h e q u a l s 0 . 1 2 5 f o r s i n u -

s o i d a l w a v e s a n d d i m i n i s h e s a s t h e w a v e s s h o a l . A s s u c h , B o c o u l d b e a p p l i e d

t o i n d i c a t e t h e d e v i a t i o n o f w a v e h e i g h t f r o m l i n e a r s h o al in g . F o r t h i s p u r -

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

a l e n t s i n u s o i d a l w a v e , d e f i n e d a s:

Heq = 8~/2azc (1 7 )

32 4 K.P. BLACKAND M.A. ROSENBERG

T h e e q u i v a l e n t h e i g h t is t h e c o u n te r p a r t o f t h e R M S h e i g h t o f a ti m e s er ie s ,

e x c e p t t h a t w e a r e t r e a ti n g th e r e c o r d o n a w a v e - b y - w a v e b a s i s. B y s u b s t i t u t io n :

Bo=0.125 ( H ~ q l H z c ) 2 (18)

Battjes and Janssen (1978) and Battjes and Stive (1984) found that lineartheory predicted the shoaling of RMS wave heights obtained from the time

series variance. Thus, if we can obtain a relationship between Hzc and Heq, itwould be possible to predict the shoaling of Heq using linear theory, and ob-tain the actual height from the known relationship. Data from three ApolloBay records (Tables 2 and 3 ) each containing 34-minute simultaneous time

T A B L E 2

D a t a u t il i se d : ( a ) r u n n u m b e r ; ( b ) d a t e ; ( c ) n u m b e r o f d a t a p o i n ts ; ( d ) s a m p l i n g r a t e ( H z ) ; ( e )n u m b e r o f w a v e p r ob e s ; ( f ) p e a k s p e c t ra l p e r i o d ( s ) ; ( g ) G o d a ' s s p e c t r a l p e a k e d n e s s p a r a m e t e r a t

t h e o f fs h o r e p r o b e ; ( h ) w i n d s t r e n g th a n d d i r e c t i o n ; ( i ) b a t h y m e t r y s u r v e y

( a ) ( b ) ( c ) ( d ) ( e ) ( f ) ( g ) ( h ) ( i )

Wave height shoaling and attenuation experimentsAP I 2 0 4 1 1 2 A p r 8 7 4 0 9 6 2 3 1 4 .2 2 .0 9 s t r o n g , o f f sh o r e 1 0 , 1 3 Ap r

AB2 2 0 3 1 2 2 M ar 8 8 4 0 9 6 2 3 1 2 .8 1 .2 3 l i g h t , o n sh o r e 2 1 M ar

A PI 0 0 4 5 1 0 Ap r 8 7 4 0 9 6 4 3 1 0 .0 0 .9 7 l i g h t , o n sh o r e 1 0 Ap r

Breaking criterion experimentsAY2 0 0 4 1 2 0 Ap r 8 9 2 0 4 8 2 3 1 3 .7 1 .1 4 ca lm 2 0 Ap r

AY 2 0 0 4 3 2 0 Ap r 8 9 2 0 4 8 2 3 1 3 .6 1 .0 6 ca lm 2 0 Ap r

AY 2 0 0 4 4 2 0 Ap r 8 9 4 0 9 6 4 3 1 2 .9 1 .4 7 ca lm 2 0 Ap r

AY 2 0 0 4 5 2 0 Ap r 8 9 4 0 9 6 4 3 1 3 .9 1 .2 8 ca lm 2 0 Ap r

T A B L E 3

W a v e h e i g h t s h o a l i n g a n d a t t e n u a t i o n e x p e r i m e n t s . W a v e p r o b e l o c a t i o n r e l a t i v e t o t h e b r e a k p o i n t,

a p p r o x i m a t e h o r i z o n t a l d i s ta n c e ( m ) o f p r o b e f r o m t h e a v e r a g e b r e a k p o i n t p o s i t i o n ( n e g a t i v e i s

o f f sh o r e o f t h e b re a k p o i n t ) , a n d m e a n w a t e r d e p t h ( m )

R u n L o c a t i on D i s t a n c e D e p t h

A P 1 2 0 4 1 P r l o f f s h o r e - 3 0 2 .3 1

AP 1 2 0 4 1 Pr 2 b r eak p t 0 1 .1 7

AP 1 2 0 4 1 Pr 3 in sh o r e 3 5 0 .9 6

A B 2 2 0 31 P r l o f fs h o r e - 2 6 2 .3 5

AB 2 2 0 3 1 Pr 2 b r eak p t 0 1 .4 9

AB 2 2 0 3 1 Pr 3 in sh o r e 4 6 1 .0 3

AP 1 0 0 4 5 Pr l o f f sh o r e - 1 5 1 .9 0

A P 1 0 0 4 5 P r 2 i n s h o r e 2 0 0 . 7 4

A P I 0 0 4 5 P r 3 i n s h o r e 5 0 0 . 48

SEMI-EMPIRICAL REATMENTOF WAVETRANSFORMATION 325

s e ri es f r o m t h r e e w a v e p r o b e s , l o c a t e d o f f s h o re , n e a r t o a n d i n s i d e t h e b r e a k e r

z o n e , w e r e a g g r e g a t e d to s e e k a u n i v e r s a l r e l a t i o n s h i p b e t w e e n H zc a n d H e q.

Equ ivalent height

Hzc/He, w a s f o u n d t o b e l i n e a r l y d e p e n d e n t o n b o t h Yzc a n d y~q ( F ig s . 5 a n d

6 ) , w h e r e Fzc=H~c/dw n d ye,=He,~/dww h i l e dw i s t h e m e a n d e p t h o v e r t h e

w a v e c y c le . A s y i s a n i n d i c a t o r o f t h e w a v e s h o a li n g , t h e r e l a t i o n s h i p w o u l d

s e e m t o b e a n e x p e c t e d re s ul t. H o w e v e r , t h e d a t a i n c l u d e s m e a s u r e m e n t s f r o m

a w i d e v a r i e t y o f l o c a t i o n s ( i n s id e , o u t s i d e a n d a t t h e b r e a k p o i n t ) a n d t h e

g e n e r a l a p p l i c a b i l i t y s e e m s s u r p r i s in g . T h e b e s t f i t l in e a r c u r v e i n v o l v i n g Yzc

w a s :

Hzc =H eq ( 1 .03 + 0 .7 1 y zc ) ( 1 9 )

w i t h c o r r e l a t i o n c o e ff i c ie n t r = 0 .6 5 a n d N = 1 75 5 d a t a p o i n t s . F o r g e n e r a l a p -

p l i c a t i o n , a r e l a t i o n s h i p a s a f u n c t i o n o f Yeq i s r e q u i r e d , a s ~'zc i s a n u n k n o w n

A L L P R O B E S : A P 1 2 0 4 1 A B 2 2 0 3 4 A P 1 0 0 4 1

B A N D P A S S F I L T E R E D : 0 . 0 3 t o 0 . 3 3 H z

* ~ , , ~ - I I ° o~

o . ._ .o , : . . . ~ : ~ , ¢ ~ _ ~ ,

1 .5 . " :~ ~ ; ~ o ~ ~

. . - ' . . . ~ ~ , ~

~ . . ~ o~ . 0 ~ ~ " . "

~.~ . . . . ' . . " •

0 . 5 . ' ~ ~ I , ~ , t

~ ~ oA

B ° ~ ~ ' o ~, . 0 , • . o o ~ 0 ~ o

. - ° . . . ~_ : ~ r o

~ ~ ~ ~ ~ ~

~ ' : . .

0 . 6 0 .8 1 . 0

H z c / d ~ ,

F i g . 5. T h e r a t i o o f z er o d o w n - c r o s s i n g h e i g h t o v e r e q u i v a l e n t h e i g h t (H=/H~)v e r s u s t h e r a t i o

o f z er o d o w n - c ro s s in g h ei g h t o v e r w a t er d e p t h ( H = / d w ) .

326 K.P. BLACKANDM.A.ROSENBERG

A L L P R O B E S : A P 1 2 0 41 A B 2 2 0 3 4 A P 1 0 0 4 1

B A N D P A S S F IL T E R E D : 0 . 0 3 t o 0 . 3 3 H z

t ==°

~o =p=

~ °=~

° ~ : . ~ .~ ,

°o~,°, o • o " eqn (2O b)o

• : o . : ' ~ . . . , . : . .e..:~ ,~: °~0 .- ~ . "=o . . o

~ . 7 . ~ , .g ~ : ; ~ . ~ , . '~ . r . . . " ,=~= A; " "~ .~ ~ 0%~ ~ • ~

= . ~ g ~ . "

. . . .

~ - ~. ~ " . ~ : . . o

0 5 0 . 0 0 . 2 0 . 4 0 . 6

H e q / d ~

Fig. 6. The ratio of zero down-cros sing height over equiva lent height (Hz¢/H~q)versus the ratio

of equivalent height over water d epth (Heq/d~,) .

w h e n t h e s h o a l i n g is p r e d i c t e d b y a p p l y i n g l in e a r t h e o r y . T h e b e s t f it e q u a t i o n

in v o lv in g ~'eq was :

Hzc = H eq ( 1 .07 + 0 .76~,eq) (2 0 a)

w i t h r = 0 . 4 5 a n d N = 1 75 5. A l t h o u g h t h e d a t a i s s c a t te r e d , th i s c o r r e la t i o n is

s i g n i f i c a n t a t th e a = 0 . 0 0 5 l e v e l o f s i g n i f i c a n c e ( F r e u n d , 1 9 74 , p . 4 2 8 ) . F o r

w a v e s o f sm a l l a m p l i t u d e w e c a n e x p e c t Hzc t o e q u a l Heq n d e e p w a t e r. T h el i n e a r c u r v e w h i c h p a s s e s t h r o u g h t h i s o r i g i n is:

H zc = H e q ( 1 - t- 1.05 ~'eq ) ( 2 0 b )

T h e s e E q s . 2 0 a a n d b r e l a t e t h e s h o a l i n g o f H ~m s, p r e d i c t a b l e w i t h l i n e a r t h e -

o r y ( B a t t je s a n d J a n s s e n , 1 97 8 ) , t o t h e a c t u a l m e a s u r e d w a v e h e i g h t, in c l u d -

i n g t h e n o n - l i n e a r e n v i r o n m e n t n e a r t h e b r e a k p o i n t .

T h e v a l i d i ty a n d t h e i m p o r t a n c e o f t h i s r e su l t a r e d e m o n s t r a t e d la t e r in t h e

p a p e r w h e n a n u m b e r o f c a s e s tu d i e s a re n u m e r i c a l l y s im u l a t e d . W e f i n d t h a t,

a l t h o u g h t h e r e is c o n s i d e r a b l e s c a t te r i n t h e d a t a ( F ig s . 5 a n d 6 ) , t h e l i n e a rb e s t f it E q . 2 0 p r o v i d e s a h i g h l y e f fe c t i v e m e a n s o f p r e d i c t i n g t h e s h o a l i n g f o r

SEMI-EMPIRICAL TREATMENT O F WAVETRANSFORMATION 327

a ll c as e s t e st e d . T h e s e c a s es i n c l u d e d s m a l l- s ca l e l a b o r a t o r y m e a s u r e m e n t s ,

f ie l d m e a s u r e m e n t s a n d s t ee p , s h o a l i n g c n o i d a l w a v e s . I t m a y b e p o s s ib l e t o

r e f in e E q . 2 0 b y a d d i n g m o r e d a t a f r o m a v a r ie t y o f c a se s . M o r e o v e r , t h e

i n t e r c e p t in E q . 2 0 a m a y b e a f u n c t i o n o f t h e d e e p - w a t e r w a v e s t e e p n e s s a n d

f u r t h e r t h e o r e t i c a l d e v e l o p m e n t , s u c h a s t h e a d d i t i o n o f a S t o k e s s te e p n e s st e r m , m a y b e p o s s i b l e . I n t h i s p a p e r , h o w e v e r , w e a s s e s s t h e a p p l i c a b i l i t y o f

t h e b e s t f i t c u r v e b y s i m u l a t i o n o f a n u m b e r o f d if f e r e n t sh o a l i n g c a se s. T h e

s u c c e s s o f th e s e t e s t s s u g g e s ts t h a t t h e e s s e n t i a l b e h a v i o u r o f s h o a l i n g w a v e s

is e m b e d d e d i n t h e e m p i r i c a l r e su l t, a n d t h e p r o c e s s e s c a u s in g t h e s c a t t e r ar e

s e c o n d a r y f ac t o rs .

S E L E C T I O N O F A B R E A K I N G C R I T E R I O N

A l t h o u g h m a n y r e s e a rc h e r s h a v e e x p e n d e d c o n s i d e r a b l e e f fo r t t o s p e c if y a

w a v e b r e a k i n g c r i t e r i o n , t h e v a r i a b i l i t y i n t h e i r s e l e c t i o n s j u s t i f i e d f u r t h e r i n -

v e st ig a ti o ns . F i e l d m e a s u r e m e n t s o f i n d i v i d u a l p l u n g i ng w a ve s , m e a s u r e d a t

A p o l l o B a y ( T a b le 2 ) w e r e e x a m i n e d w i t h l ab o r a to r y d a t a o f S e y a m a a n d

K i m u r a ( 1 98 8 ) a n d W a l k e r ( 1 9 76 ) . W a v e h e i g h t s a t t h e b r e a k p o i n t H b w e r e

c o m p a r e d w i t h t h e h e ig h t s p r e d i c t e d by s e v e ra l c o m m o n l y e n c o u n t e r e d

b r e a k i n g c r it er ia . T h e s e w e r e M c C o w a n ( 1 8 9 4 ) , M i c h e ( 1 9 4 4 ) , G o d a ( 1 97 0 ),

W e g g e l ( 1 9 7 2 ) , M a d s e n ( 1 9 7 6 ) a n d B a t t je s a n d J a n s s e n ( 1 9 78 ) , s ee T a b l e

T h e A p o l l o B a y d a t a w a s r e c o r d e d o n A p r i l 2 0 , 1 9 8 9 w h e n w i n d c o n d i t i o n sw e r e c a lm ( T a b l e 2 ) . S e a l e ve l s w e r e m e a s u r e d a t t h r e e c a p a c i t a n c e p r o b e s

s p a c e d o v e r 1 0 m , e r e c t e d a l o n g t h e d i r e c t i o n o f w a v e t r a v e l a t t h e m o s t c o m -

m o n b r e a k p o i n t , a v i d e o f i lm o f t h e p r o b e s w a s s i m u l t a n e o u s l y r e c o r d e d .

F r o m t h e m a n y w a v e s r e c o r d ed , th e i n d i v i d u a l w a v e s w h i c h b ro k e a t t h e p o -

s i ti o n o f a n y o f t h e t h r e e p r o b e s , a s i d e n t i f i e d f r o m t h e v i d e o , w e r e u s e d . Z e r o

d o w n - c r o s s i n g a n d s p e c t ra l a n a ly s e s w e r e p e r f o r m e d o n t h e t i m e s e ri es t o o b -

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

T A B L E 4

W a v e b r e a k i n g c r i t e r ia w i t h t h e i r s h a l lo w w a t e r e q u i v a l e n t ~ bs = H b l d b , f o r a b e a c h s l o p e m = 0 . 0 2 a n d

w a v e p e r i o d T = 1 4 s ( a n d f o r w a v e s o f 0 ( 1 m ) f o r t h e c a se o f W e g g e l) . H b , d b a n d L b a r e t h e w a v e

h e i g h t, w a t e r d e p t h a n d w a v e l e n g t h a t b r e a k i n g a n d C ~, C2 a n d C a a r e c o n s t a n t s

A u t h o r B r e a k i n g c r i t e r i o n Ybs

M c C o w a n H b = 0 .7 8 db 0 .7 8

M i c h e H b = 0 . 1 4 2L b t a n h ( 2 n d b / L ~ ) O . 8 9

G o d a H b = C~ Lo [ 1 - e - c2~< ~+ ~ , ~ ) ' / 3 d b / L ° ] -I- C3 0 . 8 7

W eg g e l H b = 1 . 5 6 / { [ 1 / r i b + 4 3 . 7 5 ( 1 - - e - ~ 9 m ) / g T 2 ] [ 1 + e - ~ 9 . 5 m ] } 0 . 9 2

M ad s en H b = 0 . 7 2 ri b ( 1 + 6 . 4 m ) 0 . 8 1B a t t j e s a n d J a n s s e n H b = O . 1 4 2 L b t a n h [ O . 9 0 9 ( 2 n d b / L b ) ] 0 .81

328 K.P. BLACKAND M.A. ROSENBERG

c h e c k e d a g a in s t t h e v i d e o r e c o r d , t o e n s u r e t h a t c r e s t e l e v a t io n s o f t h e s e -

l e c te d w a v e s w e r e p r o p e r ly s a m p l e d .

S e v e r a l d e f i n i ti o n s f o r w a t e r d e p t h a t b r e a k i n g ( d b ) a r e p o ss ib l e. T h e s e i n -

c l u d e ( i ) t h e m e a n d e p t h d m o v e r t h e e n t i r e t i m e s e r ie s , ( i i ) t h e m e a n d e p t h

dw o v e r t h e c y c l e o f t h e w a v e b e i n g c o n s i d e r e d , ( i i i ) t h e w a t e r d e p t h d t b e l o wt h e tr o u g h o f t h e w a v e a n d , f o r e x a m p l e , ( i v ) t h e d e p t h d s d e f i n e d b y S e y a m a

a n d K i m u r a ( 1 98 8 ) a s th e m e a n d e p t h p l u s h a l f t h e d i f fe r e n c e b e t w e e n c r e s t

a n d t r o u g h a m p l i t u d e . T h e e f f e c t iv e n e s s o f t h e b r e a k i n g c r it e r i a w a s f o u n d t o

d e p e n d o n t h e d e f i n i t i o n o f w a t e r d e p th . D i f f e r e n t f o r m u l a e r e q u i r e d d if fe r-

e n t d e f i n e d d ep t h s . T h e w a v e b r ea k i n g c r it e r ia o f M c C o w a n , M a d s e n a n d

B a t tj e s a n d J a n s s e n w e r e e f f e c t iv e w h e n dw w a s u s e d f o r db ( T a b l e 5 ) . T h e

M i c h e , G o d a a n d W e g g e l c r i t e r ia w e r e a ls o s u c c e ss f u l, b u t o n l y i f d t w a s u s e d .

T h e G o d a f o r m u l a w a s f o u n d t o o v e r - e s t i m a t e b r e a k i n g h e i g h t u s i n g d w .

S e y a m a a n d K i m u r a ( 1 9 8 8 ) s u g g e st ed n e w v a l u es o f t h e c o e ff ic ie n t s i n th eG o d a f o r m u l a a n d o u r r e s u l ts i n d i c a t e d c o e f f ic i en t s e q u a l to t h e m e a n o f t h o s e

o r i g in a l ly s u g g es te d b y G o d a a n d t h o s e b y S e y a m a a n d K i m u r a . T h e G o d a

f o r m u l a w a s c o m p a t i b l e w i th b o t h t h e s m a l l- sc a le la b o r a t o r y m e a s u r e m e n t s

a n d t h o s e f r o m A p o l l o B a y ( F i g . 7 ) w i th :

C~ = 0 . 1 6 5 , C2 = 1 .1 2, C3 = - 0 . 4 8 m + 0 . 1 ( 2 1 )

N o t a b l y , S e y a m a a n d K i m u r a u s e d d s, w h i c h w o u l d e x p la i n th e d i f f e r en c e

i n o u r r e su lts . H o w e v e r , t h e A p o l lo B a y m e a s u r e m e n t s w e r e n o t c o m p a t i b l e

w i t h t h e G o d a f o r m u l a w h e n d~ w a s u s e d . T h e a n a ly s is i n d i ca t e s a n e e d f o r ac o n s i s te n t d e f i n i ti o n o f th e w a t e r d e p t h .

O f th e b r e a k i n g c r i te r i a e x a m i n e d , t h r e e c o n t a i n a b e a c h s l op e d e p e n d e n c e

( M a ds e n , 1976; G o da , 1970 a n d W e ggel , 1972 ) a n d w e r e t he r e f o r e m o r e li ke ly

t o b e g e n e r a l l y a p p l i ca b l e . I n t h e n u m e r i c a l m o d e l , dw w a s m o r e r e a d i l y a v a il -

a b l e t h a n d r. T h e W e g g el f o r m u l a a p p e a r e d t o s y s t e m a t i c a ll y o v e r - e s t i m a t e

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

TABLE 5

Relative squared errors for comparison of predicted to measured Hb values for different breakingcriteria (N.B. Goda (1970) coefficients used in Goda). For the Apollo Bay data, bed slope=0.018

and 0.020, individual wave heights were used for Hb, and d~, was used for db. For the Walker (1976)

data, bed slope=0.033, individual wave heights were used for Hb, and dm was used for db

Breaking Apollo Bay Walker

criterion (1976)

McCowan (1894) 0.015 0.074

Miche ( 1944 ) 0.051 0.042Goda (1970) 0.036 0.034Weggel (1972) 0.070 0.016

Madsen (1976) 0.021 0.036Battjes and Janssen ( 1978 ) 0.020 0.073

SEMI-EMPIRICAL REATM ENTOF WAVETRANSFORMATION 329

1.00 f

~ 0 . 60 ~

0 .40 Io~o

0.000.001

Gode (197 0} coeffs.used in GO

[s t imete ~ c~f l s , i nGO for be st f i t to date

• . o .

• o ~ • ~ o ~ . . ° . - ~ o

,o° ° ,.o - . . . . ~ : , ° . o ° --. - . - - ~ . ~ " ¢ # ° ~ ° ~ •

i -_ . ., : ~ - ~Seyama/Kimure (198 8)coeffs, used in GO - -

0,01 0.1

d / L o

F ig . 7. D a t a f r o m A p o l l o B a y a n d S e y a m a a n d K i m u r a ( 1 98 8 ) c o m p a r e d t o t h e G o d a ( 1 9 7 0 )

w ave b reak i ng c r i t e r ion (G O ) , fo r th ree se t s o f coe f fi c ien t s . L o i s t he deep -w a t e r w ave l eng t h .

T h e G o d a f o r m u l a ( w i t h m o d i f i e d co e f fi ci e n ts ) a n d t h e M a d s e n f o r m u l a b o t h

p r o v i d e d a c c e p t a b l e p r e d ic t io n s , s o t h e s i m p l e r M a d s e n f o r m u l a w a s u ti li se d .

W h i l e th e M a d s e n b r e a k i n g c r it e r io n w a s e f fe c ti v e f o r t h e d a t a e x a m i n e d ,

o t h e r w o r k i n d i c a t e s t h a t p l u n g i n g c n o i d a l w a v e s a n d s o li ta r y w a v e s b r e a k

w h e n t h e h e i g h t t o d e p t h r a t i o i s l a r g e r t h a n t h e v a l u e s o b t a i n e d h e r e . T h e

d a t a o f B u h r H a n s e n a n d S v e n d s e n ( 1 9 7 9 ) a n d P a p a n i c o l a o u a n d R a i c h le n

( 1 98 7 ) i n d i c a te s a b r e a k i n g h e i g h t d e p e n d e n t o n U r s e l l n u m b e r a n d b e d s lo p e

w i t h ~, v a l u e s m o r e l i k e 1 . 0 - 1 . 2 ( c / f 0 . 8 - 0 . 9 ) o n b e d s l o p e s s i m i l a r t o t h o s ea t A po l l o B a y .

I n w a v e / c u r r e n t e n v i r o n m e n t s , th e b r e a k i n g h e ig h t v a ri e s w i t h th e c u r r e n t

s tr e n g th . R a t h e r t h a n s e e k a n e w f o r m u l a a n d t o m a i n t a i n c o m p a t i b i li t y ( s ee

b e l o w ) , w e u ti li se d t h e m e t h o d p r o p o s e d b y D a ll y a n d D e a n ( 1 9 8 6 ) . T h e

e q u i v a l e n t b r e a k i n g d e p t h deq w a s o b t a i n e d b y s o l v in g th e d i s p e r s i o n r e l a t io n :

c o 2 = g k t a n h ( k d eq ) ( 2 2 )

w h e r e k i s t h e w a v e n u m b e r p r e v i o u s l y d e t e r m i n e d u s i n g E q . 2 w h i c h i n -

c l u d e s t h e w a v e / c u r r e n t i n t e r a c t i o n a n d co i s t h e a b s o l u t e f r e q u e n c y , d eq r e -p l ac e s t h e a c t u a l d e p t h i n t h e M a d s e n b r e a k i n g c r it e ri o n . T h i s p r o v e d t o b e a

s e c o n d a r y m o d i f i c a t i o n fo r th e d a t a c o n s i d e r e d . F u r t h e r e v a l u a t i o n o f w a v e /

c u r r e n t i n t e r a c t i o n a t b r e a k i n g i s u n f o r t u n a t e l y b e y o n d t h e r e a lm s o f t h is

p a p e r .

P R E D I C T I O N S O F H E I G H T A T T E N U A T I O N O F B R O K E N W A V ES

D a l l y et a l. ( 1 9 8 4 ) p r o p o s e d t h a t t h e r a t e o f b r e a k i n g w a v e h e i g h t d e c a y is

d e p e n d e n t o n t h e d i f fe r e n c e b e t w e e n t h e a c t u a l en e r g y f lu x ( E ) a n d a s ta b lee n e r g y f lu x ( E s t ) . T h i s m e a n s t h a t t h e h e i g h t o f b r e a k i n g w a v e s a p p r o a c h e s a

330 K.P. BLACK AND M.A. ROSEN BERG

s t ab l e w a v e h e i g h t H ~t w h i c h i s s a i d t o b e g o v e r n e d b y t h e l o c a l w a t e r d e p t h .

T h e s t a b l e h e i g h t c a n b e d e f i n e d , t h e r e f o r e , a s:

H s t = V d (23)

w h e r e F is a d i m e n s i o n l e s s co e f f ic i e n t. A f t e r i n t r o d u c i n g E = 0 . 1 2 5 p g H 2 a n d

E st= 0.125pgH Z~ t, t he d i s s i pa t i o n due t o b r e a k i n g F b ( E q . l a ) ob t a i n e d f r o m

t h e D a l ly e t a l. ( 1 9 8 4 ) f o r m u l a , w h i c h n e g l e c t s w a v e / c u r r e n t i n t e r a c t i o n , is :

F b = K c g [H2__~,2d2]2 d H L -- _ _

w h e r e K is a d is s i p a ti o n c o e ff ic ie n t. I n a w a v e / c u r r e n t e n v i r o n m e n t , w e a p -

p l ie d E q . 9 o f D a ll y a n d D e a n ( 1 9 8 6 ) w h i c h p r e d i c ts t h e a t t e n u a t i o n o f t h e

w a v e a c ti o n . M e a s u r e m e n t s i n d i c a t e F l i es n e a r 0 .4 0 , a l t h o u g h v a l u e s a s l a rg ea s 0 .5 h a v e b e e n u t i li s e d , a n d K = 0 . 1 5 - 0 . 2 0 ( D a l l y e t a l. , 1 9 8 4 ) .

T h e n u m e r i c a l a p p r o x i m a t i o n f o r t h e h e ig h t d is s ip a t io n i n o n e m o d e l ti m e

step i s :

c ~ A t K 2 2 2A H = ~ [ H - V d ] ( 2 5)

H o n t h e R H S o f E q . 2 5 i s c e n t e r e d i n th e m o d e l s p a ce s te p b y i t e r a ti o n o f :

H 2 = H , - C [ 0 . 2 5 (H 2 + H i ) 2 - / " 2 d 2 ] / d ( H 2 + H ~ ) ( 2 6 )

C i s t he c o ns t a n t C gcA tK , w he r e cgc i s g r o up s pe e d a t t he m i d - p o i n t o f t he s te p .

H ~ a n d H 2 a r e th e o l d a n d n e w h e i g h t s, r e s p e c ti v e ly . I t e r a t io n s c e a s e w h e n H 2

va r i e s by on l y a s m a l l a m ou n t ( < 0 .0001 m ), u s ua l l y a f te r 3 - 4 i t e r a t ions .

R e f o r m i n g o f w a v es

O n c e b r o k e n , a w a v e w i ll r e m a i n s o e v e n w h e n t h e h e i g h t t o d e p t h r a ti o i s

w e l l b e l o w t h e b r e a k i n g l im i t . T h i s f a c t o r n e g a t e s t h e u s e o f a s i m i l a r i t y l a wi n s i d e th e b r e a k p o i n t , b u t i t is i m p l i c i t to t h e D a l l y e t al. f o r m u l a t i o n , w h i c h

s ugges ts t ha t w a ve s c e a s e t o b r e a k on l y a f t e r t he i r he i gh t t o d e p t h r a t i o r e a c he s

t h e s t ab l e e q u i l i b r iu m v a l u e o f /" . H o w e v e r , i n a n u m e r i c a l m o d e l , i f t h e t i m e

s te p i s sm a l l o n h o r i z o n t a l b a t h y m e t r y o r o n s lo w l y s h o a li n g b a t h y m e t r y o n a

s a n d b a n k , t h e w a v e h e i g h t a p p r o a c h e s t h e s t a b l e v a l u e H s t a s y m p t o t i c a l l y ,a n d , i f f r ic t io n is lo w , th e w a v e s n e v e r r e f o r m . A f t e r c o m p a r i s o n o f m o d e l

s i m u l a t io n s w i t h d a t a , t h e f o ll o w i n g r e f o r m a t i o n c r i t e r i o n w a s c h o s e n :

H < 1 .0 2 /" d ( 2 7 )

f o r ci n g t h e w a v e s t o r e f o r m w h e n t h e i r h e i g h t w a s w i t h i n 2 % o f nst.

S E M I - EM P I R I C A L T R E A T M E N T O F W A V E T R A N S F O R M A T I O N 3 31

B or e t heor y

B a t tj es a n d J a n s s e n ( 1 9 7 8 ) a s s u m e d t h a t th e d i s s i p a ti o n in b r o k e n w a v e s

w a s s i m i l a r t o t h a t o f a p e r i o d i c b o r e , a n d :

F b = o t f p n 2 / d ( 2 8 )

w h e r e c~ i s a c o n s t a n t o f o r d e r 1 a n d fp is t h e p e a k f r e q u e n c y o f t h e s p e c t r u m .

MOD EL CONFIRMATION

W e n e e d t o c o n f i r m t h e a p p l i c a b i li t y o f t h e a b o v e t h e o r y i n a v a r i e t y o f

c o n d i t i o n s i n c l u d i n g t h e n o n - s t e a d y c a se . T h e n u m e r i c a l m o d e l i s e m p l o y e d

f o r t h i s p u r p o s e . T h r e e t y p e s o f c h e c k a re n e e d e d t o a s se s s t h e m e r i t s o f th ee m b e d d e d p r o c e d u r e s i n a n u m e r i c a l s i m u l a t i o n . T h e s e a re m o d e l v a li d a ti o n ,

w h i c h e n t a il s c h e c k i n g t h e m o d e l f o r c o d i n g e r ro r s; m o d e l c a l ib r a t io n , w h e n

t h e i n p u t p a r a m e t e r s , p h y s i c a l c o e f f ic i e n ts a n d e m p i r i c a l r e l a ti o n s h i p s a r e e s-

t a b l is h e d ; a n d m o d e l v e r i f ic a t i o n w h i c h r e q u i r e s r u n n i n g t h e m o d e l u s in g di f-

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

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

T h e n u m e r i c a l m o d e l w a s s u c c e s sf u ll y v a l i d a t e d b y s i m u l a t i n g ca s es w i t h

k n o w n a n a l y ti c a l s o l u t io n s . T h e s e w i ll n o t b e r e p e a t e d h e r e b u t w e r e ( i ) s et -

u p a n d s e t - d o w n o n a p l a n e b e a c h , ( i i) t h e w a v e h e i g h t a t t e n u a t i o n s o l u t i o no f D a l l y e t al. ( 1 9 8 4 ) o n a p l a n e b e a c h , a n d ( i ii ) t h e s h o a l i n g p r e d i c t e d b y

l i n e ar th e o r y . T h e m o d e l w a s c a l i b ra t e d a n d v e r i f ie d b y tr e a t i n g m o r e c o m -

p le x e n v i r o n m e n t s o n a n a t u r a l b e a c h a n d f r o m t h e l a b or a to r y .

W e n e e d t o a ss e ss t h e m e r i t s o f ( i ) t h e e q u i v a l e n t h e i g h t s h o a li n g m e t h o d

( i i) t h e b r e a k i n g c ri t e r io n ( i ii ) t h e D a i l y e t a l. m e t h o d o n a w a v e - b y -w a v e

b a s is a n d ( i v ) t h e n e e d f o r i n c l u s i o n o f w a v e / c u r r e n t i n t e r a c t i o n in th e c a se s

t r e a t e d . T h e c o e f f i c i e n t s t o b e e s t a b l i s h e d w e r e t h e D a l l y c o e f f ic i e n t s K , F

( E q s . 2 3 a n d 2 4 ) , b e d f r i c t i o n c o e f f i c i e n t C f ( E q . 3 ) , e d d y v i s c o s i t y A H i n t h e

h y d r o d y n a m i c m o d e l ( B l ac k a n d H e a ly , 1 9 8 8 ), a n d s m o o t h i n g c o e ff ic i en t

N A i n t h e a n g l es s o l u t i o n ( E q . 11 ) . B e c a u s e o f t h e m u l t i p l e c o m b i n a t i o n s , w e

d e r i v e d t h e s e v a l u e s u s in g a s e n s i t i v it y a n a ly s i s b y v a r y i n g e a c h p a r a m e t e r

o n e a t a t i m e .

T h r e e d a t a s e t s f r o m A p o l l o B a y w e r e m o d e l l e d . T h e s e c o n t a i n e d t h e s i -

m u l t a n e o u s o u t p u t f r o m t h r e e w a v e p r o b e s , l o c a te d o u t s id e , n e a r t o a n d i n-

s h o r e o f t h e b r e a k p o i n t ( T a b l e s 2 a n d 3 ; F i g . 4 ) . T h e f i r s t t w o w e r e s a m p l e d

a t 2 H z , s o t h e t h i r d d a t a s e t, w i t h a 4 H z s a m p l i n g r a te , w a s s e l e c t e d t o e n s u r e

t h a t w a v e c r e st s w e r e b e i n g a d e q u a t e l y r e s o l v e d . T h e t i m e s e ri es w e r e s e l e c t ed

t o i n c l u d e a n a r r o w a n d a w i d e b a n d s w e ll ( T a b l e 2 ) , i n c o n d i t i o n s o f l o w o r

o f f s h o re w i n d s . T o a s se s s th e g o o d n e s s o f fi t o f th e m o d e l t o t h e m e a s u r e dt i m e s e r ie s , a r e l a t iv e s q u a r e d e r r o r w a s c a l c u l a t e d :

~2=~(Hp - - n m ) 2 / ~ . , ( H m ) 2 (29)

w h e r e H p is th e p r e d i c t e d h e i g h t a n d H m i s th e m e a s u r e d h e ig h t .

A s t h e s h o a l i n g E q . 2 0 w a s d e r i v e d f r o m f ie l d d a ta , i ts a p p l i c a t i o n t o l a b o -

r a t o r y m e a s u r e m e n t s s h o u l d p r o v i d e a r o b u s t g e n e r a li ty te st . A c c o r d in g l y , w em o d e l l e d th e o n e - d i m e n s i o n a l m e a s u r e m e n t s o f w a v e h e ig h t s h o al in g a n d a t-

t e n u a t io n m a d e b y W a t a n a b e a n d D i b a j n i a ( 1 98 8 ) . T h e y d e v e l o p e d a n u -

m e r i c a l m o d e l w h i c h a p p l i e d t h e m i l d -s l o p e e q u a t i o n s ( W a t a n a b e a n d M a -

r u y a m a , 1 9 8 6 ) a n d s i m u l a t e d t h r e e l a b o r a t o r y c a s e s f o r v e r i f i c a t i o n . W e

u t i li s e d t h e i r C a s e 2 c o n f i g u r a t i o n o f tw o 1 : 2 0 p l a n e b e a c h e s s e p a r a t e d b y a

h o r i z o n t a l p l a t f o r m ( F i g . 8 b ) . T h i s t e st s e rv e d a d d i t i o n a l l y t o p r o v i d e a c o m -

p a r i s o n w i t h t h e r e s u l ts o b t a i n e d u s i n g t h e m i l d - s lo p e e q u a t i o n s .

C n o i d a l w a v e s h o a l i n g is a n a d d i t i o n a l c a se w o r t h e x a m i n i n g f o r i ts h i g h e r

o r d e r. T h u s , w e s im u l a t e d t h e m e a s u r e m e n t s o f B u h r H a n s e n a n d S v e n d s e n( 1 9 79 ) f o r c n o i d a l w a v e s p r o p a g a t i n g a n d b r e a k i n g o n a p l a n e 1 : 3 4 . 26 s l o p e

( F i g . 9 ) . C a s e 0 4 1 0 7 1 w a s m o d e l l e d , w i t h i n i t ia l w a v e h e ig h t o f 0 . 07 m a n d

p e r i o d o f 2 .5 s.

F i n a l ly , w e s o u g h t a d i f f i c u lt t w o - d i m e n s i o n a l t e st w h i c h i n c l u d e d s t r o n g

r e f r a c t io n , d i f f r a c t io n a n d s h o a l in g . F o r t h i s p u r p o s e , t h e e l l ip t i ca l s h o a l m o -

d e l le d b y H a r d y a n d K r a u s ( 1 9 8 8 ) w a s s e le c te d . H a r d y a n d K r a u s ( 1 9 8 8 )

u s e d t h i s d a t a t o e x a m i n e t h e r e l a ti v e m e r i t s o f l i n e a r a n d c n o i d a l w a v e t h e -

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

w a v e t h e o r y te st s. T h e b a t h y m e t r y ( F ig . 1 0 ) w a s c o m p o s e d o f a f la t b o t t o m

( 0 .4 6 m d e e p ) o n w h i c h w a s p l a c e d a h a l f e l li p s o id w i t h m a j o r a n d m i n o r

s e m i - a x e s o f le n g t h 3 .9 6 a n d 3 .0 5 m . T h e e l e v a t i o n o f th e e l li p s o i d at i ts h ig h -

e st p o i n t a b o v e t h e b o t t o m w a s 0 .3 0 5 m , m a k i n g th e u n d i s t u r b e d w a t e r d e p t h

0 . 1 5 5 m a t th i s p o i n t . T h e w a v e s a p p r o a c h e d w i t h t h e i r c r e st s i n i ti a l ly p a r a ll e l

t o t h e l o n g e r a x is . W e m o d e l l e d t h e l a r g e s t w a v e h e i g h t C a s e M 11 c o n s i d e r e d

b y H a r d y a n d K r a u s ( 1 98 8 ) , w i t h H = 0 . 06 5 m a n d T = 1 .3 s, t o m a x i m i s e t h e

e f fe c t o f w a v e s h o a li n g , b u t t h e s e w a v e s d o n o t b r e a k o n t h e s h o a l. T h e m o d e l

p r e d i c t io n s w e r e c o m p a r e d w i t h d a t a m e a s u r e d a l o n g s e ct io n s p a ra ll el to t h e

l ong a x is o f t he s hoa l , i .e . , a c r o s s t he m i dd l e o f t he s hoa l , a nd a t 1 .5 m a nd 3 . 0

m l e e w a r d .

Application of the enhanced shoaling method and energy dissipation

T o a p p ly E q , 2 0 , t h e n u m e r i c a l m o d e l w a s u p g r a d e d a s fo llo w s . T h e s i m u -

l a ti o n s s t a r t a t t h e o f f s h o r e b o u n d a r y w i t h th e m e a s u r e d v a l u e o f H eq . T h e

s h o a l i n g o f H eq w a s s i m u l a t e d u s i n g l i n e a r t h e o r y ( E q . 7 ) . Hzc w a s o b t a i n e d

a t e a c h t i m e s t ep f r o m H eq w i t h E q . 2 0 . F r i c t i o n a l d i s s i p a t i o n w a s a p p l i e d t o

H zc a n d t h e a d j u s t e d H zc w a s e m p l o y e d t o d e t e r m i n e i f t h e b r e a k i n g h e i g h t

w a s e x c e e d e d a n d , i f so , t h e w a v e h e i g h t w a s a d j u s t e d a g a i n u s i n g E q. 2 5 .A f t e r s o m e m a n i p u l a t i o n o f E q . 2 0 , w e f in d :

SEMI-EMPIRICAL REATMEN TOF WAVETRANSFORMATION 3 3 3

. - , .12

• r 8

• M e a s u r e d D a t a ( a

m L i n e a r S h o a l i n g

- - W a t a n a b e

• • • Une xplained

~ Shoal ing

~ I , , I ~ I , , , ~ l

O I I = ~ _ I ! ,,,-,,-:¢"/ ~ ~ ¥ ' "

_ 2 0 ] ~. : . ' . . : . ' . . ' . - : . ' . . / . . ' . ' . . ' . ' / . . : . ' . . : . ' / . ' / . . / . . ' ." - ' """ 0Z ~ : : ' . ' ~ 2 0

¢ ~ , , , :J " . . . . ( b )

,._. 12

• M e a s u r e d D a t a ( C )

- - E n h a n c e d s h o a l i n g I

6 4 2 0

X ( m )

F ig . 8 . W a t a n a b e a n d D i b a j n i a ( 1 9 8 8 ) C a s e 2 . ( a ) C o m p a r i s o n w i t h m e a s u r e m e n t s o f ( i ) t h e

W a t a n a b e a n d D i b a j n i a m i l d - s lo p e e q u a t i o n p r e d i c t i o n s a n d ( i i ) l i n e a r w a v e t h e o ry , ( b ) B a t h -y m e t r y , ( c ) C o m p a r i s o n w i th m e a s u r e m e n t s o f p r e d ic t io n s u s i n g t h e e n h a n c e d s h o a l in g m e t h o d

( E q . 2 0 a ) .

W a v e G e n e r a t o r

~ 14.78m ~ :: : 12.33m ,1

F ig . 9 . T h e e x p e r i m e n t a l s e t -u p o f B u h r H a n s e n a n d S v e n d s e n ( 1 9 79 ) .

33 4 K.P. BLACKAND M.A. ROSENBERG

g ( m )

t~ 3 LeewardSide

I d=o . 45m

W A V E S

F ig . 1 0. T h e e l l i p ti c a l s h o a l m o d e l l e d b y H a r d y a n d K r a u s ( 1 9 8 8 ) .

Heq =0.5 [ -a d/ b+ ((ad/b)2+4dHzc/b) 1/2] (30)

where a, b are the gradient and intercept in Eq. 20, respectively. Heq was then

obtained from the adjusted Hzc using Eq. 30c in preparation for the next time

step.For the field data simulations, the equivalent height input at the offshore

boundary was obtained directly from the measured time series. For the labo-

ratory tests, the equivalent height was obtained by substituting the publishedheights and water depths into Eq. 30c. Linear shoaling and the equivalentheight procedure were both modelled in all the above tests for comparative

purposes.

M O D E L C A L I B R A T I O N - - -W A T A N A B E A N D D I B A J N I A L A B O R A T O R Y D A T A

The model was calibrated against the Watanabe and Dibajnia ( 1988 ) Case2 data and the most appropriate coefficients were found to be K=0.15,

F=0.35, Cf=0.01 andAH=0.1 m 2 s -1.Linear theory significantly under-estimated the measured shoaling (Fig. 8a)

but, with the addition of the enhanced shoaling (Eq. 20a), the maximumwave height was in good agreement with the measurements (Fig. 8c). TheWatanabe and Dibajnia data is periodic and measured in the laboratory, whilethe enhanced shoaling equation was derived from field data. The correspond-ence evident in the model calibration provides the first firm evidence of theusefulness of the enhanced shoaling technique for a wider variety of condi-tions than those measured. With the shoaling giving close agreement to the

measurements, breaking occurred at the measured location using the Madsenbreaking criterion.

SEMI-EMPIRICAL TREATMEN T OF WAVE TRANSF ORMATI ON 335

T h e w a v e h e i g h t a t t e n u a t i o n p r e d i c t e d b y th e D a l l y et al. m e t h o d c o m p a r e d

w e l l w i t h t h e m e a s u r e m e n t s u s i n g D a l l y c o e ff ic i en t s K = 0 .1 5 , F = 0 .3 5 . T h e s e

a r e w i t h i n th e r a n g e r e c o m m e n d e d b y D a l l y e t a l. ( 1 9 8 4 ) a l th o u g h t h e v a l u e

o f F = 0 .3 5 i s l es s t h a n t h e v a l u e 0 . 4 5 o b t a i n e d b y E b e r s o l e ( 1 9 87 ) . T h e w a v e s

r e f o r m e d o n t h e h o r i z o n t a l p l a tf o r m ( F ig . 8 b ) a n d t h e n s h o a l e d o n t h e s ec -o n d b e a c h b e f o r e b r e a k i n g f o r t h e s e c o n d t i m e , a s m e a s u r e d . T h i s d i d n o t

o c c u r w i t h o u t th e r e f o r m a t i o n c r i te r i o n ( E q . 2 7 ) . T h e i n e x p li c ab l e w a v e

s h o a li n g o n t h e p l a t fo r m ( m a r k e d i n F ig . 8 a ) w a s n o t p r e d i c t e d b y th e m o d e l .

W e f o u n d t h a t A n h a d v e r y l i t t l e e f f e c t o n t h e s o l u t i o n a n d w e c h o s e t h e

v a l u e 0 .1 m 2 s - ~, w h i c h m i l d l y s m o o t h e d t h e v e l o c it ie s o n t h e f i n e n u m e r i c a l

g r id s u s e d . T h e l i m i t e d e f fe c t o c c u r s b e c a u s e t h e e d d y v i s c o s i ty i s o n l y o p e r-

a t in g o n l o w - f r e q u e n c y w a v e s o f l o w a m p l i t u d e .

F i g u r e 8 a i n d i c a t e s t h a t t h e m i l d - s lo p e e q u a t i o n s , a s a p p l i e d b y W a t a n a b e

a n d D i b a j n i a ( 1 9 88 ) , a p p e a r t o u n d e r - e s t i m a t e t h e w a v e h e ig h t s h o a l i n g a tt h e b r e a k p o i n t. C o n s e q u e n t l y , th e b r e a k p o i n t l o c a t i o n a n d b r e a k i n g cr it e-

r i o n a re a ff e ct ed . T h e W a t a n a b e a n d D i b a j n i a a t t e n u a t i o n p r o c e d u r e i n s i d e

t h e s u r f z o n e i n d u c e s w a v e h e i g h t d e c a y a t a s i m i l a r r a t e t o t h e D a l l y e t al.

f o r m u l a .

MODEL VERIFICATION

Cnoidal waves of Buhr Hansen and Svendsen

F o r t h e d i f f i c u lt c a s e o f c n o i d a l w a v e s h o a l in g , a g o o d f it t o t h e d a t a o f B u h r

H a n s e n a n d S v e n d s e n ( 1 9 7 9 ) w a s o b t a i n e d w i th th e e n h a n c e d s h o a li ng

m e t h o d ( E q . 2 0 b ; ( F i g . 1 l a ) . N o t a b l y , a b e t t e r f it o c c u r r e d i f t h e g r a d i e n t i n

E q . 2 0 b w a s in c r e a s e d t o a b o u t 1 .3 4 ( F i g . 1 l b ) . I n a ll o t h e r s i m u l a t io n s i n -

c l u d i n g t h e e l l ip t ic a l s h o a l ( b e l o w ) , t h e b e s t p r e d i c t i o n o f s h o a l in g o c c u r r e d

w i t h E q . 2 0 a e x c e p t f o r t h i s c n o i d a l w a v e te s t, w h e n t h e s t e e p e r c u r v e ( E q .

0 . 1 2

0 . O 8

0 . 0 0

. _ ~ - - _

- - M E~SUR~. M F . J~S

- - E N h A N C e D S ~ O m J N O

, 1 ~ 1 , 1

, I ~ L i I L N

1 6 2 0 2 4

- - E N H~ NC F. D S H O A UN O

~ I ~ J I I I I I ~ I

4 8 1 2 1 6 2 0 2 4

× Im)

Fig. 11. Comparison of model predictions with cnoidal wave measurements of Buhr Hansen

and Svendscn (1979) Case 041071, (a) using Eq. 20b and (b ) with the gradient in Eq. 20bincreased to 1.34 from 1.05.

336 K .P . B L A C K A N D M . A . R O S E N B E R G

2 0 b ) w a s n e e d e d . N o d i s t in g u i s h a b l e d i f f e r e n c e b e t w e e n t h e tw o E q s . 2 0 a a n d

b w a s e v i d e n t w i t h t h e A p o l l o B a y f ie ld m e a s u r e m e n t s .

T h e c n o i d a l w a v e h e i g h t w a s u n d e r - e s t i m a t e d u s i n g li n e a r th e o r y , a s ex -

p e c te d . H o w e v e r , i t w a s f o u n d t h a t a m o d i f i c a t io n o f t h e p o w e r i n th e s h oa l-

i n g E q . 7 b t o a b o u t 1 .2 ( f r o m 0 .5 ) g a v e a s i m i l a r r e s u lt to t h e e n h a n c e d s h o a l -i n g m e t h o d . T h i s s u g ge s ts a p o s s ib l e a l t e r n a t i v e e m p i r i c a l r e l a ti o n s h i p , o f t h e

f o r m o f E q , 2 0a , w h i c h r e l at es t h e s h o a li n g e q u a t i o n p o w e r ( i n s t e a d o f

H z c / H e q ) t o y . M u n k ( 1 9 49 ) s u g g e s t e d a p o w e r 4 / 3 f o r s o l it a ry w a v e s .

T h e M a d s e n b r e a k i n g c r i te r i o n w a s n o t a p p l i ca b l e to t h e c n o i d a l w a v e s a n d

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

T h e r e su l t ( y = 1 .1 5 ) w a s i n a c c o r d a n c e w i th m e a s u r e m e n t s o f P a p a n i c o l a o u

a n d R a i c h l e n ( 1 98 7 ) . I n s i d e t h e b r e a k p o i n t , e n e r g y d e c a y a p p e a r s t o o c c u r

i n i ti a l ly f a st e r th a n t h a t p r e d i c t e d b y t h e D a l l y e t a l. f o r m u l a . T h i s i s p r e s u m -

a b ly o c c u r r i n g i n t h e w a v e h e i g h t t r a n s i ti o n z o n e i m m e d i a t e l y a f te r b r e a k in g ,r e f e r r e d t o b y a n u m b e r o f r e s e a r c h e r s ( e. g. S v e n d s e n , 1 98 4; B a s c o a n d Y a -

m a s h i t a , 1 9 8 6 ) . F u r t h e r r e f i n e m e n t o f t h e a t te n u a t i o n m a y b e r e q u i r e d in

t h i s h i g h l y t u r b u l e n t z o n e f o r p lu n g i n g w a v e s ( R o s e n b e r g a n d B l a ck , 1 99 1 ) .

H o w e v e r , t h e g e n e r a l tr e n d is s ti ll s u r p r is i n g l y w e l l p r e d i c t e d u s i n g t h e s a m e

c o e ff ic ie n ts , K = 0 . 1 5 a n d F = 0 .3 5 .

E l l i p ti c a l s h o a l

T h e t w o - d i m e n s i o n a l E u l e r i a n m o d e l ( E q . 13 ) w a s a p p l i e d t o a n e ll ip t ic a ls h o a l to i n c o r p o r a t e t w o - d i m e n s i o n a l i t y . F o r t h i s c a se , a h i g h d e g r e e o f c o r -

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

w a s o b t a i n e d a f t er a p p l ic a t io n o f t h e e n h a n c e d s h o a l in g m e t h o d ( F ig . 1 2 ).

T h e e n h a n c e m e n t o f h e i g h t w a s p a r ti c u la r l y i m p o r t a n t a c ro s s t h e m i d d l e o f

t h e e ll ip s o i d , w h e r e s h o a l in g w a s m o s t p r o n o u n c e d ( F i g . 1 2 a ) . R e f r a c t i o n

d o m i n a t e d l e e w a r d o f t h is l o c a t i o n a n d s o t h e c o r r e s p o n d e n c e b e tw e e n t h e

m o d e l a n d m e a s u r e m e n t s r e l ie d to a g r e a t e r d e g r e e o n t h e a c c u r a t e s p ec i fi c a-

t i o n o f t h e w a v e a n g l e s.

S m o o t h i n g w a s r e q u i r e d w h e n s o l v i n g f o r t h e a n g l e s i n t h i s t e s t , a n d t h e

c o e f f i c i e n t N A = 0 . 0 5 ( E q . 11 ) w a s s e l e c te d t o b e t h e m i n i m u m v a l u e n e c e s -

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

( H a r d y a n d K r a u s , 1 9 88 ) . T h i s s t a b il i se d t h e a n g l e s s o l u t io n , w i t h t h e r e s u l t

t h a t t h e w a v e h e ig h t s w e r e a ls o s m o o t h e d . N o t a b l y , d i f f ra c t io n i s a d o m i n a n t

p r o c e s s i n th e l e e o f th e s h o a l a n d t h e m o d e l a p p l i e d d o e s n o t s p e c if ic a l ly

a c c o u n t f o r w a v e d if f r a c ti o n . T h e r e s u l ts i n d i c a t e t h a t t h e s m o o t h i n g s c h e m e

( E q , 11 ) p r o v i d e d a n a c c e p t a b l e s u b s t i tu t e f o r t h e f u ll d if f r a c t io n e q u a t i o n s

in t h i s ca se .

T h e e l li p ti c a l s h o a l is a d i f fi c u l t t e st o f b o t h t h e s h o a l in g e q u a t i o n a n d t h e

r e f r a c t i o n m o d e l , p a r t i c u l a r l y a s t h e w a v e h e i g h t s d e p e n d c r i t i c a l l y o n t h es p e c i f i c a t i o n o f w a v e a n g l e a n d w a v e h e i g h t s h o a l i n g . A l t h o u g h t h e r e w a s

SEMI-EMPIRICALTREATMENTOF WAVETRANSFORMATION 337

• ( a )

0 . 0 9 ~

0 . 0 ~

, ' , ' 1~o

1 5 3 0

oZ - 5<

~ 0 .10

(c)•

- 2 - 1 - 0 1 2

- 15 _ 3 3 - 3 0 , _ ,

, I , 1 ~ 1 , 1 , 1 ,

, I , I , I , I ~ I ~ T "

- 2 - 1 - 0 1 2 3

x(m) x(m)• M e a s u r e d D a t a

F i g. 12. C o m p a r i s o n o f m o d e l p r e d i c t i o n s w i t h m e a s u r e m e n t s o v e r a n e l l ip t i c al s h oa l. ( a ) W a v e

h e i g h t a c r o s s th e c e n t r e o f t h e s h o a l . ( b ) W a v e h e i g h t s 3 .0 5 m l e e w a r d o f t h e c e n t r e o f t h e s h o a l .

( c ) W a v e a n g l e s a c r o s s t h e c e n t r e o f t h e s h o a l . ( d ) W a v e a n g l e s 1 . 5 3 m l e e w a r d o f t h e c e n t r e .

E S = e n h a n c e d s h o a l i n g ( E q . 2 0 a ) ; C N = c n o i d a l w a v e t h e o r y ; L S = l i n e a r s h o a l i n g . T h e d a t a

a n d c n o i d a l w a v e c o m p a r i s o n ( C N ) w a s e x t r a c t e d f r o m H a r d y a n d K r a u s ( 1 98 8 ) .

T A B L E 6

2R e l a t i v e s q u a r e d e r r o r s ~ c o m p a r i n g t h e p r e d i c t e d a n d m e a s u r e d w a v e h e i g h t s a t A p o l l o B a y . T h r e e

s e ts o f s i m u l a t i o n s w e r e m a d e : ( i ) a n d ( i i ) u s i n g H zc a n d H ~ q, r e s p e c t i v e l y w i t h t h e D a l l y e t a l. h e i g h t

a t t e n u a t i o n ; a n d ( i i i ) u s i n g H z c w i t h b o r e t h e o r y h e i g h t a t t e n u a t i o n

A P 1 2 0 4 1 A B 2 2 0 3 1 A P 1 0 0 4 5

(/)n~cP r l - -

P r 2 0 .0 7 0 .0 4

Pr 3 0 .1 0 0 .0 8

(ii) neq

Pr l 0 .0 2 0 .0 1

Pr 2 0 .0 6 0 .0 4

Pr3 0 .11 0 .07

( i iO Hzc B ore theo ry c¢ = 1.0Pr 2 0 .0 8

Pr 3 0 .1 4

0 .1 0

0 .0 4

338 K.P.BLACKANDM.A.ROSENBERG

c o n s i d e r a b l e s c a tt e r in F i g . 6 ( r e l a t in g H zc /H eq o 7eq ) , t h e e n h a n c e d s h o a l i n g

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

a p p l i e d b y H a r d y a n d K r a u s ( 1 9 8 8 ) to th e s a m e c o n f i g u r a t i o n ( F i g . 1 2 a ) .

F ie l d m easu r em en ts

T h e m o d e l w a s a p p l i e d to th r ee d a t a s et s f r o m A p o l l o B a y w i t h o u t m o d i -

f y in g th e m e t h o d s o r c o e f f ic i en t s . A g o o d c o r r e s p o n d e n c e b e t w e e n t h e m o d e l

a n d m e a s u r e m e n t s w a s i n d i c a t e d b y a r e l a t i v e s q u a r e d e r r o r v a r y i n g f r o m

0 . 0 1 t o 0 . 1 1 ( T a b l e 6 ; F i g s . 1 3 a n d 1 4 ) . H o w e v e r , t h e q u a l i t y o f t h e f it v a r i e d

i rr e gu la r ly b e t w e e n t h e t h r e e e x p e r i m e n t s a n d t h e d i f fe r e n t p r o b e s. S o m e o f

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

p r o g r e ss e d s h o r e w a r d s , d u e t o n o n l i n e a r i n t e r a c t i o n a n d f o r m a t i o n o f ha r -

A P 1 20 4 1 P A R T I C L E M O D E L V E R I F I C A T I O N

H = H z c , K = 0 . 1 5 , r ' = 0 . 3 5

R E F O R M A T 1 .0 2 r" c f = 0 .0 1

1.5 ~- ~ Measurements ( i ) Offsho re ~

9 . 5 , o

0 . 00 . 0 2 4 0 . 0 4 8 0 . 0 7 2 0 . 0 9 6 9 . 0 1 2 0 0 .0 1 4 4 0 . 0 1 6 8 0 0 1 9 2 0 . 0

0.00.0

O.O0.0

240.0 4-80.0 720.0 960,0 12 00 .0 1 4 -4 0 .0 16800 1920.0

( i ii ) Insho re

240.0 480,0 720 0 960.0 12000 14 40 .0 1680.0 19200

T I M E ( s)

SEMI-EMPIRICALTREATMENTOF WAVETRANSFORMATION 339

A P 1 20 41 P A R T I C L E M O D E L V E R I F I C A T I O N

H = H e q ( 1 . 0 7 + 0 . 7 6 ~ e q ) , K = 0 . 1 5 , r ' = 0 . 3 5

R E F O R M A T 1 .0 2 r ' c ! = 0 .0 1(b )

~ t . 0

0 .00 .0

t= 1.o~

0 . 0O.O

- - • o Measurements ( i) O f f s h o r e ~

2 4 0 . 0 4 8 0 . 0 7 2 0 , 0 9 6 0 . 0 1 2 0 0 , 0 1 4 4 0 . 0 1 6 8 0 , 0 1 9 2 0 . 0

( i i i ) I n s h o r e

2 4 0 . 0 4 8 0 0 7 2 0 0 9 6 0 . 6 1 2 0 0 . 0 1 4 4 0 . 0 1 6 8 0 . 0 1 9 2 0 0

T I M E ( s )

F i g . 1 3. C o m p a r i s o n o f m e a s u r e m e n t s a t A p o l l o B a y o n A p r i l 4 , 1 9 8 7 w i t h p r e d i c t io n s u s i n g

( a ) l i n ea r w a v e s h o a l in g , a n d ( b ) t h e e n h a n c e d s h o a l i n g p r o c e d u re .

m o n i e s . M o r e o v e r , t h e s m a l l w a v e s w e r e s o m e t i m e s o v e r t a k e n b y la r ge r w a v e s .

T h e m o d e l d i d n o t a t t e m p t t o t r e a t t h e s e f a c t o r s ( i t s i n p u t w a s t h e o f f s h o r e

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

v e r y e n c o u r a g i n g ( T a b l e 6 ) .

B e c a u s e o f t h e s c a t te r i n t h e r e l a t i o n s h i p b e t w e e n H z c a n d H e q ( F i g . 6 ) a n

e x a c t c o n v e r s i o n b e t w e e n t h e t w o h e i g h t s w a s n o t a c h i e v a b l e . T h e e f f e c t o f

t h e s c a tt e r is d e p i c t e d b y t h e r e c o r d s f r o m t h e o f f s h o r e p r o b e s ( F i g s . 1 3 b a n d

1 4 b ) i n w h i c h , a t P r o b e 1 , t h e d i r e c t c o n v e r s i o n f r o m t h e m e a s u r e d H eq u s i n g

E q . 2 0 a i s c o m p a r e d w i t h t h e m e a s u r e d H zc. T h e s q u a r e d e rr o r c o m p a r i n g t h e

c o n v e r t e d H e q a n d a c t u a l H z c w a s s m a l l a n d v a r i e d f r o m 0 . 0 1 t o 0 . 0 4 .

T h e s h o a l in g w a s b e t te r e s t im a t e d u s i n g t h e e q u i v a l e n t h e ig h t m e t h o d t h a nw i t h l in e a r t h e o r y ( F i g . 1 3 a a n d b ) . A t P r o b e 2 C a s e A P 1 2 0 4 1 ( F i g . 1 3 a ) ,

340 K.P. BLACK AND M.A. R OSENB ERG

w h e r e s h o a l i n g w a s m o s t e v i d e n t a t t i m e s 1 2 0 , 8 6 0 , 1 4 0 0 a n d 1 7 5 0 s , t h e

s h o a l i n g w a s u n d e r p r e d i c t e d u s i n g H z c (F i g . 1 3 a ) b u t t h is w a s m o s t l y c o r -

r e c t e d b y t h e e q u i v a l e n t h e i g h t p r o c e d u r e ( F i g . 1 3 b ) . A t o t h e r t i m e s a n d lo -

c a t i o n s , t h e d i f f e r e n c e s b e t w e e n t h e e q u i v a l e n t h e i g h t r e s u lt s a n d l i n e a r t h e -

o r y w e r e m a r g i n a l ( T a b l e 6 ) . O n c e t h e w a v e s b r o k e , th e i r h e i g h t s w e r e m o r eg o v e r n e d b y t h e D a l l y e t a l . f o r m u l a t i o n t h a n t h e s h o a l i n g , a n d t h e a d v a n -

t a g es o f t h e e q u i v a l e n t h e i g h t p r o c e d u r e b e c a m e l e s s i m p o r t a n t .

T h e la r ge r w a v e s b r o k e o n t h e s t e e p e r s l o p e o f f s h o r e o f P r o b e 2 ( F i g . 4 ) .

A t t h i s l o c a t i o n 7b v a r i e d a r o u n d 0 . 9 0 , a s p r e d i c t e d b y t h e M a d s e n f o r m u l a .

T h u s , t h e d a ta s u p p o r t e d t h e M a d s e n f o r m u l a , w i th t h e e x c l u s io n o f so m e

s te e p e r p e a k s a t P r o b e 2 , w h i c h m a y b e r e la t ed t o w a v e / w a v e o r w a v e / c u r -

r e n t i n te r a c ti o n s . T h e m o r e c o m m o n l y e m p l o y e d v a l u e o f 0 .7 8 a t t h e b r e a k

A B 22 03 1 P A R T I C L E M O D E L V E R I F I C A T IO N

H = H z c , K = 0 . 1 5 , r ' = 0 . 3 5

R E F O R M A T 1 . 0 2 r" C f = 0 .0 1

t. 5 ~ ~ Measurements ~ (i) Offsho re J

0.~0 . 0 2 4 0 . 0 4 8 0 . 0 7 2 0 . 0 9 6 0 . 0 1 2 0 0 . 0 1 4 4 .0 . 0 1 6 8 0 0 1 9 2 0 0

0 , 0@ ,o

E ~.o,~ ,

0 0o . o

( i i) Near break point

e , , o . o 4. 8 0 .0 7 2 0 0 9 6 0 . 0 ~2 0 0 . 0 1 4 . 4. 0 .° 1 6 8 0 . 0 , 9 2 0 0

(iii) Inshore

2 4 0 . 0 4 8 0 . 0 7 2 0 0 9 6 0 . 0 1 2 0 0 0 1 4 4 0 . 0 1 5 8 0 0 1 9 2 0 0

T I M E ( s )

SEMI-EMPIRICAL TREATMEN T OF WAVE TRANSF ORMATION 341

A B 2 2 0 3 1 P A R T I C L E M O D E L V E R I F I C A T I O N

H = H e q ( 1 . 0 7 + 0 . 7 6 ~ e q ) , K = 0 . 1 5, r ' = 0 . 3 5

R E F O R M A T 1 . 0 2 r ' c f = 0 . 0 1

0 . 00 . 0

0 . 0O.O

2~ 0 - -I

i : =" ~ ' 1 , 0

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( i i ) N e a r b r e ak po in t -

2 4 0 . 0 4 8 0 . 0 7 2 0 . 0 9 6 0 . 0 1 2 0 0 . 0 1 4 4 Q . 0 1 6 8 0 . 0 1 9 2 0 . 0

( i ii ) Inaho re

240 .0 4 8 0 . 0 7 2 0 . 0 9 6 0 . 0 1 2 0 0 . 0 1 4 -4 0 .0 1 6 8 0 0 1 9 2 0 . 0

T I M E ( S )

Fig. 14. Com parison of m easuremen ts at Apollo Bay on M arch 22, 198 8 with predictions using(a) l inear wave shoaling and ( b) the enhanced shoaling procedure. The inshore record in (a)and (b ) shows a l inear trend in measured height which is du e to tidal sea level rise. This factorwas neglected in the model.

p o i n t w a s t o o s m a l l . M o r e o v e r , 7b i n s i d e t h e b r e a k p o i n t w a s l es s th a n 0 . 7 8 ,

n e g a t i n g t h e u s e o f a s im p l e s i m i l a r it y l a w i n s i d e t h e s u r f z o n e .

T h e D a l l y e t al . w a v e h e i g h t a t t e n u a t i o n f o r m u l a o p e r a t e d s a t i s fa c t o r il y o n

a w a v e - b y - w a v e b a s i s in th e n a tu r a l b e a c h e n v i r o n m e n t . H o w e v e r , b e c a u s e o f

t h e i n te r a c t i o n b e t w e e n t h e D a l l y a n d f r i c t i o n c o e f f i c ie n t s , a n e x a c t s p ec i f i-

c a t i o n o f t h e b e s t v a l u e s o f b o t h w a s d i f f i c u l t t o a c h i e v e . I n o n e c a s e w i t h h i g h

f r i c t io n ( C f = 0 .1 ) , t h e w a v e h e i g h t s w e r e g r e a t e r a t t h e i n s h o r e p r o b e t h a n i n

a l o w e r f r i ct i on c a s e ( C f = 0 .0 1 ). T h i s o c c u r r e d b e c a u s e , w i t h h i g h f r i c t i o n ,

t h e w a v e s q u i c k l y r e a c h e d t h e r e f o r m a t i o n h e i g h t a n d t h e n p r o p a g a t e d u n b r o -k e n . T h i s a l l o w e d t h e m t o s h o a l a n d a t t a i n g re a te r h e i g h t s th a n t h e ir b r o k e n

c o u n t e r p a r t s . O t h e r w i s e , t h e b e d f r i c ti o n h a d l it tl e in f l u e n c e , a n d v a r i o u s a t -

t e m p t s t o i m p r o v e t h e f i t t o t h e d a t a b y v a r y i n g th e f r i c t io n h a d n o r e a l su c-

c es s. U l t i m a t e l y , w e t o o k t h e s t a n d a r d v a l u e o f C f = 0 .0 1 . B y m e a s u r i n g t h e

b o t t o m s h e a r s tr e ss u n d e r b r o k e n w a v e s , S a w a r a gi a n d I w a t a ( 1 9 7 4 ) c o n -

c l u d e d t h a t t h e b e d f r i c ti o n w a s a s e c o n d a r y t e r m a n d t h i s w a s c o n f i r m e d b yD a l ly e t al. ( 1 9 8 4 ) . W h i l e t h is i s in a c c o r d a n c e w i t h t h e m o d e l l i n g h e r e , t h e

i m p a c t o f b e d f r i c ti o n o n w a v e r e f o r m a t i o n s h o u l d n o t b e n e g le c te d . T h e w a v e s

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

c o n s i d e r a b l y b e f o r e b r e a k i n g a g a i n .

A t t h e i n s h o r e p r o b e , s o m e o f t h e m e a s u r e d v a r i a b il it y i n t h e w a v e h e i g h t

t i m e s e ri es w a s n o t e v i d e n t i n t h e m o d e l p r e d i c t i o n ( e. g. F ig . 1 4 b ) . T h e

a m o u n t o f v a r i a b il i ty i n t h e m o d e l w a s h i g h ly se n s i ti v e t o t h e r e f o r m a t i o n

c r i te r i o n . I n s o m e l o c a t i o n s n e a r t h e p o i n t o f r e f o r m a t i o n , s m a l l a d j u s t m e n t s

t o t h e r e f o r m a t i o n c r i t e r io n a l l o w e d m o r e w a v e s t o r e fo r m , w h i c h r e s u l t e d i nh i g h e r v a r i a b il i ty i n t h e n u m e r i c a l s i m u l a t i o n a t m o r e s h o r e w a r d l o c a ti o n s .

B o r e t h e o r y a t t e n u a t i o n w i t h a = 1 .0 ( E q . 2 8 ) h a v e a w o r s e f it t o th e d a t a

t h a n t h e D a l ly e t a l. i n f o r m a t i o n . T h e p r e d i c t e d h e i g h t s w e r e t o o l ar ge , p a r t ic -

u l a rl y a t th e i n s h o r e p r o b e s , w h i c h i s in a c c o r d a n c e w i t h t h e r e s ul ts o f T h o r n -

t o n a n d G u z a ( 1 9 8 3 ) a n d o f S t i v e ( 1 9 8 4 ) w h o f o u n d t h a t t h e c la s si ca l h y -

d r a u li c j u m p f o r m u l a t i o n u n d e r - e s t i m a t e d t h e d i s s ip a t io n b y 3 0 - 5 0 % . T h e

D a l l y e t a l. f o r m u l a p r o v i d e d a n a c c e p t a b l e r e s u lt , so n o f u r t h e r a t t e m p t t o

s p e c i f y a w a s m a d e ( s e e, f o r e x a m p l e E b e r s o l e , 1 9 87 ) .

A l t h o u g h t h e m o d e l i n c l u d e d t h e w a v e / c u r r e n t i n t e r a ct io n , t h e d i f fe r en c e si n t h e s o l u t i o n w i t h o u t t h e in t e r a c t i o n w e r e s m a l l ( D a l l y a n d D e a n , 1 9 8 6 ) .

T h e r e w a s n o c o n s i s te n t c h an g e in t h e s q u a r e d e r r o r fu n c t i o n w h e n w a v e /

c u r r e n t i n t e r a c t i o n w a s n e g l e c t e d a t A p o l l o B a y , a b e a c h w i t h n e a r l y p a ra l le l

c o n t o u r s w i t h n o o b v i o u s r ip c u r r e n t s . T h e t y p i c a l l o w - f r e q u e n c y c u r r e n t i n -

t e n s i ti e s o v e r m o s t o f t h e s u r f z o n e w e r e u p t o a b o u t 0 .3 m s - l w h i c h w a s

o n l y a p p r o x i m a t e l y 10% o f t h e g r o u p s p e e ds . T h e w a v e / c u r r e n t i n t e r a c t io n

p l a ys a m o r e s i g n i fi c a n t r o l e in t h e s w a s h z o n e w h e r e c u r r e n t s p e e d s a r e f a s te r.

CONCLUSIONS

W a v e h e i g h t d a t a , r e c o r d e d o n a n o c e a n b e a c h i n s o u t h e r n A u s t r a li a , w a s

a n a l y s e d f o r b r e a k e r c r i t e r io n , s h o a l i n g a n d a t t e n u a t i o n . A m e t h o d t o e n -

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

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

t h e b r e a k p o i n t . A n a n a l y s is o f b r e a k i n g c r it e r ia i d e n t i f ie d a n u m b e r o f s ui t-

a b le c h o ic e s , b u t t h e s l o p e - d e p e n d e n t M a d s e n ( 1 9 7 6 ) b r e a k e r h e i g h t c ri te -

r i o n a d e q u a t e l y p r e d i c t e d t h e b r e a k e r p o s i t i o n i n l a b o r a to r y a n d f ie ld d a ta ,

e x c l u d i n g s t e e p c n o i d a l w a v e s . T h e D a l l y e t a l. w a v e h e i g h t a t t e n u a t i o n

m e t h o d ( w i th K = 0 . 1 5 a n d F = 0 . 3 5 ) w a s f o u n d t o b e a p pl ic a bl e t o a w i d ev a r i e ty o f c o n d i t i o n s . W e u n i f i e d t h e s e f in d i n g s in a n u n s t e a d y n u m e r i c a l

SEMI-EMPIRICAL TREAT MENT OF WAVE TRANSFORMATIO N 343

s i m u l a t io n . A m i x e d L a g r a n g i a n / E u l e r i a n n u m e r i c a l s o lu t i o n s c h e m e w a s

p r e s e n te d w h i c h h as s o m e a d v a n ta g e s o v e r t h e m o r e c o m m o n l y e m p l o y e d

E u l e r i a n s c h e m e s . T h e m o d e l w a s v a l i d a t e d , c a l i b ra t e d a n d v e r i f i e d w i t h l ab -

o r a t o ry a n d f ie ld m e a s u r e m e n t s . A g o o d p r e d i c t i o n o f w a v e h e i g h t a n d w a v e

a n g l e w a s o b t a i n e d i n a ll c as e s .

A C K N O W L E D G E M E N T S

T h e a u t h o r s w o u l d li ke t o t h a n k G r a h a m S y m o n d s , P e te r N i e l se n a n d T o m

H a r d y f or t h e i r v a l ua b le c o m m e n t s o n t h e m a n u s c r i p t . T h i s w o r k w a s f u n d e d

b y t h e A u s t r a l i a n R e s e a r c h C o u n c i l a n d t h e V i c t o r i a n I n s t it u t e o f M a r i n e

S c i e n c e s .

R E F E R E N C E S

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