One aspect of the dynamics of a coast, partly protected by a row of groynes

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ONE ASPECT OF THE DYNAMICS OF A COAST. PARTLY PROTECTED BY A ROW OF GROYNES

by

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R I J K S W A T E R S T A A T

D I R E C T I E W A T E R H U I S H O U D I N G

AFD. KUSTONDERZOEK

EN W A T E R B E W E G I N G

' s - G R A V E N H A G E K O N I N G S K A D E 2 5 T E L E F O O N 1 8 3 2 8 0 Aan de V / e l e d e l g e s t r e n g e Heer I r . J.A. B a t t j e s A f d . Weg- e n Y/aterbouwkunde v a n de T e c h n i s c h e H o g e s c h o o l O o s t p l a n t s o e n 25 DELFT u w K E N M E R K : U W B R I E F V A N : O N S K E N M E R K : 67.162B

_{'s-GRAVENHAGE}

T E R U G : B I J L A G E N 3 29 m e i 196? O N D E R W E R P : Waarde J u r r i e n , , O v e r e e n k o m s t i g m i j n b e l o f t e j e op de h o o g t e t e h o u d e n met w a t i k b e s t u d e e r , zend i k j e h i e r b i j e e n e x e m p l a a r v a n h e t r a p p o r t : "The C o a s t a l Dynamics o f Sandwaves a n d t h e • I n f l u e n c e o f B r e a k w a t e r s a n d G r o y n e s " e n e e n e x e m p l a a r v a n h e t v o o r l o p i g r a p p o r t : "One A s p e c t o f t h e D y n a m i c s o f a C o a s t , p a r t l y p r o t e c t e d b y a Row o f G r o y n e s " .

Het l a a t s t e r a p p o r t , d a t v a n 17 a p r i l s t a m t , h e n i k op h e t o g e n b l i k a a n h e t omwerken, met w a t b e t e r e aannamen.

V/at meer h i e r o v e r v i n d j e i n b i j l a g e 3.

De d e f i n i t i e v e v e r s i e v a n h e t l a a t s t g e n o e m d e r a p p o r t z a l i k j e t e z i j n e r t i j d s t u r e n .

Met h a r t e l i j k e g r o e t e n ,

( i r . W.T. B a k k e r )

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B i j b r i e f n r . 67.162B, d.d.

O p m e r k i n g b e t r e f f e n d e :

"ONE ASPECT OF TPIE DYNAMICS-OF A COAST. PARTLY PROTECTED BY A ROW OF GROYT^TES"

Een b e t e r e b e n a d e r i n g v a n h e t t r a n s p o r t m e c h a n i s m e b i j s t r a n d h o o f d e n kan w o r d e n g e g e v e n d o o r een p r o f i e l a a n t e nemen v o l g e n s f i g . 1 e n aan t e . nemen d a t e e n a e e w a a r t s t r a n s p o i ' t o p t r e e d t a l s > y^

( t e s t e i l p r o f i e l ) en een l a n d w a a r t s t r a n s p o r t a l s y.| < yg» H i e r b i j i s de g e m i d d e l d e a f s t a n d v a n de . v o o r o e v e r t o t h e t r e f e r e n t i e v l a k XZ g e l i j k aan y^ + V/; W moet w o r d e n g e z i e n a l s de a f s t a n d d i e i n e e n e v e n w i c h t s -p r o f i e l b e s t a a t t u s s e n s t r a n d e n v o o r o e v e r . - y Het i s m o g e l i j k e e n t h e o r i e t e o n t w i k k e l e n , g e b a s e e r d op de t r a n s p o r t -v e r g e l i j k i n g e n : Qy = ^ ^ y ^ ^ i - ~ d w a r s t r a n s p o r t Q. = Q„., - q. T - — - l a n g s t r a n s p o r t b i j h e t s t r a n d I O I 1 o X Q„ = Q „ - q„ - — - l a n g s t r a n s p o r t b i j de v o o r o e v e r Met de aanname, d a t h e t l a n g s t r a n s p o r t o v e r h e t s t r a n d n u l i s b i j e e n s t r a n d h o o f d , v i n d t men de k u s t v o r m e n ( f i g . 3 ) .

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2 -S i ^0^2 ^ K X + . — : — ; —~ r r -O s m h K^i, • s i n h K X 0 1 + K L o q^ t g h K ^ L K L ''o"" s i n h K 1 s i n h K X 0 1 + K L 0 t g h K ^ L fig. 3 Van b e l a n g b l i j k t een r e f e r e n t i e - l e n g t e 1. O A l s de a f s t a n d 2L t u s s e n de s t r a n d h o o f d e n k l e i n e r i s dan 2L^, dan s t e l t h e t s t r a n d ( y ^ ) z i c h v r i j w e l l o o d r e c h t op de g o l f r i c h t i n g i n , t e r w i j l de vooroevex" ( y g ) geen m e r k b a r e i n v l o e d o n d e r g a a t . A l s de a f s t a n d t u s s e n de h o o f d e n g r o t e r w o r d t , g a a t de t h e o r i e g e l d e n , z o a l s o n t w i k k e l d i n h e t bovengenoemde r a p p o r t . De t r a n s p o r t v e r g e l i j k i n g w o r d t : 0 ^ ^ x Q = - q w a a r i n y op d e z e l f d e m a n i e r g e d e f i n i e e r d i s a l s i n h e t r a p p o i - t ; '^02 ^ ^0 1 1 -t g h K ^ L ÏTL O 1 + t g h K ^ L en ^1 ^ ^2 q = _{q^ t g h}_{K ^ L} P 1 + q q t g h K ^ L K L O

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3 -V o o r g r o t e v/aarden v a n K^L ( L » L^) w o r d t p; 1 + ^2 ^ o ^ De V/aarde " ^ " u i t h e t r a p p o r t i s dus g e l i j k a a n : ^ <l2 ^^o ^2 V o o r k l e i n e w a a r d e n v a n K^L ( L « L^) w o r d t q g e l i j k aan q^.

Door h e t bouwen v a n s t r a n d h o o f d e n i s de t i j d s c h a a l t dus m a x i m a a l t e v e r l e n g e n t o t m a a l de o o r s p r o n k e l i j k e ; d i t g e b e u r t r e e d s b i j ^^2 h o o f d e n op een o n d e r l i n g e a f s t a n d v a n 2L^. Het l e g g e n v a n h o o f d e n op k o r t e r e a f s t a n d v a n e l k a a r dan 2L h e e f t d u s O w e i n i g z i n . Den Haag, 24 m e i 196?.

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SYMBOIENLIJST 2 _ L o h a l v e a f s t a n d t u s s e n de h o o f d e n ^1 ^2 e v e n r e d i g h e i d s c o n s t a n t e i n v g l . v a n h e t l a n g s t r a n s p o r t b i j h e t s t r a n d . " " " " " l a n g s t r a n s p o r t b i j de v o o r -o e v e r . '* " " " d w a r s t r a n s p o r t t u s s e n s t r a n d e n v o o r o e v e r . " " " " " l a n g s t r a n s p o r t b i j de s c h i j n b a r e k u s t l i j n . l a n g s t r a n s p o r t l a n g s de k u s t l a n g s t r a n s p o r t l a n g s h e t s t r a n d l a n g s t r a n s p o r t l a n g s de v o o r o e v e r d w a r s t r a n s p o r t l a n g s t r a n s p o r t l a n g s h e t s t r a n d , a l s = O l a n g s t r a n s p o r t l a n g s de v o o r o e v e r , a l s — — = O t l a n g s t r a n s p o r t l a n g s de k u s t , a l s = O a f s t a n d t u s s e n s t r a n d en v o o r o e v e r b i j e e n e v e n w i o h t s p r o f i e l r e f e r e n t i e l i j n , o n g e v e e r i n k u s t r i c h t i n g . o v e r de d i e p t e g e m i d d e l d e a f s t a n d u i t h e t r e f e r e n t i e v l a k XZ v a n h e t s t r a n d o v e r de d i e p t e g e m i d d e l d e a f s t a n d u i t h e t r e f e r e n t i e v l a k XZ v a n de v o o r o e v e r , m i n W y-coördinaat v a n de s c h i j n b a r e k u s t l i j n

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*

ONE ASPECT OP THE DYNAMICS OF A COAST, PARTLY PROTECTED BY A ROW OF GROYNES

1. A b s t r a c t

A m a t h e m a t i c a l theory w i l l be g i v e n about the phenomena t h a t o c c u r i f on a c o a s t a l a r e a groynes a r e c o n s t r u c t e d .

The t h e o r y i s a s i m p l i f i c a t i o n : i t j u s t d e a l s - a h o t t t one a s p e c t . The s t a r t i n g p o i n t i s the c o a s t a l e q u a t i o n of Pelnard-Considère [1_ and a l i n e a r i s a t i o n between the t r a n s p o r t a l o n g the groyne and the s i z e of the " s t e p " i n the c o a s t l i n e a t the groyne.

The change of the c o a s t l i n e , caused by c h a n g i n g boundary c o n d i t i o n s , by s t a t i o n a r y t r a n s p o r t and by the - s t r a i g h t e n i n g " of the c o a s t a r e c o n s i d e r e d .

The c o n c l u s i o n i s drawn, t h a t i n the middle of a row of g r o y n e s the same p r o c e s s e s o c c u r a s w i t h o u t g r o y n e s , but on a l a r g e r t i m e s c a l e . Hear the edges of the row t h e r e a r e edge e f r e c t s . t h a t cause b i g g e r e r o s i o n , and a c c r e t i o n than without g r o y n e s .

2. D e f i n i t i o n s and a s s u m p t i o n s

m the t h e o r y of Felnard-Oonsidère only the i n f l u e n c e of waves i s t a k e n i n t o a c c o u n t .

A c c o r d i n g to t h i s t h e o r y , the p r o f i l e of the c o a s t i s s c h e m a l i s e d ( f i g . 1) a s a h o r i z o n t a l a r e a A,

a t a depth D, where no t r a n s p o r t t a k e s p l a c e , because the waves a r e not a b l e to move the m a t e r i a l a t t h i s depth, and an a r e a B, which moves to and f r o

We d e f i n e the " c o a s t l i n e " y a s the l i n e t h a t l i n k s a l l p o i n t s a l o n g the c o a s t w i t h the mean y ^ i where y^ i s the d i s t a n c e to a v e r t i c a l r e f e r e n c e l i n e

( f i g . 1 ) :

D

'If

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6

-5. I n f l u e n c e of a row of g r o y n e s

We c o n s i d e r a c o a s t , where o v e r a l o n g s t r e t c h g r o y n e s a r e c o n s t r u c t e d a t time t = 0.

What i s the i n f l u e n c e of the g r o y n e s on the c o a s t ?

We w i l l c o n s i d e r t h r e e i n f l u e n c e s , w h i c h c a n g i v e an i m p r e s s i o n about the p r o c e s s e s t h a t o c c u r ( f i g . 6 ) :

1° the i n f l u e n c e of the boundary c o n d i t i o n s ( f i g . 6 b , c ) ; 2° the i n f l u e n c e of the s t a t i o n a r y t r a n s p o r t ( f i g . 6 d , e ) ;

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5.1 The I n f l u e n c e o f the boundary c o n d i t i o n s • We c o n s i d e r a c o a s t l i n e ^¥^\s s t r a i g h t a t time t w i t h o u t s t a t i o n a r y t r a n s p o r t (Q^ = 0).(!^ ' J l ) NO STATIONARY TRANSPORT Y 0 ( y = 0 ) , MOVING BOUNDARY SAME BOUNDARY CONDITION NO STATIONARY TRANSPORT 7-U REST 7

FIC.7^ coastline at tiive t

FIG . 7" coastline at time t, if no groynes would have been constructed

We assume, t h a t the row of g r o y n e s b e g i n s a t x - 0, and t h a t t h e motion of t h e c o a s t i s g i v e n i n x = - o o .

We w i l l show, t h a t t h e f o r m a t i o n o f t h e c o a s t l i n e ( y i f o r x > 0. y^ f o r X < o f c'a!n be found from t h e c o a s t l i n e y that' would e x i s t a t t h e same t i m e , i f no g r o y n e s would have been c o n s t r u c t e d , by t h e f o l l o w i n g o p e r a t i o n : i a t h e c o a s t l i n e o f t h e p r o t e c t e d part^'ban be found by r e d u c i n g t h e x - s c a l e of y by a f a c t o r i and by m u l t i p l y i n g t h e y - s c a l e ^ - i t h a f a c t o r (1 + r ) . i n w h i c h : 1^ 1 1 + ( 9 )

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4

9

-The above mentioned p r o c e d u r e can be put i n t o f o r m u l a ae:

i f X ^ 0 t h e n y^'(x) = (1 + r ) y ( p x ) ( 1 0 a )

The r e d t t c t i o n o f - t h e x - s c a l e i s r e a s o n a ) ) l e : t h e ^ / t i m e - s c a l e of , i h e / boundary c o n d i t i o n x = 0 f o r tHe p a r ^ / o f the c o a s t w i t h x > Ö r e m a i n s

same^,dr y

/^^^^^^^^^^

( 5 ) / t h e x - s c a l e must be ^^ " 0 0 '

b the c o a s t l i n e of the u n p r o t e c t e d p a r t y^ c a n be found a s the sum of the o r i g i n a l y p l u s a " r e f l e c t e d y"; the l a t t e r one b e i n g the r e f l e c t i o n of the o r i g i n a l y ( f o r x > O) w i t h r e s p e c t to the y - a x i s and m u l t i p l i e d w i t h the r e f l e c t i o n f a c t o r r , g i v e n i n eq. ( 9 ) : i f X < 0 t h e n y^ ( x ) - y ( x ) + r . y (-x) ( l O b j I n f i g . 8 the method of c o n s t r u c t i o n i s v i s u a l i s e d . Y , Y i f i g . 8 The f o l l o w i n g p r o o f can be g i v e n . It w i l l be s e e n t h a t y ^ s u f f i c e s the c o a s t a l e q u a t i o n ( 4 ) , a s i t i s a s u p e r p o s i t i o n of two s o l u t i o n s of t h i s e q u a t i o n , it—hetealreadybeen' -shown, t h a t y ' s u f f i c e s the c o a s t a l e q u a t i o n w i t h c o a s t a l c o n s t a n t , s o We 3*94—have- t o show t h a t the boundary c o n d i t i o n s a t x = 0 a r e f u l f i l l e d .

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4

These c o n d i t i o n s a r e :

1° y^iO, t ) = y.|(0, t ) , w h i c h i s c o r r e c t : ( 0 , t ) = y\{0, t ) - (1 + r ) y (0, t )..(a)

2° Q(x - - 0 ) - Q(x = + 0 )

As » 0, t h i s g i v e s , a c c o r d i n g t o ( 1 ) and ( 8 ) :

^ i i r " ^ 5 T - ^ o r X . 0 ( b )

Prom (10a) and (10b) one f i n d s r e s p e c t i v e l y :

^ PTrL;^'

**P(1 * , so U - 1 . p(i**

^^^/x-O 4 = 0 v^xy^^Q U x ; ^ . o S u b s t i t u t i n g t h i s i n ( b ) , one f i n d s : 1 - r q' < . y " P Q • w h i c h i s c o r r e c t , a c c o r d i n g t o ( 9 ) . j'V.ii'--: ^. ^

We w i l l show now, t h a t the c o a s t l i n e o f the p r o t e c t e d p a r t i s y ' g i v e n i n ( 1 0 a ) .

The p r o t e c t e d p a r t b e g i n s a t x . 0. A c c o r d i n g t o ( a ) t h e time s c a l e of t h e boundary c o n d i t i o n a t t h i s p l a c e r e m a i n s the same f o r y ( t h e c o a s t l i n e i f np g r o y n e s were c o n s t r u c t e d ) and f o r y^'.

The c o a s t a l c o n s t a n t pf y' i s a f a c t o r ~ s m a l l e r t h a t h e c o a s t a l P

c o n s t a n t o f y, a c c o r d i n g to ( 9 ) , and so the x s c a l e must be a f a c t o r

-s m a l l e r , a c c o r d i n g to ( 5 ) . ^ T h e r e f o r e , i f y ( p x ) i s a s o l u t i o n o f t h e c o a s t a l e q u a t i o n f o r t h e

u n p r o t e c t e d c o a s t , t h e n y ( x ) i s a s o l u t i o n o f t h e c o a s t a l e q u a t i o n f o r t h e p r o t e c t e d c o a s t . As mentioned i n c h a p t e r 3, the y - s c a l e c a n be c h o s e n a u x i l i a r y and may be m u l t i p l i e d , f o r i n s t a n c e w i t h a f a c t o r (1 + r ) .

Then one g e t s y j ( x ) - (1 + r ) y ( p x ) a c c o r d i n g t o ( 1 0 a ) , w h i c h must be a s o l u t i o n of the c o a s t a l e q u a t i o n ( s ) f o r t h e p r o t e c t e d c o a s t .

As t h e u n p r o t e c t e d p a r t y^, t h e p r o t e c t e d p a r t y.,', and the p o i n t x - 0 a l l s u f f i c e the r e q u i r e d c o n d i t i o n s , t h e c o a s t l i n e g i v e n i n eq. ( 1 0 a ) and

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. - 10 - . . _ _ 5.2 The I n f l u e n c e of the s t a t i o n a r y t r a n s p o r t

We c o n s i d e r a c o a s t l i n e , w h i c h i s a s t r a i g h t l i n e a t time t = 0 ( y « o ) , and r e m a i n s a t r e s t f o r x » oo .

A l l a l o n g the c o a s t the s t a t i o n a r y t r a n s p o r t i s Q^, b e f o r e the c o n s t r u c -t i o n of -t h e g r o y n e s a -t -t = 0. C o n s i d e r f i r s t the s i t u a t i o n of f i g . 9a (computed s i t u a t i o n I l a o f f i g . 6 ) , Q o / / / / / / / / y / / / / y / / _{) / y )}

/

/'

/ I

_{/ / / :}

/'

1 1

/ I

1

C o a s t l i n e a t time t = 0 _{P i g . 9a} C o a s t l i n e a t time t P i g . 9b

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- li»

On the l e f t - h a n d s i d e a t i n f i n i t y the t r a n s p o r t i s Q^, on t h e

r i g h t - h a n d s i d e i t i s ; t h e r e f o r e the s e d i m e n t a t i o n p e r u n i t time must be Q - Q ' . With the same method a s g i v e n i n 5.1 one c a n proof, t h a t t h e

o 0

f o l l o w i n g " d e l t a " w i l l o c c u r ( f i g . 9 b ) .

ïhe b r a n c h y ^ i s the b r a n c h , t h a t would o c c u r , i f a r i v e r would debouch a t X » 0 on an u n p r o t e c t e d s h o r e , b r i n g i n g to the s h o r e p e r u n i t of time a q u a n t i t y of sediment Q j * '

• ^o •

p u t t i n g a l l i t s sediment on the l e f t - h a n d s i d e of the r i v e r - m o u t h .

The b r a n c h y ' i s the b r a n c h t h a t would o c c u r i f t h a t r i v e r debouched

H q' 3E

on a p r o t e c t e d s h o r e w i t h c o a s t a l c o n s t a n t ^ , b r i n g i n g a q u a n t i t y Q ^ j ,

I I .

P 0

p u t t i n g a l l i t s m a t e r i a l on the r i g h t - h a n d s i d e of t h e r i v e r - m o u t h . The b r a n c h y^'^ i s a r e f l e c t i o n of b r a n c h y^. w i t h r e s p e c t t o the y - a x i s , but on a s m a l l e r x - s c a l e ( f i g . 1 0 ) :

P i g . 10

The f o r m u l a s of y^. and y^'j c a n be found w i t h the t h e o r y of P e l n a r d - C o n s i d e r e and a r e g i v e n i n annex 1.

I n annex 2 the shape of the d e l t a i s g i v e n a s a f u n c t i o n of — .1, i n which 1 i s the d i s t a n c e between the g r o y n e s .

The maximum a c c r e t i o n a t x = 0 i s r e l a t i v e t o ^ ^ t :

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12

-The f o r m u l a s f o r the t r a n s p o r t a l o n g t h e c o a s t c a n be found w i t h t h e a i d of ( 1 ) and ( 8 ) . They a r e g i v e n i n annex 1.

The t r a n s p o r t a l o n g the u n p r o t e c t e d c o a s t a t a l o n g d i s t a n c e of the b e g i n n i n g of the row of g r o y n e s i s Q^, a l o n g the p r o t e c t e d c o a s t a t a l o n g d i s t a n c e of the b e g i n n i n g - S | , and j u s t a t t h e b e g i n n i n g i t i s ^ .

P ^ I f one c o n s i d e r s the r e a l c o a s t l i n e y^^. i n s t e a d o f the v i r t u a l c o a s t l i n e

^ I I * ^^^^ f i n d s f o r the s t e p a t t h e f i r s t g r o y n e :

a ^ 0

V. — V m w —

''^l ^ r M. p-tt

V ^ 1

T h i s s t e p does not change i n the c o u r s e of time.

Q'o 1 1 1 1 1 1 1 1 1 1 1 _1._J Q o / / / / / / / / / / / / / / / / / / / / / / / / _{y / ////////} Y f f _ _ _ _ _ P i g . 11b ( 1 2 )

The c o m p u t a t i o n o f the s c o u r h o l e a t t h e l e e - s i d e of a row of g r o y n e s . ( f i g . 11) i s q u i t e t h e same a s t h e ' c o m p u t a t i o n of the " d e l t a * * on the o t h e r s i d e .

B r a n c h y ^ ^ j and y ^ ^ a r e the same a s b r a n c h y^'^ and y^ r e s p e c t i v e l y , r e f l e c t e d w i t h r e s p e c t t o the x- and the y - a x i s ( s e e annex 1 ) . The maximum

s c o u r i s g i v e n by y^^^Q i n ( 1 1 ) .

By e n l a r g i n g the d i s t a n c e between t h e g r o y n e s a t the end o f t h e row, or t h e i r p e r m e a b i l i t y , one w i l l be a b l e t o s p r e a d the e r o s i o n more e v e n l y o v e r a l a r g e r a r e a .

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

boundary c o n d i t i o n s , i t was assumed, t h a t t h e phenomena n e a r A i n f i g . 6 a d i d not a f f e c t t h e phenomena n e a r B. The c o a s t l i n e was s c h e m a t i s e d a s one s e m i - i n f i n i t e u n p r o t e c t e d and one s e m i - i n f i n i t e p r o t e c t e d peo't ( f i g . 6 ) . The i n f l u e n c e of t h e phenomena w h i c h take p l a c e n e a r A w i l l a t t e n u a t e q u i c k on t h e p r o t e c t e d p a r t from A t o B and t h e r e f o r e t h e i r i n f l u e n c e c a n be

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15

-The i n f l u e n c e of the s t a t i o n a r y t r a n s p o r t a t A w i l i be, t h a t a f t e r some time the g r o y n e s a r e c o v e r e d by sand i n w h i c h c a s e the t h e o r y does not h o l d . I t i s assumed, t h a t the g r o y n e s a r e l e n g t h e n e d b e f o r e they a r e c o v e r e d and a r e s h o r t e n e d i n c a s e of heavy e r o s i o n . The f i n a l s i t u a t i o n i n the c a s e

of s t a t i o n a r y t r a n s p o r t would be *

original coastline

final c o a s t l i n e

the s i t u a t i o n g i v e n i n f i g . 12, where everywhere the t r a n s p o r t i s Q^. I t i s v e r y u n r e a l i s t i c to c o n s i d e r t h i s c a s e i n d e t a i l , because l o n g b e f o r e the s i t u a t i o n w i l l have been a l t e r e d by human i n t e r f e r e n c e ( b u i l d i n g more g r o y n e s and so o n ) .

5.3 The i n f l u e n c e of the s t r e t c h i n g of the c o a s t l i n e

I n 5.1 and 5.2 i t was assumed, t h a t the c o a s t l i n e was a s t r a i g h t l i n e a t the moment of c o n s t r u c t i o n of the g r o y n e s . We w i l l now f i r s t compare the dynamics of a p r o t e c t e d c o a s t y* and an u n p r o t e c t e d c o a s t y, i f the b o u n d a r i e s r e m a i n a t r e s t and i f t h e r e i s no s t a t i o n a r y t r a n s p o r t .

At t « 0 we assume y » y ( f i g . 13a and 13t))

no s t a t i o n a r y transport rest' s a m e cunve (t c o a s t l i n e at time t r t( no stationary transport rest no s t a t i o n a r y transport P i g . 13a P i g , 13b

The c o a s t s y and y have a d i f f e r e n t c o a s t a l c o n s t a n t (.q

The t i m e - s c a l e i s r e l a t e d w i t h the c o a s t a l c o n s t a n t , a c c o r d i n g to e q u a t i o n ( 5 ) : n n t c c c o n s t a n t . n

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As the c o a s t l i n e i s g i v e n a t time t <= 0 a s a f u n c t i o n of x, the x - s c a l e of y and y i s the same and t h e r e f o r e the t i m e - s c a l e n i s e q u a l to — ^

* " c c C o n c l u s i o n :

At a p r o t e c t e d c o a s t the same p r o c e s s e s o f the u n p r o t e c t e d c o a s t w i l l o c c u r , a t a l o n g e r t i m e - s c a l e t , i n w h i c h : t ' - p ^ t . (1 + , ^ ) t However, m o s t l y o n l y a c e r t a i n p a r t of the c o a s t l i n e i s p r o t e c t e d . IT P i g . 14 Prom the f o r e g o i n g i t w i l l be c l e a r , t h a t t h i s l o n g e r t i m e - s c a l e o n l y o c c u r s i n a r e a I V of f i g . 14, t h a t t h e r e w i l l be no i n f l u e n c e i n the a r e a s I and V I I , and t h a t t h e r e w i l l be d i s t u r b a n c e s n e a r A and B i n the a r e a s I I , I I I , V and V I . The l a s t - m e n t i o n e d a r e a s w i l l i n c r e a s e i n c o u r s e of time ( p r o p o r t i o n a l to f t ) .

A g r a f i c a l method f o r computing the c o a s t l i n e i n t h i s c a s e i s the method of Schmidt, about which w i l l d e a l a n o t h e r paper.

6, C o n c l u s i o n s 1.

2.

3.

I t depends of the c a s e whether g r o y n e s w i l l work f a v o u r a b l y o r n o t . One h a s to pay the a c c r e t i o n on one p l a c e w i t h the e r o s i o n on a n o t h e r . With the a i d of g r o y n e s one can more o r l e s s choose t h e s e p l a c e s . I f anywhere e r o s i o n o c c u r s , where i t i s not t o l e r a b l e , one f i r s t h a s to c l a s s i f y the cause of e r o s i o n .

a r e a II

One c a u s e of e r o s i o n can be a g u l l y o r a r i v e r - m o u t h , w h i c h w i t h d r a w s sand from the c o a s t . With the a i d of some g r o y n e s , one can s t o p the e r o s i o n on the s i d e of

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- 15 -

_•I

the row o f g r o y n e s , o f f the g u l l y ( a r e a I I i n f i g , 1 5 ) . One i n c r e a s e s the e r o s i o n i n a r e a I , n e a r the g u l l y .

4. Another c a u s e o f e r o s i o n c a n be the l e e - s i d e s c o u r on a row o f g r o y n e s . One c a n s p r e a d t h i s e r o s i o n o v e r a l a r g e r a r e a by i n c r e a s i n g the d i s t a n c e between the g r o y n e s o r by i n c r e a s i n g t h e i r p e r m e a b i l i t y ( e x p r e s s e d i n the p e r m e a b i l i t y f a c t o r x^, ( 2 ) ) a t the end o f t h e row.

5. A t h i r d c a u s e c a n be, t h a t the c o a s t l i n e h a s not found y e t i t s e q u i l i b r i u m . One c a n stop the e r o s i o n by p r o t e c t i n g a l l t h i s p a r t w i t h g r o y n e s and by a d d i t i o n a l a r t i f i c i a l n o u r i s h m e n t . One c a n f i n d t h e e c o n o m i c a l optimum between the number of g r o y n e s and t h e amount of a r t i f i c i a l n o u r i s h m e n t . 6. I f a c o a s t a c c r e t e s , because i t h a s not found y e t i t s e q u i l i b r i u m o r

b e c a u s e o f a d e l t a , s u p p l y i n g sand, the e f f e c t o f a row o f g r o y n e s c a n be: d i m i n i s h i n g the a c c r e t i o n i n the a r e a of the g r o y n e s . T h i s i s the i n v e r s e of the c a s e s , mentioned i n c o n c l u s i o n s 3 and 5.

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L I S T OF SYMBOLS

D » w a t e r depth a t h o r i z o n t a l p a r t of the bottom ( a r e a A i n f i g . 1 ) . 1 = d i s t a n c e between two s u c c e s s i v e g r o y n e s . n ^ s c a l e of t o a s t a l c o n s t a n t " . c c D - time s c a l e , n^ = s c a l e i n x - d i r e c t i o n . ny. = s c a l e i n y - d i r e c t i o n . n „ « v e r t i c a l s c a l e . z p = \| 1 + ^ ' x - s c a l e of phenomena w h i c h o c c u r a t an u n p r o t e c t e d s c a l e i s p t i m e s the x - s c a l e of the phenomena a t the p r o t e c t e d c o a s t , i f the boundary c o n d i t i o n s a r e the same. q - p r o p o r t i o n a l i t y c o n s t a n t i n the t r a n s p o r t e q u a t i o n ( 1 ) . L a r g e q means

l a r g e " s e n s i b i l i t y " of the c o a s t f o r changement of the c o a s t a l d i r e c t i o n . * q q - = p r o p o r t i o n a l i t y c o n s t a n t i n the t r a n s p o r t e q u a t i o n ( 8 ) ; w i t h p r e s p e c t to the v i r t u a l c o a s t l i n e . Q - l i t t o r a l d r i f t . ^ s t a t i o n a r y t r a n s p o r t » t r a n s p o r t a t p l a c e s where the c o a s t a l d i r e c t i o n i s p a r a l l e l to the x - a x i s ( = 0 ) . Qo Q = — T ' = t r a n s p o r t a t p l a c e s where the v i r t u a l c o a s t l i n e i s p a r a l l e l 0 p^; I to the X - a x i s ( § ^ = 0 ) . r = P " , ; r i s a r e f l e c t i o n c o e f f i c i e n t . P + < t = t i m e . ' 2 t * p t X ^ r e f e r e n c e l i n e , about i n c o a s t a l d i r e c t i o n . y = mean d i s t a n c e from r e f e r e n c e l i n e of a r e a B ( f i g . 1 ) . y y - c o o r d i n a t e of " v i r t u a l c o a s t l i n e " ( t h e l i n e t h a t l i n k s the m i d d l e between the g r o y n e s ) .

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. 2 -y-j^ « y on l e f t h a n d s i d e o f a g r o y n e , y^, =^ y on r i g h t h a n d s i d e of a g r o y n e . y^ « h o r i z o n t a l d i s t a n c e of c o n t o u r l i n e of depth z from r e f e r e n c e l i n e . Ai * p r o p o r t i o n a l i t y c o n s t a n t . L a r g e M- means l a r g e " p e r m e a b i l i t y " of the g r o y n e .

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s t a t i o n a r y

transport Q stationary _{transport Qo}

• X X < 0 X > 0 (x=0) x < L ( x = L ) X > L branch branch ( r e a l c o a s t l i n e ) branch y|j ( v i r t u a l c o a s t l i n e ) branch y^ ( r e a l c o a s t l i n e ) branch y' ( v i r t u a l c o a s t l i n e )

b r a n c h yj^^ (real coastline )

branch y^^ (real c o a s t l i n e )

y n ( ' < ) = y , ( - p x : y ; ( x ) = - y ^ ( L - x ) = - y , | p ( x - L ) | y t v ^ x ) = - y i ( L - x ) in w h i c h erf X 1 P-1 QQ •IK P ^ q / e - ' ^ d t erfc X = 1 - erf X 3 J L B y i ^ x •dx = - p Y o ( ^ { l - e r f ( p x y : ^ ) ) y , / n [ w e r f ( ( L - x ) \ | : ^ ) ; J i y n r . B x • ^ ^ ) ^ - ^ - H p x \ U , )

y

Aqt I N F L U E N C E OF S T A T I O N A R Y T R A N S P O R T D E V E L O P M E N T O F C O A S T L I N E R ' J K S W A T E R S T A A T D I R E C T I E y^.zn W. A F D . K U S T O N D E R Z O E K Getek. ft Gewijz Gezien

5 ^

A c c A N N E X 1

A3

6 7 . 1 0 A

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Y,Y'

X

-1.0

INFLUENCE OF STATIONARY TRANSPORT

S H A P E OF C O A S T L I N E A S FUNCTION OF D I S T A N C E B E T W E E N T H E G R O Y N E S

ANNEX 2