Some results of comparison between numerical and analytic solutions of the one-Line model for shoreline change

V i e t n a m J o u r n a l o f M e c h a n i c s , V A S T , V o l . 2 8 , N o . 2 ( 2 0 0 6 ) , p p . 9 4 - 1 0 2 S O M E R E S U L T S O F C O M P A R I S O N B E T W E E N N U M E R I C A L A N D A N A L Y T I C S O L U T I O N S O F T H E O N E - L I N E M O D E L F O R S H O R E L I N E C H A N G E L E X U A N R O A N I n s t i t u t e o f M e c h a n i c s , V A S T , 2 6 4 D o i C a n , H a n o i , V i e t n a m A b s t r a c t . A q u a l i t a t i v e u n d e r s t a n d i n g o f t h e b a s i c p r o p e r t i e s o f c o m p l i c a t e d p h y s i c a l p h e - n o m e n a c a n o f t e n b e o b t a i n e d t h r o u g h t h e s t u d y o f a n a l y t i c s o l u t i o n s d e r i v e d f r o m s i m p l i f i e d p r o b l e m s . A n a l y t i c s o l u t i o n s o f s h o r e l i n e c h a n g e m o d e l f o r s i m p l e s h o r e l i n e c o n f i g u r a t i o n s a r e d e r i v e d u n d e r i d e a l i z e d w a v e c o n d i t i o n s . B o t h a n a l y t i c a n d n u m e r i c a l m e t h o d s a r e b a s e d o n t h e o n e - l i n e t h e o r y o f s h o r e l i n e c h a n g e . I n t h i s p a p e r s o m e r e s u l t s o f c o m p a r i s o n o f t h e n u m e r i c a l w i t h a n a l y t i c s o l u t i o n s a r e p r e s e n t e d . 1 . I N T R O D U C T I O N U n d e r c e r t a i n i d e a l i z e d w a v e c o n d i t i o n s a n d s i m p l e s h o r e l i n e c o n f i g u r a t i o n s , t h e e q u a - t i o n s o f o n e - l i n e t h e o r y o f s h o r e l i n e c h a n g e c a n b e r e d u c e d t o t h e o n e - d i m e n s i o n a l e q u a t i o n o f h e a t d i f f u s i o n t y p e , w h i c h i n s o m e c e r t a i n s i m p l i f i e d c a s e s c a n b e s o l v e d a n a l y t i c a l l y . T h e a n a l y t i c s o l u t i o n s a r e o f t e n v a l u a b l e f o r g i v i n g q u a l i t a t i v e i n s i g h t s a n d i n v e s t i g a t i n g t h e p r o p e r t i e s o f s h o r e l i n e c h a n g e . H o w e v e r , i t i s i m p o r t a n t t o b e a w a r e o f t h e l i m i t a t i o n s o f a n a l y t i c s o l u t i o n s a n d e r r o r s i n t r o d u c e d b y t h e s e l i m i t a t i o n s . F o r t h e r e a l s i t u a t i o n , t h e u s e o f n u m e r i c a l m o d e l o f s h o r e l i n e c h a n g e c o u l d b e m o r e a p p r o p r i a t e . S e v e r a l a u t h o r s h a v e p r e s e n t e d a n a l y t i c s o l u t i o n s f o r c e r t a i n s i m p l i f i e d c o n d i t i o n s ( e . g . B a k k e r a n d E d e l m a n 1 9 6 5 ; B a k k e r 1 9 6 9 ; L e M e h a u t e ' a n d S o l d a t e 1 9 7 7 ; W a l t o n a n d C h i u 1 9 7 9 ; L a r s o n , H a n s o n , a n d K r a u s 1 9 8 7 ) . I n o r d e r t o d e s c r i b e m o r e r e a l i s t i c s i t u a t i o n s i n v o l v i n g g e n e r a l s h o r e l i n e c o n f i g u r a t i o n s , t o g e t h e r w i t h t i m e v a r y i n g w a v e c o n d i t i o n s , · t h e o n e - l i n e t h e o r y h a s b e e n d e v e l o p e d u s i n g n u m e r i c a l s o l u t i o n t e c h n i q u e s ( e . g . P r i c e , T o m l i n s o n , a n d W i l l i s 1 9 7 3 ; S a s a k i a n d S a k u r a m o t o 1 9 7 8 ; K r a u s , H a n s o n , a n d H a r i k a i 1 9 8 5 ; H a n s o n a n d K r a u s 1 9 8 7 ) . F o u r e x a m p l e s o f s h o r e l i n e e v o l u t i o n f o r s i m p l i f i e d c o n f i g - u r a t i o n u s i n g t h e a n a l y t i c a l s o l u t i o n ( A ) a n d t h e n u m e r i c a l f o r m u l a t i o n ( N ) a r e e x a m i n e d a n d p r e s e n t e d i n t h i s p a p e r . 2 . O N E - L I N E T H E O R Y F U N D A M E N T A L S T h e a i m o f o n e - l i n e t h e o r y i s t o d e s c r i b e l o n g - t e r m v a r i a t i o n o f s h o r e l i n e p o s i t i o n s . S h o r t - t e r m v a r i a t i o n ( e . g , c h a n g e s c a u s e d b y s t o r m s o r b y r i p c u r r e n t s ) a r e r e g a r d e d a s n e g l i g i b l e p e r t u r b a t i o n s s u p e r i m p o s e d o n t h e m a i n t r e n d o f s h o r e l i n e e v o l u t i o n . T h e f u n d a m e n t a l a s s u m p t i o n o f t h i s t h e o r y i s t h a t e r o s i o n o r a c c r e t i o n o f a b e a c h r e s u l t s i n a p u r e t r a n s l a t i o n o f b e a c h p r o f i l e . T h u s , t h e b o t t o m p r o f i l e m o v e s i n p a r a l l e l t o i t s e l f w i t h o u t c h a n g i n g s h a p e . T h e m a j o r a s s u m p t i o n o f t h e ~h,eory i s t h a t t h e l o n g s h o r e s a n d t r a n s p o r t t a k e s p l a c e u n i f o r m l y o v e r t h e b e a c h p r o f i l e d o w n t o a c e r t a i n l i m i t i n g d e p t h c a l l e d t h e d e p t h o f c l o s u r e , D e . T h u s , b e y o n d t h i s d e p t h t h e b o t t o m d o e s n o t m o v e . F o l l o w i n g t h e a b o v e a s s u m p t i o n s , m a s s c o n s e r v a t i o n o f s a n d a l o n g a n i n f i n i t e l y s m a l l l e n g t h 6 . x o f t h e s h o r e l i n e c a n b e f o r m u l a t e d a s ( s e e F i g . 1 ) [ 1 ] : Some Results of Comparison between Numerical and Analytic Solutions .... . 95 By 1 [BQ ] at + (De + DB) Bx + q = o, (2 .1) where xis the longshore coordinate (m); y is the shoreline position (m) and perpendicular to x-coordinate; t is the time ( s); Q is the longshore sand transport rate ( m3 / s); DB is the average berm height (m) ; and q represents line sources and/or sinks along the coast (m3 / s/ m). In order to solve equation (2.1), it is necessary to specify an expression for the longshore sand transport rate, Q. This quantity is considered to be generated by wave obliquely incident to the shoreline. This relationship is taken to be [2]: WATER LEVEL DATU"1 y ~ ~ 0 w u ~ c ANGLE OF INCIDENT WAVE ~~~ro-~• x Fig. 1. Definition sketch for shoreline change calculation Fig. 2. Definition of breaking wave angle Q = Qo sin(2abs), (2.2) where abs is the angle between breaking wave crest and the local shoreline and Qo is the amplitude of the longshore sand transport rate. The empirical predictive formula for the amplitude of the longshore sand transport rate is taken to be [2]: Qo ~ (H 2Cg) (P ) K '' 16 ; - 1 (1 - p) 1.4162 (2.3) where H is the significant breaking wave height (m); C9 is the wave group velocity at breaking point (m/s); K is the empirical coefficient treated as a calibration parameter; Ps is the density of sand (kg/ m3); pis the density of water (kg/m3 ) ; pis the porosity of sand on the bed. The breaking wave angle, abs, may be expressed as (see Fig. 2): -1 (f)y) O:bs = O:b - O:s = O:b - tan ax ' (2.4) where O:b is the angle between breaking wave crests and the'x-axis; as is the angle between the shoreline and the x-axis. 9 6 L e X u a n H a a n 3 . A N A L Y T I C S O L U T I O N T E C H N I Q U E ( A ) F o r b e a c h e s w i t h m i l d s l o p e , i t c a n b e a s s u m e d t h a t t h e b r e a k i n g w a v e a n g l e t o t h e s h o r e l i n e i s s m a l l . I n t h i s c a s e , s i n ( 2 a b s ) ; : : : : : : 2 a b s · I f a l s o t h e a n g l e b e t w e e n t h e s h o r e l i n e a n d t h e x - a x i s , i s a s s u m e d t o b e s m a l l . I n a c c o r d a n c e w i t h e q u a t i o n ( 2 . 4 ) , a b s ; : : : : : : a b - o y / o x , s i n c e t h e i n v e r s e t a n g e n t c a n b e r e p l a c e d b y i t s a r g u m e n t i f t h e a r g u m e n t i s s m a l l . I n t h i s c a s e t h e e q u a t i o n ( 2 . 2 ) c a n b e r e w r i t t e n a s : Q = 2 Q o ( a b - ~~) · ( 3 . 1 ) I f t h e a m p l i t u d e o f t h e l o n g s h o r e s a n d t r a n s p o r t r a t e Q o a s w e l l a s t h e b r e a k i n g w a v e a n g l e a b i s a s s u m e d i n d e p e n d e n t o f x a n d t , a n d w i t h n e g l i g i b l e c o n t r i b u t i o n s f r o m s o u r c e s o r s i n k s ( q = 0 ) , e q u a t i o n s ( 2 . 1 ) a n d ( 3 . 1 ) c a n b e r e w r i t t e n a s : o y o 2 y o t = € o x 2 ' ( 3 . 2 ) w h e r e c i s a d i f f u s i o n c o e f f i c i e n t 2 Q o € - . - ( D e + D B ) ( 3 . 3 ) E q u a t i o n ( 3 . 2 ) i s a n a l o g o u s t o t h e o n e - d i m e n s i o n a l h e a t d i f f u s i o n e q u a t i o n , i t c a n b e s o l v e d a n a l y t i c a l l y f o r v a r i o u s i n i t i a l a n d b o u n d a r y c o n d i t i o n s . 4 . N U M E R I C A L S O L U T I O N T E C H N I Q U E ( N ) S o l v i n g e q u a t i o n s ( 2 . 1 ) - ( 2 . 4 ) n u m e r i c a l l y , w e a r e n o l o n g e r c o n s t r a i n e d b y s m a l l a n g l e a s s u m p t i o n , m a k i n g p o s s i b l e t h e s o l u t i o n o f a w i d e r v a r i e t y o f s h o r e / s t r u c t u r e c o n f i g u r a - t i o n a n d a m o r e r e a l i s t i c w a v e c l i m a t e . E q u a t i o n s ( 2 . 1 ) - ( 2 . 4 ) a r e d i s c r e t i z e d o n a s t a g g e r e d g r i d , i n w h i c h s h o r e l i n e p o s i t i o n s Y i a r e d e f i n e d a t t h e c e n t r e o f t h e g r i d c e l l s a n d t r a n s p o r t r a t e s Q i a t t h e c e l l w a l l s . T h e C r a n k - N i c h o l s o n i m p l i c i t s c h e m e i s u s e d . T h e d e r i v a t i v e o Q / o x a t e a c h g r i d p o i n t i s e x p r e s s e d a s a n e q u a l l y w e i g h t e d a v e r a g e b e t w e e n t h e p r e s e n t a n d t h e n e x t t i m e s t e p s [ 3 ] : o Q i - ~ [Q~+l - Q~ + Q i + l - Q i ] o x - 2 . 6 . x . 6 . x ' ( 4 . 1 ) w h e r e t h e p r i m e ( t ) i s u s e d t o d e n o t e a q u a n t i t y a t t h e n e w t i m e l e v e l , w h e r e a s t h e u n p r i m e d q u a n t i t y i n d i c a t e s a v a l u e a t t h e p r e s e n t t i m e s t e p , w h i c h i s k n o w n . S u b s t i t u t i n g e q u a t i o n ( 4 . 1 ) i n t o e q u a t i o n ( 2 . 1 ) a n d l i n e a r i z i n g o f t h e w a v e a n g l e i n e q u a t i o n ( 2 . 2 ) i n t e r m o f o y / o x r e s u l t s i n t w o s y s t e m s o f c o u p l e e q u a t i o n s f o r t h e u n k n o w n s I d Q ' · Y i a n i · y~ = B ' (Q~ - Q~+ 1 ) + y c i , ( 4 . 2 ) Q~ = E i (Y~+l - Y D + F i , ( 4 . 3 ) w h e r e B ' = . 6 . t / 2 ( D B + D e ) . 6 . x , y c i i s t h e f u n c t i o n o f k n o w n q u a n t i t i e s , i n c l u d i n g Q i , q i a n d Y i · F i a n d E i a r e t h e f u n c t i o n s o f w a v e h e i g h t , w a v e a n g l e , a n d o t h e r k n o w n q u a n t i t i e s . Some Results of Comparison between Numerical and Analytic Solutions ..... 97 Substituting ( 4.2) into ( 4.3) results in three-diagonal linear equat ion system and it is solved by the Thomas algorithm: (4.4) .6.x .6.x [Yi+ i - Yi Fi ] where Ai = 2Fi - B' and Gi = [jl 6.x Fi - Ei - .6.x (yci - yci-1) . The initial condition is taken to be Yi = Y(xi,o) where Y(xi ,o) is the initial shoreline position. The most commonly used boundary condit ion at both lateral boundaries is 8Q/8x = O[l] . For equation (2 .1 ), if 8Q/8x = 0 at the boundaries and with negligible sources or sinks, then 8y/8t = 0, indicating t hat y does not change with time. The above boundaries should be located far away from a project to assure that the conditions in the vicinity of the boundary are unaffected by changes t hat take place in the project. The numerical stabilily of the calculation scheme is governed by stability parameter: R - c; .6.t s - (6.x)2 · (4.5) Kraus and Harikai showed that t he numerical accuracy of the solution depends on the value of Rs . For the implicit sheme, t he values of Rs less than 0.26 are suggested [3) . 5. SIMULATIONS In order to investigate the agreement between the numerical and the analytic solutions, the breaking wave height, Hb , the wave group velocity at breaking point, Cg, and the breaking wave angle, O:&, are held constant (H& = 1.0 m , Cg = 4.0 m/s), in which the breaking wave angle is taken a small value (o:&= 5 degrees) to satisfy the assumption of small angle in equation (3 .1 ). In addition, in order to estimate the differences between analytic and numerical solutions in case of larger breaking wave angle, the numerical model was applied for two values of breaking wave angle (o:&= 15 and 30 degrees) . In all cases, the total calculation t ime was an year . Relative errors (3) between numerical (Rf) and analytic (R~) solutions at computed point i are calculated as follow: (5 .1) where Ri is the shoreline position at point i. The other parameters were used for simulating here: the empirical coefficient, K =0. 77, the depth of closure, D e= 4 m , average berm height, D s = 1 m. 5.1 Rectangle-shaped beach fill A beach fill (or a natural cape) shape is treated by approximating its shape with a rectangle. The init ially distance from the outer side of the rectangle to t he local shoreline is taken y = y0 (see Fig. 3). The analytic solution describing the shoreline positions is [4] 00 [ (n7ra)2 ] n7ra Y(x,t) = ~an exp - L t sin y x, (5.2) 9 8 L e X u a n H a a n L 2 2 J n - r r x a n = L Y o s i n L d x , n = 1 , 2 , 3 , . . . L 1 w h e r e L i s t h e b e a c h l e n g t h , L 1 , L 2 a r e t h e p o s i t i o n s o f t h e i n i t i a l b e a c h f i l l o n t h e x - a x i s . T h e s h o r e l i n e c a l c u l a t e d b y a n a l y t i c s o l u t i o n ( 5 . 2 ) i s s h o w n a s l i n e 1 i n F i g . 3 . T h e s h o r e l i n e s o b t a i n e d b y n u m e r i c a l m e t h o d a r e s h o w n a s l i n e 2 , 3 , a n d 4 , c o r r e s p o n d i n g t o b r e a k i n g w a v e a n g l e s 5 , 1 5 , a n d 3 0 d e g r e e s , r e s p e c t i v e l y . T h e l i n e n u m b e r 0 s h o w s t h e i n i t i a l s h o r e l i n e . oo.--~~~....---~~~~~~~~~~~....-~~~~ 7 0 e l l g 0 0 ~ E 4 J ~ 0 1 l 3 J c " : E 2 0 0 1 0 N o . s o l u t i o n o:. 1 ~(deg . ) 1 A 0 - Q J 2 H 5 0 3 H 1 5 4 H 3 J O i n i t i a l s h o r e l i n e 0 I rrl"'W'--·~ I I - . . . . . - ? ' > - I · 1 0 ' - - - - ' - - - - - - ' - - - - ' - - - - - ' - - - - - - ' 0 4 6 8 1 0 D i s t a n c e a l o r g s h o r e ( k m ) F i g . 3 . S p r e a d i n g o f a r e c t a n g l e - s h a r p e d b e a c h f i l l 1 6 0 1 4 0 1 2 0 ' E ; - 1 0 0 ·: g ~ s o f ~ 8 0 .. . 0 4 0 2 0 0 0 ~resG 4~ / \ f \ 3 !~ 2 - - f j \ \ _ 1 ! T \ \ ; i ' · 4 6 N o . s o l u t i o n a b ( d e g . ) 0 i n i t i a l s h o r e l i n e I A s m a l l 2 N S 3 N I S 4 N 3 0 D i s t a n c e a l o n g s h o r e ( k m ) 1 0 F i g . 4 . D e l t a e v o l u t i o n c a u s e d b y d i s c h a r g i n g s a n d f r o m a r i v e r T h e d i f f e r e n c e s b e t w e e n t h e n u m e r i c a l a n d a n a l y t i c s o l u t i o n s f o r t h e s m a l l b r e a k i n g w a v e a n g l e a r e m u c h s m a l l e r t h a n t h o s e f o r t h e l a r g e r b r e a k i n g w a v e a n g l e s , b e c a u s e t h e a s s u m p t i o n a p p l i e d t o o b t a i n t h e e q u a t i o n ( 3 . 1 ) i s v i o l a t e d . T h e n u m e r i c a l r e s u l t s s h o w t h a t e v e n i f u n d e r a c t i o n s o f d i f f e r e n t b r e a k i n g w a v e a n g l e s , t h e b e a c h f i l l i s a l w a y s s p r e a d e d s y m m e t r i c a l l y t o w a r d s b o t h s i d e s o f i n i t i a l b e a c h f i l l . T h i s c a n b e i n t e r p r e t e d b y t h a t t h e b r e a k i n g w a v e a n g l e a s w e l l a s w a v e b r e a k i n g h e i g h t a r e a s s u m e d t o b e i n d e p e n d e n t o f x , t h e r e f o r e t h e d i s t r i b u t i o n o f a l o n g s h o r e s a n d i s c o n t r o l l e d b y o n l y s h o r e l i n e c o n f i g u r a t i o n w h i c h h a s t h e s y m m e t r i c s h a p e w i t h r e s p e c t t o t h e c e n t r e o f t h e r e c t a n g l e . T h e m a x i m u m r e l a t i v e e r r o r s c a u s e d b y t h e a n g l e s o f 5 , 1 5 , a n d 3 0 d e g r e e s a r e 5 . 2 , 1 1 . 0 , a n d 3 8 . 0 % , r e s p e c t i v e l y ( s e e T a b l e 1 ) . T h a t m e a n s t h e t o o l a r g e b r e a k i n g w a v e a n g l e w i l l i m p a i r t h e a n a l y t i c s o l u t i o n , o v e r e s t i m a t i n g t h e s p e e d o f t h e s h o r e l i n e r e s p o n s e . 5 . 2 R i v e r d i s c h a r g j . n g s a n d I f a r i v e r m o u t h i s s m a l l i n c o m p a r i s o n w i t h t h e s t u d i e d a r e a , t h e s a n d d i s c h a r g e s o u r c e f r o m t h e r i v e r m a y b e a p p r o x i m a t e d b y a p o i n t s o u r c e . A s s u m p t i o n t h a t a c o n s t a n t s a n d s o u r c e q 0 [ m 3 / s ] f r o m t h e r i v e r i s m a i n t a i n e d a t x = x 8 , t h e i n i t i a l s h o r e l i n e i s a s t r a i g h t l i n e y = y 0 . A c c o r d i n g t o C a r l a w a n d J a e g e r ( 1 9 5 9 ) , t h e a n a l y t i c s o l u t i o n m a y b e e x p r e s s e d Some Results of Comparison between Numerical and Analytic Solutions . .... 99 as [2]. ( t) qo [~ (-(xs-x)2 ) lx-xsl f (lx-xsl)] y x, = - exp - er c De+ DB 7rc 4ct 2c 2y'd, ' (5.3) fort > Oand - oo < x < oo, where the symbol erfc denotes the error function which is defined as [2] z erfc(z) = 1- 5rr j exp (-e) dz. 0 In this example, qo was set to 0.02 m3 /s. The shorelines calculated by using the analytic solution (5 .3) and numerical solutions are shown in Fig. 4. In case of a small breaking wave angle, the numerical solution produces an almost identical shoreline (line 2) to the analytic (line 1). Sand from the river will be transported away from the river mouth much faster when the breaking wave angle approaches a large value, (line 4) . The Fig. 4 shows that even if under actions of varying breaking wave angles, sand is symmetrically distributed towards both sides of the river mouth. This is explained as for the rectangle-shaped beach fill above. The details of wave parameters in the surf zone as well as river flow are ignored, thus, the breaking wave angle are only playing the role in as a diffusion coefficient. The maximum relative errors caused by the angles of 5, 15, and 30 degrees are 0.7, 7.5, and 43.53, respectively (see Table 1). 5.3 Groin interrupting sand transport Initially, the beach is in equilibrium (parallel to the x-axis) . At time t = 0 a thin groin is instantaneously placed at x = 0, blocking all transport (Q = 0) . Mathematically, by the equation (2.4) , this boundary condition can be formulated as [2] 8y ax = tan O'.b' x= 0. (5.4) This equation states that the shoreline at the groin is instant parallel to the wave crests. A groin interrupts the transport of sand alongshore, causing an accumulation on the updrift side and erosion on the downdrift side. The analytic solution describing the accumulation part on updrift side of the groin is [2] y(x, t) ~ 2 tanao [ J¥ exp (-x2 /4£t) - ~erfc C~) l · (5 .5) In the numerical solution, the boundary condition at the groin which is totally blocking the transport of sand alongshore, is taken to be Q = 0. The shoreline positions calculated by analytic solution (5.5) and numerical solution are shown in Fig. 5. The comparison between the analytic and numerical solutions is only implemented on the updrift side, since the analytic solution on the downdrift side has not been considered in this paper. 1 0 0 L e X u a n H a a n I n t h i s e x a m p l e , t h e w a v e d i f f r a c t i o n a t t h e g r o i n a n d s a n d b y p a s s i n g o v e r t h e g r o i n a r e n o t t a k e n i n t o a c c o u n t . H o w e v e r , t h e r e s u l t o b t a i n e d b y n u m e r i c a l m o d e l s h o w s a q u a l i t a t i v e a g r e e m e n t w i t h r e a l i t y t h a t a c c r e t i o n a p p e a r e d o n u p d r i f t s i d e w h e r e a s e r o s i o n o n d o w n d r i f t s i d e . A s e x p e c t e d , f o r t h e s m a l l a n g l e , t h e n u m e r i c a l s o l u t i o n ( l i n e 2 ) g i v e s a n a l m o s t i d e n t i - c a l s h o r e l i n e t o t h e a n a l y t i c ( l i n e 1 ) . L i k e i n t h e p r e v i o u s c a s e s , a t o o l a r g e b r e a k i n g w a v e a n g l e w i l l i m p a i r t h e a n a l y t i c s o l u t i o n , o v e r e s t i m a t i n g t h e s p e e d o f s h o r e l i n e r e s p o n s e o n u p d r i f t s i d e . T h e f i g u r e 5 s h o w s t h a t t h e a c c u m u l a t i o n r a t e o f s a n d o n u p d r i f t s i d e i s m u c h f a s t e r w h e n t h e b r e a k i n g w a v e a n g l e i n c r e a s e s . T h e m a x i m u m r e l a t i v e e r r o r s c a u s e d b y t h e a n g l e s o f 5 , 1 5 , a n d 3 0 d e g r e e s a r e 4 . 4 , 7 . 2 , a n d 1 6 . 0 % , r e s p e c t i v e l y ( s e e T a b l e 1 ) . T h e e r r o r c o r r e s p o n d i n g t o t h e a n g l e o f 3 0 d e g r e e s i s r a t h e r s m a l l i n c o m p a r e w i t h t w o e x a m p l e s a b o v e . 5 . 4 S i n e - s h a p e d b e a c h T h e i n i t i a l s h o r e l i n e s h a p e i s t r e a t e d b y a p p r o x i m a t i n g i t s s h a p e w i t h a r h y t h m i c f o r m o f a s i n e w a v e . T h e a n a l y t i c s o l u t i o n t o t h i s c a s e i s f o u n d t o b e [ 4 ] n 7 l ' a 2 . n 7 l ' X 0 0 [ ] Y ( x , t ) = ~anexp - ( L ) t s m L , ( 5 . 6 ) L 2 J . . n 7 l ' X a n = L A s m w x s m L d x , n = 1 , 2 , 3 , . . . , 0 w h e r e A i s a n i n i t i a l a m p l i t u d e o f t h e b e a c h w a v e , w i s a n a n g l e f r e q u e n c y o f t h e b e a c h w a v e . 1 6 0 0 1 1 4 0 0 * - - N o . s o l u t i o n < l b ( d e g . ) O i n i t i a l s h o r e l i n e I A S 2 N S ~ 1 2 0 0 6 3 A I S . § . z 4 N I S S A 3 0 6 N 3 0 ~ 1 0 0 0 ~ ( . ' ) € ~ / I g -··'-"-~· . , 4 f t , _ . _ . . s s o o l ~-- / , ? ' - , a I :,,~<2 4 0 0 2 0 0 0 0 I , I ' " I 4 i / " '" " ' I I e " " D i s t a n c e a l o n g s h o r e ( k m ) F i g . 5 . S h o r e l i n e e v o l u t i o n a t a g r o i n 1 0 400,-~~~~~~~~~~~~~~~~~~ 3 5 0 f 1 : 1 ~ 2 0 0 g s 1 5 0 . ! ! ! c 1 0 0 5 0 0 ~~- 3 4 N o . s o l u t i o n a • ( d e g . ) 0 i n i t i a l s h o r e l i n e I A s m a l l 2 N s 3 N 4 N O '-·---~----"------'------',----~ 0 3 6 9 1 2 1 5 D i s t a n c e a l o o g s h o r e ( k m ) F i g . 6 . E v o l u t i o n o f a n i n i t i a l s i n e - s h a r p e d s h o r e l i n e T h e s h o r e l i n e s c a l c u l a t e d b y t h e a n a l y t i c s o l u t i o n ( 5 . 6 ) a n d n u m e r i c a l s o l u t i o n a r e s h o w n i n F i g . 6 . L i k e i n t h r e e p r e v i o u s e x a m p l e s , a s m a l l a n g l e w i l l g i v e s a g o o d a g r e e m e n t o f t h e n u m e r i c a l w i t h t h e a n a l y t i c s o l u t i o n . U n d e r w a v e a c t i o n s , t h e i n i t i a l a m p l i t u d e o f t h e b e a c h w a v e i s a t t e n u a t e d w i t h t i m e b u t m a i n t a i n e d i t s r h y t h m i c c h a r a c t e r . Some Results of Comparison between Numerical and Analytic Solutions ..... 101 The maximum relative errors caused by the angles of 5, 15, and 30 degrees are 1.5, 11.4, and 44.9%, respectively (see Table 1). The analytic solution (5.6) shows that when the time approaches to a great value, the amplitude of the beach wave will be reduced to zero, that means the beach wave becomes a straight line, and then the stable shoreline situation is established. Thus, under the action of waves having the constant parameters, the stable shoreline shape will be a straight line. This is reflected in Fig. 7. No. simulation time (years) 350 0 initial shoreline I I 300 2 3 3 II 50 0 00:;----~~~3~~~~6~~~~9~~---,1~2~~--"15 Distance alongshore (km) Fig. 7. Attenuation of an initial sin-sharped shoreline to straight shoreline Table 1. Maximum relative error (%) between numerical and analytic solutions for different breaking wave angles Cases Wave breaking angles, ab (degrees) O'.b = 5 O'.b = 15 O'.b = 30 Rectangle-shaped beach fill 5.2 11.0 38.0 River discharging sand 0.7 7.5 43.5 Groin interupting sand transport 4.4 7.2 16.0 Sine-shaped beach 1.5 11.4 44 .9 6. CONCLUSION The comparison between the analytic and numerical solutions for four different shore- line configurations under idealized wave condition are presented. The obtained results show that the agreement of the analytic with the numerical solution is only well if the breaking wave angle is small (kept within 15 degrees with maximum relative error of about 10%) . If the details of wave parameters near the structure as well as effects of river flow (in the case of river discharging sand) are ignored and the parameters of breaking waves are assumed to be independent of x and t, the shoreline will have the symmetric shape with respect to the centre of the shoreline/ structure configuration. 1 0 2 L e X u a n H o a n I f t h e b r e a k i n g w a v e a n g l e i s t o o l a r g e , t h e a n a l y t i c s o l u t i o n s w i l l g i v e a n o v e r e s t i m a t i o n o f t h e s p e e d o f s h o r e l i n e r e s p o n s e ( e x c e p t t h e c a s e o f r i v e r d i s c h a r g i n g s a n d ) . I n t h e c a s e o f r i v e r d i s c h a r g i n g s a n d , t h e a n a l y t i c s o l u t i o n w i l l g i v e a n u n d e r e s t i m a t i o n o f t h e s p e e d o f s h o r e l i n e r e s p o n s e T h e a u t h o r a c k n o w l e d g e P r o f . P h a m V a n N i n h a n d P r o f . T r a n G i a L i c h f o r t h e i r u s e f u l c o m m e n t s . T h e p a p e r w e r e p a r t l y s u p p o r t e d b y f u n d a m e n t a l r e s e a r c h p r o j e c t " M a r i n e H y d r o d y - n a m i c s a n d E n v i r o n m e n t , C o d e 3 2 1 5 0 1 " . R E F E R E N C E S 1 . H a n s H a n s o n a n d N i c h o l a s C . K r a u s , G E N E S I S : G e n e r a l i z e d M o d e l f o r S i m u l a t i n g S h o r e l i n e C h a n g e , R e p o r t 1 , D e p a r t m e n t o f t h e A r m y , U S A r m y C o r p s o f E n g i n e e r s , W a s h i n g t o n , D C 2 0 3 1 4 - 1 0 0 0 , D e c e m b e r 1 9 8 9 . 2 . M a g n u s L a r s o n , H a n s H a n s o n a n d N i c h o l a s C . K r a u s , A n a l y t i c a l S o l u t i o n s o f t h e O n e - l i n e M o d e l o f S h o r e l i n e C h a n g e , F i n a l R e p o r t , D e p a r t m e n t o f t h e A r m y , U S A r m y C o r p s o f E n g i n e e r s , W a s h i n g t o n , D C 2 0 3 1 4 - 1 0 0 0 , O c t o b e r 1 9 8 7 . . 3 . H a n s H a n s o n , G e n e r s i s - A G e n e r a l i z e d S h o r e l i n e C h a n g e N u m e r i c a l M o d e l , J o u r n a l o f C o a s t a l R e s e a r c h 5 ( 1 ) ( 1 9 8 9 ) 1 - 2 7 . 4 . A . G . W e b s t e r , P a r t i a l D i f f e r e n t i a l E q u a t i o n s o f M a t h e m a t i c a l P h y s i c s , D o v e r P u b l i - c a t i o n s , I N C . 1 9 5 5 R e c e i v e d A u g u s t 8 , 2 0 0 5 R e v i s e d A p r i l 1 8 , 2 0 0 6 M Q T s 6 K E T Q U A s o s A N H G I U A NGHI~M G I A I T i C H v A NGHI~M , , . ! . . . . . . , , . ! , , . . . . . . . . . . . . . . . . . . . . S O V E S V B I E N D O I D U O N G B C T D V A T R E N M O H I N H M Q T D U O N G Vi~c hi~u b i e t n h u n g d~c t f n h w b a n c u a c a c hi~n t m ; m g v~t l y p h u c t~p t h m ' : m g nh~n d t r Q ' C n h a v a o vi~c n g h i e n C U U c a c nghi~m g i a i t f c h d t r Q ' C d a n r a t u C a C b a i t o a n d a d t r Q ' C d a n g i a n h o a . N h u n g nghi~m g i a i t f c h c u a m o h l n h b i e n d 6 i d u a n g b a d o i v & i c a c d~ng d m ' m g b a d a n g i a n d u q c d a n r a d u & i g i a t h i e t c a c d i e u ki~n s o n g d u q c l y t u & n g h o a . C a nghi~m g i a i t i c h v a m o h l n h S O v e S \ l ' b i e n d o i d m ' m g b a d e u d v a t r e n l y t h u y e t m Q t d u a n g ( o n e - l i n e ) . B a i b a o t r l n h b a y m 9 t s o k e t q u a s o s a n h g i u a nghi~m g i a i t f c h v a nghi~m s o .