ª Œª Œ 29ƒ 4B Á œ pp. 377 ~ 384 «ª ª ww w x q x Computation of Wave Propagation over Multi-Step Topography by Partition Matrix Method û SeoSGSeu
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1 29ƒ 4B Á œ pp. 377 ~ 384 «ª ª ww w x q x Computaton of Wave Propagaton over Mult-Step Topography by Partton Matrx Method û SeoSGSeung-Nam Abstract In order to reduce computng tme sgnfcantly for a large matrx n EFEM of lnear waves propagaton over rpple beds each of whch s approxmated to a mult-step topography a partton method s presented to calculate reflecton coeffcents. By use of 10 evanescent modes n the model the most accurate numercal solutons have been obtaned up to date whch show dfferent behavors of computed reflecton coeffcent n some cases aganst the exstng results. Both computng tme and memory of the present partton model for solvng a large matrx are stll so much demandng that t s needed to develop an effcent method. Keywords : wave propagaton mult-step topography egenfuncton expanson method partton method reflecton coeffcent w x ù x q q w š w j»ƒ j w t w» w ww w w. x 10 q w x ¾ ƒ w ew w š w». j»ƒ j w t ww» f z w. w : q q x š w ww 1. x x w š w (EFEM egenfuncton expanson method w q x w. w q (Berkhoff 1972; Smth 1975; Massel 1993; Athanassouls 1999 Laplace x l š w w w w š w wq(propagatng wave q(evanescent wave w q w w. w p EFEM w q x p n w» (Takano 1960; Krby 1983; Guazzell 1992; 1999; Bender 2003; û 2008a. wr EFEM w» w x x w w q x w w. ƒ w w. EFEM w w w ƒƒ w wq q N w N1 ƒ ¼. w» w ü w w l 2N2 x. x M ü w 2M(N1 ƒ. M N ƒw ƒ w ƒw š» 3 ƒƒ w» EFEM ƒ j. yw» w (Mles 1967; Devllard q wq w» w xk w wq j w ƒ. ù» z Áw w (E-mal : snseo@kord.re.kr 29ƒ 4B œ 377
2 x q sww x ¾ w. wr wq w x w O'Hare ( û(2008b. q w w w qƒ EFEM sw w. xqƒ x w w Guazzell ( q sww q w s q w ƒ. Guazzell l xqƒ x O'Hare x w yw. w w xq 4 x w 800 w ( w û(2008b q 4 w w š p q q O'Hare w.» ww EFEM w» q 4 w w. w» ( ƒ q w j» q w w w w. (1999 û (2008b w q ƒ w w w w. w q wù ƒ j ƒ w y. wr w w q w. q y e x w. EFEM l» LU decomposton w wù w w w» w. wr Guazzell (1992 yw» w x w» x w tš ww w. ù x w s ù w w w» wš w. 2. EFEM w x x w w š w q x w (e ωt w sl φ ( xyz w Laplace w ùký. t w e w w w w t w. x -1 x ew h t»w. 2 φ = 0 h< z< 0 φ zz (1 ω 2 φ z φ = 0 z = 0 g φ z = 0 z = h (2a (2b φ x = 0 (2c» ƒ q ω = 2π T w» T w w g ƒ. (2 w w ƒƒ. š w j» sl» w ƒ. 1 q k 1 t»wš» ƒ θ 1 ƒ y w q b = k 1 sn θ 1. x yƒ b w y w w wq xk e by ( b > 0 ƒ (3 x. 0 = φ xx φ yy φ zz = φ xx b 2 φ φ zz (3 (2 φ = φ( xz e by w w xk w. (3 w (2 w w w š x w š w w ùký. 2a x 0 a x 0 ( p p e e coshk ( 0 z h a j ( x x a j ( x x [ s j e s j e ]cosk ( j z h k 0 b j = 1 φ = ( p p xcoshk ( 0 z h a j ( x x a j ( x x [ s j e s j e ]cosk ( j z h k = 0 b j = 1 gk ω 2 0 tanhk 0 h = gk n tank n h n = 1 2 (3 (4a (4b» q k n (n=012ã q (4b l w. (4a w wq p q s w ƒƒ t»w. w w w ëš w w w. š (4a q w ù e N ww. w r w w w ƒƒ w» ü ƒ w (knematcs (dynamcs w w w w. w w» w w (5a š w w x Bernoull w (5b ƒƒ w. φ φ = x x φ φ = (5a (5b q d d w w ƒ w. q d d 378
3 ww d p 1 = 1 s 1 j = 0 ƒƒ w. M1 ww q w» p M 1 = 0 s M 1 j = 0 ± ±. w p p j (5 w w w. M ü x M(22N ƒ w. w» w (5a ¾ š w wš (5b û š w w û» w w ¾ w Krby (1983 w. w w (5a ü n w ww ¾ y w. (5 l 2N2 š ü w. w. l ƒ n K T K R w. n wq n qš qš qš ƒƒ ù š (4 tx sl w w η ùkü (6ƒ. ( = φ ( x 0 η x (6 w s. d w q n (7 (N r =M1. w» w w w. l n.» w. ω g p K R = K T = K R K k = T n = 2 3. w w a Nr 0 n Nr 2 k Nr 0 a 1 0 n 1 Cg C h Nr h Im a Nr Re( a Nr 0 coshk Nr 0 p Nr coshk 1 0 ( = 0 = 0 (6 (7 (8 Cg C q 2 w A l x š l l r ùkü (9ƒ. Ax=r (9 (9 w w» w A LU wwš ey z ey w». wr (9 r 3(N1 2(N1 x ü ƒ wù ƒw 4(N1 2(N1. ƒ ww w. (9 w A w w (block matrx (10 txwš w B ww wš w. A A 11 B B A 11 B = = = 12 A 21 A 22 B 22 (10» w w A n w A 22 ƒƒ pq w š n=pq. A 21 ƒƒ p qw q pw š B j A j w w w w. w ƒ l AB=BA š n w. AB=I l (11.a (11.b š B 11 = I p B 12 B 22 = 0 BA = I l (11.c (11.d. B 22 A 21 = 0 B 22 A 22 = I q (11.a (11.b (11.c (11.d (11 w w. r q w C wš B 22 =C 1 txw (11.b B 12 = ( C š (11.c (12.bƒ. C A A 1 = ( w (11.a l (12.c B 11 ( C A A 1 = = ( (12.a (12.b (12.c (11.d w (12.d. C = A 22 ( A 21 (12.d (12.d w w B 22 ƒ w B j ƒ A j. (12 w w A w B w w w. 1. w D 21 D 21 A 21 w w. 2. (12.d w D 21 w w C w š l w C w 1 B 22 ƒ. 3. B 22 D 21 (12.b w w w. 4. w D 12 D 12 A w w. 5. D 12 w (12.a l B 12 w. 6. B 12 (12.a w B 11 w. 6 e ü w A w w B w 1-6¾ ü ¾ w. 2(N1 w C w C w 1 29ƒ 4B œ 379
4 Table 1. Expermental setup of Daves (1984 Case A (cm λ (cm x r (cm h 0 (cm DH DH DH Fg. 1 Elapsed CPU tmes on a PC wth Intel Core2 Extreme (3.0 GHz between two matrx solvng methods for dfferent number of steps; Number of evanescent modes = 10. w w w j» w. ü w w w. z 1-6 mw w w. x ù w l 2(N1 (Ν1w» e w w.» ü ƒ w š 1-6 ww w A w 1. z w A l r w wƒ 1. w w» j w» w w A 1 w š l r w. LU w w» eƒ v wù w f w.» w Booj(1983 x w x ww. Fg. 1 q 10 ü M 22M ƒ w w l w w» w» LU w ww w š l Core2 Extreme (3.0 GHz w PC CPU ùkü. l LU ww ƒ ƒw w w. w ww ww LU w w w w e x y w. 4. e x ww w Booj(1983 x w w xq x(daves 1984 š xq x(guazzell 1992 w e x w. e x q» 10 w»» j w. e x ƒ w» ( 8G w x x x w» 50» w š w. x x 0.6 m n 0.2 m š s( S q» 2. Daves (1984 x w wù xqƒ 2 4 š 10 š (13 tx h 0 x r ƒ w ¼ š A λ xq x s q ƒƒ w. š Table 1 (13 ùkü. h 0 x 0 hx ( = h 0 Asn( 2πx λ 0 x x r h 0 x x r (13 w xq Guazzell (1992 x (14 ùký š» λ 1 λ 2 q Table 2 x ùkü. h 0 x 0 hx ( = h 0 A[ sn( 2πx λ 1 sn( 2πx λ 1 ] 0 x x r h 0 x x r (14 Table 2 Guazzell (1992 x 3 w h 0 3 ƒ w 9 x. x 101 w e x» w w ( û 2008b. x w e x x w» w w š» w q 10 w.» x» ƒ wš ƒ PC Workstaton xw Table 3 ù kü. Table 2. Expermental setup of Guazzell (1992 Case A λ 1 λ 2 x r h 0 (cm (cm (cm (cm (cm a b c G G G
5 Table 3. Elapsed CPU tme for the present numercal experments Case No. of steps No. of wave condtons Ave. CPU tme per a wave condton (sec Total CPU tme DH hours DH hours DH hours/ 12.6 days G hours/ 0.37 days G hours/ 8.75 days G hours/ 8.79 days Fg. 2 Convergence test of reflecton coeffcent for waves normally ncdent on a double rpple patch G1a. Table 3 e w w ùkü. w (DH wù q l n w 1.5 v w. e x q w s q w (Plane-wave approxmaton PA ƒ w x w. PA w w» w y (nversonw. ù û(2008a w ü w w w w w w w. w w w x y w ùkü. q w» w e x Fg. 2 ùkü. w šw q w w xq 4 x G1a w. q 1 w w q x w wq q y q w». q 3 w x 16 w ù x» w w. Fg. 3 Comparson of results from PA EFEM(N=10 and laboratory data of Daves and Heathershaw(1984: depth of case (a h1=15.6 cm; (b h0=15.6 cm; (c h0=31.3 cm. 29ƒ 4B œ 381
6 Fg. 4 Comparson of results from PA EFEM(N=10 and laboratory data of Guazzell et al. (1992: depth of case (a h0=2.5 cm; case (b h0=3 cm; case (c h0=4 cm 10 q x x w q. Fg. 3 Daves (1984 x w. ww ( PA w( q (k w xq (k r ƒ 1 w.» ewù q 2k w /k r =2 0. wr w w» w stream functon l q w Km (2004 DH2 0. ù x 0 j x PA w O'Hare ( x w» x w w ù sƒ w. 382
7 PA x O'Hare q w yw (transfer matrx method. EFEM q ew w wƒ w š q w» ƒ. š x ü qƒ 10% w š š d w w. ww O'Hare w x mw w w xq w ww ƒ w q. Fg. 4 Guazzell (1992 x w ùkü. w xqƒ x ww PA w» w. Guazzell x ü q w w w. G1 2k w k r 1 G2 G3 2k w k r 0.5 j w ù 0.1~0.2 w w š x. ww e x z G1 1.8 G3 1.4» ƒwš j w w. w Fg. 4 z t wš PA ww wƒ x ew. w x Guazzell (1992 Athanassouls (1999 e x w. 5. EFEM l w w w LU w ww w ƒ w wš s q w x w. w ew» w q 10 w» w. w f ww LU w w 50% ( 400 w ù w xq x ww 2k w k r = 2 0. w s q w ww w š Booj w x s q w ww w. 29ƒ 4B œ 383
8 ù w xq x ww w s q w x w ƒ. l EFEM w q w w. x p q p w e ƒ w ƒ wùƒ w e w ù w j yƒ. w v w wù». (Magne 2007 w EFEM w ƒ ù EFEM w» mw w qš w» sƒ. ww w ƒ ù x x w w w w»» w» w w» w ƒ v w. w w» w x y» w e d» w (Hazard Map» : w q d w. š x û(2008a w x q x. w w w œwz w w w œwz 20«5y pp û(2008b w q x. w w w œwz w w w œwz 20«6y pp z(1998 w x m w q m. wm wz wm wz 18 «II-4y pp Athanassouls G.A. and Belbassaks K.A. (1999 A consstent coupled-mode theory for the propagaton of small-ampltude water waves over varable bathymetry regons. J. Flud Mech. Vol. 389 pp Bender C.J. and Dean R.G. (2003 Wave transformaton by twodmensonal bathymetrc anomales wth sloped transtons. Coastal Eng. 50 pp Berkhoff J.C.W. (1972 Computaton of combned refracton-dffracton. Proc. 13th Coastal Eng. Conf. Vol. 1 pp Booj N. (1983 A note on the accuracy of the mld-slope equaton. Coastal Eng. Vol. 7 pp Chamberlan P.G. and Porter D. (1995 The modfed mld-slope equaton. J. Flud Mech. Vol. 291 pp Daves A.G. and Heathershaw A.D. (1984 Surface-wave propagaton over snusodally varyng topography. J. Flud Mech. Vol. 144 pp Devllard P. Dunlop F. and Soullard B. (1988 Localzaton of gravty waves on a channel wth a random bottom. J. Flud Mech. Vol. 186 pp Guazzell E. Rey V. and Belzons M. (1992 Hgher-order Bragg reflecton of gravty surface waves by perodc beds. J. Flud Mech. Vol. 245 pp Km J.W. and Ba K.J. (2004 A new complementary mld-slope equaton. J. Flud Mech. Vol pp Krby J.T. (1986 A general wave equaton for waves over rppled beds. J. Flud Mech. Vol. 162 pp Krby J.T. and Dalrymple R.A. (1983 Propagaton of oblquely ncdent water waves over a trench. J. Flud Mech. Vol. 133 pp Magne R. Belbassaks K.A. Herbers T.H.C Arhun F. O'Relly W.C. and Rey V. (2007 Evoluon of surface gravty waves over a submarne canyon J. Geophys. Res. 112 C01002 do: /2005jc Massel S.R. (1993 Extended refracton-dffracton equaton for surface waves. Coastal Eng. Vol. 19 pp Mles J.W. (1967 Surface-wave scatterng matrx for a shelf. J. Flud Mech. Vol. 28 pp O'Hare T.J. and Daves A.G. (1992 A new model for surfacewave propagaton over undulatng topography. Coastal Eng. Vol. 18 pp O'Hare T.J. and Daves A.G. (1993 A comparson of two models for surface-wave propagaton over rapdly varyng topography. Appled Ocean Res. Vol. 15 pp Porter D. (2003 The mld-slope equatons. J. Flud Mech. Vol. 494 pp Porter D. and Stazker D.J. (1985 Extenson of the mld-slope equaton. J. Flud Mech. Vol. 300 pp Smth R. and Sprnks T. (1975 Scatterng of surface waves by a concal sland. J. Flud Mech. Vol. 72 pp Takano K. (1960 Effets d'un obstacle paralllppdque sur la propagaton de la houle. La Houlle Blanche Vol. 15 pp ( : / : / :
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