29ƒ 4B Á 2009 7œ 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 1988. q wq w» w xk w wq j w ƒ. ù» z Áw w (E-mal : snseo@kord.re.kr 29ƒ 4B 2009 7œ 377
x q sww x ¾ w. wr wq w x w O'Hare (1992 1993 û(2008b. q w w w qƒ EFEM sw w. xqƒ x w w Guazzell (1992 3 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 (1999 401 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
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 = 1 2 1 K R K k 2 1 0 = 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 0 1 0 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 = ( 21 11 w (11.a l (12.c B 11 ( C A A 1 = = ( 21 11 (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 2009 7œ 379
Table 1. Expermental setup of Daves (1984 Case A (cm λ (cm x r (cm h 0 (cm DH1 5 100 200 15.6 DH2 5 100 400 15.6 DH3 5 100 1000 31.3 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. 220 220 w 2200 2200 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 101-501 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 G1 1 12 6 48 2.5 3 4 G2 0.5 6 4 48 2.5 3 4 G3 1 6 4 48 2.5 3 4 380
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 DH1 101 201 9.462 0.53 hours DH2 201 201 82.359 4.60 hours DH3 501 201 5421.926 302.72 hours/ 12.6 days G1 201 391 82.05 8.91 hours/ 0.37 days G2 401 391 1933.892 210.04 hours/ 8.75 days G3 401 391 1942.135 210.94 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 (DH3 11022 11022 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 10 16 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 2009 7œ 381
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 (1992 1993 x w» x w w ù sƒ w. 382
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 2009 7œ 383
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