The Two Densonal Nueal Modelng Of Aous Wave Poagaon n Shallow Wae Ahad Zaaa John Penose Fan Thoas and Xung Wang Cene fo Mane Sene and Tehnology Cun Unvesy of Tehnology. CSIRO Peoleu. Absa Ths ae desbes ogess on a wo densonal nueal sulaon of aous wave oagaon ha has been develoed o vsuale he oagaon of aous wave fons and o ovde e-doan sgnal eesenaon n shallow wae. I s nended ha an eenson of he wo esened hee o aoun fo hee-densonal effes wll lae be oaed wh feld esuls. The nueal sulaon of shallow wae aous oagaon has gven se o a wde vaey of odelng ehnques wh vaous degees of auay. One ehnque nvolvng fne dffeene ehods s oe oonly used n he deson of ses oagaon bu also ous n he shallow wae oagaon leaue. Ths wod eoed hee nvolves he alaon of fne dffeene ehnques o odel oagaon n he e doan ogehe wh assoaed ode o allow wave fon vsualaon. Inoduon Many eseahes have develoed nueal neeaons of he wave equaons sued o aous and ses oagaon (Alfod Kelly and Booe 974; Kelly Wad Sven Teel and Alfod 976; Cean Kosloff and Reshef 985; Wllas Rehen Andeson 996; Wu Lnes and Lu 996 Keswee Bla and Shessne 996; Alesev Mhaleno 999). The nueal odelng of ses daa has been used o suo neeaons of feld daa o ovde synhe daa fo esng oessng ehnques and aquson aaees and o enhane sesologss undesandng of wave oagaon (Keswee Bla and Shessne 996). Fo hese alaons fne-dffeene ehods have ofen been used. Ths eo es he wave equaons sued o waves n fluds aous waves and wave n solds suh ha boh shea and oessonal defoaons ae aouned fo ae eed elas waves. Mos ses odelng neessaly uses he elas wave equaons. (Kelly Wad Sven Teel and Alfod 976) bu he aous wave equaons have also been used fo geohysal odelng ehnques (Alfod Kelly and Booe 974). The elas wave equaons ae needed o fully aoun fo wave oagaon n he seabed bu an aous wave aoaon s ofen used fo seabed sedens when shea veloes ae low. Ths ae eos on ogess n develong a oue oga whh deals wh he wodensonal nueal odelng of aous wave oagaon n shallow wae. Key feaues of he odel a esen ae: () () () Theoy The use of aous wave equaon Te doan odellng A oason of he use of nd and 4 h ode auay Aous wave equaon A wo-densonal aous wave equaon an be found usng Eule s equaon and he equaon of onnuy (Behovsh 960). v u 0 Connuy () u v 0 Eule () Aouss 000
Aouss 000 Whee u s he ale veloy s he aous essue s he densy and s he veloy of he aous wave n he aous eda. Subsuon of he dvegene of he Eule equaon and he e devave of he equaon of onnuy yeld f δ v v (3) f δ (4) Whee δ s he Da dela funon assoaed wh he oson of he soue n sae and f s he soue funon. Fo hoogenous eda he aous wave equaons an be slfed as follows f δ (5) Fne-dffeene soluon Aous wave equaon Fne-dffeene ehods an be aled o he sala aous wave equaon. The seond e devave and fs saal devave of he wave equaon an be aoaed usng a seond ode enal dffeene aoaon as follows. (6) (7) (8) Whee ± ± ± ± An aous wave equaon fo hoogenous eda an be aoaed n eangula oodnaes syse by he seond-ode and fouh-ode enal dffeene (Alfod Kelly Booe 974; Wang Pesonal Counaon 000) as follows (9) Whee h s he gd se n he and deons esevely and s he e se. Anohe alenae eesson fo hghe auay uses he fouh-ode enal dffeene shee of he aous wave equaon. I s oe auae han seond-ode enal dffeene shee. 3 4 5 (0) Whee: h A fne-dffeene shee wll be sable f / fo equaon (9) and 8 / 3 fo equaon (0) (Alfod e. al. 974) Bounday ondons Whee ansaen bounday ondons ae nvolved we use he ehod due o Reynolds (978). Tansaen bounday ondon Lef sde bounday 3 ()
Rgh sde bounday n n ( ( ) n n n n n n () Sufae sde bounday ( ( ) Boo sde bounday 3 ( ( ) Nonefleng bounday ondon (3) (4) Fgue.a: ( nd ode) We ae a esen nvesgang he aoah due o Cean e al. (985) whh ay be suased as follows The essue aludes ousde he bounday lnes us be ulled by G fao (Cean e.al.985). { [ 0.05( 0 ) ] } G EXP (5) Fgue.b: (4 h ode) Whee: 0 Ths gves a value of fo 0 o a he neaes boundaes wh bounday lnes and a value of abou /50 fo o a he oue boundaes. Soue funon As he soue funon f() a sngle yle snusod was used. Fgue.a: ( nd ode) Resuls We esen hee soe of he esuls of he nueal odelng osng a oason beween he fnedffeene esuls fo seond ode (7) and fouh ode aoaons (0) usng he ansaen bounday ondon. The wave fon esuls ae also oaed wh soe ognal esuls fo elaed aous wave sulaon wo ha has been develoed by Wang. Fgue.b: (4 h ode) Aouss 000 3
Fgue 3: 5 se ( nd ode) Fgue 5.b: 85 se (4 h ode) Fgue 4.a: 50 se ( nd ode) Fgue 6.a: 50 se (Wang) Fgue 6.b: 60 se (Wang) Fgue 4.b: 50 se (4 h ode) Fgue 5.a: 85se ( nd ode) Fgue.a b a and.b. show wave fon sulaon esuls n a hoogeneous sae eesened usng 0 0 gd ons wh 500 /s 05 3 /g 0.05se wh a sngle snusod sgnal of soue alude A and fequeny (f) 00H. Soe efleons ae obseved. Fgue 3 shows he soue oson used n he sulaons eesened n fgues 4 and 5. Fgues 4.a 4.b 5.a and 5.b show he esuls usng 00 gd ons n a wo-laye envonen. Veloy and densy n he ue and lowe layes ae 4000 /s 300 3 /g and 6000 /s 800 /s esevely. Hee.5 0.05 se wh soue alude A and fequeny (f) 400H. Fgues 6.a and 6.b show aous wave fon sulaon esuls usng 5 5 gd ons due o Wang. The aous veloes n ue and 4 Aouss 000
lowe layes ae 4000/s and 6000/s esevely bu wh onsan densy houghou. Fgues 4.a 5.b show evdene of dseson esenly abued o gd se effes. The effes of nolee bounday ansaeny ae sll aaen. The 4 h ode esuls show soewha less dseson han hose asng fo he nd ode ouaons. The esuls also an be oaed wh Wang s esuls n fgues 6.a. and 6.b..In all ases de efleed efaed and head waves ae obseved. Alude sgnals fo eeves fo he hoogeneous eda ase of fgues and ae shown as follows Fgue C.: oo C. Fgue C.: A oson S55;R5 Fgue D.: Zoo D. Fgue D: A oson S55;R530 Fgue E.: Zoo E. Fgues C. C. D. D. E. E. show n he e doan he nfluenes of dsesed waves and ansaen boundaes. Fgue E.: A oson S55; R50 The ange deendene of aous essue s shown n fgue F. Ths shows he aveage of he aludes of he nal osve and negave essue eusons P Aouss 000 5
as a funon of ange. The elaonsh P() ay be eessed by equaon (6) P a. b (6) Cuve fng yelds a b and assoaed oelaon oeffen as shown n he able Ode a b R nd.57-0.483 0.998 4 h.6650-0.447 0.9993 Aveage 9589-0.465-0.5 0.9993 Table.: Coeffens a and b Coeffen b s lose o he 0.5 value eeed fo ylndal seadng. Conlusons We have develoed -D aous fne-dffeene odes fo seond ode n e and seond o fouh ode n sae o odel aous wave oagaon n heeogenous eda. We have esened a oason beween wave fons develoed usng nd ode and 4 h ode aoaons o he aous wave equaons shown dsesed wave obles and aal ansaen bounday effes. The aous odelng shows he eeed de efleed and efaed and head waves aens n a wo-laye sae. Refeenes Alfod R. M. Kelly K. R. and Booe D. M. 974 Auay of fne-dffeene odelng of aous wave oagaon: Geohyss v. 39 no. 6. 834-84. Kelly K. R. Wad R. W. Sven Teel and Alfod R. M. 976 Synhe Sesogas: A fne-dffeene Aoah: Geohyss v. 4 no..-7. Reynold A. C. 978 Bounday ondons fo he nueal soluon of wave oagaon obles: Geohyss v.43. 099-0. Wllas R. S. Rehen Rhad D. and Andeson Nel L. 996 The one-densonal elas wave equaon: A fne-dffeene foulaon fo anaed oue alaons o full wavefo oagaon: Coues & Geosenes v. no. 3.53-66. Cean C. Kosloff D. Kosloff R. and Reshef M. 985. A nonefleng bounday ondon fo dsee aous and elas wave equaons: Geogyss v. 50 no. 4. 705-708. Behovsh L. M. 960 Waves n layeed eda: Aade Pess New Yo.7. 6 Aouss 000