Wave Free Potentials in the Theory of Water Waves Having Free Surface Boundary Condition with Higher Order Derivatives

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1 Itetiol Joul of Egieeig Sciece Ivetio (IJESI) ISSN (Olie): , ISSN (Pit): Volue 7 Iue Ju 8 PP Wve Fee Potetil i the Theo of Wte Wve Hvig Fee Sufce Boud Coditio with Highe Ode Deivtive Nitd Thku d Dilip D * Deptet of Mthetic, Suedth Eveig College, Clcutt Uiveit, 4/ MG Rod, olkt- 79, Idi Deptet of Mthetic, Diod Hbou Woe' Uiveit, Diod Hbou Rod, Sih, South 4 Pg , Idi Coepodig Autho: Dilip D, Eil: dilipd99@gilco, Phoe: Abtct: The ethod of cotuctig wve-fee potetil i tetic e fo ube of itutio uch two-dieiol o-oblique d oblique wve, thee dieiol wve i fluid with fee ufce coditio with highe ode ptil deivtive e peeted hee I pticul, thee e obtied tkig ito ccout of the effect of the peece of fee ufce, ufce teio t the fee ufce d i the peece of ice-cove odelled thi eltic plte e Wod: wve-fee potetil, o-oblique d oblique wve, fee ufce, ice-cove, Lplce opeto Dte of Subiio: --8 Dte of cceptce: I Itoductio Poble ivolvig geetio o ctteig of ufce wte wve b bod of geoeticl cofigutio peet i wte e of iee ipotce i oce elted idut d e geell ivetigted theticll uig lie theo The poble of hevig otio of log, hoizotl cicul clide o the ufce of wte w ivetigted b Uell (949) uig the ethod of ultipole epio of the tiehoic te fuctio The coepodig velocit potetil lo h iil epio Ifct, fo ifiitel log hoizotl clide of bit co ectio flotig o the ufce of wte, the potetil fuctio i geel c be epeed i te of egul wve, wve ouce, dipole d wve-fee potetil (Uell (968), Atholi (984))The wve-fee potetil e igul t oe poit d ted to zeo pidl t ifiit Obvioul thee tif the fee-ufce coditio Epio i te of the wve ouce d ifiite et of wve-fee potetil wee itoduced fo the thee-dieiol poble ivolvig flotig phee hlfieed d kig peiodic hevig ocilltio b Hvelock (955) Two d thee-dieiol ultipole epio i the theo of ufce wve i ifiitel deep wte d lo i wte of uifo fiite depth h bee give b Thoe (953) Rhode-Robio (97) cotucted wve-fee potetil i the peece of ufce teio t the fee ufce Mdl d Gowi (984) tudied oblique ctteig b hlf-ieed cicul clide b uig two ethod, oe bed o itegl equtio foultio d othe bed o epio of the ctteed velocit potetil b the ethod of ultipole Thu fo viou cle of wte wve poble eeche ue the wve-fee potetil i the theticl li I ot of thee wok the epeio of wve fee potetil e ol give without thei ethod of deivtio Howeve, Lito d McIve () idicted biefl how thee c be cotucted i ce of wte with fee ufce Recetl thee i coideble iteet i the theticl ivetigtio of ice-wve itectio poble due to icee i the cietific ctivitie i pol oce Ited of fee ufce, pol oce i coveed b ice The ice cove i odelled thi uifo heet of ice of which till lle pt i ieed i wte, d i copoed of teil hvig eltic popetie Aled, quite ube of eeche hve coideed viou tpe of wte wve poble i pol oce with ice-cove odelled thi eltic plte D d Mdl (9) ivetigted wve ctteig b cicul clide hlf-ieed i wte with ice-cove The eploed the ethod of ultipole b uig the geel epio theoe fo the wve potetil ivolvig wve-fee potetil whoe epeio wee ol give Recetl D d Mdl () ivetigted cotuctio of wve-fee potetil i the lieized theo of wte wve The ethod of cotuctig thee wve-fee potetil w peeted thee i tetic e fo ube of itutio uch deep wte with fee ufce, eglectig o tkig ito ccout the effect of ufce teio, o with ice-cove odelled thi eltic plte flotig o wte 33 Pge

2 Wve Fee Potetil I The Theo Of Wte Wve Hvig Fee I thee ce the highe-ode boud coditio ivolve thid ode ptil deivtive (ufce teio) d fifth-ode ptil deivtive (ice-cove) wee peeted Howeve, the boud vlue poble ivolvig highe-ode boud coditio oe th fifth ode ptil deivtive (M et l (6), D et l (8), D (5)) hve ot bee eteivel tudied with view to etblih the wve-fee potetil i igle le fluid I thi ppe, we eted the poble of D d Mdl (), Dhillo d Mdl (4) ivetigted the poble of wve-fee potetil i wte wve theo fo fee ufce boud coditio with highe-ode deivtive d peeted i tetic e Whe the highe-ode ptil deivtive educe to t ode (fee ufce) o, thid ode (ufce teio) o fifth ode (ice-cove), wve fee potetil ectl coicide with the wve fee potetil fo two dieio (cf D d Mdl ()) d lo fo thee dieio (cf Dhillo d Mdl (4)) II Foultio Of The Poble The uul uptio of icopeible, hoogeeou d ivicid fluid, iottiol d iple hoic otio with gul fequec ude gvit ol, e de A ectgul ctei co-odite te i choe with it oigi o the e hoizotl poitio of the uppe ufce of the fluid tke (, ) ple d -i i tke to be veticll dowwd ito the fluid egio We fit coide olutio of Lplce f ', equtio which e igul t (, ) Let t, f t ' d let, ' (, f) ( f ) epectivel No Oblique wve deote the dil ditce of the poit be the gle defied b (, ) f fo the poit (, f ) iwt If Re{ (, ) e } deote the velocit potetil decibig the otio i the fluid, the (, ) tifie deote the two-dieiol Lplce opeto The botto coditio i give b z, i the fluid egio, () (, ) () Alo (, ) behve outgoig wve The potetil fuctio (, ) tifie (), () d lo lieized boud coditio fo highe-ode deivtive h bee itoduced b M et l(6) d h the fo o, (3) i lie diffeetil opeto of the fo I (4) fluid tuctue itectio, ol the eve ode ptil deivtive i Let d boud coditio (), (3) d d C (4) C (,,, ) e kow cott eepig i id viou phicl poble ivolvig e coideed i the diffeetil opeto deote the etic d ti-etic ultipole tifig () ecept t (, ) f with 34 Pge

3 Alo the epeet outgoig wve Ak ( ) Bk ( ) Wve Fee Potetil I The Theo Of Wte Wve Hvig Fee co i co i, The uitble ultipole e, k A(k) e co k dk, k B( k)e i k dk, d e fuctio of k to be foud uch tht the itegl eit i oe ee d boud coditio (3) i tified The ukow cott e obtied (cf D d Mdl ()) ( ) ( ( ) C ) k kf A( k) k e (9) ( )! Hk ( ) ( ( ) C ) k ( ) B( k) k e ( )! Hk ( ) H( k) ( ) Ck Thu we hve ( ( ) ) co ( ) C k k ( f ) kf (5) (6) (7) (8) () k e co k dk, ( )! H( k) ( ( ) ) i ( ) C k k ( f ) () k e i k dk, ( )! H( k) -i, el poitive oot of the dipeio equtio ( ) ( ) the cotou of the itegl i ideted below the pole k o the el k () beig the ol H k C k (3) The f-field fo of the ultipole e give b H '( ) i e ( f ) e i, (4) 35 Pge

4 Wve Fee Potetil I The Theo Of Wte Wve Hvig Fee H '( ) ib e ( f ) e i, (5) ( ) ( )!, (6) ( ) ( )! b (7) Uig (6) d (7), we fid Thu,, b b (8) ( ) ( ) co co( ) ( ) ( ) ( ) ( )! k g ( k) e co kdk, (9) k ( f ) i i( ) ( ) ( ) ( ) ( )! g ( k) k ( f ) k g( k) e i kdk, () ( ( ) C ) k g( k) g k c k k k () ( ) ( ) ( ) 3 ( ) c ( k k k ) c ( k ) c Mkig f, the fo (9) d () we get etic d ti etic wve-fee potetil e give b ( ) ( )! co co( ) ( ) ( ) k k g k e kdk ( ) co,,,3,, () 36 Pge

5 Wve Fee Potetil I The Theo Of Wte Wve Hvig Fee d i i( ) ( ) (o) ( ) ( )! k k g k e kdk ( ) i,,,3, (3) c, c, i,,,, the boud vlue poble (BVP) becoe the BVP I pticul, chooe i fo wte with fee ufce (cf D d Mdl ()) d the wve-fee potetil becoe the wve-fee c, c, c D, potetil fo igle le fluid with fee ufce Siill, if chooe c, i 3,4,5,,, the the BVP becoe the BVP fo fluid with ice cove boud coditio i d obti wve-fee potetil (cf D d Mdl ()) Oblique wve Ude the uul uptio of lie wte wve theo velocit potetil c be defied fo oblique wve i the fo (, ) i t i z (,, z, t) Re{ (, ) e } i cople vlued potetil fuctio, i the wve ube copoet log the z-diectio tifie Helholtz equtio ( ), i the fluid egio (4) O the uppe ufce hvig the e poitio, tifie the fee-ufce coditio with highe-ode deivtive of the fo (cf M et l (6)), o (5) i lie diffeetil opeto of the fo c (6) c (,,, ) e kow cott eepig i id viou phicl poble ivolvig I (6) fluid tuctue itectio, ol the eve ode ptil deivtive i e coideed i the diffeetil opeto Let d boud coditio (5), () d deote the etic d ti-etic ultipole tifig (4) ecept t (, ) f with ( )co, (7) 37 Pge

6 Wve Fee Potetil I The Theo Of Wte Wve Hvig Fee ( )i (8) (z) deote the odified Beel fuctio of ecod kid The ultipole e cotucted (cf Thoe (953)) A k k e dk (9) coh k ( )co ( )co( ih ), ( )i B ( k)i( ih k) e dk (3) coh k A( k) d B ( k) boud coditio (5) i tified e fuctio of k The ufce coditio (5) i tified if A( k) to be obtied uch tht the itegl eit i oe ee d the d B ( k) e choe ( ( ) c ) f ( ) ( ) coh, A k k e H( ) ( ( ) c ) f ( ) ( ) ih, B k k e H ( ) Thu we c cotuct the ultipole e give b coh k (3) (3) ( )co ( ( ) C ) ( f ) ( ) coh k co( ih k) e dk H( ) (33) ( )i ( ( ) C ) ( f ) ( ) ih k i( ih k) e dk H ( ) the cotou i ideted below the pole k o the el k, coh (34) -i to tke ce of the outgoig tue beig the ol el poitive oot of the dipeio equtio H( ) ( ( ) c ) (35) 38 Pge

7 Wve Fee Potetil I The Theo Of Wte Wve Hvig Fee The f-field fo of the ultipole e give b H '( ) co () ( f ) i co i e e, (36) ib e e H '( ) co () ( f ) i co (37), ( ) coh, (38) () b ( ) ih (39) () Uig (38) d (39), we fid () () (), (4) b b b () () () (4) Thu ( )co( ) ( )co( ) ( ) ( )co ( f ) g( ) coh( ) k co( ih k) e dk (4) d ( )i( ) ( )i ( ) ( )i( ) Thee e wve-fee potetil with igulit t (, f ) ( f ) g( )ih k i( ih k) e dk (43) 39 Pge

8 f Wve Fee Potetil I The Theo Of Wte Wve Hvig Fee Mkig i (4) d (43) we obti the etic d ti-etic wve-fee potetil with igulit i the fee ufce d e give b d ( )co( ) ( )co( ) ( )co ( ) g ( )coh( ) k co( ih k) e dk,,,3, ( )i( ) ( )i ( )i( ) ( ) g ( )ih k i( ih k) e dk,,,3, (44) (45) Thee hve bee ued b D d Mdl (9) i the tud of wve ctteig b log cicul clide hlfieed i wte with ice-cove Hee lo i pticul, the eult fo wte with fee ufce well icecove ufce e iil to the ce of o oblique wve potetil (cf D d Mdl ()) 3 Thee-Dieiol Wve-Fee Potetil With the oigi t the e fee ufce, the with depth, we defie the gle ', d z d b the eltio -e hoizotl d the ' t, t, t -i veticl, R R z f f R z Let poit (, f,) d (,,) ' d deote the dil ditce of the poit (,, ) f epectivel iwt If Re{ (,, z) e } deote the velocit potetil igul t (, f,) i the fluid, the (,, z) tifie z The botto coditio fo wte of ifiite depth i give b iceig z fo the decibig the otio i the fluid egio ecept t (, f,) (46) (,, z) (47) Alo (,, z) behve outgoig wve R The potetil fuctio (,, z) tifie (46), (47) d the lieized coditio give b C, z o (48) 4 Pge

9 Wve Fee Potetil I The Theo Of Wte Wve Hvig Fee C (,,,, ) e kow cott, I (48) R, z R R ( R ) R R R (49) I thi ce (cf Dhillo d Mdl (4)), p (co ) k A (k)e J ( kr) dk, (5) d A (k) i fuctio of k to be obtied uch tht the itegl eit d the boud coditio (48) i tified The coditio (48) i tified if A (k) i choe ( ) C k kf ( ) A ( k) k e, ( )! H( k) (5) Thu we get (co ) ( ) ( ) p C k k( f ) ( )! H( k) k e J ( kr) dk, (5) the cotou of the itegl i defied below the pole k o the el k i, beig the ol el poitive oot of the dipeio equtio H ( k) ( ) Ck (53) The f-field fo of the ultipole i give b 4 i g( ) e e R, H'( ) R i( R ) () ( f ) 4 Fo (54) we get, ( ) ( )! i () e (54) 4 Pge

10 Wve Fee Potetil I The Theo Of Wte Wve Hvig Fee ( ) () () (55) Theefoe, the cobitio doe ot cotibute thig R ( ), o tht the e wve-fee Thu, p (co ) p (co ) ( ) ( ) ( ) ( )! Thi i the wve-fee potetil hvig igulit t (,,) g k k J kr dk k ( f ) ( ) e ( ), f (56) Mkig i (56) we fid the wvefee potetil hvig igulit i the fee ufce d i give b p (co ) p (co ) ( ) ( ) ( )! f k g ( k) k e J ( kr) dk (57) c, c, i,,,, the boud vlue poble (BVP) becoe the BVP I pticul, chooe i fo wte with fee ufce (cf Dhillo d Mdl (4)) d the wve-fee potetil becoe the wve-fee c, c, c D, potetil fo igle le fluid with fee ufce Siill, if chooe c, i 3,4,5,,, the the BVP becoe the BVP fo fluid with ice-cove boud coditio i d obti wve-fee potetil (cf Dhillo d Mdl (4)) III Cocluio Wve fee potetil i igle-le fluid with fee ufce coditio with highe ode deivtive fo o-oblique d oblique wve (two dieio) d lo thee-dieio e cotucted i etic e Appopite odifictio of the wve-fee potetil c be de i the cicutce whe the fluid e of uifol fiite depth hvig fee ufce coditio with highe ode deivtive I pticul, thee e obtied tkig ito ccout of the effect of the peece of fee ufce, ufce teio t the fee ufce d lo i the peece of ice-cove odelled thi eltic plte Ackowledgeet The utho thk the Edito d the Reviewe fo thei coet d uggetio to evie the ppe Refeece [] Atholi G A, A epio theoe fo wte-wve potetil J Egg Mth 8(984) 8-94 [] D Dilip, Cotuctio of wve-fee potetil d ultipole i two le fluid hvig fee-ufce boud coditio with highe-ode deivtive J Mie Sci d Appl 4(5) 7-8 [3] D Dilip d Mdl B N, Cotuctio Wve-fee potetil i the lieized theo of wte wve J Mie Sci d Appl 9 () [4] D Dilip d Mdl B N, Wve ctteig b cicul clide hlf ieed i wte with ice-cove Itetiol J Egg Sci 47 (9) [5] D Dilip, Mdl B N d Chkbti A, Eeg idetitie i wte wve theo fo fee-ufce boud coditio with higheode deivtive Fluid Dic Reech, 4 (8) 53-7 [6] Dhillo, H d Mdl, B N, Thee dieiol wve-fee potetil i the theo of wte wve ANZIAM J 55(4)75-95 [7] Hvelock T H, Wve due to flotig phee kig peiodic hevig ocilltiopoc R Soc Lod A 3(955) -7 4 Pge

11 Wve Fee Potetil I The Theo Of Wte Wve Hvig Fee [8] Lito, C Md McIve, P Hdbook of Mtheticl Techique fowve/stuctue Itoductio Chp d Hll/CRC Boc Rto, p43-68 [9] M S R, Bhttchjee J, Shoo T Epio foul i wve tuctue itectio poble Poc R Soc A46 (6)63-87 [] Mdl, B N d Gowi, S Sctteig of ufce wve obliquel icidet o fied hlf-ieed cicul clide Mth Poc cb Phil Soc 96(984) [] Rhode-Robio, Fudetl igulitie i the theo of wte wve with ufce teio Bull Autl Mth Soc (97) [] Thoe R C, Multipole epio i the theo of ufce wve Poc CbPhill Soc 49(953)77-76 [3] Uell F, O the hevig otio of cicul clide o the ufce of fluidq J Mech Appl Mth (949)8-3 [4] Uell F, The epio of wte wve potetil t get ditce Poc CbPhill Soc64(968),8-86 Itetiol Joul of Egieeig Sciece Ivetio (IJESI) i UGC ppoved Joul with Sl No 38, Joul o 433 Nitd Thku d Dilip D Wve Fee Potetil I The Theo Of Wte Wve Hvig Fee Sufce Boud Coditio With Highe Ode Deivtive Itetiol Joul of Egieeig Sciece Ivetio (IJESI), vol 7, o, 8, pp Pge