A Note on the Approximation of the Wave Integral. in a Slightly Viscous Ocean of Finite Depth. due to Initial Surface Disturbances

Transcription

1 Applied Mahemaical Sciences, Vl. 7, 3, n. 36, HIKARI Ld, A Ne n he Apprximain f he Wave Inegral in a Slighly Viscus Ocean f Finie Deph due Iniial Surface Disurbances Arghya Bandypadhyay Deparmen f Mahemaics Khalisani Cllege, Chandannagar, India b.arghya@gmail.cm Aviji Mandal Baidyapur Ramkrishna Vidyapih Burdwan, India Cpyrigh 3 Arghya Bandypadhyay and Aviji Mandal. This is an pen access aricle disribued under he Creaive Cmmns Aribuin License, which permis unresriced use, disribuin, and reprducin in any medium, prvided he riginal wrk is prperly cied. Absrac The difficuly f analyzing he muliple-inegral expressin fr he surface displacemen ζ fr small min in a viscus cean f finie deph due an arbirary iniial surface disurbance is well-knwn. I is shwn here ha under cerain cndiins his expressin is reducible a simpler ne which is easily amenable analysis, when erms crrec rder O(ν / and pssibly rder O(ν /n/ / 3, n >, are reained where ν = ν gh, he cefficien f viscsiy ν being suppsed be small, g is he graviy and h is he deph f he cean. The mahemaical mehd adped here fr he apprximain prcess succeed faciliaes he sudy f he classical Cauchy-Pissin (C-P prblem in a mre accurae and lgical manner wih necessary resricins n he exernal disurbance mdel. Needless menin ha he inclusin f viscsiy fr a waer bdy f finie deph will in urn widen he range f applicabiliy f C-P prblem make i mre general and enhance is relaive imprance in differen regimes f fluid min. Keywrds: De la Vallée-Pussine Tes, The mehd f sainary phase

2 778 A. Bandypadhyay and A. Mandal Inrducin Fr many physical prblem bain an accurae expressin f he wave prfile generaed by arbirary disurbances we face he challenge f evaluaing he scillary muliple inegrals which frces us g fr he numerical evaluain. Bu wih he numerical evaluain here is he bvius prblem f increasing errr as he waves ravel lnger in space and ime. The siuain wrsens as he scillain f inegrals ge faser. A his sage we apply asympic apprximain f he inegrals bu even fr ha he cmplicaed expressin f he inegrals sands as he main hindrance. Here in ur sudy we have ried prvide a unique mehd which can be applied he sluin f he C-P prblem wih viscsiy and finieness f he waer bdy included fr he surface inegral expressin ζ [, ] in a w-dimensinal min which will help us analyze he expressins easily. The mehd applied he exac muliple-inegral sluin f ζ is a majr advanage by which i becmes rue fr all ime and space and helps us analyze hem fr any r r and. The echnique is spil he inegrand in w pars: firs ne cnains erms in pwers f ν / and he secnd par cnains erms f rder O(ν 3/ and i is shwn ha he cnribuin f he secnd par is negligible and can be discarded under cerain cndiins impsed n he applied iniial surface displacemen (A and n he applied iniial pressure mdel (B. The nvely f he mehd lies in he spliing up f he quie cmplicaed and large inegrand in w pars which is n nly difficul bu als has be dne keeping in mind he several cnvergence crierin invlved in he prf. In a w-dimensinal fluid min, fr he prf, he mehd f sainary phase is used and handle he cnvergence crierin f several infinie inegrals De la Vallée-Pussine es is adped. In he fllwing secin we place his apprximain as a saemen and in he subsequen secin we place he prf. Saemen In he hery f small-ampliude waves due an arbirary ime-dependen surface pressure p (x, y, H(, H( being he Heaviside uni funcin, geher an iniial elevain ζ (x, y f he surface in a viscus fluid f cnsan deph h, he wave-heigh ζ is bained as ( [] ζ = ζ where iρπ ρgh i s s e ci F( ν / / ds P (k, k, τdτ ke i s(τ e ci -i(k x k y F( ν dk dk / /. ds (

3 Apprximain f wave inegral 779 F( ν / / = (s νk = (s νk (s νk kν / { / / (s νk / / kν anh{ (s νk / ν} h νk νk sec h kh sech{ ( νk s / ν} {(s νk ν k σ } anh kh (s / νk (s νk gkch kh} s νk anh kh anh ν / h. / h ( Here (k, k and s are duble Furier ransfrm and Laplace ransfrm parameers respecively, k = k k. H, P dene he duble Furier ransfrms f p, ζ respecively and σ = [ σ( k ] = gk anh kh. In wha fllws i is shwn ha his expressin ( f ζ can be reduced he fllwing ζ = ζ iρπ ke i ( kx k y dk dk [ ρgh i s e s ci T ds Δ i ( T s τ P ( k,k, d e ds O( 3 / τ τ ν, as ν. c i Δ prvided as k, H (k, k and P (k, k, τ are bh f rder 7 n O(k = ksinψ. Here T T Δ Δ,n >, unifrmly in ψ and in ψ and τ, respecively, wih k = kcsψ, k ( s,k = s anh kh k( νs / νk anh kh, 3 ( s,k = s sσ ( νs / g ( s σ g k νk ( 9s σ. The basic wave inegral ( is hen apprximable by he firs w erms f he r.h.s f (3 wih an accuracy f rder O(ν. Evidenly hese erms are much easier analyze and als his resul hlds fr all x, y and. Our aim in his ne is prvide a prf f he abve resul. ( (3 Prf: Firs, he funcins s / F(ν / and s / f(ν / are expanded in pwers f ν / by Yung s herem, and he w sums f erms up he secnd pwer f ν /

4 78 A. Bandypadhyay and A. Mandal are dened by T and Δ respecively. The fllwing ideniy is inrduced separae he erms f rder O(ν 3/ : F f ( ν / ( ν / T Δ { Δ ( ( } {( } [ ( { / / / ( }] [ { / F ν Tf ν νk s kν f ν ( }] / kν / νk s / Δ / / νk s kν T = R ( say. (5 Δ Le ν 3/ ζ R dene he par f ζ ( / F ν ζ in ( when / herein is replaced by R. T esablish (3, i hen suffices deermine cndiins under which ζ R exiss finiely fr all x, y,. lim ν We firs shw ha he funcin f ν, ν R, is bunded in < ν < ν <, fr each fixed k. Fr his, we ne he fllwing (i If f(ν is a cmplex funcin f ν and f denes is cnjugae = Re f( ν. ν ν 3 [ Δ F( Tf( ]/[( k s k] (iii As, ν ν ν 3 ν ν ν ν / / / / / 7 / 3 3/ θ ~ 8k ν anh kh (8anh kh 9( c y cs < where c iy = s and -π/ < θ < π/. Therefre he funcin derived abve decreases as ν increases in a righ neighburhd f ν =. Hence when < ν < ν < [ Δ F( ν ] [ ] / T / / ( νk s / ν / k ν 3/ < is wn value fr ν = (iv As ν, / / / kν ν = 8k 3 anh kh s / / / { ( νk s } ~ ν k( anh kh.re[ s (s σ (s gk ] > fr large y and fixed psiive k where s = c iy, < y < Hence as argued in (iii / / kν / ( νk s / > is wn value fr ν = (v As ν, Δ / ( νk s ν { } = s σ in < ν < ν <. / / / / { kν } ~ ν k( anh kh. Re[s (s σ (s gk]

5 Apprximain f wave inegral 78 > fr large y and fixed psiive k, Therefre, / / Δ / ( νk s kν / s (s in < ν < ν <. { } σ By (i - (v, i fllws ha 3 / 8k s anh kh ν 3/ R fr < ν < ν <. s σ Frm he De la Vallée-Pussine Tes [6] will nw fllw he unifrm cnvergence f ζ R in < ν < ν < prvided he fllwing inegral is absluely cnvergen fr all x, y, ; his inegral in ha even will be he expressin fr lim ζ R : ν c i / s i( k x k y s e lim ζ R ~ k anh kh e dk dk gh ds ρ ν iρπ ci ( s σ (6 i / s( τ s e P ( k,k, d ( ds τ τ ci s σ The s-inegrals are evaluaed by he help f w resuls frm ( [3] which give 7 ( i kx k y lim ζ R ~ k anh kh e dkdk ρgh ν πρ F ; 9 /,/ ; σ Γ( 9 / σ dτ ( ( ( ( P k, k, τ τ 5 / F ; 7/; 9/, τ Γ 7 / (7 When σ > > we have ( [] 3 / 5 / F ; 9/,/ ; - σ ~ Γ(9 / ( σ (cs σ sin σ (8 3 / 3 / F ; 7/, 9/ ; - σ ~ Γ(7 / ( σ (cs σ sin σ Afer a change f variables in (7 he plar crdinaes (r, θ and (k, ψ in he (x, y and (k, k planes respecively, we see by he firs resul f ( ha he abslue cnvergence f he par f (7 invlving H depends n ha f / π dk d.k 5 / anh 3/ -ikr cs( ψ-θ ψ kh H (k csψ, ksinψ e.(csσ sin σ / πg (9 k Fr kr >>, he ψ-inegral f (9 is O((kr -/ as seen by an applicain f he mehd f sainary phase. I fllws ha (9 is absluely cnvergen if, 7 n H = O k, n > unifrmly in ψ, as k... (A

6 78 A. Bandypadhyay and A. Mandal Applied he k-inegral, he sainary phase mehd als shws ha (9 is bunded fr large values f. In he remaining par f (7, we apply he secnd mean value herem f Inegral Calculus he τ-inegral. The resul is 7 / π 5 ikr cs( ψθ ( r. h.s f (7 Par in P = ψ πργ dk d k anh kh e (7 / δ 3 / σ ( τ P (k csψ, ksinψ, τ( τ F ; 7 /, 9 / ; dτ < δ < ( Is asympic value fr large σ is bained by he use f he secnd resul f (8 / π δ 7/ 3/ ikr cs( ψθ ψ ψ ψ τ πρ dk d. k anh kh e P (k cs, k sin, 3/ g [ σ( τ sinσ( τ ] τ cs d ( Fr large values f k which make σ >> and kr >>, he mehd f inegrain by pars and he sainary phase principle ( [] shw ha he τ-inegral and he ψ-inegral cnribue he inegrand abve a facr O(k - P k as k. I fllws ha (, and hence (, is absluely cnvergen if 7 / n / P = O k, n >, unifrmly in ψ and τ, as k.... (B In he same prcess, we find ha (, and hence (, remains bunded fr large values f if P is bunded as. This las cndiin is hwever sricly n necessary as he ples f R have negaive real pars. When he cndiins (A and (B are fulfilled, (7 gives he value f lim ζ which als hen exiss finiely fr all x, y and ; relain (3 nw fllws. ν R 3 Cnclusin The apprximain prcess develped here impses a sufficien cndiin n he pressure disribuin and helps us replace he cmplicaed expressin f he wave inegral wih a simpler ne. This kind f inegral expressin arises in many cases because f is cnnecins wih many impran physical prblems such as aerial blas waves n he cean, srm surge r ship waves; s he abve resul may prve fruiful fr he analyical inegral expressin exis in his kind f cases. The classical C-P prblem as inrduced by Lamb[8] deals nly wih ne-dimensinal sanding waves in an cean wih infinie deph and is hardly resemblance sunami sudy bu he linear waves generaed by iniial surface disurbances exhibis same qualiaive behaviur ha f sunami (cf. Wehausen and Laine [9]. S ur analysis may prve quie helpful in his direcin als. In he sudy f wave min we frequenly seek he asympic naure f wave

7 Apprximain f wave inegral 783 prfile a lng disances and imes whse evaluain, which wuld becme much faser wih his apprximain f he inegrals. Acknwledgemen. Auhrs are deeply indebed Prfessr A.R. Sen f he Deparmen f Mahemaics, Jadavpur Universiy, Calcua, fr his help and suggesins during he preparain f his paper. References [] Bandypadhyay.A, Effecs f viscsiy n linear graviy waves due surface disurbances in waer f finie deph. ZAMM, 83, 3( [] Bandypadhyay A., A sudy f he waves and bundary layers due a surface pressure n a unifrm sream f a slighly viscus liquid f finie deph. JAM, (6, - [3] Bleisein, N. and Handelsman, R.A., Asympic expansins f inegrals. Hl, Rinehar & Winsn, New Yrk, 975. [] Erdélyi, A. e al., Tables f Inegral Transfrms. Vl. - I, McGraw-Hill, New Yrk, 95. [5] Luke, Y. L., The Special Funcins and Their Apprximain. Vl. II, Academic Press, 969, p. 9. [6] Nikiin, A. K. & Peyunk, E. N., The hree dimensinal Cauchy - Pissn prblem fr waves n he surface f a viscus liquid f finie deph. Svie Physics, Dklady,, N. - 5(967, -. [7] Whiaker, E.T. & Wasn, G. N., Mdern Analysis, Cambridge, 9. [8] Lamb, H., Hydrdynamics, 6 h edn, Dver, New Yrk, 95, 738 p. [9] Weahausen, JV. and Laine, EV., Surface waves, Handbuch der In Physik, Vl. 9, Par. 3, Springer, Berlin, 96, pp Received: January, 3