Finite Difference Analysis of 2-Dimensional Acoustic Wave with a Signal Function

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1 Fiite Differece Aalysis of -Dimesioal Acostic Wave with a Sigal Fctio Opiyo Richard Otieo 1, Alfred Mayoge 1, Owio Marice & Ochieg Daiel 1 richardopiyo08@gmailcom 1,wmayoge@gmailcom 1 & maricearaka@yahoocom 1 Dept of Pre & Applied Mathematics Maseo UiversityKeya) Dept of Mathematics & Compter Scieces Uiversity of Kabiaga Keya) December 9, 015 Abstract This paper describes progress o a two dimesioal merical simlatio of acostic wave propagatio that has bee developed to visalize the propagatio of acostic wave frots ad to provide time-domai sigal I this exercise, we have simlated propagatio of sod i sch a medim sig both explicit ad Crak Nicolso fiite differece schemes, we have also tested for stability of the developed schemes sig Vo Newma ad Matrix stability aalysis together with its associated code i matlab The stability aalyses of the developed schemes revealed that Explicit scheme was coditioally stable while the Hybrid oe Crak Nicolso Scheme) was coditioally stable, for all vales of corat mber r The rate of covergece of the algorithms deped o the trcatio error itrodced whe approximatig the partial derivatives, the Crak-Nicolso method coverged at the rate of k + h ), which is a faster rate of covergece tha either the explicit method, or the implicit method Keywords: Acostic wave, Fiite differece approximatio, Sigal fctio, Crak Nicolso, Vo Newma, Matrix stability aalysis 1 Itrodctio Whe determiig the acostic properties of a eviromet, we are actally iterested i the propagatio of sod, give the properties ad locatio of a sod sorce Propagatio of light or sod wave is of log stadig iterest i several braches of basic ad applied physics, from old disciplies sch as x-ray diffractio i crystallography, to the moder sciece of photoic crystals May problems i atral eviromet so ivolve wave propagatio i periodic media For example, early periodic sad bars are freqetly fod i shallow seas otside the srf zoe; their presece chages the wave climate ear the coast The techology of remote-sesig, either by derwater sod or by radio waves from a satellite, depeds o or derstadig of scatterig by the wavy sea srface Fiite differece method is a key tool i merical aalysis ad the motivatio to stdy ad lear this method is the fact that i Flid dyamics, thermodyamics, solid mechaics etc a large mber of differetial eqatios are fod Ad to solve all of them aalytically is very difficlt ad at times impossible As a reslt Fiite Differece Methods provide sfficietly satisfactory accrate merical soltios to sch eqatios Fiite-differece modellig of wave propagatio i heterogeeos media is a sefl techiqe i a mber of disciplies, icldig seismology ad ocea acostics Sod is a logitdial wave that is, waves of alteratig pressre deviatios from eqilibrim casig local regios of compressio ad rarefactio as a reslt of vibratig objects Sod is a wave which ca be described as a distrbace that travels throgh a medim, trasportig eergy from oe locatio to aother locatio May researchers have developed merical iterpretatios of the wave eqatio sited to acostics ad seismic propagatio Hgh ad Pat [13], developed secod order fiite differece scheme for modellig the acostic wave eqatio i Matlab bt their major limitatio was, isfficiet cosideratio of bodary coditios Alford, Kelly ad Boore [], proposed that acostic wave eqatio for homogeeos media ca be approximated i rectaglar co-ordiate system by the secod ad forth order cetral differece Althogh, oe-way wave eqatio method i ihomogeeos media has bee extesively stdied i the literatre, few detailed stdies have bee made o the implemetatio of sorce term ad free bodary coditios For this reaso, Xie ad W [9] itegrated free srface bodary coditio ad the sorce term for oe way elastic waves for decompositio of plae wave Charara ad Taratola [7], i their pblicatio co- 1

2 Iteratioal Joral of Mltidiscipliary Scieces ad Egieerig, Vol 6, No 10, October 015 sidered bodary coditios ad sorce term for oeway acostic depth extrapolatio ad they sed a mber of fiite differece schemes ad techiqes amely, implicit fiite differece scheme, cetral fiite differece schemes ad splittig methods Seogjai [4], came p with forth order implicit time steppig scheme for merical soltio of the acostic wave eqatio as a variat of the covetioal modified eqatio method, the scheme icorporated a locally oe-dimesioal LOD) procedre with splittig error of O t 4 ) Walstij ad Kowalczyk [19], focsed o compact stecil fiite differece time domai FDTD) scheme for approximatig D wave eqatio i the cotext of digital adio This preset work is a fiite differece aalysis of two dimesioal acostic wave eqatio with a sigal fctio Frther, Vo Nema ad matrix stability aalyses criterio is doe 11 Fiite Differece Method The mathematical modellig of practical problems ofte ivolves the se of Partial Differetial Eqatios Very few of these eqatios ca be solved aalytically For the acostic wave eqatio described by a Partial Differetial Eqatio, aalytical soltios do exist bt oly for special or simple cases like the homogeeos case However, for complex or sfficietly realistic models, it is ecessary to resort to merical methods The fiite differece method is oe of several techiqes for obtaiig merical soltios to practical problems govered by Partial Differetial Eqatios PDE) I all merical soltios the cotios partial differetial eqatio PDE) is replaced with a discrete approximatio I this cotext, the word discrete meas that the merical soltio is kow oly at a fiite mber of poits i the physical domai The mber of those poits ca be selected by the ser of the merical method I geeral, icreasig the mber of poits ot oly icreases the resoltio, bt also the accracy of the merical soltio The discrete approximatio reslts i a set of algebraic eqatios that are evalated or solved) for the vales of the discrete kows Figre 1 is a schematic represetatio of the merical soltio The mesh is the set of locatios where the discrete soltio is compted These poits are called odes, ad if oe were to draw lies betwee adjacet odes i the domai the resltig image wold resemble a et or mesh Two key parameters of the mesh are x& z, the local distace betwee adjacet poits i space, ad t, the local distace betwee adjacet time steps For the case cosidered i this article x ad z are iform throghot the mesh The core idea of the fiite differece method is to replace cotios derivatives with differece formlas that ivolve oly the discrete vales associated with positios o the mesh Applyig the fiite differece method to a differetial eqatio ivolves replacig all derivatives with differece formlas I the wave eqatio there are derivatives with respect to time, ad derivatives with respect to space Usig differet combiatios of mesh poits i the differece formlas reslts i differet schemes I the limit as the mesh spacig x, z) ad t) go to zero, the merical soltio obtaied with ay sefl scheme will approach the tre soltio to the origial differetial eqatio However, the rate at which the merical soltio approaches the tre soltio varies with the scheme I additio, there are some practically sefl schemes that ca fail to yield a soltio for bad combiatios of x, z ad t 1 Discretizatio Procedre I developig the schemes, comptatioal domai Ω is discretized with iform grid with assmptio that with iform grid, both the space ad time are adeqate for the soltio, it implies that x = z = h) Dividig the domai ito a grid of Nx by Nz poits, where x ad z are the distace betwee poits i the grid i the x ad z axes respectively, to yield x = x x ad z = z z, where x = 1,,, Nx ad z = 1,,, Nz Also, if t is the icremet i time, the t = k t where k is the time step with k = 1,,, Deotig the discrete approximatio of x, z, t) at the grid poit differet poits i space ad time) as x i = i x, z j = j z, t = t), the the acostic wave field merical soltio) ca be specified as x, z, t) = ih, jh, k), for all i = 1,, 3,, x, j = 1,, 3,, z ad = 0, 1,, 13 Fiite Differece Approximatios Fiite differece formlas are first developed with the depedet variable φ as a fctio of oly oe idepedet variable, x, ie φ = φx) The resltig formlas are the sed to approximate derivatives with respect to either space or time By iitially workig with φ = φx), the otatio is simplified withot ay loss of geerality i the reslt 131 First Order Forward Differece Cosider a Taylor series expasio φx) abot the poit x i φx+ x) = φx i )+ x + x φ x x i + x3 3 φ + 1) where ) x is a chage i x relative to x i Solvig for yields x x i = φx + x) φx i) x φ x x x i x 3 φ + ) Notice that the powers of x mltiplyig the partial derivatives o the right had side have bee redced [ISSN: ] wwwijmseorg

3 Iteratioal Joral of Mltidiscipliary Scieces ad Egieerig, Vol 6, No 10, October 015 by oe Let the approximate soltio for the exact soltio, ie φ i φx i ) ad φ i+1 φx i + x), the eqatio ) becomes; φ i+1) φ i x x φ x x i x 3 φ + 3) From the mea vale theorem we ca have for higher order derivatives x φ x x i + x)3 3 φ + = x φ x ɛ 4) where x i ɛ x i+1, therefore or eqivaletly; φ i+1) φ i + x φ x x ɛ φ i+1) φ i x φ x x ɛ 5) x φ x ɛ = O x ) The eqals sig i this expressio is tre i the order of magitde sese I other words its ot a strict eqality, bt rather, meas that the left had side is a prodct of a kow costat ad x Althogh the expressio does ot give s the exact magitde of x φ x )x i )ɛ, it tells s how qickly that term approaches zero as x is redced Usig big O otatio, Eqatio 3) ca be writte = φ i+1) φ i + O x) 6) x Eqatio 6) is called the forward differece formla for x x i sice it ivolves odes x i ad x i+1, hece, forward differece approximatio has a trcatio error that is O x) The size of the trcatio error is mostly) der or cotrol becase we ca choose the mesh size x The part of the trcatio error that is ot der or cotrol is x ɛ 13 First Order Backward Differece A alterative first order fiite differece formla is obtaied if the Taylor series like that i Eqatio 1) is writte with a backward shift x) Usig the discrete mesh variables i place of all the kows, oe obtais φ i 1 = φ i x + x φ x x i x)3 3 φ + Notice i this case the alteratig sigs of terms o the right had side Solvig for, we arrive at = φ i φ i 1 x O sig big O otatio we get + x φ x x i x) 3 φ + = φ i φ i 1 + O x) 7) x This is called the backward differece formla becase it ivolves the vales of φ at x i ad x i 1 The order of magitde of the trcatio error for the backward differece approximatio is the same as that of the forward differece approximatio The term o the right had side of Eqatio 5) is called the trcatio error of the fiite differece approximatio It is the error that reslts from trcatig 133 First Order Cetral Differece the series i Eqatio 3) I geeral, otice that ɛ is ot kow Frthermore, Cosider the Taylor series expasios for φ i+1 ad φ i 1 as below; sice the fctio φx, t) is also kow, φ x caot be compted We apply the big O otatio to express φ i+1 = φ i + x + x φ x the depedece of the trcatio error o the mesh x i + x3 3 φ + 8) spacig Note that the right had side of Eqatio 5) cotai the mesh parameter x, which is chose by the perso sig the fiite differece simlatio φ i 1 = φ i x + x φ x x i x)3 3 φ + 9) Sice this is the oly parameter der the ser s cotrol that determies the error, the trcatio error is Sbtractig Eqatio 9) from Eqatio 8) yields simply writte φ i+1 φ i 1 = x + x)3 3 φ Solvig for x ) x i gives = φ i+1 φ i 1 x 3 φ x + which reslts to; = φ i+1 φ i 1 + O x ) 10) x which is the cetral differece approximatio to x ) x i To get good approximatios to the cotios problem geerally, small x is chose Whe x << 1, the trcatio error for the cetral differece approximatio goes to zero mch faster tha the trcatio error i Eqatio 6) or Eqatio 7) 134 Secod Order Cetral Differece Fiite differece approximatios to higher order derivatives ca be obtaied with the additioal maiplatios of the Taylor Series expasio abot φx i ) Addig Eqatio 9) ad Eqatio 8) yields φ i+1 +φ i 1 = φ i + x) φ x x i + x)4 4 φ 4! x 4 x i + [ISSN: ] wwwijmseorg 3

4 Iteratioal Joral of Mltidiscipliary Scieces ad Egieerig, Vol 6, No 10, October 015 Solvig for φ x )x i gives; φ x x i = φ i+1 φ i + φ i 1 x x) 4 φ 1 x 4 x i + Usig order otatio φ x x i = φ i+1 φ i + φ i 1 x + O x ) 11) This is also called the cetral differece approximatio, to the secod derivative, whereas Eqatio 11) is the cetral differece approximatio to the first derivative 14 Discretizig the acostic eqatio Geerally i mathematical approach, the cotios formlatio is trasformed to a discrete formlatio by replacig derivatives by say fiite differece approximatios while discretizig The idea is to discretize the problem by choosig a step size h i both x ad z ad a step size k i t as i the soltio procedre above The we try to approximate the acostic potetial pressre) o a grid of poits Therefore, we replace the cotios problem domai by a grid, or mesh, of discrete locatios o Ω Figre 11: x,z,t) space Comptatioal molecle stecil) i The figre below clearly shows schematic represetatio of D x,z,t) operator for discrete domai : Figre 1: Represetatio i Grid stecil) i x,z,t) space [ISSN: ] wwwijmseorg 4

5 Iteratioal Joral of Mltidiscipliary Scieces ad Egieerig, Vol 6, No 10, October 015 Nmerical schemes I this sectio, we develop the two merical schemes that we shall se i this stdy, that is Cetral Differece Scheme explicit) ad Crak-Nicolso schemes Hybrid) for the model eqatio x + z 1 c = Sx, z, t) x, z) t which is a hyperbolic PDE, therefore we first discretize this eqatio by sig the cetral differece approximatio to the secod derivative i xx, zz ad tt 1 Cetral Differece SchemeCDS) Explicit) Costrctio of the simple explicit scheme for the homogeeos -dimesioal acostic wave eqatio i rectaglar coordiate is a fairly straight forward matter Namely; tt = c xx + zz ), 1) where S = 0 which meas that there is o spply of eergy from the sorce To develop explicit scheme for this eqatio, we discretize the terms i the homogeeos eqatio govered by 1) i the stadard way by defiig the cetral differece operators as follows: D x = i+1,j + i 1,j x) D z = z) Sbstittig these operators i eqatio 1), we arrive at: = 1 Systematic sbstittio yields; + c k D x + c k D z + 1 t) = c x) i+1,j + ) i 1,j therefore, σ ) 4) = 4σ) + σ i+1,j + σ i 1,j+ σ +1 + σ 1 1 5) Usig the same reasoig we ca exted this cocept to o-homogeeos case below as xx + zz 1 c x, z) tt = Sx, z, t) c x) i+1,j + ) i 1,j + c z) +1 + ) ) 1 t) = c S 6) ) c t Agai by lettig σ = x ad sbscripts, i, j ad sperscript to represet the x, z ad time coordiates respectively for a discrete grid of iform spacig that is x = z the, collectig the kow terms that is o the left had side gives; = 4σ) + σ i+1,j + i 1,j ) 1 1 c t S, 7) which is the explicit scheme for the two dimesioal acostic wave with sorce term for all i = 1,, 3,, M 1; j = 1,, 3,, N 1 Crak - Nicolso scheme I Crak-Nicolso scheme, we replace the spatial coordiates xx ad zz by the average of each cetral differece approximatios at th time level ad at + 1) th time level These yields i 13) as + c z) +1 + ) 1 ) We the express i terms of other terms to give; = 1 + c t) x) i+1,j + ) i 1,j + c t) z) +1 + ) 1 3) sbscripts, i, j ad sperscript represet the x, z ad time co-ordiates respectively for a discrete grid of iform spacig that is x = z ad for coveiece, we itrodce the sbstittio σ = ) c t, x this yields; c [ +1 x) i+1,j +1 + i 1,j)] + c [ x) i+1,j + )] i 1,j + c [ +1 z) )] + c [ z) +1 + )] ) 1 S k = c + S +1 ) 8) = 1 + σ i+1,j + i 1,j) + [ISSN: ] wwwijmseorg 5

6 Iteratioal Joral of Mltidiscipliary Scieces ad Egieerig, Vol 6, No 10, October 015 I order to redce the comptig time, we adopt iform grid spacig that is x = z = h ad t = k, ow lettig r = c t x = c k h to give c k [ +1 h i+1,j +1 + ) i 1,j )] + c k [ h i+1,j + ) i 1,j )] 1 S + 1 = c k + S +1 ) 9) o collectig kow terms o the left had side gives the Implicit Crak-Nicolso scheme r i+1,j +1+4r)+1 r i 1,j r+1 +1 r+1 1 = r i+1,j+ 4r) +r i 1,j+r +1+r 1 1 ad S c k + S +1 ), 10) for all i = 1,, 3,, M 1 ad j = 1,, 3,, N 1 Takig S to be a space fctio of x ad z bt ot a fctio of timet), the S = S+1, or Implicit Crak-Nicolso eqatio redces to r i+1,j +1+4r)+1 r i 1,j r+1 +1 r+1 1 = r i+1,j+ 4r) +r i 1,j+r +1+r 1 1 c k S 11) for all i = 1,, 3,, M 1; ad j = 1,, 3,, N 1 3 Reslts Accracy ad Stability aalysis 31 Matrix stability of Explicit scheme Matrix stability method cosiders the fiite differece represetatio of both the PDE ad bodary coditio i a matrix form for which eigevale aalysis is sed to stdy stability, the theory behid this method is that the modls of the eigevales of the amplificatio matrix shold be less tha ity Employig matrix method to aalyze stability of the scheme 7)ad expadig this scheme by takig i = 1,, 3,, M 1; j = 1,, 3,, N 1, ad r = σ = c t x ), geerates the system of eqatios see appedix) which ca be expressed i matrix form as where U +1 = = AU U 1 + b, 1,1,1 1, M 1,N r) r r r 4r) r r A = r r 4r) 1,1,1 b = = 1, M 1,N 1 r 0,1 + r 1,0 + r 0,1 c 1,1k S 1,1 r,0 c,1k S,1 1 = 1,1,1 1, M 1,N 1 1 We realise some patter ad the resltig matrix [M 1) N 1)] [M 1) N 1)] is of blocktridiagoal form as C G = D B C D B C, where B, C ad D are M 1) M 1) matrices, ad there are N sch C matrices o the diagoal For this case, B ad D are diagoal matrices whereas C is tridiagoal, C = 4r) r r 4r) r r B = D = r r r 4r), Sice C is tridiagoal matrix which is symmetric positive defiite ad is diagoally domiat, the C is o-siglar ths there is a iqe soltio The symmetry the implies that we have both a ecessary coditio for stability, therefore this scheme will always be stable for restricted vales of r 3 Vo Nema stability of Explicit scheme CDS) The vo Nema stability aalysis is a way to determie whe a particlar merical method is stable [ISSN: ] wwwijmseorg 6

7 Iteratioal Joral of Mltidiscipliary Scieces ad Egieerig, Vol 6, No 10, October 015 It looks at soltios of the form a j = ξ e ijkh, where i = 1, j is or spatial idex, k is the time idex, ad h is the spatial step To do the aalysis sig this method, we simply sbstitte the above soltio ito the discretized form of the merical method ad determie where ξ 1 This tells s whether the amplitde of the wave is less tha or eqal to oe If the amplitde is greater tha oe, the the amplitde is icreasig ad will therefore evetally become stable Ths the method is stable at the vales where ξ 1 I geeral, the Vo Nemas procedre itrodces a error represeted by a fiite Forier series ad examies how this error propagates drig the soltio Stability beig idepedet of sorce term, ow gettig the stability of explicit scheme sig Vo Nema s method, we set S = 0 i the explicit scheme 7) to give the homogeeos eqatio; = 4σ) + σ i+1,j + i 1,j ) ) The sig the fact that the soltio of this costat coefficiet differetial eqatio is satisfied by the Forier harmoics U = ξ e iβmh e iγlh where β is time idex i x γ is time idex i z h is spatial step i x ad z m is spatial idex i x ad l is spatial idex i z Sbstittig i the homogeeos scheme 31), we get ξ +1 e iβmh e iγlh = 4σ)ξ e iβmh e iγlh + [ σ ξ e iβm+1)h e iγlh + ξ e iβm 1)h e iγlh] + [ σ ξ e iβmh e iγl+1)h + ξ e iβmh e iγl 1)h] ξ 1 e iβmh e iγlh 3) so that o dividig eqatio 3) by ξ e iβmh e iγlh, we have; ξ = 4σ) + σ [ e iβh + e iβh + e iγh + e iγh] ξ 1 Bt cos θ = e iθ + e iθ which redces this eqatio to [ ξ = 4σ)+σ 1 si βh ) + 1 γh ] si ) the; ξ ξ 1 33) [ 1 σsi βh + γh ] si ) ξ + 1 = 0 [ we the let g = 1 σsi βh + si γh ], ) to get; ξ gξ + 1 = 0, where the i th eigevale is give by ξ i = g ± g 1 Therefore, for stability, ξ i 1; i = 1,,, N, this implies 1 1 σsi βh + γh si ) 1 which has o-trivial soltio whe 1 σsi βh + si γh ) 1, for this we get; σsi βh + si γh ) 1, sice the maximm vale of si βh is ity, or eqatio redces to σ 1 as stability coditio Therefore, covergece of the scheme follows the Corat et al 198) CFL) coditio for covergece, which applies to explicit differece replacemet of hyperbolic eqatios It reqires that 31) to be coverget whe 0 σ 1 Ths, the stability coditio coicides with the CFL coditio Crak-Nicolso scheme Hybrid) 33 Matrix stability of Crak-Nicolso scheme Similarly, we adopt the matrix method to aalyze stability of the Crak-Nicolso scheme 11) We expad this scheme by takig i = 1,, 3,, M 1; j = 1,, 3,, N 1, to get the system of eqatios which we ca express i matrix form as 1 + 4r) r r r 1 + 4r) r r r 1 + 4r) r 1 + 4r) 1,1,1 1, M 1,N r) r r r 4r) r r r r 4r) r r r 4r) = [ISSN: ] wwwijmseorg 7

8 Iteratioal Joral of Mltidiscipliary Scieces ad Egieerig, Vol 6, No 10, October 015 1,1,1 1, M 1,N 1 r 0,1 + r+1 1,0 + r 0,1 + r 1,0 1 1,1 k S1,1 r,0 + r,0 1,1 k S,1 + r 0, + r 0, 1 1, k S1, Which we ca express i matrix form as U +1 AU +1 = BU + C = A 1 B)U + A 1 C 33a) A ad B are block tridiagoal matrices Ths, eqatio 33a) may be pt i the form where I ra N 1 )U +1 = I + ra N 1 )U + D, A N 1 = I is a N 1) N 1) idetity matrix Ths, U +1 = [ I + ra N 1 )I ra N 1 ) 1] U + E, where D = A 1 C ad E = CI ra N 1 ) 1 I simpler form we write this eqatio as U +1 = P U + E I this case, P = I + ra N 1 )I ra N 1 ) 1 is the amplificatio matrix, ad the stability coditio is that absolte vale of the eigevales of the amplificatio matrix shold be less tha or eqal to 1, that is λ i 1 Sice or Eqatio 11) is implicit ad A ad B are block tridiagoal matrices which are symmetric positive defiite ad are weakly diagoally domiat, the A ad B are o-siglar ths there is a iqe soltio, the symmetry the implies that we have both ecessary ad sfficiet coditio for stability, therefore this scheme will always be stable for all vales of r sice r has o restrictios coditioally stable) 34 Vo Nema stability of Crak- Nicolso scheme To get stability of Crak Nicolso via this method, we set S = 0 sice stability is idepedet of sorce term, the sbstitte U = ξ e iβmh e iγlh i the homogeeos eqatio 11) r i+1,j +1+4r)+1 r i 1,j r+1 +1 r+1 1 = r i+1,j+ 4r) +r i 1,j+r +1+r 1 1, 34) which yields rξ +1 e iβm+1)h e iγlh r)ξ +1 e iβmh e iγlh rξ +1 e iβm 1)h e iγlh rξ +1 e iβmh e iγl+1)h rξ +1 e iβmh e iγl 1)h = rξ e iβm+1)h e iγlh + 4r)ξ e iβmh e iγlh + rξ e iβm 1)h e iγlh + rξ e iβmh e iγl+1)h +rξ e iβmh e iγl 1)h ξ 1 e iβmh e iγlh 35) Agai dividig 35) by ξ e iβmh e iγlh, we obtai 1 + 4r)ξ rξe iβh + e iβh ) rξe iγh + e iγh ) = 4r)+re iβh +e iβh )+re iγh +e iγh ) ξ 1 36) Recall that cos θ = 1 si θ = eiθ +e iθ, therefore sig this fact i 36), yields 1+4r)ξ rξ 1 si βh ) + 1 γh ) si ) = 4r)+r 1 si βh ) + 1 γh ) si ) 1 ξ After rearragemet, we get [ ξ 1 + 4rsi βh + γh ] si ) [ ξ 4rsi βh + γh ] si ) + 1 = 0 which has a o-trivial soltio whe 1 ξ i 1, where ξ i is the magificatio factor correspodig to eigevale, ths; ξ i = βh 4rsi + si γh ) + 8rsi βh + si γh ) 1 Now for ξ i 1, we have si βh = 1 ad = 1, therefore si γh ξ i = 1 4r 1 + 8r Hece for stability r > 0, which makes ξ i less tha ity for all vales of r implyig coditioal stability throghot [ISSN: ] wwwijmseorg 8

9 Iteratioal Joral of Mltidiscipliary Scieces ad Egieerig, Vol 6, No 10, October 015 Aalysis ad Software I this sectio we preset a aalysis of the merical experimets We also preset ad discss the reslts obtaied from these methods We shall display these reslts sig three- dimesioal figres ad graphs From the iitial coditio t x, z, 0) = 0 Bt sice t is approximated sig cetral differece ie??), the cetral differece aaloge of t yields t +1 1 k = 0 Takig = 0, from iitial coditio we fid where Implyig that t k k = t 1 = 1 = 0, Agai, from the iitial coditio, x, z, 0) = si πx)si πz), we get that x, z, 0) 0 = si πx)si πz) At this poit we developed a Matlab program that cold give the pressre field as a fctio of x ad z at varyig time levels ad reslts have bee plotted for both eqatio 7) ad 11) Figre 31: Nmerical soltio explicit scheme at c=1500,dt=05 Figre 3: Nmerical soltio Crak Nicolso scheme at c=1500,dt=05 [ISSN: ] wwwijmseorg 9

10 Iteratioal Joral of Mltidiscipliary Scieces ad Egieerig, Vol 6, No 10, October 015 Figre 33: merical soltio of explicit scheme at t=10,c=1500dt=05 Figre 35: Nmerical soltio explicit scheme at c=1000,dt=08 Figre 34: merical soltio of Crak Nicolso scheme at t=10,c=1500dt=05 Figre 36: Nmerical soltio Crak Nicolso scheme at c=1000,dt=08 4 Discssios I reality, sod propagatio i elastic medim is damped, the amplitde of the pressre of the sod wave decreases with icreasig distace from the sod sorce Or reslts from the two merical schemes CDS) ad CNS) are cofirmig this sice the displacemet of the particles give by x, z, t) is decreasig with a icrease i the distace from the sorce i this case t=0) The efficacy of a fiite differece scheme is achieved with the icrease of the grid poits ivolved hece the icrease i the accracy of a fiite differece scheme I additio, the speed of sod redces with icrease i the distace from the sorce, this is evideced by the redctio of the ripples as the propagatio advaces away from sorce see figre 413 Coclsios This stdy focssed o the secod order acostic eqatio with a sigal fctio Two merical schemes amely Cetral Differece Scheme Explicit scheme) ad Hybrid scheme Crak Nicolso Scheme) were developed ad sed i this stdy The stability aalyses of the developed schemes revealed that Explicit scheme was coditioally stable while the Hybrid oe Crak Nicolso Scheme) was coditioally stable, for all vales of corat mber r The rate of covergece of the algorithms depeds o the trcatio error itrodced whe approximatig the partial derivatives, the Crak-Nicolso method coverges at the rate of k + h ), which is a faster rate of covergece tha either the explicit method, or the implicit method Frther, sice c is a fctio of x, z), from the reslts it sffices to se the maximm sod velocity i the model The smaller the mesh sizes, the more fiely the reslts, this makes the grid more fier ths improvig the approximatio arod the bodary bt at the cost of strogly icreased comptatioal time as evideced by figres 415, 416) 5 Recommedatios We wish to recommed that frther research ca be dertake to; [ISSN: ] wwwijmseorg 10

11 Iteratioal Joral of Mltidiscipliary Scieces ad Egieerig, Vol 6, No 10, October 015 [4] Borthe, J 010): A merical approximatio of the wave eqatio, Mathematics ad Statistics, Georgetow Uiversity [5] Brekhovskikh, L M, ad Y Lysaov, P 003): Fdametals of Ocea Acostics, 3rd editio, Spriger-Verlag, NY [6] Cerja, C, Kosloff, D, Kosloff, R, ad Reshef, M 1985): A oreflectig bodary coditio for discrete acostic ad elastic wave eqatios, Geopgysics, vol 50, o 4, pp Figre 37: Nmerical soltio explicit scheme at t=5,c=05,dt=05 [7] Charara, M ad Taratola, A 1996): Bodary coditios ad the sorce term for oe-way acostic depth extrapolatio, Geophysics,vol 61, pp 44-5 [8] Clay, C S, Medwi, H 1977): Acostical Oceaography: Priciples ad Applicatios, Joh Wiley ad Sos, New York, NY, pp 88 ad [9] Daiel, R R 009): The sciece ad Applicatio of acostics secod editio, The City College of the City Uiversity of New York [10] David, H ad Robert, R 1989): Fdametal of physics, d Editio, Joh Wiley ad sos, New York 1)pg Figre 38: Nmerical soltio Crak Nicolso scheme at t=5,c=05,dt=05 i) Explore merical soltio to this problem sig other methods like fiite elemet ad compare reslts ii) Try ot a aalytical method via gree s fctio Refereces [1] Alford, R M, Kelly, K R, ad Boore, D M 1974): Accracy of fiite-differece modelig of acostic wave propagatio, Geophysics, vol 39, pp [] Alford, R M, Kelly, K R, ad Boore, D M 1974): Accracy of fiite-differece modelig of acostic wave propagatio, Geophysics, vol 39, pp [3] Beratz, R 010): Forier series ad merical methods for partial differetial eqatios, Lther College, Joh Wiley & Sos, ic, pblicatio [11] Evas, C L 1997): Partial Differetial Eqatios, America Mathematical Society [1] Frak J F 000): Fodatios of egieerig acostics, Istitite of sod ad vibratio research, Uiversity of Sothampto, UK [13] Hgh D G ad Pat, F D 003): Fiite differece modellig of the fll acostic wave eqatio i Matlab, CREWES Research Report vol 15 [14] Jai, M K 1991): Nmerical soltio of Differetial Eqatios, Wiley Easter Limited, New Delhi [15] Jese, F, Kperma, W, Porter, M, ad Schmidt, H 000): Comptatioal Ocea Acostics, spriger, New York, pp11-1 ad 5-54 [16] Joh, H M 001): Nmerical Methods for Mathematics, Sciece ad Egieerig,d Editio, Pretice Hall of Idia New Delhi [17] Kelly, K R, Ward, R W, Sve T ad Alford, R M 1976): Sythetic Seismograms: A fiite-differece Approach, Geophysics, vol 41, o 1, p-7 [18] Lies, L R Slawiski, R ad Bordig, R P 1999): A recipe for stability of fiite differece wave eqatio comptatios, Geophysics, vol 64, pp [ISSN: ] wwwijmseorg 11

12 Iteratioal Joral of Mltidiscipliary Scieces ad Egieerig, Vol 6, No 10, October 015 [19] Maarte, V W ad Kowalczyk, K 008): O the merical soltio of the D wave eqatio with compact FDTD schemes, Soic Arts Research Cetre, Qee s Uiversity Belfast, Uited Kigdom [0] Radall J L 005): Fiite differece Methods for Differetial Eqatios, lectre otes for Amath Uiversity of Washigto [1] Rao, S K 004): Itrodctio to Partial Differetial Eqatios, Pretice Hall of Idia, New Delhi [] Richtmyer, R D ad Morto, K W 1967): Tridiagoal Algorithm, Lectre otes [3] Rodey, F W C 1990): Uderwater acostic systems, Joh Wiley ad sos, New York [4] Seogjai, K 00): Higher-Order Schemes for Acostic Waveform Simlatio, Acostic Waveform Simlatio, Departmet of Mathematics, Uiversity of Ketcky, Lexigto, Ketcky USA, March [5] Serway, R A ad Fagh, J S 199): Stdet Soltios, Maal ad Stdy Gide to accompay College Physics, 4th Editio Saders college Pblishig [6] Strikwerda, J C 1989): Fiite Differece Schemes ad Partial Differetial Eqatios, Wadsworth ad Brook/Cole [7] Vetreo, J R 007): Aalytic Models for Acostic Wave Propagatio i Air, Master s Thesis, North Carolia State Uiversity [8] William, F A 199): Nmerical methods for partial differetial eqatios, third editio, Academic press, New York [9] Xie, X B, ad W, RS 1997): Free srface bodary coditio ad the sorce term for oeway elastic wave method, Istitte of Tectoics, Uiversity of Califoria, Sata Crx [ISSN: ] wwwijmseorg 1

13 Iteratioal Joral of Mltidiscipliary Scieces ad Egieerig, Vol 6, No 10, October Appedix 61 Matrix Geeratio Set oe, j = 1; 1,1 = r,1 + 4r) 1,1 + r 0,1 + r 1, + r 1,0 c 1,1 t) S1,1 1 1,1,1 = r 3,1 + 4r),1 + r 1,1 + r, + r,0 c,1 t) S,1 1,1 3,1 = r 4,1 + 4r) 3,1 + r,1 + r 3, + r 3,0 c 3,1 t) S3,1 1 3,1 = M 1,1 = r M,1 + 4r) M 1,1 + r M,1 + r M 1, + r M 1,0 c M 1,1 t) S M 1,1 1 M 1,1 I set two, we set j = to geerate the systems of eqatios 1, = r, + 4r) 1, + r 0, + r 1,3 + r 1,1 c 1, t) S1, 1 1,, = r 3, + 4r), + r 1, + r,3 + r,1 c, t) S, 1, 3, = r 4, + 4r) 3, + r, + r 3,3 + r 3,1 c 3, t) S3, 1 3, = M 1, = r M, + 4r) M 1, + r M, + r M 1,3 + r M 1,1 c M 1, t) S M 1, 1 M 1, Cotiig i the same tred, we set j = 3 to give 1,3 = r,3 + 4r) 1,3 + r 0,3 + r 1,4 + r 1, c 1,3 t) S1,3 1 1,3,3 = r 3,3 + 4r),3 + r 1,3 + r,4 + r, c,3 t) S,3 1,3 3,3 = r 4,3 + 4r) 3,3 + r,3 + r 3,4 + r 3, c 3,3 t) S3,3 1 3,3 = M 1,3 = r M,3 + 4r) M 1,3 + r M,3 + r M 1,4 + r M 1, c M 1,3 t) S M 1,3 1 M 1,3 Settig j = 4 yields 1,4 = r,4 + 4r) 1,4 + r 0,4 + r 1,5 + r 1,3 c 1,4 t) S1,4 1 1,4,4 = r 3,4 + 4r),4 + r 1,4 + r,5 + r,3 c,4 t) S,4 1,4 3,4 = r 4,4 + 4r) 3,4 + r,4 + r 3,5 + r 3,3 c 3,4 t) S3,4 1 3,4 4,4 = r 5,4 + 4r) 4,4 + r 3,4 + r 4,5 + r 4,3 c 4,4 t) S4,4 1 4,4 = M 1,4 = r M,4 + 4r) M 1,4 + r M,4 + r M 1,5 + r M 1,3 c M 1,4 t) S M 1,4 1 M 1,4 [ISSN: ] wwwijmseorg 13

14 Iteratioal Joral of Mltidiscipliary Scieces ad Egieerig, Vol 6, No 10, October 015 This process is cotied til i = M 1), j = N 1) as below 1,N 1 = r,n 1 + 4r) 1,N 1 + r 0,N 1 + r 1,N + r 1,N c 1,N 1 t) S1,N 1 1 1,N 1,N 1 = r 3,N 1 + 4r),N 1 + r 1,N 1 + r,n + r,n c,n 1 t) S,N 1 1,N 1 3,N 1 = r 4,N 1 + 4r) 3,N 1 + r,n 1 + r 3,N + r 3,N c 3,N 1 t) S 3,N 1 1 3,N 1 = M 1,N 1 = r M,N 1 + 4r) M 1,N 1 + r M,N 1 + r M 1,N + r M 1,N c M 1,N 1 t) S M 1,N 1 1 M 1,N 1 6 Matlab Programme [ISSN: ] wwwijmseorg 14