A Mathematical Study of Electro-Magneto- Thermo-Voigt Viscoelastic Surface Wave Propagation under Gravity Involving Time Rate of Change of Strain

Transcription

1 Thortical Mathmatics & Applicatios vol.3 o ISSN: (prit) (oli) Sciprss Ltd 3 A Mathmatical Study of Elctro-Magto- Thrmo-Voigt Viscolastic Surfac Wav Propagatio udr Gravity Ivolvig Tim Rat of Chag of Strai Rajsh akar ad Aru umar Abstract This rsarch is a mathmatical ivstigatio of th propagatio of surfac wav i a Voigt viscolastic mdium. A mathmatical modl for wav propagatio i lctro-magto-thrmo htrogous viscolastic isotropic half spac udr gravity ivolvig tim rat of chag of strai of th ordr is purposd. A solutio to th partial diffrtial quatio of motio is assumd ad is show to satisfy th two cssary boudary coditios. Th frqucy quatios for surfac wavs (Lov Rayligh ad Stoly wavs) ar obtaid with th hlp of Biot s thory of icrmtal dformatios. Htrogitis i th mdium ar assumd to vary xpotially with dpth. Th problm ivstigatd by Bull [Cambridg ivrsity Prss pp ] has b rducd as particular cas. Pricipal DIPS Polytchic Collg Tada Hoshiarpur Idia. HOD Dpartmt of Physics SPN Collg Mukria Hoshiarpur Idia. Articl Ifo: Rcivd: Ju 3. Rvisd: August 5 3. Publishd oli : Sptmbr 3

2 88 Mathmatical Study of Elctro-Magto-Thrmo-Voigt Viscolastic Mathmatics Subjct Classificatio: 74H45; 74F; 74J5; 74J; 745 ywords: viscolastic solid mdium; gravity; ihomogitis; variabl dsity; tmpratur; magtic fild; lctric fild Itroductio I th arlir tims sufficit itrst has b ld o wav propagatio i that matrial whos mchaical charactristics ad dsity ar fuctios of spac i.. htrogous girig matrial. It is prhaps du to lack of iformatio (xprimtal or mpirical) availabl i th litratur cocrig th prcis mchaical bhaviour of htrogous mdia that ot much work has b rportd i this ara of applid mchaics. Wav propagatio i ihomogous mdium is a challg for both thortical rsarch ad girig practic. With th rapid dvlopmt of scic ad tchology wav motio study of th htrogous mdium (atmosphr oca arth-crust fuctioally gradd matrials ad cycl grid structur tc.) sms much mor importat. Although most wav quatios assum propagatio i a lastic mdium it is wll kow that may solids do ot xactly oby th laws of th thory of lasticity. Th purpos of this rsarch thrfor is to assum a o-lastic mdium that rprstd by a Voigt viscolastic lmt ad ivstigat th coditios cssary for th propagatio of a surfac wav. To th arthquak sismologist ad to thos cocrd with prdictig th ffcts of xplosivs i solids th surfac wav is o of th most importat typs of wavs that hav b obsrvd. With accurat arthquak sismograms of surfac wavs th thickss of th suprficial layr of th arth (th crust) may b dtrmid. O a smallr scal i sismic xploratio kowldg of th thickss of th wathrd surfac layr is of primary importac

3 Rajsh akar ad Aru umar 89 Th ivstigatio of viscolastic wav propagatio was iitiatd by Szawa [] who was cocrd primarily with purly dilatatioal pla wavs ad obtaid his solutio usig Fourir itgrals. May yars ago Bromwich [] dtrmid th ifluc of gravity o wav propagatio i a lastic solid mdium. Lov [3] ivstigatd th ffct of gravity o Rayligh wav vlocity. Earlir Thomso [4] discussd trasmissio of lastic wavs through a stratifid solid mdium. Haskll [5] Ewig Jardtzky ad Prss [6] studid wav propagatio. D ad S [7] prstd ot o lastic wavs. S ad Acharya [8] ivstigatd th ffct of gravity o wavs i a thrmolastic layr. Das t al. [9] studid magto-visco-lastic surfac wavs i strssd coductig mdia. Rctly akar t al. [-3] prstd may paprs o surfac wavs i viscolastic mdia. I this papr w hav cosidrd that th surfac wavs ar propagatig i isotropic viscolastic htrogous mdium udr th ffct of tmpratur lctric fild magtic fild ad gravity. Th problm of th ordr viscolastic lctro-magto-thrmo surfac wavs (Lov Stoly ad Rayligh wavs) udr gravity ivolvig tim rat of strai i htrogous mdium is studid i dtail. Govrig quatios Th govrig quatios of motio for 3D viscolastic solid mdium i Cartsia co-ordiats with Eq. () ar [3] τ τ τ3 w u ρg = ρ x y z x t τ x y z y t τ τ 3 w ν ρg = ρ (a) (b)

4 9 Mathmatical Study of Elctro-Magto-Thrmo-Voigt Viscolastic whr τij ji τ τ τ ρg + = x y z x y t u v w ρ. (c) = τ i j ar th strss compots ad ρ is th dsity of th mdium. W hav assumd both th mdiums ar prfct lctric coductor thrfor th govrig quatios of motio for such mdiums ar Ε= Β= Ε= Bt Β= ε E t. whr Ε Β ad ε ar lctric fild magtic fild iductio prmability ad prmittivity of th mdium. Th valu of magtic fild itsity is Η Η =Η +Ηi ( ) whr magtic fild Η is actig alog y-axis. magtic fild itsity. Η i () (3) is th prturbatio i th Th strss-strai rlatios for viscolastic mdium accordig to Voigt ar [5] τ ij = D ij + (D λ ) δ ij Thrfor th strss-strai rlatios for gral isotropic thrmo magto lctro viscolastic mdium ca b writt as (4a) τ ij = D ij + (D λ D β T + + E D + H D ) δ ij (4b) E whr = u v + + w D λ D D β D m x y z D ar lastic costats. E Lt iitial tmpratur of both th mdium is kpt at costat absolut tmpratur T. Fourir s law of hat coductio is usd to calculat T ad it is m giv by T t + (5) t p T = Cν TG L ( φ ) whr b th thrmal coductivity ad obys th law as giv by = mz p = ρ ad C ν b th spcific hat of th body at costat volum.

5 Rajsh akar ad Aru umar 9 3 Formulatio of th problm Cosidr a htrogous thrmo-lctro-magto-voigt viscolastic isotropic smi-fiit mdia M ad M (as show i Figur ). M ( u ν wt ) z= M ( u' ν ' w' T' ) z Figur : Gomtry of th problm Th mchaical proprtis of M ar diffrt from thos of M. Lt th compots of displacmts ar u v w. Th Cartsia co-ordiat systm (x y z) is locatd at th itrfac sparatig th two layrs at z =. Th z-axis is actig dowward. Itroducig Eq. (4a) i Eq. (a) Eq. (b) Eq. (c) w gt Dλ Dλ + + D x x u x + u D x x Dm D E w u ρ ρ x x x t D T u w β + D + x z z x + u w D D u w D D + H Dm + + E DE z x z x z x z x + H D + E D + g = D (6a) D D ν ν v+ + =ρ ν (6b) x x z z t

6 9 Mathmatical Study of Elctro-Magto-Thrmo-Voigt Viscolastic u w u w D w w D + D x z x + z x x z z Dλ + Dλ + z z D β T z D T z β D + H D + z m H D z D D z D DE u + E DE + ED ρg z z x = ρ w. (6c) t W assum that th htrogitis for th mdia M ad M ar giv by mz D λ = λ t D = mz t D β = mz β t ρ = ρ mz = = = m D m ad = mz ( ) t D = η = N = η mz t D E = mz ( ε ) t (7) = D' λ = = λ ' lz t D = lz ' t ρ' = ρ' lz D β = = = β lz t = mz = t D' ( ') D' η N lz = η ' D' t = ε = mz ( ε ') t (8) whr λ M λ' ' ε' ar lastic costats whras β β' ar thrmal paramtrs ar ρ ρ' m ar costats. λ ε ( =... ) ar th = paramtrs associatd with th ordr viscolasticity ad β (ε ) ad ( ) ( =... ) ar th thrmal lctric ad magtic paramtrs associatd with th ordr. T is th absolut tmpratur ovr th iitial tmpratur T. I a thrmo viscolastic solid th thrmal paramtrs β ( =... ) ar giv by β = (3λ + ) α t whr α t b th cofficit of liar xpasio of solid. G G HG EG G u λ x u w T w u + mg + Gβ + ρ g = z x x x t ( m E ) ρ (9a)

7 Rajsh akar ad Aru umar 93 G v v + mg z = v ρ (9b) t (G λ + G + m EG E H G + ) z + G w + G λ m + G m w z whr G T β z mg B T+ u w mh DG + me m D G ρ g = ρ. E x t G λ = λ t G = t = = (9c) G β = β t = G ε = ( ε ) t = G = ( ) = t = x + = z x + z. () To ivstigat th surfac wav propagatio alog th dirctio of Ox w itroduc displacmt pottial φ (x z t) ad ψ (x z t) which ar rlatd to th displacmt compots as follows: u= φ ψ w = φ x z z + ψ x. () Th displacmt pottial φ (x z t) ad ψ (x z t) i Eq. () satisfy th followig Laplac quatio (kow as dilatio ad rotatio ad ar associatd with P ad SV wavs) φ = u + w = ψ = w u = Ω. (a) x z x z Substitutig Eq. () i Eqs (9a) (9b) ad (9c) w gt G R φ + mg s ( φ z ψ x ) G S v + mg v z = v tt + G T L + gψ x = φ tt (a) (b) Whr G S ψ + mg P φ x + m G s ψ z G q 4 ψ gφx = ψ tt (c)

8 94 Mathmatical Study of Elctro-Magto-Thrmo-Voigt Viscolastic λ + + H(m ) + E( E ) R = ρ S = ρ L = β ρ ad P = λ + H (m ) + E (E ) ρ G R = R t G S = S t G = P P t = G L = L t G = = q q = t = =. (3) By usig Eq. (5) tmpratur T ca b calculatd. Furthr similar rlatios i mdium M ca b foud out by rplacig λ β ε ρ by λ' ' β' ε' ρ' ad so o. 4 Solutio of th problm Now our mai objctiv to solv Eq. (a) Eq. (b) Eq. (c) ad Eq. (5) for this w sk th solutios i th followig forms. (φ ψ T v)= [f (z) V (z) T (z) h (z)] iα(x ct) (4) sig Eq. (4) i Eq. (a) Eq. (b) Eq. (c) ad Eq. (5) w gt a st of diffrtial quatios for th mdium M as follows: d f df + mf + h f + ( iαmf + iαgj ) j g T= d h dh m h + + = d g dg ( α α ) + m + j + i ml i gn f =

9 Rajsh akar ad Aru umar 95 whr dt d f + AT + B α f = (5) f = = = S ( α ) i c R ( α ) i c h = = α R c ( α ) i c α = = α S c ( α ) i c α J = = T ( iα c) l = N = = = = P ( α ) i c S ( α ) i c S ( iα c) g = ad thos for th mdium M ar giv by whr d f df f + l h + = = L ( α ) i c R ( α ) i c A= C ν iα c α B = i ct p p ' f ' ' ' ' ( α α ) ' d h dh + l + h = + i l f + i gj j g T = d g dg ' ' ' ( α. α ) d T + l + j + i l l i gn f = α G L (6) A T B d f + ' + ' α f = (7)

10 96 Mathmatical Study of Elctro-Magto-Thrmo-Voigt Viscolastic f ' = = = ' S ' R h ' = = α ' R c α ' = = α ' S c α J / = = / T ( iα c) N / = = / S ( iα c) l ' = = = ' P ' S g ' = = = ' L ' R α B' = i ct p' G L ' (8) Eq. (5) ad Eq. (7) must hav xpotial solutios i ordr that f j T h will dscrib surfac wavs ad thy must bcom varishig small as z. Hc for th mdium M λ φ (x z t)= z λz λ3z iα ( x ct) { } A + B + C λ ψ (x z t)= z λz λ3z iα ( x ct) { } A + B + C λ T (x z t)= z λz λ3z iα ( x ct) { } A + B + C v (x z t)= C 4 z iα ( x ct) (9a) λ + For fiit disturbacs as z for mdium M must hold R (λ i )> for i=345. Similarly for th mdium M ar giv by λ' φ (x z t)= z λ' z λ' 3z iα ( x ct) { } A' + B' + C'

11 Rajsh akar ad Aru umar 97 λ' ψ (x z t)= z λ' z λ' 3z iα ( x ct) { } A' + B' + C' λ' T (x z t)= z λ' z λ' 3z iα ( x ct) { } A' + B' + C' v (x z t)= C' ' 4 z iα ( x ct) (9b) λ + For fiit disturbacs as z for mdium M must hold R(λ' i )< for i=345. Whr λ j ad λ' j (j = 3) ar th ral roots of th Eqs. λ 6 + ξ λ 5 + ξ λ 4 + ξ 3 λ 3 + ξ 4 λ + ξ 5 λ + ξ 6 = () whr ξ = m { + f } ξ = + A + 4m + h + Bg ξ 3 = ma + f m ( + A) + mh + mbg ξ 4 = A + 4m A f + ( + A) h + α m l f + B g α Bg ξ 5 = ma f + mah m α Bg ξ 6 = A h + Aα m l l f α B g. () λ' 6 +ξ' λ' 5 +ξ' λ' 4 +ξ' 3 λ' 3 +ξ' 4 λ' +ξ' 5 λ'+ξ' 6 = () whr ξ' = l { + f ' } ξ' = ' + A' + 4l + h ' + B'g ' ξ' 3 = la' + lf ' ( ' + A) + lh ' + l B'g ' ξ' 4 = A' ' + 4l A' f ' + ( ' + A') h ' + α l l ' f ' + B' ' g ' α B' g ' ξ' 5 = la' ' f ' + la'h ' l α B'g ' ξ' 6 = A' ' h ' + A'α l l l ' f ' α B' ' g '. (3) λ 4 = {m + (m - 4 ) ½ }/ λ' 4 = {l + (l - 4 ' ) ½ }/

12 98 Mathmatical Study of Elctro-Magto-Thrmo-Voigt Viscolastic whr th symbol usd i qs. () ad (3) ar giv by qs. (6) ad (8). Th costats A j B j C j (j = 3) ar rlatd with A' j B' j C' j (j = 3) i Eq. (9a) ad Eq. (9b) by mas of first quatios i Eq. (5) ad Eq. (7). Equatig λz λz λ3z λ' z λ' z λ' 3 th cofficits of z to zro aftr substitutig Eq. (9a) ad Eq. (9b) i th first ad 3rd quatios of Eq. (5) ad Eq. (7) rspctivly w gt A = γ A B = γ B C = γ 3 C ad A 3 = δ A B 3 = δ B C 3 = δ 3 C (4) whr γ j = λ j iα m l mλ + j (j = 3) δ j = g [λ j m f λ j + h + i α m f γ j ] j = 3. Similar rsult holds for mdium M ad usual symbols rplacig by dashs rspctivly. 5 Boudary coditios Thr ar two boudary coditios (i) Th displacmt compots tmpratur ad tmpratur flux at th boudary surfac btw th mdia M ad M must b cotiuous at all tims ad positios. i.. u ν w T p T z = u ν w T p' M T z M (ii) Th strss compots τ 3 τ 3 τ 33 must b cotiuous at th boudary z =.

13 Rajsh akar ad Aru umar 99 i..[ τ τ τ ] whr M = [ τ3 τ3 τ33] M at z = rspctivly τ 3 = D φ ψ ψ + x z x z τ 3 = D ν z φ φ τ 33 = Dλ φ+ D + D B T + Dm H φ+ DE E φ. (5) z xz Applyig th boudary coditios w gt A ( i γ ζ ) + B ( i γ ζ ) + C ( i γ 3 ζ 3 ) A' ( i γ' ζ' ) B' ( i γ' ζ' ) C' ( i γ ' 3 ζ' 3 ) = (6a) C = C' A (γ + iζ ) + B (γ + iζ ) + C (γ 3 + iζ 3 ) A' (γ' + iζ' ) (6b) B' (γ ' + iζ' ) C' (γ' 3 + iζ' 3 ) = (6c) δ A + δ B + δ 3 C = δ' A' + δ' B' + δ' 3 C' (6d) pλ δ A + pλ δ B + pλ 3 δ 3 C p' λ' δ' A' + p' λ' δ' B' p'λ' 3 δ' 3 C' = (6) [(i ζ + γ + ζ γ ) A + (i ζ + γ + ζ γ ) B + (i ζ 3 + γ 3 + ζ 3 γ 3 ) C ] = ' [(i ζ' + γ' + ζ ' γ' ) A' + (i ζ' + γ' + ζ ' γ' ) B' +(i ζ' 3 + γ ' 3 + ζ 3 ' γ' 3 ) C' ] (6f) [ λ 4 C]= ' [ λ' 4 C'] (6g) A [( λ + ( ) H + B [( λ + ( ) + C [( λ + ( ) = A' [( λ' + ( ') + B' [( λ' + ( ') +C' [( λ' + ( ') ( ε ') H H H ( ε ') E ) (ζ ) + (ζ iζ ) β δ ] ( ε ') ( ε ') H H ( ε ') ( ε ') E ) (ζ ) + (ζ iζ ) β δ ] E ) (ζ 3 ) + (ζ 3 iζ 3 ) β δ 3 ] E )(ζ ' )+ ' (ζ ' iζ' ) β' δ' ] E ) (ζ ' ) + ' (ζ ' iζ' ) β' δ' ] E ) (ζ 3 ' ) + ' (ζ 3 ' iζ' 3 ) β' δ' 3 ] (6h)

14 Mathmatical Study of Elctro-Magto-Thrmo-Voigt Viscolastic whr ζ j = λ j α ζ'j = λ α ' j j = 3 λ = λ ( ) = = = = ( ) ( ε) = ( ε) ad = λ' = λ' ( ') = = ( ') = = ' = ' = ( ') β = β = ε = ( ε') = β' = β' = From Eq. (6b) ad Eq. (6h) w hav C = C' =. Thus thr is o propagatio of displacmt v. Hc SH-wavs do ot occur i this cas. Fially limiatig th costats A B C A' B' C' from th rmaiig quatios w gt a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a = (7) whr a = iγ ζ a = iγ ζ a 3 = iγ 3 ζ 3 a 4 = (i γ' ζ' ) a 5 = (i γ ' ζ' ) a 6 = (i γ' 3 ζ' 3 ) a = γ + iζ a = γ + iζ a 3 = γ 3 + iζ 3 a 4 = (γ' + i ζ' ) a 5 = (γ' + iζ' ) a 6 = (γ' 3 + iζ' 3 ) a 3 = δ a 3 = δ a 33 = δ 3 a 34 = δ' a 35 = δ' a 36 = δ' 3

15 Rajsh akar ad Aru umar a 4 = pλ δ a 4 = pλ δ a 43 = pλ 3 δ 3 a 44 = p' λ' δ' a 45 = p' λ' δ' a 46 = p' λ' 3 δ' 3 a 5 = (i ζ + γ + γ ζ ) a 5 = (i ζ + γ + γ ζ ) a 53 = (i ζ 3 + γ 3 + γ 3 ζ 3 ) a 54 = ' (i ζ' + γ' + γ' ζ ' ) a 55 = ' (i ζ' + γ' + γ' ζ ' ) a 56 = ' (i ζ' 3 + γ' 3 + γ' 3 ζ 3 ' ) a 6 = ( λ + ( ) a 6 = ( λ + ( ) ε ε E ) (ζ ) + (ζ iζ ) β δ E ) (ζ ) + (ζ iζ ) β δ a 63 = ( λ + ( ) + ( ) ε E H ) (ζ 3 ) + (ζ 3 iζ 3 ) β δ 3 a 64 = ( λ' + ( ') a 65 = ( λ' + ( ') ε ε E ) (ζ ' ) + ' (ζ ' iζ' ) β' δ' E ) (ζ ' ) + ' (ζ ' iζ' ) β' δ' a 66 = ( λ' + ( ') ε E ) (ζ 3 ' ) + ' (ζ 3 ' iζ' 3 ) β' δ' 3. (8) From Eq. (7) w obtai vlocity of surfac wavs i commo boudary btw two viscolastic htrogous solid mdia udr th ifluc of thrmal lctric ad magtic fild whr th viscosity is of gral th ordr ivolvig tim rat of chag of strai. 6 Particular cass Stoly Wavs: Eq. (7) dtrmi th wav vlocity quatio for Stoly wavs i th cas of gral magto-thrmo viscolastic htrogous solid mdia of th ordr ivolvig tim rat of strai. Clarly from Eq. (7) it is follows that th wav vlocity quatio for Stoly wavs dpds upo th htrogity of th matrial mdium tmpratur lctric magtic ad viscous fild. This quatio of cours is i wll agrmt with th corrspodig classical rsult wh th

16 Mathmatical Study of Elctro-Magto-Thrmo-Voigt Viscolastic ffcts of thrmal lctric magtic ad viscous fild ad htrogity ar abst. Rayligh Wavs: To ivstigat th possibility of Rayligh wavs i a lctro magto thrmo viscolastic htrogous lastic mdia w rplac mdia M by vacuum i th procdig problm; w also ot th SH-wavs do ot occur i this cas. Sic th tmpratur diffrc across th boudary is always small so thrmal coditio giv by T + ht = at z = rspctivly (8) z Thus Eq. (6f) ad Eq. (6h) rducs to (i ζ + γ + γ ζ ) A + (i ζ + γ + γ ζ ) B + (i ζ 3 + γ 3 + γ 3 ζ 3 ) C = (9a) [( λ + ( ) ε E ) (ζ ) + (ζ iζ ) β δ ] A + [( λ + ( ) ε E ) (ζ ) + (ζ iζ ) β δ ] B + [( λ + ( ) From Eq. (8) w hav ε E ) (ζ 3 ) + (ζ 3 iζ 3 ) β δ 3 ] C = (9b) (λ h) δ A + (λ h) δ B + (λ 3 h) δ 3 C = (9c) Elimiatig A B ad C from Eqs. (9a) (9b) ad (9c) w gt whr dt (b ij )= i j = 3. (3) b = (i ζ + γ + γ ζ ) b = (i ζ + γ + γ ζ ) b 3 = (i ζ 3 + γ 3 + γ 3 ζ 3 ) b = [( λ + ( ) b = [( λ + ( ) b 3 = [( λ + ( ) ε ε ε E ) (ζ ) + (ζ iζ ) β δ ] E ) (ζ ) + (ζ iζ ) β δ ] E ) (ζ 3 ) + (ζ 3 iζ 3 ) β δ 3 ]

17 Rajsh akar ad Aru umar 3 b 3 = (λ h) δ b 3 = (λ h) δ b 33 = (λ 3 h) δ 3. (3) Thus Eq. (3) givs th wav vlocity quatio for Rayligh wavs i a htrogous lctro magto-thrmo viscolastic solid mdia of th ordr ivolvig tim rat of strai. From Eq. (3) it is follows that disprsio quatio of Rayligh wavs dpds upo th htrogity th viscous magtic ad thrmal filds. Wh th ffcts of thrmal lctro magtic viscous fild ad htrogity ar abst this quatio of cours is i complt agrmt with th corrspodig classical rsult by Bull [4]. Lov Wavs: To ivstigat th possibility of lov wavs i a htrogous viscolastic solid mdia w rplac mdium M is obtaid by two horizotal pla surfacs at a distac H-apart whil M rmais ifiit. For mdium M th displacmt compot ν rmais sam as i gral cas giv by quatio (9). For th mdium M w prsrv th full solutio sic th displacmt compot alog y-axis i.. v o logr dimiishs with icrasig distac from th boudary surfac of two mdia. Thus v' = C λ' z + iα( x ct ) λ' z + iα( x ct ) C (3) I this cas th boudary coditios ar (i) v ad τ 3 ar cotiuous at z = (ii) τ' 3 = at z = H. Applyig boudary coditios (i) ad (ii) ad usig qs (9) ad (6) w gt C= C + C (33)

18 4 Mathmatical Study of Elctro-Magto-Thrmo-Voigt Viscolastic λ 4 C = (' ) [λ' 4 C λ' 4 C ] (34) C C = (35) λ' 4H λ' 4H O limiatig th costats C C ad C from Eqs. (33) (34) ad (35) w gt λ 4 tah (λ' 4 H) =-. (36) λ ' ' 4 ( ) Thus Eq. (36) givs th wav vlocity quatio for Lov wavs i a htrogous lctro magto thrmo viscolastic solid mdium of th ordr ivolvig tim rat of strai. Clarly it dpds upo th htrogity magtic ad viscous filds ad idpdt of thrmal fild. 7 Coclusios Th tim rat of strai paramtrs ifluc th wav vlocity of surfac wavs to a xtt dpdig o th corrspodig costats charactrizig th lctro-magto thrmo ad viscolasticity of th matrial. So th rsults of this aalysis bcom usful i circumstacs whr ths ffcts caot b glctd. Ths vlocitis dpd upo th wav umbr α cofirmig that ths wavs ar affctd by htrogity of th matrial mdium. It has b obsrvd that tmpratur has o ffct o Lov wavs. Howvr viscosity gravity magtic filds lctric filds ad htrogity of th mdium ffcts th propagatio of Lov wavs. Th disprsio of Rayligh wavs is obsrvd du to th prsc of htrogity tmpratur gravity magtic fild lctric fild ad viscosity of th mdium. Th wav vlocity quatio of Stoly wavs is vry similar to th corrspodig problm i th classical thory of lasticity. Th disprsio of wavs is du to th prsc of htrogity gravity magtic fild lctric

19 Rajsh akar ad Aru umar 5 fild tmpratur ad viscolasticity of th solid. Also wav vlocity quatio of this gralizd typ of surfac wav is i complt agrmt with th corrspodig classical rsult i th absc of all filds ad htrogity. Th solutio of wav vlocity quatio for Stoly wavs caot b dtrmid by asy aalytical mthods howvr w ca apply umrical tchiqus to solv this dtrmiatal quatio by choosig suitabl valus of physical costats for both mdia M ad M. Ackowldgmts. Th authors ar thakful to rfrs for thir valuabl commts. Rfrcs []. Szawa O th dcay of wavs i visco-lastic solid bodis Bul. Earthq. Rs. Ist. (Japa) 3 (97) [] T.J. Bromwich O th ifluc of gravity o lastic wavs ad i particular o th vibratios of a lastic glob Proc. Lodo Math. Soc. 3 (998) 98-. [3] AEH Lov Som problms of Godyamics Cambridg ivrsity Prss Lodo 965. [4] W.T. Thomso Trasmissios of lastic wavs through a stratifid solid mdium J. Appl. Phys. (95) [5] N.A. Haskll Th disprsio of surfac wavs i multilayrd mdia Bull. Sis. Soc. Amr. 43 (953) [6] W.M. Ewig W.S. Jardtzky ad F. Prss Elastic wavs i layrd mdia McGraw-Hill Nw York 957.

20 6 Mathmatical Study of Elctro-Magto-Thrmo-Voigt Viscolastic [7] S.. D ad P.R. Sgupta Ifluc of gravity o wav propagatio i a lastic layr J.Acast. Soc. Am. 55 (974) 5-3. [8] P.R. Sgupta ad D. Acharya Th ifluc of gravity o th propagatio of wavs i a thrmolastic layr Rv. Romm. Sci. Tchmol. Mch. Appl. Tom 4 (979) [9] S.C. Das D.P. Acharya ad P.R. Sgupta Magto-visco-lastic surfac wavs i strssd coductig mdia Sadhaa 9 (994) [] R. akar S. akar ad.c. Gupta Surfac wav propagatio i o homogous gral magto-thrmo viscolastic mdia Itratioal Joural of IT Egirig ad Applid Scics Rsarch () () [] R. akar ad S. akar Thrmolastic Surfac Wavs Propagatio i No- Homogous Viscolastic Mdia of Highr Ordr JECET () () [] R. akar ad S. akar Magtolastic surfac wavs propagatio i ohomogous viscolastic mdia of highr ordr J. Chm. Bio. Phy. Sci. Sc. B 3() () [3] R. akar ad S. akar Rayligh wavs i a o-homogous thrmo magto prstrssd graular matrial with variabl dsity udr th ffct of gravity Amrica Joural of Modr Physics () (3) 7-. doi:.648/j.ajmp.3. [4].E. Bull Thory of Sismology Cambridg ivrsity Prss 965. [5] W. Voigt Abh. Gsch Wiss [6] M.A. Biot Mchaics of icrmtal Dformatios J. Willy 965.