EFFECT OF GRADIENTS IN INHOMOGENEOUS MEDIA ON PLANE WAVE REFLECTION COEFFICIENT RESONANCE

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1 EFFECT OF GRADIENTS IN INHOMOGENEOUS MEDIA ON PLANE WAVE REFLECTION COEFFICIENT RESONANCE PACS efeece: 43.3 Ma. Fokia Magaia; Fokia Vadimi Isiue of Appied Pysics Russia Academy of Scieces 46, Uyaov S., Nizy Novgood, 639, Russia Te: , Fax: E-mai: fok@ydo.app.sci-ov.u ABSTRACT. Tis pape cosides iomogeeous soid sedime, wose popeies vay coiuousy wi dep, o easic subsae. Pae wave efecio coefficie fo is media ave bee obaied by geeaizaio of maix meod. Fo eac iomogeeous aye may be foud exac expessio fo caaceisic maices i fom of egua covege seies. Tese maices wi maix of af-space deemie efecio coefficie. Refecio of pae waves a iomogeeous sedime wi gadie of paamees ave bee sudied. Depedecies of efecio coefficie esoace vesus gadie of sedime paamees ave bee eveaed. INTRODUCTION Te pupose of is sudy is o examie e effec o efecio coefficie of desiy, compessio ad sea wave veociies gadies i maie soid iomogeeous sedimes. Te effec of desiy vaiaio o wave popagaio oug easic soids as eceived i pape [1]. Te maemaica mode fo efecio of soud fom e ocea boom a icudes a iea compessio veociy gadie i wo ayeed absobig iquid sedimes was peseed i [11]. Tus, wie e equaios fo wave moio i iomogeeous easic media ave bee cosideed by a umbe of auos [1,9]. Te esuig equaio ae compex, ad do o ead emseves o easy pysica iepeaio. Aso, souios ae obaiabe oy fo a imied umbe of cicumsaces. Hook [7] ooks fo foms of desiy ad Lamé paamee vaiaio wic pemi e equaios fo compessio ad sea compoes o be decouped, bu assumes a e Lamé paamees ad desiy a ave a simia depedece o dep, wie e simpe modes of sedimes deveoped by oe auos [7] do o coside desiy vaiaios. Desiy effecs ave bee cosideed by [13,14], bu mos of ose sudies ave coceaed o fuid sedimes oy. Te objecive of is pape is o povide a uified eame of easic wave popagaio i oizoay ayeed media fo wic e paamees i e paia diffeeia equaios ae coiuous fucios of oy oe spaia vaiabe. I is ayeed media sea speed, soud speed, dumpig faco ad desiy ae vayig wi dep simuaeousy. By appyig a combiaio of Fouie-Besse asfom o e paia diffeeia equaios descibig e easic wave popagaio a sysem of iea odiay diffeeia equaios wee obaied, wee e coefficies ae fucios of e dep coodiae ad e asfomed ime ad oizoa space coodiaes. I addiio, pope bouday codiios wiig o e bouday of eac ayes. Te popagao maix [6,1] ad maices appoac Tomso ad Haske [6,16] eabes o avoid compex ecique of bouday codiios saisfyig. Howeve, e domai of e vaidiy of e Tomso-Haske compuaioa sceme us ou o be basicay esiced. I is coecio, i pape [4] we passed fom e 4 -ode caaceisic maices o e 6 -ode maices fis suggesed by Duki [] ad Towe [17], wic iceased e accuacy of compue cacuaio of pae wave efecio coefficie. Fo eac iomogeeous aye may be foud e exac expessio fo e caaceisic maices i e fom of a egua covege seies [6]. Tese maices wi e maices descibig e afspace deemie e efecio coefficie. Refecio of e pae waves fom e easic soid aye wi e gadie of e paamees as bee sudied i is pape. Depedecies of e

2 efecio coefficie esoace sucues (posiio, ampiude, af-wid) vesus gadie of e desiy, e dumpig faco, e ogiudia ad sea veociies i e sedimes ave bee eveaed ad pysica iepeaio of e esus ave bee pefomed. Veociy ad desiy gadies ude deep-sea sedimes ave bee measued i [8,]. Mos measued gadies ae bewee. ad.6sec -1 fo compessio wave veociy ad bewee.4 ad 1.19 fo desiy fo e fis m, ese gadies used i compuaioa of efecio coefficies esoaces. Te iees i cosideig e pocess of e efecio ad popagaio of soud i waveguides wi a easic ayeed boom is eaed o e deveopme of e meods ad meas fo e diagosics ad ecosucio of e boom caaceisics, as we as o e ugecy of miea, oi, ad gas pospecig a a sea sef by acousic meods. THE MATHEMATICAL AND PHYSICAL MODEL Te pysica mode of a iomogeeous medium cosiss of a easic aye <z<, wose popeies vay coiuousy wi dep, yig bewee wo omogeeous media a uppe fuid aye epeseig e ocea ad a semi-ifiie, omogeeous, soid subsae. Te mai goa is obaiig of e maix fo iomogeeous aye. Te saig poi fo aaysis is Lamé equaios: U U U U U µ U U ( λ + µ ) ( + ) =, µ ρ λ µ (1) z z z µ U z U U z ( + ) + ( ) ( ) + =, + + λ λ µ U ρ λ µ U () z z z z ad e geeaized Hook s aw: U z U U U U z σ zz = ( λ + µ ) + λ +, σ z = µ + (3) z z wee λ ad µ ae e Lamé paamees. Te veociies c ad c of e compessio ad sea waves ae give by e we-kow eaios c = ( λ + µ ) / ρ ad c = µ / ρ. If oy waves of veica poaizaio ae ake io accou, e dispaceme fieds U=( U U =, U ) ca be expessibe as e sum of a ioaioa compoe, wic x, y z epeses a compessio wave, ad a soeoida compoe, wic epeses a sea wave. Te veco poeia ψ epeses e compoe of moio wi ozeo voiciy. Te efeece axis of e sysem ca be cose suc a ψ as oy oe compoe aog e y-axis, a is o say ψ=(, ψ,). Repeseig compoes fo dispaceme ad sess eso i isoopic y media i ems of e poeias ϕ ad ψ i fom of Fouie-Besse ad Mei iegas ad usig bouday codiios, e equaios (1)-(3) ae epeseed by e maix equaio: dw = k χ W, (4) dz wee W is e ow veco W = [ U ] T, U z,σ z, σ zz (T deoes e opeaio of asposiio), k = ω si( θ ) / c = ω si( θ ) / c ad χ = AA 1, ee = [ α, α, β, β ] is diagoa maix o e bouday of ayes, α ξ = k, β = k ξ ae veica compoes of ogiudia ad asvese compoes of wave umbe, ξ k si θ = k si( θ ) = si( θ ), A ad A -1 ae = ig ad iveed maix of omogeeous aye, e χ wie i maix fom: 1 d b c χ =, () a + ρη b ρη 1 k

3 4µ ( λ + µ ) λ 1 1 ee a =, b =, c =, d =, pois of bacig λ + µ λ + µ λ + µ µ η = ic / si( θ ) = ic / si( θ ) coec wi poi η= by e bac cus, wic pass ove e ef af-pae. Te diffeeia equaio (4) is aamou o Voe s iega equaio: z W( z) = W() + k χ ( z1 ) W ( z1) dz1 (6) souios of e equaio (6) fidig by e meod of sequeia appoacig, wee zeo appoximaio is assumed by e expessio W ( z) = (). If i is souio z=, e we W obai expessio: W ( ) = C( ) W(), (7) ad aso oe may expad maix C() io powe seies: C( ) = E + k χ ( z) dz + k χ ( z1) dz1 χ ( z ) dz +..., (8) z1 C=E (if k=), wee E is ui maix, C() is caaceisic maix 4 -ode fo iomogeeous aye ad i covege i domai of ow fequecies, ad e Lamé paamees λ ad µ, desiy ρ, compessio ad sea soud speeds c, c ae fucio of dep coodiae z. Usig Duki-Towe maix appoac [,17] we passed o mio maix 6 -ode, e diec asiio fom oe appoac o e oe was eaized o e basis of e eoem o e popeies of e associae maices, we e caaceisic maix 4 -ode is se i coespodig wi e maix 6 -ode, wi e eemes beig d -ode mios of e maix 4 -ode. I meas a if e maix 4 -ode W goves equaio (4), e e appopiae mio maix Wˆ is e souio of e equaio: dwˆ = B Wˆ, (9) dz b ij ae eemes of maix kχ. Maix C() saisfies by equaio dc()/d=kχ()c() wi iiia codiio C()=E. Appyig e eoem o e popeies of e associae maices oe may be obaied aaogous maix diffeeia equaio: dcˆ ( ) = kχ ( ) Cˆ( ), Ĉ () = Ê, (1) d i is case e expessio fo maix χ () wie o e base of expessios () ad maix B as b 1 χ 1 1 = b b b c 1 d ρη 1/ µ. (11) ρη a c ρη + a ρη Maix 6 -ode C ˆ ( ) caaceizig e iomogeeous aye foows fom e expessio: 1 ˆ ˆ C( ) = E + k χ ( z) dz + χ ( z1) dz1 χ ( z ) dz (1) Refeced ad efaced waves ae caaceized by equaios: { Q K Dˆ } V S ˆ = 11 {, 33 Q K Dˆ W = D }1 Q K Dˆ S z { S } 1, Dˆ W = 31 { Q K Dˆ S } 1 wee K S is asiio maix bewee iquid af-space ad se of easic ayes, Q is maix of iquid af-space epeseig e ocea, Dˆ = Aˆ 1 Cˆ Cˆ... Cˆ Aˆ 1 1 is maix popagao fo e sack of easic iomogeeous ayes ad a semi-ifiie, omogeeous, easic afspace epeseig subsae, wic obaiig by muipicaio i pope ode. Usig e 6 - ode maices as some advaages, because of aows oe o pefom coec cacuaios. (13)

4 NUMERICAL RESULTS Tis secio peses some iusaive souio fo efecio coefficies. Maix meod was eaized as a compue code. Fo esig code was used e compuaioa daa fo efecio oss obaied i e papes [1,13]. Te es cacuaios agee we wi e pubised esus. Te esus of cacuaio of e moduus of e efecio coefficie fo sige iomogeeous aye, coveig easic af-space, is poed i a ee-dimesioa gap simuaeousy vesus e fequecy ickess poduc Fd ad age of icidece (Fig.1). Coside compaiso bewee e esus of umeica modeig of soud efecio fom e iomogeeous easic aye wi gadies of desiy, compessio ad sea veociies ad fom omogeeous easic aye (e pysica paamees fo e omogeeous aye ae equa o aveaged paamees). As e ue, we ave a umbe pubicaio ad simia compuaios oy fo sige fequecy o age of icidece. Tese cicumsaces impede pysica iepeaio of obaiig Figue 1. Refecio coefficie fom aye wi gadies of paamees. esus. Tis secio peses compaiso of umeica esus i e boad fequecy age a a ages of icidece. Tis fac aows o add e eve of compexiy o cosideed pobem ad o pefom compaiso omogeeous ad iomogeeous easic ayes o e pae of e Fd-Age of Icidece. Te esus of compuaio efecio coefficies fo ese cases ae peseed i e fom of coous o e Fig. a, b, accodigy. Te suface as a compicaed sucue cosisig of egua sequeces of miima ad maxima. Te Fig. sows a effec of gadies of paamees i sedime fis of a obsevig i modifyig of e posiios, ampiude ad af-wid of e efecio coefficie esoaces. Tus, i is easoabe o sudy Fd Fd Age of icidece, gad Age of icidece,gad. Figue. Refecio coefficie: a) omogeeous aye; b) gadie aye. effecs gadies of paamees of sedime o e caaceisic of esoace (posiio, ampiude ad af-wid) ad o compae obaiig esus wi compuaios pefomig fo omogeeous sedimes. Te basic assumpio of e esoace appoac is a i e viciiy of e esoace e ampiude of a pocess is descibed esseiay by e Bei-Wige esoace fom wi e addiio of a sowy vayig backgoud [3]. Foowig is idea, e exac expessio fo e efecio coefficie ca be expaded i a Tayo seies wi espec o e powes (δ-δ ) ad (η-η ) aoud e esoace posiios δ ad η, wic wee aayicay obaied fo omogeeous easic aye i, ee δ = αd, η = βd ae e pase ems []. Tus, e esoace expessio fo e efecio coefficie ca be obaied as e sum of esoace ems, bo i e fequecy ad agua vaiabes. Te sum ove esoaces mus be ake oy symboicay sice e expasio is assumed o be vaid oy i e immediae viciiy of eac esoace posiio. Afe e usua maipuaios, e efecio coefficie may be wie i e suggesive fom: Γ P + F1 ( δ δ ) + F( η η ) + ig V =, (14) Γ Q W1 ( δ δ ) W ( η η ) + i wee Γ / is e af-wid of esoace measued ea e oca miima of e efecio coefficie we e ampiude of e pocess eaces af is vaue A/, meas a is em

5 . soud be esimaed fo - oo, F 1, W 1, W ad F aio of impedaces, P,Q ad G ae deemied by paamees of medias ieacig wi soud. Te aayica ad umeica ivesigaios of iedepedece of Γ ad maeia paamees of wo ayeed mode was pefomed. Haf-wid of esoaces was obaied i is pape aayicay. I is easy o obai ad id esoace em pocess ampiude. Fom equaio (14) i esoace codiios δ = δ, η = η we go esoace ampiude oo. Depedece of esoace caaceisics fom boom paamees was cosideed fo omogeeous ad iomogeeous sedime aye. Fo iomogeeous easic aye wi gadies of paamee, coveig a easic subsae, ivesigaio of gadies ifuece o esoace sucue of e efecio coefficie was pefomed. Veociy gadies i deep-sea sedimes ave bee measued i [8,]. Mos measued gadies ae bewee. ad. 1/sec wi a aveage gadie of 1. 1/sec. Beavio of esoace ampiude ad af-wid o e age of icidece gadie of compessio soud speed i e aye ae sow i e Fig Ampiude of esoaces Haf-Wid of esoace Age of Icidece Gadie CL Age of Icidece Figue 3. Ampiude of efecio coefficie esoaces Gadie CL Figue 4. Haf-wid of efecio coefficie esoaces Beavio of esoace ampiude ad af-wid o e age of icidece gadie of compessio soud speed i e aye ae sow i e Fig. -6. Aaysis of ese depedecies pemi o make cocusios abou possibiiies of esoace caaceisics use fo deemiaio of sedimes paamees a i's gadies by esoace caaceisics. Compaisos of eoeica cuves wi expeimeay measued dae wi be e ex sage of ivesigaios. Age of Icidece Ampiude of esoace Age of Icidece Gadie C Figue. Ampiude of efecio coefficie esoaces. Haf-Wid of esoace Gadie C Figue 6. Haf-wid of efecio coefficie esoaces. SUMMARY Te acousic efecio coefficie fo pae waves, i a omogeeous fuid af-space, icide a abiay age, as bee cacuaed fo a mode cosisig of easic iomogeeous aye upo a udeyig easic af-space. Popeies of a iomogeeous soid sedime vay coiuousy wi dep, easic af-space omogeeous, soid subsae. Te pae wave

6 efecio coefficie fo is media ave bee obaied by geeaizaio of a maix meod ad is give i geea by Eq.(13). Fo eac iomogeeous aye ave bee foud e exac expessio fo e caaceisic maices i e fom of a egua covege seies. Refecio of e pae waves a e easic soid aye wi e gadie of e paamees ave bee sudied. Depedecies of e efecio coefficie esoace sucues (posiio, ampiude, af- wid) vesus gadie of sedime paamees ave bee eveaed ad pysica iepeaio of e esus ave bee pefomed. Tis wok was suppoed by e ga of RFBR No REFERENCES [1] L.M. Bekovskik, Wave-guide effecs i soid ayeed media wi coiuousy vayig paamees Sov. Pys. Acous., 14, (1968). [] I.W. Duki, Compuaio of moda souios i ayeed easic media a ig fequecies Bu. Seismo. Soc. Am.,, (196). [3] L. Fax, G.C. Gauad, H. Übea, Te esoace Scaeig eoy Pysica Acousics, (1981). [4] M.S. Fokia, V.N. Foki, Numeica Modeig of e Refecio Coefficies fo e Pae Soud Wave Refecio fom a Layeed Easic Boom Acousica Pysics, 46, (). [] M.S. Fokia, V.N. Foki, Resoaces of acousic waves ieacig wi a easic seabed J. Compuaioa Acousics, 9, (1). [6] N.A. Haske, Te dispesio of suface waves o muiayeed media Bu. Seismo. Soc. Am., 43, (193). [7] J.F. Hook, Sepaaio of e veco wave equaio of easic fo ceai ypes of iomogeeous, isoopic media J. Acous. Soc. Am., 33, (1961). [8] R.E.Houz, J.I.Ewig, Deaied sedimeay veociies fom seismic efacio pofies i e Wese No Aaic J. Geopys. Res., 68, (1963). [9] F.C.Kaa, J.B.Kee, Easic-wave popagaio i omogeeous ad iomogeeous media J. Acous. Soc. Am., 31, (199). [1] L.A. Mokov, Te Maix Meod i e Teoy of Wave Popagaio i Easic Layeed ad Fuid Layeed Media, (Nauka, Leigad, 1984). [11] H.E. Mois, Boom-Refecio-Loss Mode wi a Veociy Gadie J. Acous. Soc. Am., 48, (197). [1] A.J. Robis, Geeaio of sea ad compessio waves i a iomogeeous easic medium J. Acous. Soc. Am., 96, (1994). [13] A.J. Robis, Refecio of pae acousic waves fom a aye of vayig desiy J. Acous. Soc. Am., 87, 46-3 (199). [14] S.R. Ruefod, K.A. Hawke, Effecs of desiy gadies o boom efecio oss fo a cass of maie sedimes J. Acous. Soc. Am., 63, 7-77 (1978). [] G.A. Semeov, Seismic modes of e oceaic sedime aye (SIO RAS, Moscow, 199). [16] W.T. Tomso, Tasmissio of easic waves oug a saified soid medium J. App. Pysics, 1, (19). [17] E.N. Towe, Te compuaio of e dispesio of easic waves i ayeed media J. Soud Vib.,, 1-6 (196).