Theory of Propagation of Elastic Waves in a Fluid-Saturated Porous Solid. I. Low-Frequency Range

THEJOURNALOFTHEACOUSTICALSOCIETYOFAMERICAVOLUME28.NUMBER2MARCH.1956TheoryofPropagationofElasticV•ravesinaFluid-SaturatedPorousSolid.I.Low-FrequencyRangeM.A.B•OT*ShellDevelopmentCompany,RCAB1dlding,NewYork,NewYork(ReceivedSeptember1,1955)Atheoryisdevelopedforthepropagationofstresswavesinaporouselasticsolidcontainingacompressibleviscousfluid.Theemphasisofthepresenttreatmentisonmaterialswherefluidandsolidareofcomparabledensitiesasforinstanceinthecaseofwater-saturatedrock.ThepaperdenotedhereasPartIisrestrictedtothelowerfrequencyrangewheretheassumptionofPoiseuilleflowisvalid.TheextensiontothehigherfrequencieswillbetreatedinPartII.Itisfoundthatthematerialmaybedescribedbyfournondimen-sionalparametersandacharacteristicfrequency.Therearetwodilatationalwavesandonerotationalwave.Thephysicalinterpretationoftheresultisclarifiedbytreatingfirstthecasewherethefluidisfrictionless.Thecaseofamaterialcontainingaviscousfluidisthendevelopedanddiscussednumerically.Phasevelocitydispersioncurvesandattenuationcoefficientsforthethreetypesofwavesareplottedasafunctionofthefrequencyforvariouscombinationsofthecharacteristicparameters.1.INTRODUCTIONThisisobtainedifa"dynamiccompatibility"relationisverifiedbetweentheelasticanddynamicconstants.ThisURofelasticpurposewavesistoinestablishasystematheorycomposedofpropagationofaporousisacasewheredissipationduetofluidfrictionwillelasticsolidsaturatedbyaviscousfluid.Itisassumeddisappear.InSec.6thedynamicrelationsarederivedthatthefluidiscompressibleandmayflowrelativetowiththeadditionoffluidviscosityandSec.7derivesthesolidcausingfrictiontoarise.PartIwhichisthepropertiesofthepropagationofthewaveswhenpresentedhereassumesthattherelativemotionofthedissipationispresent.ItisfoundthatthephasevelocityfluidintheporesisofthePoiseuilletype.AsalreadyofrotationalwavesincreasesslightlywiththefrequencypointedoutbyKirchhoffthisisvalidonlybelowafwhiletheabsorptioncoefficientisproportionaltothecertainfrequencywhichwedenotebyf•,andwhichsquareofthefrequency.Allplotsarepresentednon-dependsonthekinematicviscosityofthefluidandthedimensionallybyreferringtoacharacteristicfrequencysizeofthepores.Extensionofthetheoryinthefre-f•whichdependsonthekinematicviscosityofthefluidquencyrangeabovef•willbepresentedinPartII.Weandtheporediameter.Thecharacteristicfrequencyfchaveinmindparticularlytheapplicationtocaseswheremaybeconsideredasafrequencyscaleofthematerial,thefluidisaliquid,andwehavethereforedisregardedthenondimensionalabscissaoftheplotsbeingtheratiothethermoelasticeffect.Weincludeonlymaterialssuchf/ft.Therearetwodilatationalwavesdenotedaswavesthatthewallsofthemainporesareimperviousandforofthefirstandsecondkind.Thewavesofthesecondwhichtheporesizeisconcentratedarounditsaveragekindarehighlyattenuated.Theyareinthenatureofavalue.Extensiontomoregeneralmaterialswillbediffusionprocess,andthepropagationiscloselyanalo-consideredalongwiththethermoelasticeffectinalatergoustoheatconduction.Thewavesofthefirstkindareandmorecompletetheory.truewaves.ThedispersionispracticallynegligiblewithDevelopmentofthetheoryproceedsasfollows.Sec-aphasevelocityincreasingordecreasingwithfrequencytion2introducestheconceptsofstressandstraininthedependingonthemechanicalparameters.Theabsorp-aggregateincludingthefluidpressureanddilatation.tioncoefficientisproportionaltothesquareofthefre-Relationsareestablishedbetweenthesequantitiesforquencyasfortherotationalwaves.Incasesclosetothestaticdeformationinanalogywi.thaprocedurefolloweddynamiccompatibilitycondition,thedispersionandinthetheoryofelasticityforporousmaterialsdevelopedattenuationofthewavesofthefirstkindtendtovanish.inreference1.Section3considersthedynamicsoftheTheattenuationofthiswavemaythereforevarywidelymaterialwhenthefluidisassumedtobewithoutvis-formaterialsofsimilarcompositionandmaybelargercosity.Thiscaseisofpracticalinterestsincethisrepre-orsmallerthantheattenuationoftherotationalwaves.sentsthelimitingbehaviorofwavepropogationatveryAbeginningalongthepresentlineswasmadebyhighfrequency.ItintroducestheconceptsofapparentFrenke12Hediscussestherotationalanddilatationalmassesanddynamiccouplingbetweenfluidandsolid.waves,butthesubjectissummarilytreatedandimpor-Thewavepropogationintheabsenceoffrictionistantfeaturesareneglected.analyzedinSecs.4and5.ItisfoundthatthereisoneSoundabsorptioninmaterialcontainingairwastherotationalwaveandtwodilatationalwaves.Aremark-objectofextensiveworkbyZwikkerandKosten.aablepropertyisthepossibleexistenceofawavesuchRotationalwavesarenotconsidered,andsimplifiedthatnorelativemotionoccursbetweenfluidandsolid.equationsareusedforthedilatationalwaves.The2j.Frenkel,J.Phys.(U.S.S.R)8,230(1944).*Consultant.aC.ZwikkerandC.W.Kosten,SoundAbsorbingMaterialstM.A.Biot,J.Appl.Phys.26,182(1955).(ElsevierPublishingCompany,Inc.,NewYork,1949).168ELASTICWAVESINPOROUSSOLIDS.I169applicationsarepresentedmainlythroughtheconceptwithofimpedance.Thesameapproachisalsoemphasizede•=audax,etc.,byBeranek.4TheproblemwasalsodiscussedbyMorseunderthesimplifyingassumptionofarigidsolid.6In%=+--,etc.,thepresentseriesofpapersweshallattempttoestab-OyOxlishamorefundamentalapproachaimedatincludingsuccessivelyallpertinentphysicalmechanismsinawhereu:,u,,andu,arethecomponentsofthedis-theoreticMandquantitativewayasfarasitseemsplacementvectorofthesolid.Wemustkeepinmindpossible.thatthematerialisofporousorgranularstructureand2.STRESS-STRAINRELATIONSINAPOROUSthatweassumethesizeoftheunitelementstobelargeELASTICSOLIDCONTAININGAFLUIDcomparedtothepores.Thedisplacementvectorisdefinedhereasthedisplacementofthematerialcon-Consideravolumeofthesolid-fluidsystemrepre-sideredtobeuniformandaveragedovertheelement.sentedbyacubeofunitsize.ThestresstensorisInasimilarwaywemaytalkoftheaveragefluidseparatedintotwoparts.displacementvectorLI,U,U,definedsothattheprod-(1)Theforcecomponentactingonthesolidpartsofuctofthisdisplacementbythecross-sectionalfluidareaeachfaceofthecubeisonepart.Thisisdenotedastherepresentsthevolumeflow.ThestraininthefluidistensordefinedbythedilatationTitr.•it•'•.(2.1)au,au,au,,=++--(2.5)OxOya•(2)TheforcesactingonthefluidpartofeachfaceofthecubearerepresentedbyWenowproceedtoestablishtherelationbetweenthestressandstraincomponentsofthesolid-fluidaggregate0asdefinedintheforegoing.Forthetimebeingweshall$disregardalldissipativeforcesandassumethatweare{iøodealingwithaconservativephysicalsystemwhichisinThescalarsisproportionaltothefluidpressurepequilibriumwhenatrest.Anydeformationisthereforeaaccordingtodeparturefromastateofminimumpotentialenergy.--s=•p,(2.3)Inthefirstapproximation,therefore,thispotentialenergywillbeexpressedbyapositivedefinitequadraticwhere18isthefractionoffluidareaperunitcrosssection.form.Thesevenstresscomponentsa•r•r•%z,swillNotethatsistakennegativewhentheforceactingonthenbelinearfunctionsofthesevenstraincomponentsthefluidisapressurewhilea•azarepositivewhenthee•eitcz•%•,e.ThepotentialenergyWperurdtvolumeforceinthesolidisatension.ofaggregateisgivenbyItisimportanttocallattentiontothesignificanceofthisfactor•.Inthepresentanalysisweassumethatwe2W=•e•+½•e,+•,e,+r•%+r•%-•r•.(2.6)aredealingwithastatisticallyisotropicporousmaterialThisquadraticformexpressedintermsoftheseveninsuchawaythatforallcrosssectionswealwaysob-servethesameratioofthefluidareatosolidarea.straincomponentsinvolvesinthemoregeneralcaseHencethevolumeoffluidinathinslabofthicknessdxtwenty-eightindependentcoefficients.Thestress-strainrelationsmaybeexpressedasisalwaysafractiont8ofthetotalvolume.Thismeansthat•1isidenticalwiththequantityusuallydesignatedasporosityanddenotedintheliteraturebythesymbolf.(Thissymbolisusedheretodenotefrequency.)Itshouldbementionedthattheporosityconsideredhereisthatwhichisconnectedwithbulkmotionofthe(2.7)fluidrelativetothesolid,i.e.,thatrepresentedbytheinterconnectingvoidspace.Thisissometimesdesig-natedas"effectiveporosity."Sealedporespaceiscon-sideredaspartofthesolidforourpresentpurpose.WedenotethestraintensorinthesolidbyTheseven-by-sevenmatrixofcoefficientinthefore-goinglinearrelationsconstitutesasymmetricmatrixwithtwenty-eightdistinctcoefficients?*L.L.Beranek,J.Acoust.Soc.Am.19,556(1947).Theserelationsareverymuchsimplifiedifweintro-sR.W.Morse,J.Acoust.Soc.Am.24,696(1952).ducetheassumptionthatthesolid-fluidsystemis170M.A.BIOTstatisticallyisotropic.Inthiscasetheprincipalstressrequiredonthefluidtoforceacertainvolumeoftheandprincipalstraindirectionscoincide.Referringthefluidintotheaggregatewhilethetotalvolumeremainsstressesandstrainstothesedirectionsandtakingintoconstant.Itmustbeofpositivesign.ThecoefficientQaccountthepropertiesofsymmetryofthematerialtheisofthenatureofacouplingbetweenthevolumechangestress-strainrelationsreducetoofthesolidandthatofthefluid.Itssignificanceisillustratedbyputtingthefluidpressureequaltozero,at=Be•+C(eu+em)+O•inwhichcaseon=Benq-C(emq-e•)q-Qe•=-Qe/R.(2.13)(2.8)am=Beznq-C(etq-ett)q-Q•Sinceapressureonthesolidmusttendtoproduceas=Q'eq-R•decreaseintheporosityofthesolid,,andemustbeofwithoppositesignandQispositive.e•+ea+em=e.Onecandeviseexperimentsbywhichthefourelasticcoefficientsmaybemeasuredstatically.TheshearBecauseoftheexistenceofapotentialenergythemodulusmaybemeasureddirectly.Theothercoeffi-matrixofcoefficientsmustbesymmetric.Thismatrixiscientsarefoundbytestingforstressversusstraincon-finingthematerialbyanimperviousboundaryinoneBC.testandaperviousboundaryintheother.IntheformerCB(2.9)testthefluidpressuremustalsobemeasured.Theques-tionremains,ofcourse,todeterminewhethertheseQ,Q,Q,staticcoefficientsshouldbesubjecttoacorrectionforHence,wemusthaveapplicationtodynamicphenomena.Weshallinvestigatethisquestionmorethoroughlyinalaterpaper.Q'=Q.(2.10)Intheisotropiccasetherearethereforefourdistinct3.DYNAMICRELATIONSINTHEABSENCEelasticconstants.OFDISSIPATIONItispossibletowritethestress-strainrelationssome-WenowcallourattentiontotherelationbetweenwhatdifferentlybyintroducingnewconstantsNandstressesandaccelerationintheabsenceofdissipativeAinsteadofBandC.Wewriteforces.ItisconvenientheretointroducetheLagrangianviewpointandtheconceptofgeneralizedcoordinates.at=2Ne•+Ae+QeWeagainconsideraunitcubeoftheaggregateasanan=2Nen+Ae+Q•dement.Theelementisassumedtobesmallrelativetocr•'n'=2Neiii+Ae+Q•thewavelengthoftheelasticwavesandinturnthesizeoftheporesisassumedsmallcomparedtothesizeofthes=Qeq-R•.element.ThelimitationoffrequencieswhichisherebyInthisformitiseasytoexpressthemforarbitraryintroducedwillturnouttobeacademicformostprac-directionsofthecoordinateaxesbyusingtheinvarianceticalproblems.oftensorrelations.WefindOneimportantconsequenceofthisassumptionontherelativesizeofwavelengthandporesisthefactthattheaz=2Nezq-Ae+Qemicroscopicvelocitypatternisthesameasifthefluidav=2N%+Aeq-Q•wereincompressible.Thisfollowsfromageneralprin-ciplethatavelocityfieldincompressiblefluidsapproxi-a,=2Ne,+Ae+Q•matesthatofanincompressiblefluidforobstacleswhich(2.12)aresmallcomparedtothewavelength.Hence,themicro-scopicflowpatternofthefluidrelativetothesoliddependsonlyonthedirectionoftherelativeflowandnotonitsmagnitude.Thisassumptionisvalidifwes=Qe+Reneglecttheviscosityandconsiderthatwearedealingwithwithaperfectfluid.Inthiscase,themicroscopicveloc-e=e•q-%q-ityfieldwillbealinearfunctionofthesixaveragevelocitycomponentsofthesolidandthefluid.TheInexaminingthesignificanceoftheseconstantsweLagrangiancoordinatesarechosenasthesixaveragenoticethatAandNcorrespondtothefamiliarLam•displacementcomponentsofthesolidandthefluid,i.e.,coefficientsinthetheoryofelasticityandareofpositiverespectively,sign.u•,uy,u,,U•,U•,Uj.(3.1)ThecoefficientNrepresentstheshearmodulusofthematerial.ThecoefficientRisameasureofthepressureThekineticenergyTofthesystemperunitvolumemayELASTICWAVESINPOROUSSOLIDS.I171beexpressedasfluidperunitlengthis[/Ou•\2iOuu\•(Ou,'••1Opaxor(3.0)ro,,ou.o,,,ou,OpO2u,OfOtOtatOtJOxOt2Theleft-handsideistheforceQ•actingonthefluidper(3.2)unitvolume.Hence,takingintoaccountEq.(3.7),wemaywriteThisexpressionisbasedontheassumptionthattheQ•=o•ß(3.10)materialisstatisticallyisotropichencethedirectionsx,y,zareequivalentanduncoupleddynamically.LetNow,inthecaseusdiscussthesignificanceoftheexpressionforthekineticenergy.ThecoefficientsmlmWnaremasscoefficientswhichtakeintoaccountthefactthatthethesecondequation(3.3)becomesrelativefluidflowthroughtheporesisnotuniform.(o•2+o22)u,•=Q,•.(3.11)Indiscussingthesignificanceofexpression(3.2)wemay,withoutlossofgenerality,consideramotionComparingEqs.(3.11)and(3.10)wederiverestrictedtothex-direction.Ifwedenotebyq,theP•=PI2"•P=•.(3.12)totalforceactingonthesolidperunitvolumeinthex-directionandbyQ•,thetotalforceonthefluidperCombiningthisresultwithrelations(3.5)and(3.8)weunitvolume,wederivefromLagrange'sequationshavealsom=On+On.(3.13)O(OT]02Thecoefficiento•2representsamasscouplingparameter(3.3)betweenfluidandsolid.Thisisillustratedbyassumingthatinsomewaythefluidisrestrainedsothattheaver-=--(m•u,+•U•)=Q,.agedisplacementofthefluidiszero,i.e.,U•=0.Equations(3.3)arethenwritten,Beforeapplyingtheserelations(3.3)letusfurtherdiscussthenatureofthecoefficients•n•2•0•2.Letusq•=p•r--assumethatthereisnorelativemotionbetweenfluid(3.14)andsolid.Inthiscase02uzQz=pxzotand2r=(o,•+2m•+o•)u•.(3.4)Thesecondequation(3.14)showsthatwhenthesolidisWeconcludethatacceleratedaforceQ•mustbeexertedonthefluidtopreventanaveragedisplacementofthelatter.Thism•+2m,+o•=o(3.5)effectismeasuredbythe"coupling"coefficientp•2.TheforceQ•necessarytopreventthefluiddisplacementisrepresentsthetotalmassofthefluid-solidaggregateobviouslyinadirectionoppositetotheaccelerationofperunitvolume.Wemayalsoexpressthisquantitybythesolid;hence,wemustalwayshavemeansoftheporosity•andthemassdensitiesp•ando•forthesolidandfluid,respectively.Themassofsolidm2<0.(3.15)perunitvolumeofaggregateisThesameconclusionisreachedbyconsideringthefirstm=(1--•)p,,(3.6)equation(3.10)inwhichmlrepresentsthetotaleffectivemassofthesolidmovinginthefluid.Thistotalmassandthemassoffluidperunitvolumeofaggregatemustbeequaltothemassproperofthesolidp•plusanm=Ot,•.(3.7)additionalmassp•duetothefluid.Hence,Oil=pl-'{-Oa.(3.16)p=pi-{-p2=p,-{-l•(.ol--p•).(3.8)FromEqs.(3.13)and(3.16)wederiveLetusnowagainassumethatthereisnorelativemotionbetweensolidandfluid.Thepressuredifferenceinthem2=-o•.(3.17)172M.A.BIOTHence,m2istheadditionalapparentmasswithachangedirection,insign.Therefore,thedynamiccoefficientsmaybeOeOe02writtenNv2u+(.4+2v)--+Q--=OxOx(•.•)OeOe0•m2=•2+o•(3.18)P12=mpa.OxOxFurtherconditionsmustbesatisfiedbythesedynamicandtwoothersimilarequations,respectively,forthecoefficientsifthekineticenergyistobeapositivedirectionsyandz.Withthevectornotationdefinitequadraticform.Thecoefficientspuandp22mustu=bepositivePll>0P22>0u=andEquations(5.1)arewritten0•pno22--m22>O.(3.19)mV•u+grad[(A+N)e+Qe•=Theseinequalitiesarealwayssatisfiedifthecoefficients(4.2)aregivenbytherelations(3.18)wherepw•andp•are0•positivebytheirphysicalnature.grad[Qe+Re•=•(mgu+p•U).Ot2Intermsofstressestheforcecomponentsareex-pressedasstressgradients,i.e.,Thesesixequationsforthesixu•nowncomponentsofthedisplacementsuandUcompletelydeterminetheOa•Or,Or,propagation.q•=--+--q---OxOyOz'(3.20)Becauseofthestatisticalisotropyofthematerial,itcanbeshownthattherotationalwavesareuncoupledQ•=Os/Ox,etc.fromthedilatat/onalwavesandobeyindependentequationsofpropagation.ThisisdoneintheusualHence,wehavethedynamicequationswaybyintroducingtheoperationsdivandcud0•,Or•0%02divu=edivU=•---½---[-(4.3)OxOyOzOFcurlu=tocurlU=(3.21)0502Applyingthedivergenceoperationtobothequations--=--(o•2u•+p22U•),etc.(4.2),weobtainOxOFW(Pe-VQO=Strictlyspeaking,thegeneralizedforcesaredefinedasthevirtualworkofthemicroscopicstressesperunit(4.4)valueofthedisplacementvectoruandUandnotastheaverageofthemicroscopicstressesasusedinexpressions(3.20).However,forallpracticalpurposesitisjustifiedwiththedefinitiontouseeitherdefinition.ItisinterestingtonotethatbecauseofthecouplingP=A+2N.(4.5)coefficientanaccelerationofthesolidwithoutaverageThesetwoequationsgovernthepropagationofmotionofthefluidproducesapressuregradientinthedilatationalwaves.Asdiscussedinmoredetailinthefluid.Thisisphysicallycausedbyanapparentmassnextsection,itcanalreadybeseenthat,ingeneral,effectofthefluidonthesolid.therewillbetwosuchdilatationalwavesandthateachEquations(3.21)arereferredtothex-direction.ofthesewavesinvolvescoupledmotioninthefluidandIdenticalequationsmaybewrittenforthey~andthesolid.z-directions.Similarly,applyingthecurloperationtoequations(4.2)weobtain4.EQUATIONSOFPROPAGATIONOF0=PURELYELASTICWAVESEquationsforthewavepropagationareobtainedbysubstitutingexpressions(2.12)forthestressesintothe02(4.6)dynamicalrelat/ons(3.21).Weobtainforthex-=o.ELASTICWAVESINPOROUSSOLIDS.I173Theseequationsgovernthepropagationofpurerota-definedbytionalwaves.ItisseenthattheseequationsimplyalsoV?=H/o,(5.4)acouplingbetweentherotationtoofthesolidandthatofthefluid•.Theseequationswillnowbediscussed.withH=P+Rq-2Q.Referringtothedefinitionofoasrepresentingthemassperunitvolumeofthefluid-5.PROPERTIESOFTHEPURELYELASTICWAVESsolidaggregateitisseenbyaddingEqs.(4.4)thatV,representsthevelocityofadilatationalwaveintheConsiderfirst,Eqs.(4.6)fortherotationalwaves.aggregateundertheconditionthate=e,i.e.,iftheByeliminatingflintheseequations,wefindrelativemotionbetweenfluidandsolidwerecompletelypreventedinsomeway.ThefollowingnondimensionalNv%=,,,{1-.(5.1)parametersarefurtherintroduced\PnP•Ot•PJtQThereisonlyonetypeofrotationalwave.ThevelocityO'11•--0'22•--ffi2•--HHHofpropagationofthesewavesis(5.5)N«pll,o22(5.2)P ppt 关于艾滋病ppt课件精益管理ppt下载地图下载ppt可编辑假如ppt教学课件下载triz基础知识ppt he•o.-parametersdefinetheelasticpropertiesoftheTherotationtoofthesolidiscoupledproportionallytomaterialwhilethe%fparametersdefineitsdynamictherotation•2ofthefluidaccordingtotherelationproperties.Sincewehavetheidentitiesp12rl....to.(5.3)*n+a•q-2a•==•'n+•'•+2•=1,(5.6)P•gthereareonlyfourindependentparameters.WenoteSincep,•isnegativeandp•positive,therotationofthethatthepositivecharacterofthekineticandelasticfluidandthesolidareinthesamedirection.Thismeansenergiesimplythat•n•22--•22and-•n-•22--Tt2aarethatarotationofthesolidcausesapartialrotationalpositive.entrainmentofthefluidthroughaninertiacoupling.Withtheseparameters,Eqs.(4.4)areItthiscouplingdidnotexistandthefluidwouldstayat1O•restontheaveragetherotationalwavevelocitywould=--beV,=(N/pn){,wherepnrepresentsthemassoftheV••Ot•solidplustheapparentmassduetotherelativemotion(5.7)ofthesolidinthefluid.Actually,thepartialrotational103entrainmentofthefluidbythesoliddecreasestheVfOFapparentmasseffectwithacorrespondingincreaseinthewavevelocity.ThisdecreaseoftheapparentmassSolutionsoftheseequationsarewrittenintheformisexpressedbythefactor[-1-(m•/pno•O]informula(5.2).e=C•exp[i(Ix+at)']Theexistenceofarotationinthefluidseemsatfirst(5.8)sighttobeincontradictionwithKelvin'stheoremthate=C•exp[-i(Ix+at)'].inafrictionlessfluidwithoutbodyforcesnocirculationcanbegenerated.However,thevelocityfieldUhereThevelocityVofthesewavesisconsideredisnottheactualmicroscopicvelocitybuttheV=a/l.(5.9)averagevolumeflow.ThecirculationoftheformerremainszeroinconformitywithKelvin'stheoremThisvelocityisdeterminedbysubstitutingexpressionswhilethelineintegralofthevolumeflowcanbedifferent(5.8)into(5.7).fromzero.ThedistinctionisthesameasmadewhenwePuttingconsidertheapparentrotationalinertiaofabodyz=V•=/¾•,(5.10)immersedinafluid.Arotationofthebodyinthefluidweobtainproducesanangularmomentuminthefluidwhilethecirculationremainszero.Wenowconsiderthedilatationalwavesdefinedby(5.11)•(o-•=C•-t-•==C=)='ti=Ci-I-T==C=.Eqs.(4.4).AllessentialfeaturesarebroughtoutbydiscussingthepropagationofaplanewaveparallelwithEliminatingC•andC=yieldsanequationforz,theyz-planeandofnormaldisplacementuxandUsinthex-directionforthesolidandthefluid,respectively.ItisconvenienttointroduceareferencevelocityV,+(5.12)174M.A.BlOTThisequationhastworootsZlZ2correspondingtotwosignalwavewillbedesignatedasthewaveofthefirstkindvelocitiesofpropagationV•,V2,andthelow-velocitywaveasthewaveofthesecondkind?Vx•=vc2/z•6.EQUATIONSOFPROPAGATIONWI-IEN(5.13)DISSIPATIONISINTRODUCEDV22--V$/z•.ItwillbeassumedthattheflowofthefluidrelativetoTherearethereforetwodilatationalwaves.TherootsthesolidthroughtheporesisofthePoiseuilletype.Thatz•,z•arealwayspositive,sincethematricesofcoeffi-thisassumptionisnotalwaysvalidiswellknown,e.g.,cientsaand7ofEqs.(5.11)aresymmetricandarewhentheReynoldsnumberoftherelativeflowexceed