Theory-of-Propagation-of-Elastic-Waves-in-a-Fluid-

THEJOURNALOFTHEACOUSTICALSOCIETYOFAMERICAVOLUME28.NUMBER2MARCH,19.56TheoryofPropagationofElasticWavesinaFluid-SaturatedPorousSolid.H.HigherFrequencyRangeM.A.BIOS*ShellDevdopmentCompany,RCABuilding,grewYork,grewYork(ReceivedSeptember1,1955)ThetheoryofpropagationofstresswavesinaporouselasticsoliddevelopedinPartIforthelow-frequencyrangeisextendedtohigherfrequencies.ThebreakdownofPoiseuilleflowbeyondthecriticalfrequencyisdiscussedforporesofflatandcircularshapes.AsinPartItheemphasisofthetreatmentisoncaseswherefluidandsolidsareofcomparahledensities.Dispersioncurvesforphaseandgroupvelocitiesalongwithattenuationfactorsareplottedversusfrequencyfortherotationalandthetwodilationalwavesandforsixnumericalcombinationsofthecharacteristicparametersoftheporoussystems.Asymptoticbehaviorathighfrequencyisalsodiscussed.1.INTRODUCTIONAPREVIOUSpaperdealingwiththesubjectofpropagationofelasticwavesinafluidsaturatedporoussolidwasrestrictedtothelow-frequencyrange.BythisitwasmeantafrequencyrangebetweenzeroandacertainvaluebeyondwhichtheassumptionofPoiseuilleflowbrokedown.Thepurposeofthispaperistoextendthetheorytothefullfrequencyrangewithoutthelimitationoftheforegoingassumption.Thereremainshoweveranupperboundforthefre-quency,namely,thatatwhichthewavelengthbecomesoftheorderoftheporesize.Suchacasemust,ofcourse,betreatedbyadifferentmethod.AtheoreticalstudyofthebreakdownofPoiseuilleflowispresentedinSecs.2and3,byconsideringtheflowofaviscousfluidunderanoscillatorypressuregradienteitherbetweenparallelwallsorinacirculartube.ThecaseofthecirculartubewasoriginallytreatedbyKirkhoff.Thisstudyyieldsacomplexviscositycorrectionfactorfunctionofthefrequencythroughthedimensionlessratiof/fwherefisacharacteristicfrequencyofthematerial.Thecaseofflowbetweenparallelwallsandthatofthecirculartubeindicatethattheeffectofporecross-sectionalshapeiswellrepre-sentedbytakingthesamefunctionofthefrequencyfortheviscositycorrectionandsimplychangingthefre-quencyscale.AsinPartIweareprimarilyconcernedwithapplicationstoliquidsandwehaveneglectedthethermoelasticeffects.ApplicationoftheseresultstofluidfrictioninaporousmaterialisdiscussedinSec.4andastructuralfactorisintroducedwhichrepresentstheeffectofsinuosityandshapeofthepores.ThepropagationofrotationalwavesisdiscussedinSec.5.Fournumericalcombinationsofparametersareconsidered.Groupvelocity,phasevelocity,andattenua-tionareplottedforthesefourcasesasafunctionofthefrequencyratiof/ft.Thereisonlyonetypeofrotationalwave.Theinfluenceofthestructuralfactorisalso*Consultant.M.A.Blot,J.Acoust.Soc.Am.28,168(1956),precedingpaper.evaluatedbycalculatingphasevelocityandattenua-tionforatypicalcase.ThepropagationofdilatationalwavesisdiscussedinSec.6.Groupvelocity,phasevelocity,andattenuationcurvesareplottedforsixnumericalcombinationsoftheparameters.Therearetwotypesofsuchwaves,desig-natedaswavesofthefirstandsecondkind.Thelatterarecharacterizedbyhighattenuation.Aninterestingplotisthatoftheattenuationpercycle.Boththerotationalwavesandthewavesofthefirstkindexhibitamaxi-mumvalueofthisattenuationinarangeoff/fcnearunity.Inthisrangetheinertiaandviscousforcesareofthesameorder.AsdiscussedinPartIwhenthedynamiccompati-bilityconditionissatisfiedornearlysatisfied(z=''l)thewaveofthefirstkindhasaverysmallattenuation.Thisisshownbycases2and5.Theothertwowaves,however,retainmuchhigherattenuation.Insuchacaseonlyonetypeofwavemaybeobservedunlessspecialattentionisgiventotheothers.Anotheraspectofthisphenomenonwillbeexhibitedwhenadilatationalwaveisreflectedatasurfaceofdiscontinuity.Thereflectedenergyissplitupintothreetypesofwaves,twoofwhichmaybeunobservedbecauseoftheirhighattenuation.Thephenomenonthenappearsasthepropagationofasingle-typebodywavewithsmallattenuationinthebodyandahighabsorptionatthereflectionsurface.Certainassumptionsuponwhichthepresenttheoryisbased,suchasperfectelasticityofthesolid,limitationsonthenonuniformityofporesize,andtheneglectionofthermaleffectswilldeterminethecategoriesofmaterialsandfrequencyrangesforwhichitisapplicable.Itshould,however,beofvaluebeyonditsstrictapplica-bilitybyindicatingordersofmagnitudesorqualitativetrends.Inapplicationstowavepropagationinsuchma-terialsasclay,silts,ormuds,oneshouldnotethattherotationalwaveisdeterminedentirelybytheshearingrigidityofthesolid.Sincethelattermaybesmall,therotationalwavesmay,inthiscase,propagatewithavelocitywhichisconsiderablylowerthanthatofthedilatationalwavesoffirstandsecondkind.179180M.A.BIOT2.OSCILLATORYFRICTIONFORCEINATWO-DIMENSIONALDUCTWeareinterestedinthemotionofafluidinatwo-dimensionalduct,i.e.,aspacelimitedbytwoparallel-planeboundarieswhentheseboundariesaresubjecttoanoscillatorymotionandwhenanoscillatorypres-suregradientactsatthesametimeonthefluid.Weconsideronlythetwo-dimensionalmotionandneglectallpressuregradientsandvelocitycomponentsnormaltotheboundaries.Thez-directionisparalleltotheboundariesandthey-axisisnormaltoitwiththeboundariesrepresentedbyy=4-a.Thex-componentofthevelocityoftheboundaryis/andthatofthefluid0(Fig.1).Thelattercomponenthasadistributionalongywhichistobedetermined.Theequa