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Electronicmail:vach@leonardo.polytechnique.frJ.Acoust.Soc.Am.119,June2006©2006AcousticalSocietyofAmerica37930001-4966/2006/119 thatthebest-Þtvalueofwillbecloserto5thanto7.AsshowninFig.2,thebest-Þtvaluederivedfromthemicro-phoneexperimentsisinfactabout5.5.Thesameanaly-sishasbeendoneforthemodewhichleadstoandtherefore8.210.2.Thisisingoodagreementwiththevalue8takenfromtherespectivehologram.About25darkfringesarevisibleinFig.4givesanapproximateamplitudeofm;forFigs.4and4sevendarkfringesconsideringtheem-ployedrecordinggeometry.Theamplitudeclearlydecreaseswiththeliquidlevel,presumablyduetotheincreasedinertiaandthelessefÞcientcouplingoftheloudspeakertotheglassasthefrequencydecreases.TheparameterinEq.isproportionaltothedensityoftheliquidÞlledintotheglass.Inordertocheckformulawemeasuredtheresonancefrequenciesforliquidswithdifferentdensities.Figure6showsthedataforliquidswiththreedifferentdensities,dichloromethane1.32kg/l,water1kg/l,andisopropanol0.79kg/l.FromaÞtaccordingtoformulaweobtaintheratios=1:0.80:0.59,whichisingoodagreementwiththerealden-sityratiosof1:0.75:0.60.WecandirectlyconcludefromFig.4thatthepresenceoftheliquiddoesnotmarkedlymodifythevibrationalshapeofthequadrupolareigenmodes.Itis,forinstance,impossibletodeterminefromthephotographsuptowhichleveltheglasswasÞlled.AplotofthepositionsofthebrightfringesinthepatternsofofFig.4forapartlyandacompletelyÞlledglassusingthelogarithmicscaleofFig.5infactgivesstraightlinescloselyparalleltothatdrawnformodeofanemptyglass,showingthattheassump-tionoflittlechangeinthevibrationalmodefunctionisjus-tiÞed.Themaindifferencearisingfromtheliquidisthefactthatthenodallinenolongerreachestherimoftheglass.Inspiteoftherelativelyhighvibrationalamplitudefortheemptyglass,thebrightwhitenodallineclearlyreachestherimoftheglass,whileinthecaseofthehalf-fullglasstheupperpartappearsentirelydark.InthecaseofacompletelyÞlledglass,thisphenomenonisevenmorepronounced.Weattributethiseffecttothefactthatthecouplingofthetwodegeneratemodesshouldleadtoanenergyexchangebe-tweenthem.Theresultingmodulationofthevibrationalmodes,whichisexpectedtohaveafrequencyofafewHz,however,darkenstheimageoftheglasscompletelyovertheintegrationtimeof50sifthemodulationamplitudeislarger/2.Splittingoftheresonancesindegeneratemodes,asshowninFig.3,isalmostcertainlyduetoasymmetryinthethicknessoftheglasssothatthesineandcosineversionsofEq.haveslightlydifferentfrequencies.Whenliquidisaddedtotheglass,thisreducestheasymmetrybecausethereisanequalliquidmassaddedtoallpartsofthewall.Con-sequently,theadditionalmassappearstoshiftthetworeso-nancepeakstoapointwherethetwonearlydegeneratemodescannotberesolvedanymorecompareFig.3leadingtotheirsimultaneousexcitation.ThisphenomenoncanalsobenicelydemonstratedbytheuseoftherotatingÞngerexcitation:TheamplitudeofthesignalemittedbytheemptyglassasrecordedwithourmicrophoneisstronglymodulatedsincetheexcitationfollowstheÞnger.Asthenodepassesinfrontofthemicrophonetheamplitudebe-comesnearlyzerowhenasufÞcientlydirectionalmicro-phoneisemployed.WhenÞlledwithwater,however,theamplitudenevervanishesentirely.Tentatively,weattributethischangeinmodulationamplitudetothesimultaneousex-citationofthetwoorthogonalquadrupolarmodeswhichbe-comespossibleduetothepresenceoftheliquid.Asacon-sequence,thecoupledliquid-glasssystemclearlybehavesdifferentlyfromanemptyglass.Asimilarbehaviorhasal-readybeenreportedforcylindricalsteeltanksÞlledwithwater.11Ð13 FIG.5.Vibrationamplitudeinunitsof/2,asmeasuredbythepro-gressionofthebrightfringescorrectedforwallangle,plottedagainstheightabovethebaseinarbitraryunitsasmeasuredfromanenlargedphoto-forthefundamentalmodewithdifferentliquidlevelsandformode.Thestraightlinesarepower-lawregressionsasin.Notethattheslopeofallthemoderegressionsisthesame,indicatingthatadditionoftheliquiddoesnotsigniÞcantlymodifythemodeshape.DifferingheightsforcorrespondingbandsforthemodearisefromdifferentexcitationefÞciencies. FIG.6.TheplotshowstheresonancesoftheglassÞlledwithdichlo-romethane,water,andisopropanol,respectively.ThedensityoftheliquidinuseisproportionaltotheparameterJ.Acoust.Soc.Am.,Vol.119,No.6,June2006Jundtetal.:Vibrationalmodesofwineglasses3797