FUNDAMENTALS OF ACOUSTICS(9)

FUNDAMENTALSOFACOUSTIC(9)THEENERGYRELATIONSHIPSINSOUNDFIELD3-4-1SOUNDENERGYDENSITYTheenergytransportedbyacousticwavesthroughafluidmediumisoftwoforms:•Thekineticenergyofthemovingparticles;•Thepotentialenergyofthecompressedfluid.ConsiderasmallfluidelementthatmoveswiththefluidandoccupiesvolumeV0oftheundisturbedfluid.ThekineticenergyofthiselementisThechangeinpotentialenergyassociatedwithavolumechangefromV0toVisWhere200)(21uVEK00VThemassoftheelement,iscalculatedusingthedensityandvolumeoftheundisturbedfluid.VVVVppdVPdVE00•Wherethenegativesignindicatesthatthepotentialenergywillincreaseasworkisdoneontheelementwhenitsvolumeisdecreasedbyactionofapositiveacousticpressurep.•Fromconservationofmasswehave)(,0200cpPVddV00VV(theequationofstate)Soweobtain:•Thetotalacousticenergyofthevolumeelementis02222000002000()()22pVVpEcdcVc22000201122kppEEEVuVcTheinstantaneousenergydensity:2022002121cpuVETheenergyperunitvolume,Injoulespercubicmeter(J/m3)2022002121cpuVETheinstantaneousparticlespeedandacousticpressurearefunctionsofbothpositionandtime,andconsequentlytheinstantaneousenergydensityisnotconstantthroughoutthefluid.Soundenergyfluxdensitysoundenergytransmittedthroughaunitareaandaunittimenormaltothedirectionofsoundtransmission.FdpdSdpdSddppudSdtdtSoundenergySoundenergyfluxdensity3-4-3Soundintensityandsoundpower•TheacousticintensityIofasoundwaveisdefinedastheaveragerateofflowofenergythroughaunitareanormaltothedirectionofpropagation.•Itsfundamentalunitsarewattspersquaremeter(W/m2).•Thesoundintensityisthetimeaverageofsoundenergyfluxdensity01TIpudtTItcanbewritteninanotherform•Wheretheintegrationistakenoveratimecorrespondingtotheperiodofonecompletecycle.Forharmonicwavesthisyields01Re()Re()TIpudtTcos21mmupITotalsoundpoweremittedbyasourceinalldirectionsMeasuredinwatts(joules/second)SoundpowerScWaasWIdS•Theamountofpowerproducedbysoundgeneratorisverysmallcomparedwithusualelectricalpowers.•Theaveragepowerproducedbyapersontalkinginanordinaryconversationaltoneisabout10-5watt,or100ergspersec;•Therangeofpowerthatcanbeproducedbythevoicevariesfromabout1ergpersecforveryweakspeechtoabout104ergspersecforloudspeech.PressureandintensitylevelItiscustomarytodescribesoundpressureandintensitiesthroughtheuseoflogarithmicscalesknownassoundlevels.Onereasonfordoingthisisthathumansjudgetherelativeloudnessesoftwosoundsbytheratiooftheirintensities,alogarithmicbehavior.•Theresponseofearisnotproportionaltotheintensity,however;itismuchmorenearlyproportionaltothelogarithmoftheintensity.•Ifweincreasetheintensityofasoundinstepsofwhatseemtobeequalincrementsofloudness,wefindthattheintensitiesformasequenceofthesort1,2,4,8,16…or1,10,100,1000…andnotofthesort1,2,3…or1,19,28….•Asecondreasonisthattheverywiderangeofsoundpressureandintensitiesencounteredinouracousticenvironment,audiblesoundpowerrangefromapproximately10-5to109.•Theuseoflogarithmicscalecompressestherangeofnumbersrequiredtodescribethiswiderangeofintensities.•Thehumanearisaremarkablyruggedyetsensitiveorgan.•Itrespondstoafrequencyrangeofabouttenoctaves.•Itrespondstoairvibrationswhoseamplitudeishardlymorethanmolecularsize;•Consequently,alogarithmicscaleisoftenchoseninwhichtoexpressacousticenergiesandintensities.Theunitisthedecibel(abbreviateddB)•Thedifferenceinleveloftwosoundsindecibelsisequaltotentimesthelogarithmtothebase10oftheratiobetweentheintensities:10010log(/)xLxxWherex0isareferencequantityNotethatthedecibelisnotanabsolutemeasurebutisreferencedtoaselectedquantityx0Soundintensitylevel0SIL10lg(/)II20/1mpWIReferenceintensityThereferencestandardforairbornesoundsis10-12W/m2,whichisapproximatelytheintensityofa1000Hzpuretonethatisjustbarelyaudibletoapersonwithunimpairedhearing.(thequietestsoundwecanhear)Thesoundintensityofthethresholdofhearingis10-12watts/m2•Whilstthesoundintensityatthethresholdofpainisabout10watts/m2SoundPowerlevel0SWL10lg(/)aWWReferencepowerpWW10MediumReferenceNearlyequivalenttoAir10-12W/m220uPaWater6.76*10-9W/m21uPaTheearismostsensitiveatabout1000to3000cps(cyclespersecond)0SPL20lg(/)eppPap200SoundpressurelevelReferenceeffectivepressurePap10InairInwaterWherepe=rmssoundpressureinPaThesoundpressurelevelatthethresholdofhearingisthus5521020log20log10210SPLdBTableofSoundPressureLevelsandCorrespondingSoundPressureandSoundIntensityExamplesSoundPressureLeveldBSoundPressurePaSoundIntensitywatts/m²30mfromjetaircraft140200100Thresholdofpain13010120201Chainsaw1100.1Disco10020.01900.001Busyroad800.20.0001700.00001Conversationalspeech600.020.000001500.0000001400.0020.00000001Quietbedroomatnight300.000000001BackgroundinTVstudio200.00020.0000000001100.00000000001Thresholdofhearing00.000020.0000000000013-6HARMONICPLANEWAVES•Ifalltheacousticvariablesarefunctionsofonlyonespatialcoordinate,thephaseofanyvariableisaconstantonanyplane,suchawaveiscalledaplanewave.•Ifthecoordinatesystemischosensothatthisplanewavepropagatesalongthexaxis,2222(34)pcpt0zpyp222022ppctx•Thewaveequationanditssolution00,,xctxct02