FUNDAMENTALS OF ACOUSTICS(2)

Chapter1ParticlesVibratingSystemsIntroductionAcousticsmaybedefinedasthegeneration,transmission,andreceptionofenergyintheformofvibrationalwavesinmatter.•Thisfirstchapterintroducesthebasicmathematicsofsimplevibratingsystems.•Thetreatmentislimitedtoelectromechanicaldevicesbecausethesearebyfarthemostwidelyusedandlendthemselvestomanypurposes.•Accordingtofunction,wemaygroupanelectromechanicalsourceintothreesections:•anelectricalsectionwithtwoormoreterminals;•amechanicalsectionwithtwoormoreterminals;•andasectionbetween,whichhasthecapabilityofconvertingeventsatonesetofterminalsintorelatedeventsattheotherset.•Theacousticalengineerisinterestedinthefidelityofreproductionofsound•Theconversionofmechanicalandelectricalenergyintoacousticalenergy,•Thedesignofacousticaltransducers.•Thearchitectismoreinterestedintheabsorptionandisolationofsoundinbuildings,andincontrolledreverberationandechopreventioninauditoriums.•Themusicianlikestoknowhowtoobtainrhythmiccombinationsoftonesthroughvibrationsofstrings,aircolumns.•Sounddoesnotconsistofairmovingtowardsusinbulk;ittravelsthroughtheairasasoundwave.•Asavibratingbodymovesforwardfromitsstaticequilibriumposition,itpushestheairbeforeitandcompressesit.Atthesametime,ararefactionoccursimmediatelybehindthebody,andairrushesintofillthisemptyspaceleftbehind.Inthiswaythecompressionofairistransferredtodistantpartsandairissetintoamotionknowassoundwaves..1.WAVES•Soundwaves(influids)arelongitudinalwaves---theparticlesmoveinthedirectionofthewavemotion.•Propagationofsoundwavesinvolvesthetransferofenergythroughspace.•Whilesoundwavesspreadoutinalldirectionsfromthesource,theymaybereflectedandrefracted,scatteredanddiffracted,interferedandabsorbed.•Amediumisrequiredforthepropagationofsoundwave,thespeedofwhichdependsonthedensityandtemperatureofthemedium.THESOUNDWAVEPRODUCESSOUNDPRESSURECHANGESThepressurechangesproducedbyasoundwaveareknownasthesoundpressure.Comparedwithatmosphericpressure(about100000pascals(1Pa=1N/m2)theyareverysmall(between20micropascalsand200pascals)andaresuperimposedonitSoundpressuresuperimposedonatmosphericpressureThisprocesscanbevisualizedifyoucangetholdofacoiledspring.Vibrateoneendandwatchthecoilscompressandstretchout.PERIODANDFREQUENCYPeriodisthetimetakenforonevibrationcycle.ItssymbolisTanditsunitisseconds(s).Frequencyisthenumberofvibrationcyclespersecond.Itssymbolisfanditismeasuredinunitscalledhertz(Hz)(namedafterHeinrichHertz1857-1894theGermanphysicistwhostudiedelectromagneticwaves).FREQUENCYRATESoundsoveralargerangeoffrequenciescanbeproduced.Thelowestnoteonthepianois28Hz,thehighestis4186HzTherangeaudibletoyoungpeoplewithundamagedhearingisabout20Hzto20000Hzandforadults20to16000Hz.Soundwithafrequencybelow20Hziscalledinfrasound.Soundwithafrequencyabove20000Hziscalledultrasound.Asoundmaycontainwavesofonlyonefrequency,inwhichcaseitiscalledapuretone.Noiseismadeupofwavesofmanydifferentfrequenciesandmagnitudessuperimposedononeanother.SPEEDOFSOUNDDEPENDSONTHEMEDIUMOFTRANSMISSIONThespeedofsounddependsonthemedium(itselasticityanddensity)anditstemperature.Inairthespeedofsoundis340m/s,Inwaterthespeedofsoundis1470m/sandinsteel,5050m/s.Thecowboywillhearthetrainnoiseviatherailsbeforehehearsitthroughtheair2SIMPLEHARMONICMOTION(SHM)•Instudyingvibrationsitisadvisabletobeginwiththesimplesttype:aone-dimensionalvibrationthathasonlyasinglefrequencycomponent(apuretone)•Ifanyobjectisdisplacedslightlyfromequilibriumitwilloscillateaboutitsequilibriumpositioninwhatiscalledsimpleharmonicmotion(SHM).SimpleHarmonicMotionThisfundamentalvibratingsystemhasbeenchosenasaninstrumentfortheintroductionofthebasicconceptsutilizedinthedescriptionofsound.xmD•Simpleharmonicmotionisanabstractionconventionallyrepresentedbyamassmconcentratedatapointandattachedtoaspringwhichhastheotherendfixed.Inadditiontobeingweightless,thespringobeysHooke’slaw.•whenitissubjecttothelinearelasticrestoringforcegivenbyHooke'sLaw.Themotionissinusoidalintimeanddemonstratesasingleresonantfrequency.Hooke'sLaw:Anelongationperunitlengthbearsalinearrelationtotheforceproducingit.Letusassumethattherestoringforcefinnewtons(N)canbeexpressedbytheequationFs=-kx(1)wheretheequilibriumpositionischosentohavex-coordinatex=0,xbeingpositivewhenthespringisextended,andnegativewhenthespringisshortened.andkisaconstantthatdependsonthesystemunderconsideration.TheunitsofDare:f=-Dx(1-1)D=Note:•ThenegativesigninHooke'slawensuresthattheforceisalwaysoppositetothedirectionofthedisplacement.•TheconstantkinHooke'slawistraditionallycalledthespringconstantforthesystem,evenwhentherestoringforceisnotprovidedbyasimplespring.Ifthedisplacingforcefisjustsufficienttoholdthemassstationaryatthepointx,thesumofforcesonmiszero:f+Dx=0f=-Dxortheforceexertedbythespringisopposingthedisplacingforce.Thisconditionisdescribedbythelinearsecond-orderdifferentialequation:22(1.2)dxmDxdtBothkandmarepositive,sothatwecandefineaconstantmk20,theequationmaybewrittenas)3.1(222oxdtxdoThisisanimportantlineardifferentialequationwhosesolutioniswell-Knownandmaybeobtainedbyseveralmethods.Onemethodistoassumeaso