Thusfarwehavedevelopedasetofdifferentialequationsthatdescribethelocalbehaviorofsoundwaves.Webecomeacquaintedwithsomeotherimportanttypesofwavebehavior-----sphericalwaves.Ouremphasishereisonthewavesthemselvesandinthefollowingchapterwewillpursuetheinteractionofthesewaveswiththeirsources.Sphericalspreadingdescribesthedecreaseinlevelwhenasoundwavepropagatesawayfromasourceuniformlyinalldirections.Soundlevelsarethereforeconstantonsphericalsurfacessurroundingthesoundsource.Soundlevelsdecreaserapidlyassoundspreadsoutfromaspherewitharadiusofr0toalargerspherewitharadiusr.SphericalwavesarerepresentedincomplexformbyTheAcousticimpedanceTheacousticimpedanceofsphericalwaveReal,imaginarypartsandphaseWeemphasizeagainthattheacousticimpedancedependsonthefrequency,aswellasonthemedium,thewavetype,andthegeometricposition.Neitherthefrequencynorthedistancealoneissoimportantasisthedimensioncombinationkr.Forkr>>1,Zaispredominantlyresistive,meaningthatenergyfedintosuchwavespropagatesawayfromthesource,nevertoreturn.Thelargerthevalueofkr,thelesscurvedarethewavefrontsandthemorenearlydothesewavesactjustlikeplanewaves.Butforkr<<1thereactivecomponentislarge,anditispositive(masslike)Thismeansthatpushingonsuchawaveismainlyamatterofstoringkineticenergyinmotionoftheliquid,whereitcanberecoveredlaterbythesource.(likeenergyinanelectricalinductor)TheintensityandpowerofsphericalwavesForaharmonicsphericalwave,theacousticintensityis:Theacousticintensitycanbeexpressedinanotherform:Theaveragerateofenergyflowthroughanysphericalsurfacesurroundingtheoriginisindependentoftheradiusofthesurface,aconclusionthatisconsistentwithconservationofenergyinalosslessmedium.Adivergingsphericalwavehasapeakacousticpressureof2Paatadistanceof1mfromthesourceatstandardatmosphericpressureandtemperature.Whatisitsintensityatadistanceof10mfromthesource?Assumethesourceisemittingaconstantamountofenergytothesoundwave.Fordivergingsphericalwaves,theareaofthewavefrontincreasesasthewavesaretravelingfartherandfartherfromthesource.Henceint