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1、Solution-TheoreticalQuestion2APiezoelectricCrystalResonatorunderanAlternatingVoltagePartA(a)RefertoFigureA1.TheleftfaceoftherodmovesadistancevDtwhilethepressurewavetravelsadistanceuDtwith.Thestrainattheleftfaceis(A1a)*Anequationsmarkedwithanasteriskcontainsanswerto
2、theproblem.FromHooke’slaw,thepressureattheleftfaceispuDtt=0Dt/2ppFigureA1vDtDtpp(A1b)*(b)Thevelocityvisrelatedtothedisplacementxasinasimpleharmonicmotion(orauniformcircularmotion,asshowninFigureA2)ofangularfrequency.Therefore,if,then.(A2)*Thestrainandpressurearerel
3、atedtovelocityasinProblem(a).Hence,(A3)*(A4)*xkx-wtvxFigureA2-------------------------------------------------------------------------------------------------Alternatively,theanswersmaybeobtainedbydifferentiations:,,.------------------------------------------------
4、-------------------------PartB(c)Sincetheangularfrequencywandspeedofpropagationuaregiven,thewavelengthisgivenbyl=2p/kwithk=w/u.Thespatialvariationofthedisplacementxisthereforedescribedby(B1)Sincethecentersoftheelectrodesareassumedtobestationary,g(b/2)=0.Thisleadsto
5、B2=0.Giventhatthemaximumofg(x)is1,wehaveA=±1and(B2)*Thus,thedisplacementis(B3)(d)Sincethepressurep(orstressT)mustvanishattheendfacesofthequartzslab(i.e.,x=0andx=b),theanswertothisproblemcanbeobtained,byanalogy,fromtheresonantfrequenciesofsoundwavesinanopenpipeoflen
6、gthb.However,giventhatthecentersoftheelectrodesarestationary,allevenharmonicsofthefundamentaltonemustbeexcludedbecausetheyhaveantinodes,ratherthannodes,ofdisplacementatthebisectionplaneoftheslab.Sincethefundamentaltonehasawavelengthl=2b,thefundamentalfrequencyisgiv
7、enby.Thespeedofpropagationuisgivenbym/s(B4)and,giventhatb=1.00´10-2m,thetwoloweststandingwavefrequenciesare,(B5)*-------------------------------------------------------------------------------------------------[AlternativesolutiontoProblems(c)and(d)]:Alongitudinals
8、tandingwaveinthequartzslabhasadisplacementnodeatx=b/2.Itmayberegardedasconsistingoftwowavestravelinginoppositedirections.Thus,itsdisplacementandv
