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WavesZNormalMode s

MattJarvis

February334s⌘

Contents

sOs cillations3 s4sSimpl eHarmonicMotionrevision 3

3Norm alModes⇣

3sThec oupledpend ulum ⌘

3ssTheDec ouplingMet hod (

3s3TheMat rixMethod s4

3s5Initi alconditionsandexamples s5

3spEnergyof acoupledpendulu m s⇣

33Unequ alCoupledPendula s)

35TheHor izontalSpri ngMasssystem 33

35sDecoupl ingmethod 33

353TheMat rixMethod 35

355Energy ofthehorizontalspri ngm asssystem 3⇣

35pInitial Condition 3⌘

3pVert icalspringmasssystem 3⌘

3psThemat rixmethod 3(

3⇣Inte rludeSolvinginhomogeneous3ndorderd i

erentialequations 3)

3⌘Horizon talspringmasssystem withadrivingterm 5s

3(TheF orcedCoupl edPendulumwitha DampingFactor 55

5Norm almodesII towardsthecontin uouslim it5⇢

5sNcoupledoscillators 5⇢

5ssSpeci alcases p4

5s3General case p3

5s5Nverylarge pp

5spLongitudi nalOscillations p(

pWavesIp) psThew aveequati on p) pssTheStr etchedStr ing p) p3dÕAlam bertÕssolutiontothewaveequation ⇣4 p3sInter pretationofdÕAlambertÕssolution ⇣s p33dÕAlambe rtÕssolutionwithboundaryconditions ⇣3 p5Solvi ngthewaveequationby separationof variables ⇣p p5sNegative C ⇣⇣ i s p53Positiv eC ⇣⌘ p55C=4 ⇣⌘ ppSinu soidalwaves ⇣⌘ p⇣Phase Di erences ⇣( p⌘Ener gystoredinamechan icalwave ⇣) p⌘sKinet icEnergy ⇣) p⌘3Potenti alEnergy ⇣⇢ ⇣Wa vesII⌘3 ⇣sReße ctionZTransmissionofwaves ⌘3 ⇣ssBoundary Conditions ⌘5 ⇣s3Particu larcases ⌘p ⇣3Powe rßowataboundary ⌘p ⇣5Stand ingWaves ⌘⇣

Chapters

Oscillations

Beforewegointoth emainbo dyof thecourseonwa vesandnormal modesiti susefulto haveasmallrec aponw hatweknowabout simplesystem swh ereweonly haveas ingle massonapen dul umf orexampleThiswouldallcomeun derthe remitofsimpleharmonic motionwhichforms thebasisofsomeofthe problemsthatw ewillencou nterinthisc ours e s4sSimpleHarmoni cMotionrev ision

FirstconsiderHookeÕs Law

F=⇥kx"ss

whereFisthef orcexisthed isplaceme ntwithrespecttotheequilibriumposi tionandk isthec onstantofprop ortionalityrelatingth etw o Theusualai mistosolveforxasaf unc tionoftimetintheos cillationoft hespring orpe ndulumforexample

Weknow thatF=math erefore

F=⇥kx=m

d 3 x dt 3 "s3 Thisequationt ellsusthatweneedtoÞ ndasolutionforwhi chthes econdderi vative ispropor tionaltothenegativeofitself Wekno wthatfunct ionsthatobeythisareth e sinecosineandexpon entialsSowecantryafairlygeneralsoluti onsofth eform i t "s5 3 5 orcon tractsthecurveonthetim eaxisandthe constantAgivestheamplit udeofthec urve Tochec kthatthisallworksw ecansubst ituteEqs5in toEq s3obtaining 3 "sp 3 Sincethisequation mustholdatallt imestwe mustth ereforehave k⇥m⇥ 3 r k m "s⌘ Youcanals odoth isslightly morerigou rouslybywrit ingthedi erentialequationas ⇥kx=m dv dt "s( butthisi sawkwardasitcon tainsth reevariablesxvvandtSo youcanÕ tuseth estandard strategyofseparationofvari abl esonthetwosidesoftheequ ati onandthenintegrateBut wecanwr ite

F=ma=m

dv dt =m dx dt dv dx =mv dv dx "s)

Whichthenleadst o

v dv dx Z kxdx= Z mvdv"s⇢

IntegratingweÞnd

s 3 kx 3 s 3 mv 3 #E"ss4 wheretheconstant ofintegration Ehappenstobetheener gyItfol lowsth at v=± r 3 m r

E⇥

s 3 kx 3 "sss whichcanbewritt enas dx r E q s⇥ kx 3 3E r 3 m Z dt"ss3 p Atr igsubstituti onturnstheLHSintoanarcsinorarccosfuncti onandtheresultis r k m "ss5 whichisthesamer esultgi veninEqs 5 GeneralsolutionstoHook eÕslawcanobviouslyalsoen compassc ombinationsoftrig functionsand2orexponentialsf orexample therefore x"t=A s sin⇥t#A 3 cos⇥t"ss⌘ isalsoa solution

Finallyforthecomplexex ponenti alsolution

x"t=Ce i t "ss(

F=⇥kx=Ce

i t =m d 3 x dt 3 =⇥mr 3 Ce i t "ss) 3 k m 3 "ss⇢ e i t #B e "s34

UsingEulerÕsf ormula

e i r =cos ✓#isin✓"s3s x"t=A cos⇥t#A isin⇥t#B cos⇥t⇥B isin⇥t"s33

IfA=i"A

⇥B and B="A #B then x"t=Asin⇥t#Bcos⇥t"s35

Chapter3

NormalModes

Manyphysi calsystemsrequiremoreth anonevariabletoquantifyth eirconÞguration;for exampleacircuitmay hav etwoconnectedcurrentl oopssoonen eedstoknoww hatcurrent isßowi ngineachloopateachmome ntAnoth erexamplei sasetofNcoupledpendulaeach ofwh ichisaonedimen sionalosc ill atorAsetofdi erentialequationsoneforeac hvariable willdetermin ethedynamicsofsuchasystemFor asystemof NcoupledsDoscillators thereexistNnormalmodesin whichalloscillators movew iththesamefreque ncyandt hus haveÞxedampl ituderatios"if eachoscillatorisallowedtomov einx⇥dimensionsthen xNnormalmodesexi stThenormalmod eisforwholesystemEven thoughuncoup led angularfrequen ciesoftheoscillatorsarenotthesamethee ectofcoupli ng isthatall bodiescanmovewitht hesamef requencyIf theinitialstat eofthesys temcorresp ondsto motioninanormalm odethe nt heoscillationsc ontinueinthe normalmodeHowe verin generalthemotionis described byalinearcomb inationof allthenormalmodes;sincethe di erentialequationsareline arsuchalinearcombinationi salsoas olutiontothecoupl ed linearequations Anor malmodeofanosci llatingsystem ist he motioninwhichallpartsofthesyste m movesinusoidal lywiththesamefrequency andwith aÞxedphaserel ationquotesdbs_dbs20.pdfusesText_26