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Vibrations and Waves

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Vibrations and Waves
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Science, Physics, Vibrations, Resonance, Free Waves, Bounded Waves, Period, Frequency, Amplitude, Harmonic Motion, Wave Motion, Waves on a String, Sound Waves, Light Waves, Periodic Waves, Doppler Effect, Energy in Vibrations, Reflection, Transmission, Absorption, Quantitative Treatment of Reflection, Inverted Reflections, Uninverted reflections in general, …
Physics, Scientific Concepts
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This is a text on vibrations and waves for an introductory college physics class. The treatment is algebra-based, with applications of calculus discussed in optional sections. Contents: 1) Vibrations. 2) Resonance. 3) Free Waves. 4) Bounded Waves. This is book 3 in the Light and Matter series of free introductory physics textbooks.
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Community College, Higher Education
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Adobe PDF Reader.
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Benjamin Crowell, Fullerton College, CA
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Textbook
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www.lightandmatter.com
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http://florida.theorangegrove.org/og/file/982c5ac1-a779-dec5-fb3d-1a2a166f5f27/1/Vibrations.pdf

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University of Florida
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Copyright 1998-2008 Benjamin Crowell. This book is licensed under the Creative Commons Attribution-ShareAlike license, version 1.0, http://creativecommons.org/licenses/by-sa/1.0/, except for those photographs and drawings of which I am not the author, as listed in the photo credits. If you agree to the license, it grants you certain privileges that you …
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Book3intheLightandMatterseriesoffreeintroductoryphysicstextbooks www.lightandmatter.com

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The LightandMatter seriesof introductoryphysicstextbooks: 1NewtonianPhysics 2ConservationLaws 3VibrationsandWaves 4ElectricityandMagnetism 5Optics 6TheModernRevolutioninPhysics

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BenjaminCrowell www.lightandmatter.com

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Fullerton,California www.lightandmatter.com copyright1998-2008BenjaminCrowell rev.May14,2008 ThisbookislicensedundertheCreativeCommonsAttribution-ShareAlikelicense,version1.0, http://creativecommons.org/licenses/by-sa/1.0/,except forthosephotographsanddrawingsofwhichIamnot theauthor,aslistedinthephotocredits.Ifyouagree tothelicense,itgrantsyoucertainprivilegesthatyou wouldnototherwisehave,suchastherighttocopythe book,ordownloadthedigitalversionfreeofchargefrom www.lightandmatter.com.Atyouroption,youmayalso copythisbookundertheGNUFreeDocumentation Licenseversion1.2,http://www.gnu.org/licenses/fdl.txt, withnoinvariantsections,nofront-covertexts,andno back-covertexts. ISBN0-9704670-3-6

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ToDizandBird.

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BriefContents 1Vibrations13 2Resonance25 3FreeWaves47 4BoundedWaves73

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Contents 1Vibrations 1.1Period,Frequency,andAmplitude.14 1.2SimpleHarmonicMotion.....17 Whyaresine-wavevibrationssocommon?, 17.|Periodisapproximatelyindependent ofAmplitude,iftheAmplitudeissmall., 18. 1.3 ? Proofs............19 Summary.............22 Problems.............23 2Resonance 2.1EnergyinVibrations.......26 2.2EnergyLostFromVibrations...28 2.3PuttingEnergyIntoVibrations...30 2.4 ? Proofs............38 Statement2:maximumAmplitudeat resonance,39.|Statement3:Amplitude atresonanceproportionalto q ,39.| Statement4:fwhmrelatedto q ,40. Summary.............41 Problems.............43 3FreeWaves 3.1WaveMotion..........49 1.superposition,49.|2.themediumis nottransportedwiththewave.,51.|3.a wave'svelocitydependsonthemedium., 52.|Wavepatterns,53. 3.2WavesonaString........54 Intuitiveideas,54.|Approximate treatment,55.|Rigorousderivationusing calculusoptional,56. 3.3SoundandLightWaves.....57 Soundwaves,57.|Lightwaves,58 3.4PeriodicWaves.........59 Periodandfrequencyofaperiodicwave, 59.|Graphsofwavesasafunctionof position,60.|Wavelength,60.|Wavevelocityrelatedtofrequencyandwavelength, 60.|Sinusoidalwaves,62. 3.5TheDopplerEffect.......63 TheBigbang,66.|WhattheBigbangis not,67. Summary.............69 10

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Problems.............71 4BoundedWaves 4.1Reection,Transmission,and Absorption.............74 Reectionandtransmission,74.| Invertedanduninvertedreections,77.| Absorption,77. 4.2 ? QuantitativeTreatmentofReection80 Whyreectionoccurs,80.|Intensityof reection,81.|Invertedanduninvertedreectionsingeneral,82. 4.3InterferenceEffects.......83 4.4WavesBoundedonBothSides..86 Musicalapplications,88.|Standing waves,88.|Standing-wavepatternsofair columns,90. Summary.............92 Problems.............93 Appendix1:Exercises 95 Appendix2:PhotoCredits 97 Appendix3:HintsandSolutions 98 11

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Thevibrationsofthiselectricbass stringareconvertedtoelectrical vibrations,thentosoundvibrations,andnallytovibrationsof oureardrums. Chapter1 Vibrations Dandelion.Cello.Readthosetwowords,andyourbraininstantly conjuresastreamofassociations,themostprominentofwhichhave todowithvibrations.Ourmentalcategoryofdandelion-ness"is stronglylinkedtothecoloroflightwavesthatvibrateabouthalfa millionbilliontimesasecond:yellow.Thevelvetythrobofacello hasasitsmostobviouscharacteristicarelativelylowmusicalpitch |thenoteyouarespontaneouslyimaginingrightnowmightbe onewhosesoundvibrationsrepeatatarateofahundredtimesa second. Evolutionhasdesignedourtwomostimportantsensesaround theassumptionthatnotonlywillourenvironmentbedrenchedwith information-bearingvibrations,butinadditionthosevibrationswill oftenberepetitive,sothatwecanjudgecolorsandpitchesbythe rateofrepetition.Grantingthatwedosometimesencounternonrepeatingwavessuchastheconsonantsh,"whichhasnorecognizablepitch,whywasNature'sassumptionofrepetitionnevertheless sorightingeneral? Repeatingphenomenaoccurthroughoutnature,fromtheorbits ofelectronsinatomstothereappearanceofHalley'sCometevery75 years.Ancientculturestendedtoattributerepetitiousphenomena 13

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b / Aspringhasanequilibriumlength,1,andcanbe stretched,2,orcompressed,3.A massattachedtothespringcan besetintomotioninitially,4,and willthenvibrate,4-13. a / IfwetrytodrawanonrepeatingorbitforHalley's Comet,itwillinevitablyendup crossingitself. liketheseasonstothecyclicalnatureoftimeitself,butwenow havealessmysticalexplanation.SupposethatinsteadofHalley's Comet'strue,repeatingellipticalorbitthatclosesseamlesslyupon itselfwitheachrevolution,wedecidetotakeapenanddrawa whimsicalalternativepaththatneverrepeats.Wewillnotbeableto drawforverylongwithouthavingthepathcrossitself.Butatsuch acrossingpoint,thecomethasreturnedtoaplaceitvisitedonce before,andsinceitspotentialenergyisthesameasitwasonthe lastvisit,conservationofenergyprovesthatitmustagainhavethe samekineticenergyandthereforethesamespeed.Notonlythat, butthecomet'sdirectionofmotioncannotberandomlychosen, becauseangularmomentummustbeconservedaswell.Although thisfallsshortofbeinganironcladproofthatthecomet'sorbitmust repeat,itnolongerseemssurprisingthatitdoes. Conservationlaws,then,provideuswithagoodreasonwhy repetitivemotionissoprevalentintheuniverse.Butitgoesdeeper thanthat.Uptothispointinyourstudyofphysics,Ihavebeen indoctrinatingyouwithamechanisticvisionoftheuniverseasa giantpieceofclockwork.Breakingtheclockworkdownintosmaller andsmallerbits,weendupattheatomiclevel,wheretheelectrons circlingthenucleusresemble|well,littleclocks!Fromthispoint ofview,particlesofmatterarethefundamentalbuildingblocks ofeverything,andvibrationsandwavesarejustacoupleofthe tricksthatgroupsofparticlescando.Butatthebeginningof the20thcentury,thetabledwereturned.Achainofdiscoveries initiatedbyAlbertEinsteinledtotherealizationthattheso-called subatomicparticles"wereinfactwaves.Inthisnewworld-view, itisvibrationsandwavesthatarefundamental,andtheformation ofmatterisjustoneofthetricksthatwavescando. 1.1Period,Frequency,andAmplitude Figurebshowsourmostbasicexampleofavibration.Withno forcesonit,thespringassumesitsequilibriumlength,b/1.Itcan bestretched,2,orcompressed,3.Weattachthespringtoawall ontheleftandtoamassontheright.Ifwenowhitthemasswith ahammer,4,itoscillatesasshownintheseriesofsnapshots,4-13. Ifweassumethatthemassslidesbackandforthwithoutfriction andthatthemotionisone-dimensional,thenconservationofenergy provesthatthemotionmustberepetitive.Whentheblockcomes backtoitsinitialpositionagain,7,itspotentialenergyisthesame again,soitmusthavethesamekineticenergyagain.Themotion isintheoppositedirection,however.Finally,at10,itreturnstoits initialpositionwiththesamekineticenergyandthesamedirection ofmotion.Themotionhasgonethroughonecompletecycle,and willnowrepeatforeverintheabsenceoffriction. Theusualphysicsterminologyformotionthatrepeatsitselfover 14 Chapter1Vibrations

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c / Example1. andoverisperiodicmotion,andthetimerequiredforonerepetition iscalledtheperiod, T .Thesymbol P isnotusedbecauseofthe possibleconfusionwithmomentum.Onecompleterepetitionofthe motioniscalledacycle. Weareusedtoreferringtoshort-periodsoundvibrationsas high"inpitch,anditsoundsoddtohavetosaythathighpitches havelowperiods.Itisthereforemorecommontodiscusstherapidityofavibrationintermsofthenumberofvibrationspersecond, aquantitycalledthefrequency, f .Sincetheperiodisthenumber ofsecondspercycleandthefrequencyisthenumberofcyclesper second,theyarereciprocalsofeachother, f =1 =T Acarnivalgameexample1 Inthecarnivalgameshowningurec,therubeissupposedto pushthebowlingballonthetrackjusthardenoughsothatitgoes overthehumpandintothevalley,butdoesnotcomebackout again.Iftheonlytypesofenergyinvolvedarekineticandpotential,thisisimpossible.Supposeyouexpecttheballtocomeback toapointsuchastheoneshownwiththedashedoutline,then stopandturnaround.Itwouldalreadyhavepassedthroughthis pointoncebefore,goingtotheleftonitswayintothevalley.It wasmovingthen,soconservationofenergytellsusthatitcannotbeatrestwhenitcomesbacktothesamepoint.Themotion thatthecustomerhopesforisphysicallyimpossible.Thereis aphysicallypossibleperiodicmotioninwhichtheballrollsback andforth,stayingconnedwithinthevalley,butthereisnoway togettheballintothatmotionbeginningfromtheplacewherewe start.Thereisawaytobeatthegame,though.Ifyouputenough spinontheball,youcancreateenoughkineticfrictionsothata signicantamountofheatisgenerated.Conservationofenergy thenallowstheballtobeatrestwhenitcomesbacktoapoint liketheoutlinedone,becausekineticenergyhasbeenconverted intoheat. Periodandfrequencyofay'swing-beatsexample2 AVictorianparlortrickwastolistentothepitchofay'sbuzz,reproducethemusicalnoteonthepiano,andannouncehowmany timesthey'swingshadappedinonesecond.Ifthey'swings ap,say,200timesinonesecond,thenthefrequencyoftheir motionis f =200 = 1s=200s )]TJ/F39 7.9701 Tf 6.587 0 Td [(1 .Theperiodisone200thofa second, T =1 = f = = 200s=0.005s. Section1.1Period,Frequency,andAmplitude 15

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d / 1.Theamplitudeofthe vibrationsofthemassonaspring couldbedenedintwodifferent ways.Itwouldhaveunitsof distance.2.Theamplitudeofa swingingpendulumwouldmore naturallybedenedasanangle. Unitsofinversesecond,s )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 ,areawkwardinspeech,soanabbreviationhasbeencreated.OneHertz,namedinhonorofapioneer ofradiotechnology,isonecyclepersecond.Inabbreviatedform, 1Hz=1s )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 .Thisisthefamiliarunitusedforthefrequencieson theradiodial. Frequencyofaradiostationexample3 KKJZ'sfrequencyis88.1MHz.Whatdoesthismean,andwhat perioddoesthiscorrespondto? ThemetricprexM-ismega-,i.e.,millions.Theradiowaves emittedbyKKJZ'stransmittingantennavibrate88.1milliontimes persecond.Thiscorrespondstoaperiodof T =1 = f =1.14 10 )]TJ/F39 7.9701 Tf 6.587 0 Td [(8 s. Thisexampleshowsasecondreasonwhywenormallyspeakin termsoffrequencyratherthanperiod:itwouldbepainfultohave torefertosuchsmalltimeintervalsroutinely.Icouldabbreviate bytellingpeoplethatKKJZ'speriodwas11.4nanoseconds,but mostpeoplearemorefamiliarwiththebigmetricprexesthan withthesmallones. Unitsoffrequencyarealsocommonlyusedtospecifythespeeds ofcomputers.Theideaisthatallthelittlecircuitsonacomputer chiparesynchronizedbytheveryfastticksofanelectronicclock,so thatthecircuitscanallcooperateonataskwithoutgettingahead orbehind.Addingtwonumbersmightrequire,say,30clockcycles. Microcomputersthesedaysoperateatclockfrequenciesofabouta gigahertz. Wehavediscussedhowtomeasurehowfastsomethingvibrates, butnothowbigthevibrationsare.Thegeneraltermforthisis amplitude, A .Thedenitionofamplitudedependsonthesystem beingdiscussed,andtwopeoplediscussingthesamesystemmay notevenusethesamedenition.Intheexampleoftheblockonthe endofthespring,d/1,theamplitudewillbemeasuredindistance unitssuchascm.Onecouldworkintermsofthedistancetraveled bytheblockfromtheextremelefttotheextremeright,butit wouldbesomewhatmorecommoninphysicstousethedistance fromthecentertooneextreme.Theformerisusuallyreferredtoas thepeak-to-peakamplitude,sincetheextremesofthemotionlook likemountainpeaksorupside-downmountainpeaksonagraphof positionversustime. Inothersituationswewouldnotevenusethesameunitsforamplitude.Theamplitudeofachildonaswing,orapendulum,d/2, wouldmostconvenientlybemeasuredasanangle,notadistance, sinceherfeetwillmoveagreaterdistancethanherhead.Theelectricalvibrationsinaradioreceiverwouldbemeasuredinelectrical unitssuchasvoltsoramperes. 16 Chapter1Vibrations

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f / Theforceexertedbyan idealspring,whichbehaves exactlyaccordingtoHooke'slaw. e / Sinusoidalandnon-sinusoidal vibrations. 1.2SimpleHarmonicMotion Whyaresine-wavevibrationssocommon? Ifweactuallyconstructthemass-on-a-springsystemdiscussed intheprevioussectionandmeasureitsmotionaccurately,wewill ndthatits x )]TJ/F20 10.9091 Tf 9.259 0 Td [(t graphisnearlyaperfectsine-waveshape,asshown inguree/1.Wecallitasinewave"orsinusoidal"evenifitis acosine,orasineorcosineshiftedbysomearbitraryhorizontal amount.Itmaynotbesurprisingthatitisawiggleofthisgeneral sort,butwhyisitaspecicmathematicallyperfectshape?Whyis itnotasawtoothshapelike2orsomeothershapelike3?Themysterydeepensaswendthatavastnumberofapparentlyunrelated vibratingsystemsshowthesamemathematicalfeature.Atuning fork,asaplingpulledtoonesideandreleased,acarbouncingon itsshockabsorbers,allthesesystemswillexhibitsine-wavemotion underonecondition:theamplitudeofthemotionmustbesmall. Itisnothardtoseeintuitivelywhyextremesofamplitudewould actdierently.Forexample,acarthatisbouncinglightlyonits shockabsorbersmaybehavesmoothly,butifwetrytodoublethe amplitudeofthevibrationsthebottomofthecarmaybeginhitting theground,e/4.Althoughweareassumingforsimplicityinthis chapterthatenergyisneverdissipated,thisisclearlynotavery realisticassumptioninthisexample.Eachtimethecarhitsthe grounditwillconvertquiteabitofitspotentialandkineticenergyintoheatandsound,sothevibrationswouldactuallydieout quitequickly,ratherthanrepeatingformanycyclesasshowninthe gure. Thekeytounderstandinghowanobjectvibratesistoknowhow theforceontheobjectdependsontheobject'sposition.Ifanobject isvibratingtotherightandleft,thenitmusthavealeftwardforce onitwhenitisontherightside,andarightwardforcewhenitison theleftside.Inonedimension,wecanrepresentthedirectionofthe forceusingapositiveornegativesign,andsincetheforcechanges frompositivetonegativetheremustbeapointinthemiddlewhere theforceiszero.Thisistheequilibriumpoint,wheretheobject wouldstayatrestifitwasreleasedatrest.Forconvenienceof notationthroughoutthischapter,wewilldenetheoriginofour coordinatesystemsothat x equalszeroatequilibrium. Thesimplestexampleisthemassonaspring,forwhichforce onthemassisgivenbyHooke'slaw, F = )]TJ/F20 10.9091 Tf 8.485 0 Td [(kx Wecanvisualizethebehaviorofthisforceusingagraphof F versus x ,asshowninguref.Thegraphisaline,andthespringconstant, k ,isequaltominusitsslope.Astierspringhasalargervalueof k andasteeperslope.Hooke'slawisonlyanapproximation,but itworksverywellformostspringsinreallife,aslongasthespring Section1.2SimpleHarmonicMotion 17

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g / Seenfromcloseup,any F )]TJ/F102 9.9627 Tf 9.963 0 Td [(x curvelookslikealine. isn'tcompressedorstretchedsomuchthatitispermanentlybent ordamaged. Thefollowingimportanttheorem,whoseproofisgiveninoptionalsection1.3,relatesthemotiongraphtotheforcegraph. Theorem:Alinearforcegraphmakesasinusoidalmotion graph. Ifthetotalforceonavibratingobjectdependsonlyonthe object'sposition,andisrelatedtotheobjectsdisplacement fromequilibriumbyanequationoftheform F = )]TJ/F20 10.9091 Tf 8.484 0 Td [(kx ,then theobject'smotiondisplaysasinusoidalgraphwithperiod T =2 p m=k Evenifyoudonotreadtheproof,itisnottoohardtounderstand whytheequationfortheperiodmakessense.Agreatermasscauses agreaterperiod,sincetheforcewillnotbeabletowhipamassive objectbackandforthveryrapidly.Alargervalueof k causesa shorterperiod,becauseastrongerforcecanwhiptheobjectback andforthmorerapidly. Thismayseemlikeonlyanobscuretheoremaboutthemass-ona-springsystem,butguregshowsittobefarmoregeneralthan that.Figureg/1depictsaforcecurvethatisnotastraightline.A systemwiththis F )]TJ/F20 10.9091 Tf 9.96 0 Td [(x curvewouldhavelarge-amplitudevibrations thatwerecomplexandnotsinusoidal.Butthesamesystemwould exhibitsinusoidalsmall-amplitudevibrations.Thisisbecauseany curvelookslinearfromverycloseup.Ifwemagnifythe F )]TJ/F20 10.9091 Tf 11.903 0 Td [(x graphasshowningureg/2,itbecomesverydiculttotellthat thegraphisnotastraightline.Ifthevibrationswereconnedto theregionshowning/2,theywouldbeverynearlysinusoidal.This isthereasonwhysinusoidalvibrationsareauniversalfeatureof allvibratingsystems,ifwerestrictourselvestosmallamplitudes. Thetheoremisthereforeofgreatgeneralsignicance.Itapplies throughouttheuniverse,toobjectsrangingfromvibratingstarsto vibratingnuclei.Asinusoidalvibrationisknownassimpleharmonic motion. PeriodisapproximatelyindependentofAmplitude,ifthe Amplitudeissmall. Untilnowwehavenotevenmentionedthemostcounterintuitiveaspectoftheequation T =2 p m=k :itdoesnotdependon amplitudeatall.Intuitively,mostpeoplewouldexpectthemass-ona-springsystemtotakelongertocompleteacycleiftheamplitude waslarger.Wearecomparingamplitudesthataredierentfrom eachother,butbothsmallenoughthatthetheoremapplies.In factthelarger-amplitudevibrationstakethesameamountoftime asthesmall-amplitudeones.Thisisbecauseatlargeamplitudes, theforceisgreater,andthereforeacceleratestheobjecttohigher speeds. 18 Chapter1Vibrations

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h / Theobjectmovesalong thecircleatconstantspeed, buteventhoughitsoverall speedisconstant,the x and y componentsofitsvelocityare continuouslychanging,asshown bytheunequalspacingofthe pointswhenprojectedontothe linebelow.Projectedontothe line,itsmotionisthesameas thatofanobjectexperiencinga force F = )]TJ/F102 9.9627 Tf 7.749 0 Td [(kx LegendhasitthatthisfactwasrstnoticedbyGalileoduring whatwasapparentlyalessthanenthrallingchurchservice.Agust ofwindwouldnowandthenstartoneofthechandeliersinthe cathedralswayingbackandforth,andhenoticedthatregardless oftheamplitudeofthevibrations,theperiodofoscillationseemed tobethesame.Upuntilthattime,hehadbeencarryingouthis physicsexperimentswithsuchcrudetime-measuringtechniquesas feelinghisownpulseorsingingatunetokeepamusicalbeat.But aftergoinghomeandtestingapendulum,heconvincedhimselfthat hehadfoundasuperiormethodofmeasuringtime.Evenwithout afancysystemofpulleystokeepthependulum'svibrationsfrom dyingdown,hecouldgetveryaccuratetimemeasurements,because thegradualdecreaseinamplitudeduetofrictionwouldhaveno eectonthependulum'speriod.Galileoneverproducedamodernstylependulumclockwithpulleys,aminutehand,andasecond hand,butwithinagenerationthedevicehadtakenontheform thatpersistedforhundredsofyearsafter. Thependulumexample4 Comparetheperiodsofpendulahavingbobswithdifferentmasses. Fromtheequation T =2 p m = k ,wemightexpectthatalarger masswouldleadtoalongerperiod.However,increasingthe massalsoincreasestheforcesthatactonthependulum:gravity andthetensioninthestring.Thisincreases k aswellas m ,so theperiodofapendulumisindependentof m 1.3 ? Proofs Inthissectionweprovethatalinear F )]TJ/F20 10.9091 Tf 11.995 0 Td [(x graphgives sinusoidalmotion,thattheperiodofthemotionis2 p m=k andthattheperiodisindependentoftheamplitude.Youmay omitthissectionwithoutlosingthecontinuityofthechapter. Thebasicideaoftheproofcanbeunderstoodbyimagining thatyouarewatchingachildonamerry-go-roundfromfaraway. Becauseyouareinthesamehorizontalplaneashermotion,she appearstobemovingfromsidetosidealongaline.Circularmotion viewededge-ondoesn'tjustlooklikeanykindofback-and-forth motion,itlookslikemotionwithasinusoidal x )]TJ/F20 10.9091 Tf 8.691 0 Td [(t graph,becausethe sineandcosinefunctionscanbedenedasthe x and y coordinates ofapointatangle ontheunitcircle.Theideaoftheproof,then, istoshowthatanobjectactedonbyaforcethatvariesas F = )]TJ/F20 10.9091 Tf 8.485 0 Td [(kx hasmotionthatisidenticaltocircularmotionprojecteddownto onedimension.The v 2 =r expressionwillalsofalloutattheend. Section1.3 ? Proofs 19

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ThemoonsofJupiter.example5 Beforemovingontotheproof,weillustratetheconceptusing themoonsofJupiter.TheirdiscoverybyGalileowasanepochal eventinastronomy,becauseitprovedthatnoteverythinginthe universehadtorevolvearoundtheearthashadbeenbelieved. Galileo'stelescopewasofpoorqualitybymodernstandards,but gureishowsasimulationofhowJupiteranditsmoonsmight appearatintervalsofthreehoursthroughalargepresent-dayinstrument.Becauseweseethemoons'circularorbitsedge-on, theyappeartoperformsinusoidalvibrations.Overthistimeperiod,theinnermostmoon,Io,completeshalfacycle. i / Example5. Foranobjectperforminguniformcircularmotion,wehave j a j = v 2 r The x componentoftheaccelerationistherefore a x = v 2 r cos where istheanglemeasuredcounterclockwisefromthe x axis. ApplyingNewton'ssecondlaw, F x m = )]TJ/F20 10.9091 Tf 9.68 7.381 Td [(v 2 r cos ,so F x = )]TJ/F20 10.9091 Tf 8.485 0 Td [(m v 2 r cos Sinceourgoalisanequationinvolvingtheperiod,itisnaturalto eliminatethevariable v =circumference =T =2 r=T ,giving F x = )]TJ/F15 10.9091 Tf 9.681 7.38 Td [(4 2 mr T 2 cos 20 Chapter1Vibrations

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Thequantity r cos isthesameas x ,sowehave F x = )]TJ/F15 10.9091 Tf 9.681 7.38 Td [(4 2 m T 2 x Sinceeverythingisconstantinthisequationexceptfor x ,wehave provedthatmotionwithforceproportionalto x isthesameascircularmotionprojectedontoaline,andthereforethataforceproportionalto x givessinusoidalmotion.Finally,weidentifytheconstant factorof4 2 m=T 2 with k ,andsolvingfor T givesthedesiredequationfortheperiod, T =2 r m k Sincethisequationisindependentof r T isindependentofthe amplitude,subjecttotheinitialassumptionofperfect F = )]TJ/F20 10.9091 Tf 8.485 0 Td [(kx behavior,whichinrealitywillonlyholdapproximatelyforsmall x Section1.3 ? Proofs 21

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Summary SelectedVocabulary periodicmotion.motionthatrepeatsitselfoverandover period.......thetimerequiredforonecycleofaperiodic motion frequency.....thenumberofcyclespersecond,theinverseof theperiod amplitude....theamountofvibration,oftenmeasuredfrom thecentertooneside;mayhavedierentunits dependingonthenatureofthevibration simpleharmonic motion...... motionwhose x )]TJ/F20 10.9091 Tf 10.909 0 Td [(t graphisasinewave Notation T .........period f ..........frequency A .........amplitude k ..........theslopeofthegraphof F versus x ,where F isthetotalforceactingonanobjectand x istheobject'sposition;Foraspring,thisis knownasthespringconstant. OtherTerminologyandNotation ..........TheGreekletter ,nu,isusedinmanybooks forfrequency. ..........TheGreekletter ,omega,isoftenusedasan abbreviationfor2 f Summary Periodicmotioniscommonintheworldaroundusbecauseof conservationlaws.Animportantexampleisone-dimensionalmotion inwhichtheonlytwoformsofenergyinvolvedarepotentialand kinetic;insuchasituation,conservationofenergyrequiresthatan objectrepeatitsmotion,becauseotherwisewhenitcamebackto thesamepoint,itwouldhavetohaveadierentkineticenergyand thereforeadierenttotalenergy. Notonlyareperiodicvibrationsverycommon,butsmall-amplitude vibrationsarealwayssinusoidalaswell.Thatis,the x )]TJ/F20 10.9091 Tf 9.816 0 Td [(t graphisa sinewave.Thisisbecausethegraphofforceversuspositionwillalwayslooklikeastraightlineonasucientlysmallscale.Thistype ofvibrationiscalledsimpleharmonicmotion.Insimpleharmonic motion,theperiodisindependentoftheamplitude,andisgivenby T =2 p m=k 22 Chapter1Vibrations

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Problem4. Problems Key p Acomputerizedanswercheckisavailableonline. R Aproblemthatrequirescalculus. ? Adicultproblem. 1 Findanequationforthefrequencyofsimpleharmonicmotion intermsof k and m 2 Manysingle-celledorganismspropelthemselvesthroughwater withlongtails,whichtheywigglebackandforth.Themostobvious exampleisthespermcell.Thefrequencyofthetail'svibrationis typicallyabout10-15Hz.Towhatrangeofperiodsdoesthisrange offrequenciescorrespond? 3 aPendulum2hasastringtwiceaslongaspendulum1.If wedene x asthedistancetraveledbythebobalongacircleaway fromthebottom,howdoesthe k ofpendulum2comparewiththe k ofpendulum1?Giveanumericalratio.[Hint:thetotalforce onthebobisthesameiftheanglesawayfromthebottomarethe same,butequalanglesdonotcorrespondtoequalvaluesof x .] bBasedonyouranswerfromparta,howdoestheperiodofpendulum2comparewiththeperiodofpendulum1?Giveanumerical ratio. 4 Apneumaticspringconsistsofapistonridingontopofthe airinacylinder.Theupwardforceoftheaironthepistonis givenby F air = ax )]TJ/F18 7.9701 Tf 6.587 0 Td [(1.4 ,where a isaconstantwithfunnyunitsof N m 1.4 .Forsimplicity,assumetheaironlysupportstheweight, F W ,ofthepistonitself,althoughinpracticethisdeviceisusedto supportsomeotherobject.Theequilibriumposition, x 0 ,iswhere F W equals )]TJ/F20 10.9091 Tf 8.484 0 Td [(F air .NotethatinthemaintextIhaveassumed theequilibriumpositiontobeat x =0,butthatisnotthenatural choicehere.Assumefrictionisnegligible,andconsideracasewhere theamplitudeofthevibrationsisverysmall.Let a =1N m 1.4 x 0 =1.00m,and F W = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1.00N.Thepistonisreleasedfrom x =1.01m.Drawaneat,accurategraphofthetotalforce, F ,asa functionof x ,ongraphpaper,coveringtherangefrom x =0.98m to1.02m.Overthissmallrange,youwillndthattheforceis verynearlyproportionalto x )]TJ/F20 10.9091 Tf 11.181 0 Td [(x 0 .Approximatethecurvewitha straightline,nditsslope,andderivetheapproximateperiodof oscillation. p 5 Considerthesamepneumaticpistondescribedinproblem 4,butnowimaginethattheoscillationsarenotsmall.Sketcha graphofthetotalforceonthepistonasitwouldappearoverthis widerrangeofmotion.Forawiderrangeofmotion,explainwhy thevibrationofthepistonaboutequilibriumisnotsimpleharmonic motion,andsketchagraphof x vs t ,showingroughlyhowthecurve isdierentfromasinewave.[Hint:Accelerationcorrespondstothe Problems 23

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Problem7. curvatureofthe x )]TJ/F20 10.9091 Tf 11.557 0 Td [(t graph,soiftheforceisgreater,thegraph shouldcurvearoundmorequickly.] 6 Archimedes'principlestatesthatanobjectpartlyorwholly immersedinuidexperiencesabuoyantforceequaltotheweight oftheuiditdisplaces.Forinstance,ifaboatisoatinginwater, theupwardpressureofthewatervectorsumofalltheforcesof thewaterpressinginwardandupwardoneverysquareinchofits hullmustbeequaltotheweightofthewaterdisplaced,because iftheboatwasinstantlyremovedandtheholeinthewaterlled backin,theforceofthesurroundingwaterwouldbejusttheright amounttoholdupthisnewchunk"ofwater.aShowthatacube ofmass m withedgesoflength b oatinguprightnottiltedina uidofdensity willhaveadraftdepthtowhichitsinksbelow thewaterline h givenatequilibriumby h 0 = m=b 2 .bFindthe totalforceonthecubewhenitsdraftis h ,andverifythatplugging in h )]TJ/F20 10.9091 Tf 11.227 0 Td [(h 0 givesatotalforceofzero.cFindthecube'speriodof oscillationasitbobsupanddowninthewater,andshowthatcan beexpressedintermsofand g only. 7 Thegureshowsasee-sawwithtwospringsatCodornicesPark inBerkeley,California.Eachspringhasspringconstant k ,anda kidofmass m sitsoneachseat.aFindtheperiodofvibrationin termsofthevariables k m a ,and b .bDiscussthespecialcase where a = b ,ratherthan a>b asintherealsee-saw.cShowthat youranswertopartaalsomakessenseinthecaseof b =0. ? 8 Showthattheequation T =2 p m=k hasunitsthatmake sense. 9 Ahotscienticquestionofthe18thcenturywastheshape oftheearth:whetheritsradiuswasgreaterattheequatorthanat thepoles,ortheotherwayaround.Onemethodusedtoattackthis questionwastomeasuregravityaccuratelyindierentlocations ontheearthusingpendula.Ifthehighestandlowestlatitudes accessibletoexplorerswere0and70degrees,thenthethestrength ofgravitywouldinrealitybeobservedtovaryoverarangefrom about9.780to9.826m = s 2 .Thischange,amountingto0.046m = s 2 isgreaterthanthe0.022m = s 2 eecttobeexpectediftheearth hadbeenspherical.Thegreatereectoccursbecausetheequator feelsareductionduenotjusttotheaccelerationofthespinning earthoutfromunderit,butalsotothegreaterradiusoftheearth attheequator.Whatistheaccuracywithwhichtheperiodofa one-secondpendulumwouldhavetobemeasuredinordertoprove thattheearthwasnotasphere,andthatitbulgedattheequator? 24 Chapter1Vibrations

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Top: Aseriesofimagesfrom almoftheTacomaNarrows Bridgevibratingonthedayitwas tocollapse. Middle: Thebridge immediatelybeforethecollapse, withthesidesvibrating8.5metersfeetupanddown.Note thatthebridgeisoveramilelong. Bottom: Duringandafterthenalcollapse.Theright-handpicturegivesasenseofthemassive scaleoftheconstruction. Chapter2 Resonance Soonafterthemile-longTacomaNarrowsBridgeopenedinJuly 1940,motoristsbegantonoticeitstendencytovibratefrighteningly inevenamoderatewind.NicknamedGallopingGertie,"thebridge collapsedinasteady42-mile-per-hourwindonNovember7ofthe sameyear.Thefollowingisaneyewitnessreportfromanewspaper editorwhofoundhimselfonthebridgeasthevibrationsapproached thebreakingpoint. JustasIdrovepastthetowers,thebridgebegantoswayviolentlyfromsidetoside.BeforeIrealizedit,thetiltbecameso violentthatIlostcontrolofthecar...Ijammedonthebrakesand 25

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gotout,onlytobethrownontomyfaceagainstthecurb. AroundmeIcouldhearconcretecracking.Istartedtogetmy dogTubby,butwasthrownagainbeforeIcouldreachthecar.The caritselfbegantoslidefromsidetosideoftheroadway. Onhandsandkneesmostofthetime,Icrawled500yardsor moretothetowers...Mybreathwascomingingasps;myknees wererawandbleeding,myhandsbruisedandswollenfromgripping theconcretecurb...Towardthelast,Iriskedrisingtomyfeetand runningafewyardsatatime...Safelybackatthetollplaza,I sawthebridgeinitsnalcollapseandsawmycarplungeintothe Narrows." Theruinsofthebridgeformedanarticialreef,oneofthe world'slargest.Itwasnotreplacedfortenyears.Thereasonfor itscollapsewasnotsubstandardmaterialsorconstruction,norwas thebridgeunder-designed:thepierswerehundred-footblocksof concrete,thegirdersmassiveandmadeofcarbonsteel.Thebridge wasdestroyedbecauseofthephysicalphenomenonofresonance, thesameeectthatallowsanoperasingertobreakawineglass withhervoiceandthatletsyoutuneintheradiostationyouwant. Thereplacementbridge,whichhaslastedhalfacenturysofar,was builtsmarter,notstronger.Theengineerslearnedtheirlessonand simplyincludedsomeslightmodicationstoavoidtheresonance phenomenonthatspelledthedoomoftherstone. 2.1EnergyinVibrations Onewayofdescribingthecollapseofthebridgeisthatthebridge kepttakingenergyfromthesteadilyblowingwindandbuildingup moreandmoreenergeticvibrations.Inthissection,wediscussthe energycontainedinavibration,andinthesubsequentsectionswe willmoveontothelossofenergyandtheaddingofenergytoa vibratingsystem,allwiththegoalofunderstandingtheimportant phenomenonofresonance. Goingbacktoourstandardexampleofamassonaspring,we ndthattherearetwoformsofenergyinvolved:thepotentialenergy storedinthespringandthekineticenergyofthemovingmass.We maystartthesysteminmotioneitherbyhittingthemasstoputin kineticenergybypullingittoonesidetoputinpotentialenergy. Eitherway,thesubsequentbehaviorofthesystemisidentical.It tradesenergybackandforthbetweenkineticandpotentialenergy. Wearestillassumingthereisnofriction,sothatnoenergyis convertedtoheat,andthesystemneverrunsdown. Themostimportantthingtounderstandabouttheenergycontentofvibrationsisthatthetotalenergyisproportionaltothe 26 Chapter2Resonance

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a / Example1. squareoftheamplitude.Althoughthetotalenergyisconstant,it isinstructivetoconsidertwospecicmomentsinthemotionofthe massonaspringasexamples.Whenthemassisallthewayto oneside,atrestandreadytoreversedirections,allitsenergyis potential.Wehavealreadyseenthatthepotentialenergystored inaspringequals = 2 kx 2 ,sotheenergyisproportionaltothe squareoftheamplitude.Nowconsiderthemomentwhenthemass ispassingthroughtheequilibriumpointat x =0.Atthispointit hasnopotentialenergy,butitdoeshavekineticenergy.Thevelocityisproportionaltotheamplitudeofthemotion,andthekinetic energy, = 2 mv 2 ,isproportionaltothesquareofthevelocity,so againwendthattheenergyisproportionaltothesquareofthe amplitude.Thereasonforsinglingoutthesetwopointsismerely instructive;provingthatenergyisproportionalto A 2 atanypoint wouldsucetoprovethatenergyisproportionalto A 2 ingeneral, sincetheenergyisconstant. Aretheseconclusionsrestrictedtothemass-on-a-springexample?No.Wehavealreadyseenthat F = )]TJ/F20 10.9091 Tf 8.485 0 Td [(kx isavalidapproximationforanyvibratingobject,aslongastheamplitudeissmall.We arethusleftwithaverygeneralconclusion:theenergyofanyvibrationisapproximatelyproportionaltothesquareoftheamplitude, providedthattheamplitudeissmall. WaterinaU-tubeexample1 IfwaterispouredintoaU-shapedtubeasshowninthegure,it canundergovibrationsaboutequilibrium.Theenergyofsucha vibrationismosteasilycalculatedbyconsideringtheturnaround pointwhenthewaterhasstoppedandisabouttoreversedirections.Atthispoint,ithasonlypotentialenergyandnokinetic energy,sobycalculatingitspotentialenergywecanndtheenergyofthevibration.Thispotentialenergyisthesameasthe workthatwouldhavetobedonetotakethewateroutoftherighthandsidedowntoadepth A belowtheequilibriumlevel,raiseit throughaheight A ,andplaceitintheleft-handside.Theweight ofthischunkofwaterisproportionalto A ,andsoistheheight throughwhichitmustbelifted,sotheenergyisproportionalto A 2 Therangeofenergiesofsoundwavesexample2 Theamplitudeofvibrationofyoureardrumatthethresholdof painisabout10 6 timesgreaterthantheamplitudewithwhich itvibratesinresponsetothesoftestsoundyoucanhear.How manytimesgreateristheenergywithwhichyourearhastocope forthepainfullyloudsound,comparedtothesoftsound? Theamplitudeis10 6 timesgreater,andenergyisproportional tothesquareoftheamplitude,sotheenergyisgreaterbyafactor Section2.1EnergyinVibrations 27

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b / Frictionhastheeffectof pinchingthe x )]TJ/F102 9.9627 Tf 12.256 0 Td [(t graphofa vibratingobject. of10 12 .Thisisaphenomenallylargefactor! Weareonlystudyingvibrationsrightnow,notwaves,soweare notyetconcernedwithhowasoundwaveworks,orhowtheenergy getstousthroughtheair.Notethatbecauseofthehugerangeof energiesthatourearcansense,itwouldnotbereasonabletohave asenseofloudnessthatwasadditive.Consider,forinstance,the followingthreelevelsofsound: barelyaudiblewind quietconversation....10 5 timesmoreenergythanthe wind heavymetalconcert..10 12 timesmoreenergythanthe wind Intermsofadditionandsubtraction,thedierencebetweenthe windandthequietconversationisnothingcomparedtothedierencebetweenthequietconversationandtheheavymetalconcert. Evolutionwantedoursenseofhearingtobeabletoencompassall thesesoundswithoutcollapsingthebottomofthescalesothatanythingsofterthanthecrackofdoomwouldsoundthesame.Sorather thanmakingoursenseofloudnessadditive,mothernaturemadeit multiplicative.Wesensethedierencebetweenthewindandthe quietconversationasspanningarangeofabout5/12asmuchasthe wholerangefromthewindtotheheavymetalconcert.Although adetaileddiscussionofthedecibelscaleisnotrelevanthere,the basicpointtonoteaboutthedecibelscaleisthatitislogarithmic. Thezeroofthedecibelscaleisclosetothelowerlimitofhuman hearing,andadding1unittothedecibelmeasurementcorresponds to multiplying theenergyleveloractuallythepowerperunitarea byacertainfactor. 2.2EnergyLostFromVibrations Untilnow,wehavebeenmakingtherelativelyunrealisticassumptionthatavibrationwouldneverdieout.Forarealisticmass onaspring,therewillbefriction,andthekineticandpotential energyofthevibrationswillthereforebegraduallyconvertedinto heat.Similarly,aguitarstringwillslowlyconvertitskineticand potentialenergyintosound.Inallcases,theeectistopinch"the sinusoidal x )]TJ/F20 10.9091 Tf 11.282 0 Td [(t graphmoreandmorewithpassingtime.Friction isnotnecessarilybadinthiscontext|amusicalinstrumentthat nevergotridofanyofitsenergywouldbecompletelysilent!The dissipationoftheenergyinavibrationisknownasdamping. self-checkA Mostpeoplewhotrytodrawgraphslikethoseshownontheleftwill tendtoshrinktheirwiggleshorizontallyaswellasvertically.Whyisthis wrong? Answer,p.98 Inthegraphsingureb,Ihavenotshownanypointatwhich 28 Chapter2Resonance

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c / Theamplitudeishalved witheachcycle. thedampedvibrationnallystopscompletely.Isthisrealistic?Yes andno.Ifenergyisbeinglostduetofrictionbetweentwosolid surfaces,thenweexpecttheforceoffrictiontobenearlyindependentofvelocity.Thisconstantfrictionforceputsanupperlimiton thetotaldistancethatthevibratingobjectcanevertravelwithout replenishingitsenergy,sinceworkequalsforcetimesdistance,and theobjectmuststopdoingworkwhenitsenergyisallconverted intoheat.Thefrictionforcedoesreversedirectionswhentheobjectturnsaround,butreversingthedirectionofthemotionatthe sametimethatwereversethedirectionoftheforcemakesitcertain thattheobjectisalwaysdoingpositivework,notnegativework. Dampingduetoaconstantfrictionforceisnottheonlypossibilityhowever,oreventhemostcommonone.Apendulummay bedampedmainlybyairfriction,whichisapproximatelyproportionalto v 2 ,whileothersystemsmayexhibitfrictionforcesthat areproportionalto v .Itturnsoutthatfrictionproportionalto v isthesimplestcasetoanalyzemathematically,andanyhowallthe importantphysicalinsightscanbegainedbystudyingthiscase. Ifthefrictionforceisproportionalto v ,thenasthevibrations diedown,thefrictionalforcesgetweakerduetothelowerspeeds. Thelessenergyisleftinthesystem,themoremiserlythesystem becomeswithgivingawayanymoreenergy.Undertheseconditions, thevibrationstheoreticallyneverdieoutcompletely,andmathematically,thelossofenergyfromthesystemisexponential:thesystem losesaxedpercentageofitsenergypercycle.Thisisreferredto asexponentialdecay. Anon-rigorousproofisasfollows.Theforceoffrictionisproportionalto v ,and v isproportionaltohowfartheobjectstravelsin onecycle,sothefrictionalforceisproportionaltoamplitude.The amountofworkdonebyfrictionisproportionaltotheforceandto thedistancetraveled,sotheworkdoneinonecycleisproportional tothesquareoftheamplitude.Sinceboththeworkandtheenergy areproportionalto A 2 ,theamountofenergytakenawaybyfriction inonecycleisaxedpercentageoftheamountofenergythesystem has. self-checkB Figurecshowsanx-tgraphforastronglydampedvibration,whichloses halfofitsamplitudewitheverycycle.Whatfractionoftheenergyislost ineachcycle? Answer,p.98 Itiscustomarytodescribetheamountofdampingwithaquantitycalledthequalityfactor, Q ,denedasthenumberofcycles requiredfortheenergytofallobyafactorof535.Theorigin ofthisobscurenumericalfactoris e 2 ,where e =2.71828 ::: isthe baseofnaturallogarithms.Choosingthisparticularnumbercauses someofourlaterequationstocomeoutniceandsimple.Theterminologyarisesfromthefactthatfrictionisoftenconsideredabad Section2.2EnergyLostFromVibrations 29

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d / 1.Pushingachildona swinggraduallyputsmoreand moreenergyintohervibrations. 2.Afairlyrealisticgraphofthe drivingforceactingonthechild. 3.Alessrealistic,butmore mathematicallysimple,driving force. thing,soamechanicaldevicethatcanvibrateformanyoscillations beforeitlosesasignicantfractionofitsenergywouldbeconsidered ahigh-qualitydevice. Exponentialdecayinatrumpetexample3 Thevibrationsoftheaircolumninsideatrumpethavea Q of about10.Thismeansthatevenafterthetrumpetplayerstops blowing,thenotewillkeepsoundingforashorttime.Iftheplayer suddenlystopsblowing,howwillthesoundintensity20cycles latercomparewiththesoundintensitywhileshewasstillblowing? Thetrumpet's Q is10,soafter10cyclestheenergywillhave fallenoffbyafactorof535.Afteranother10cyclesweloseanotherfactorof535,sothesoundintensityisreducedbyafactor of535 535=2.9 10 5 Thedecayofamusicalsoundispartofwhatgivesititscharacter,andagoodmusicalinstrumentshouldhavetheright Q ,butthe Q thatisconsidereddesirableisdierentfordierentinstruments. Aguitarismeanttokeeponsoundingforalongtimeafterastring hasbeenplucked,andmighthavea Q of1000or10000.Oneofthe reasonswhyacheapsynthesizersoundssobadisthatthesound suddenlycutsoafterakeyisreleased. Q ofastereospeakerexample4 Stereospeakersarenotsupposedtoreverberateorringafteran electricalsignalthatstopssuddenly.Afterall,therecordedmusic wasmadebymusicianswhoknewhowtoshapethedecaysof theirnotescorrectly.Addingalongertailoneverynotewould makeitsoundwrong.Wethereforeexpectthatstereospeaker willhaveaverylow Q ,andindeed,mostspeakersaredesigned witha Q ofabout1.Low-qualityspeakerswithlarger Q values arereferredtoasboomy. Wewillseelaterinthechapterthatthereareotherreasonswhy aspeakershouldnothaveahigh Q 2.3PuttingEnergyIntoVibrations Whenpushingachildonaswing,youcannotjustapplyaconstantforce.Aconstantforcewillmovetheswingouttoacertain angle,butwillnotallowtheswingtostartswinging.Norcanyou giveshortpushesatrandomlychosentimes.Thattypeofrandompushingwouldincreasethechild'skineticenergywheneveryou happenedtobepushinginthesamedirectionashermotion,butit wouldreduceherenergywhenyourpushinghappenedtobeinthe oppositedirectioncomparedtohermotion.Tomakeherbuildup herenergy,youneedtomakeyourpushesrhythmic,pushingatthe samepointineachcycle.Inotherwords,yourforceneedstoforma repeatingpatternwiththesamefrequencyasthenormalfrequency ofvibrationoftheswing.Graphd/1showswhatthechild's x )]TJ/F20 10.9091 Tf 11.18 0 Td [(t 30 Chapter2Resonance

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e / Theamplitudeapproachesa maximum. graphwouldlooklikeasyougraduallyputmoreandmoreenergy intohervibrations.Agraphofyour force versustimewouldprobablylooksomethinglikegraph2.Itturnsout,however,thatitis muchsimplermathematicallytoconsideravibrationwithenergy beingpumpedintoitbyadrivingforcethatisitselfasine-wave,3. Agoodexampleofthisisyoureardrumbeingdrivenbytheforce ofasoundwave. Nowweknowrealisticallythatthechildontheswingwillnot keepincreasingherenergyforever,nordoesyoureardrumendup explodingbecauseacontinuingsoundwavekeepspumpingmoreand moreenergyintoit.Inanyrealisticsystem,thereisenergygoing outaswellasin.Asthevibrationsincreaseinamplitude,thereisan increaseintheamountofenergytakenawaybydampingwitheach cycle.Thisoccursfortworeasons.Workequalsforcetimesdistance or,moreaccurately,theareaundertheforce-distancecurve.As theamplitudeofthevibrationsincreases,thedampingforceisbeing appliedoveralongerdistance.Furthermore,thedampingforce usuallyincreaseswithvelocityweusuallyassumeforsimplicity thatitisproportionaltovelocity,andthisalsoservestoincrease therateatwhichdampingforcesremoveenergyastheamplitude increases.Eventuallyandsmallchildrenandoureardrumsare thankfulforthis!,theamplitudeapproachesamaximumvalue,e, atwhichenergyisremovedbythedampingforcejustasquicklyas itisbeingputinbythedrivingforce. Thisprocessofapproachingamaximumamplitudehappensextremelyquicklyinmanycases,e.g.,theearoraradioreceiver,and wedon'tevennoticethatittookamillisecondoramicrosecond forthevibrationstobuildupsteam."Wearethereforemainly interestedinpredictingthebehaviorofthesystemonceithashad enoughtimetoreachessentiallyitsmaximumamplitude.Thisis knownasthesteady-statebehaviorofavibratingsystem. Nowcomestheinterestingpart:whathappensifthefrequency ofthedrivingforceismismatchedtothefrequencyatwhichthe systemwouldnaturallyvibrateonitsown?Weallknowthata radiostationdoesn'thavetobetunedinexactly,althoughthereis onlyasmallrangeoverwhichagivenstationcanbereceived.The designersoftheradiohadtomaketherangefairlysmalltomake itpossibleeliminateunwantedstationsthathappenedtobenearby infrequency,butitcouldn'tbetoosmalloryouwouldn'tbeable toadjusttheknobaccuratelyenough.Evenadigitalradiocan betunedto88.0MHzandstillbringinastationat88.1MHz. Theearalsohassomenaturalfrequencyofvibration,butinthis casetherangeoffrequenciestowhichitcanrespondisquitebroad. Evolutionhasmadetheear'sfrequencyresponseasbroadaspossiblebecauseitwastoourancestors'advantagetobeabletohear everythingfromalowroarstoahigh-pitchedshriek. Theremainderofthissectiondevelopsfourimportantfactsabout Section2.3PuttingEnergyIntoVibrations 31

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theresponseofasystemtoadrivingforcewhosefrequencyisnot necessarilythesameasthesystem'snaturalfrequencyofvibration. Thestyleisapproximateandintuitive,butproofsaregiveninthe subsequentoptionalsection. First,althoughweknowtheearhasafrequency|about4000 Hz|atwhichitwouldvibratenaturally,itdoesnotvibrateat 4000Hzinresponsetoalow-pitched200Hztone.Italwaysrespondsatthefrequencyatwhichitisdriven.Otherwiseallpitches wouldsoundlike4000Hztous.Thisisageneralfactaboutdriven vibrations: Thesteady-stateresponsetoasinusoidaldrivingforceoccursatthefrequencyoftheforce,notatthesystem'sownnatural frequencyofvibration. Nowlet'sthinkabouttheamplitudeofthesteady-stateresponse. Imaginethatachildonaswinghasanaturalfrequencyofvibration of1Hz,butwearegoingtotrytomakeherswingbackandforthat 3Hz.Weintuitivelyrealizethatquitealargeforcewouldbeneeded toachieveanamplitudeofeven30cm,i.e.,theamplitudeislessin proportiontotheforce.Whenwepushatthenaturalfrequencyof 1Hz,weareessentiallyjustpumpingenergybackintothesystem tocompensateforthelossofenergyduetothedampingfriction force.At3Hz,however,wearenotjustcounteractingfriction.We arealsoprovidinganextraforcetomakethechild'smomentum reverseitselfmorerapidlythanitwouldifgravityandthetension inthechainweretheonlyforcesacting.Itisasifwearearticially increasingthe k oftheswing,butthisiswastedeortbecausewe spendjustasmuchtimedeceleratingthechildtakingenergyout ofthesystemasacceleratingherputtingenergyin. Nowimaginethecaseinwhichwedrivethechildatavery lowfrequency,say0.02Hzoraboutonevibrationperminute.We areessentiallyjustholdingthechildinpositionwhileveryslowly walkingbackandforth.Againweintuitivelyrecognizethatthe amplitudewillbeverysmallinproportiontoourdrivingforce. Imaginehowharditwouldbetoholdthechildatourownheadlevelwhensheisattheendofherswing!Asinthetoo-fast3Hz case,wearespendingmostofoureortinarticiallychangingthe k oftheswing,butnowratherthanreinforcingthegravityand tensionforcesweareworkingagainstthem,eectivelyreducing k Onlyaverysmallpartofourforcegoesintocounteractingfriction, andtherestisusedinrepetitivelyputtingpotentialenergyinon theupswingandtakingitbackoutonthedownswing,withoutany long-termgain. Wecannowgeneralizetomakethefollowingstatement,which istrueforalldrivenvibrations: 32 Chapter2Resonance

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f / Thecollapsedsectionof theNimitzFreeway. Avibratingsystemresonatesatitsownnaturalfrequency. Thatis,theamplitudeofthesteady-stateresponseisgreatestin proportiontotheamountofdrivingforcewhenthedrivingforce matchesthenaturalfrequencyofvibration. Anoperasingerbreakingawineglassexample5 Inordertobreakawineglassbysinging,anoperasingermust rsttaptheglasstonditsnaturalfrequencyofvibration,and thensingthesamenoteback. CollapseoftheNimitzFreewayinanearthquakeexample6 IledoffthechapterwiththedramaticcollapseoftheTacoma NarrowsBridge,mainlybecauseaitwaswelldocumentedbya localphysicsprofessor,andanunknownpersonmadeamovie ofthecollapse.ThecollapseofasectionoftheNimitzFreeway inOakland,CA,duringa1989earthquakeishoweverasimpler exampletoanalyze. Anearthquakeconsistsofmanylow-frequencyvibrationsthatoccursimultaneously,whichiswhyitsoundslikearumbleofindeterminatepitchratherthanalowhum.Thefrequenciesthatwe canheararenoteventhestrongestones;mostoftheenergyis intheformofvibrationsintherangeoffrequenciesfromabout1 Hzto10Hz. Nowallthestructureswebuildarerestingongeologicallayers ofdirt,mud,sand,orrock.Whenanearthquakewavecomes along,thetopmostlayeractslikeasystemwithacertainnatural frequencyofvibration,sortoflikeacubeofjelloonaplatebeing shakenfromsidetoside.Theresonantfrequencyofthelayer dependsonhowstiffitisandalsoonhowdeepitis.TheillfatedsectionoftheNimitzfreewaywasbuiltonalayerofmud, andanalysisbygeologistSusanE.HoughoftheU.S.Geological Surveyshowsthatthemudlayer'sresonancewascenteredon about2.5Hz,andhadawidthcoveringarangefromabout1Hz to4Hz. Whentheearthquakewavecamealongwithitsmixtureoffrequencies,themudrespondedstronglytothosethatwerecloseto itsownnatural2.5Hzfrequency.Unfortunately,anengineering analysisafterthequakeshowedthattheoverpassitselfhadaresonantfrequencyof2.5Hzaswell!Themudrespondedstronglyto theearthquakewaveswithfrequenciescloseto2.5Hz,andthe bridgerespondedstronglytothe2.5Hzvibrationsofthemud, causingsectionsofittocollapse. CollapseoftheTacomaNarrowsBridgeexample7 Let'snowexaminethemoreconceptuallydifcultcaseofthe TacomaNarrowsBridge.Thesurprisehereisthatthewindwas steady.Ifthewindwasblowingatconstantvelocity,whydidit Section2.3PuttingEnergyIntoVibrations 33

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shakethebridgebackandforth?Theanswerisalittlecomplicated.Basedonlmfootageandafter-the-factwindtunnelexperiments,itappearsthattwodifferentmechanismswereinvolved. Therstmechanismwastheoneresponsiblefortheinitial,relativelyweakvibrations,anditinvolvedresonance.Asthewind movedoverthebridge,itbeganactinglikeakiteoranairplane wing.Asshowninthegure,itestablishedswirlingpatternsofair owarounditself,ofthekindthatyoucanseeinamovingcloud ofsmoke.Asoneoftheseswirlsmovedoffofthebridge,there wasanabruptchangeinairpressure,whichresultedinanupor downforceonthebridge.Weseesomethingsimilarwhenaag apsinthewind,exceptthattheag'ssurfaceisusuallyvertical.Thisback-and-forthsequenceofforcesisexactlythekindof periodicdrivingforcethatwouldexcitearesonance.Thefaster thewind,themorequicklytheswirlswouldgetacrossthebridge, andthehigherthefrequencyofthedrivingforcewouldbe.Atjust therightvelocity,thefrequencywouldbetherightonetoexcite theresonance.Thewind-tunnelmodels,however,showthatthe patternofvibrationofthebridgeexcitedbythismechanismwould havebeenadifferentonethantheonethatnallydestroyedthe bridge. Thebridgewasprobablydestroyedbyadifferentmechanism,in whichitsvibrationsatitsownnaturalfrequencyof0.2Hzsetup analternatingpatternofwindgustsintheairimmediatelyaround it,whichthenincreasedtheamplitudeofthebridge'svibrations. Thisviciouscyclefeduponitself,increasingtheamplitudeofthe vibrationsuntilthebridgenallycollapsed. Aslongaswe'reonthesubjectofcollapsingbridges,itisworth bringingupthereportsofbridgesfallingdownwhensoldiersmarchingoverthemhappenedtostepinrhythmwiththebridge'snatural frequencyofoscillation.Thisissupposedtohavehappenedin1831 inManchester,England,andagainin1849inAnjou,France.Many modernengineersandscientists,however,aresuspiciousoftheanalysisofthesereports.Itispossiblethatthecollapseshadmoretodo withpoorconstructionandoverloadingthanwithresonance.The NimitzFreewayandTacomaNarrowsBridgearefarbetterdocumented,andoccurredinanerawhenengineers'abilitiestoanalyze thevibrationsofacomplexstructureweremuchmoreadvanced. Emissionandabsorptionoflightwavesbyatomsexample8 Inaverythingas,theatomsaresufcientlyfarapartthattheycan actasindividualvibratingsystems.Althoughthevibrationsareof averystrangeandabstracttypedescribedbythetheoryofquantummechanics,theyneverthelessobeythesamebasicrulesas ordinarymechanicalvibrations.Whenathingasmadeofacertainelementisheated,itemitslightwaveswithcertainspecic frequencies,whicharelikeangerprintofthatelement.Aswith 34 Chapter2Resonance

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g / Thedenitionofthefull widthathalfmaximum. allothervibrations,theseatomicvibrationsrespondmoststrongly toadrivingforcethatmatchestheirownnaturalfrequency.Thus ifwehavearelativelycoldgaswithlightwavesofvariousfrequenciespassingthroughit,thegaswillabsorblightatprecisely thosefrequenciesatwhichitwouldemitlightifheated. Whenasystemisdrivenatresonance,thesteady-statevibrationshaveanamplitudethatisproportionalto Q Thisisfairlyintuitive.Thesteady-statebehaviorisanequilibriumbetweenenergyinputfromthedrivingforceandenergyloss duetodamping.AlowQ oscillator,i.e.,onewithstrongdamping, dumpsitsenergyfaster,resultinginlower-amplitudesteady-state motion. self-checkC Ifanoperasingerisshoppingforawineglassthatshecanimpressher friendsbybreaking,whatshouldshelookfor? Answer,p.98 Pianostringsringinginsympathywithasungnoteexample9 Asufcientlyloudmusicalnotesungnearapianowiththelid raisedcancausethecorrespondingstringsinthepianotovibrate. Apianohasasetofthreestringsforeachnote,allstruckbythe samehammer.Whywouldthistrickbeunlikelytoworkwitha violin? Ifyouhaveheardthesoundofaviolinbeingpluckedthepizzicatoeffect,youknowthatthenotediesawayveryquickly.In otherwords,aviolin's Q ismuchlowerthanapiano's.Thismeans thatitsresonancesaremuchweakerinamplitude. Ourfourthandnalfactaboutresonanceisperhapsthemost surprising.Itgivesusawaytodeterminenumericallyhowwide arangeofdrivingfrequencieswillproduceastrongresponse.As showninthegraph,resonancesdonotsuddenlyfallotozerooutsideacertainfrequencyrange.Itisusualtodescribethewidthofa resonancebyitsfullwidthathalf-maximumFWHMasillustrated ingureg. TheFWHMofaresonanceisrelatedtoits Q anditsresonant frequency f res bytheequation FWHM= f res Q Thisequationisonlyagoodapproximationwhen Q islarge. Why?Itisnotimmediatelyobviousthatthereshouldbeany logicalrelationshipbetween Q andtheFWHM.Here'stheidea.As wehaveseenalready,thereasonwhytheresponseofanoscillator issmallerawayfromresonanceisthatmuchofthedrivingforceis Section2.3PuttingEnergyIntoVibrations 35

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beingusedtomakethesystemactasifithadadierent k .Roughly speaking,thehalf-maximumpointsonthegraphcorrespondtothe placeswheretheamountofthedrivingforcebeingwastedinthis wayisthesameastheamountofdrivingforcebeingusedproductivelytoreplacetheenergybeingdumpedoutbythedamping force.Ifthedampingforceisstrong,thenalargeamountofforce isneededtocounteractit,andwecanwastequiteabitofdriving forceonchanging k beforeitbecomescomparabletothedamping force.If,ontheotherhand,thedampingforceisweak,thenevena smallamountofforcebeingwastedonchanging k willbecomesignicantinproportion,andwecannotgetveryfarfromtheresonant frequencybeforethetwoarecomparable. Changingthepitchofawindinstrumentexample10 Asaxophoneplayernormallyselectswhichnotetoplayby choosingacertainngering,whichgivesthesaxophoneacertainresonantfrequency.Themusiciancanalso,however,change thepitchsignicantlybyalteringthetightnessofherlips.This correspondstodrivingthehornslightlyoffofresonance.Ifthe pitchcanbealteredbyabout5%upordownaboutonemusicalhalf-stepwithouttoomucheffort,roughlywhatisthe Q ofa saxophone? Fivepercentisthewidthononesideoftheresonance,sothe fullwidthisabout10%,FWHM/ f res =0.1.Thisimpliesa Q ofabout10,i.e.,oncethemusicianstopsblowing,thehornwill continuesoundingforabout10cyclesbeforeitsenergyfallsoffby afactorof535.Bluesandjazzsaxophoneplayerswilltypically chooseamouthpiecethathasalow Q ,sothattheycanproduce thebluesypitch-slidestypicaloftheirstyle.Legit,i.e.,classically orientedplayers,useahigherQ setupbecausetheirstyleonly callsforenoughpitchvariationtoproduceavibrato. Decayofasaxophonetoneexample11 Ifatypicalsaxophonesetuphasa Q ofabout10,howlongwill ittakefora100-Hztoneplayedonabaritonesaxophonetodie downbyafactorof535inenergy,aftertheplayersuddenlystops blowing? A Q of10meansthatittakes10cyclesforthevibrationstodie downinenergybyafactorof535.Tencyclesatafrequencyof 100Hzwouldcorrespondtoatimeof0.1seconds,whichisnot verylong.Thisiswhyasaxophonenotedoesn'tringlikeanote playedonapianooranelectricguitar. Q ofaradioreceiverexample12 AradioreceiverusedintheFMbandneedstobetunedinto withinabout0.1MHzforsignalsatabout100MHz.Whatisits Q ? Q = f res = FWHM=1000.Thisisanextremelyhigh Q compared 36 Chapter2Resonance

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h / Example14.1.Acompassneedlevibratesaboutthe equilibriumpositionunderthe inuenceoftheearth'smagnetic forces.2.Theorientationofa proton'sspinvibratesaroundits equilibriumdirectionunderthe inuenceofthemagneticforces comingfromthesurrounding electronsandnuclei. i / Amemberoftheauthor's family,whoturnedouttobe healthy. j / Athree-dimensionalcomputerreconstructionoftheshape ofahumanbrain,basedon magneticresonancedata. tomostmechanicalsystems. Q ofastereospeakerexample13 Wehavealreadygivenonereasonwhyastereospeakershould havealow Q :otherwiseitwouldcontinueringingaftertheendof themusicalnoteontherecording.Thesecondreasonisthatwe wantittobeabletorespondtoalargerangeoffrequencies. Nuclearmagneticresonanceexample14 Ifyouhaveeverplayedwithamagneticcompass,youhaveundoubtedlynoticedthatifyoushakeit,ittakessometimetosettle down,h/1.Asitsettlesdown,itactslikeadampedoscillatorof thetypewehavebeendiscussing.Thecompassneedleissimply asmallmagnet,andtheplanetearthisabigmagnet.Themagneticforcesbetweenthemtendtobringtheneedletoanequilibriumpositioninwhichitlinesupwiththeplanet-earth-magnet. EssentiallythesamephysicsliesbehindthetechniquecalledNuclearMagneticResonanceNMR.NMRisatechniqueusedto deducethemolecularstructureofunknownchemicalsubstances, anditisalsousedformakingmedicalimagesoftheinsideofpeople'sbodies.IfyoueverhaveanNMRscan,theywillactuallytell youyouareundergoingmagneticresonanceimagingorMRI, becausepeoplearescaredofthewordnuclear.Infact,the nucleibeingreferredtoaresimplythenon-radioactivenucleiof atomsfoundnaturallyinyourbody. Here'showNMRworks.Yourbodycontainslargenumbersof hydrogenatoms,eachconsistingofasmall,lightweightelectron orbitingaroundalarge,heavyproton.Thatis,thenucleusofa hydrogenatomisjustoneproton.Aprotonisalwaysspinning onitsownaxis,andthecombinationofitsspinanditselectrical chargecauseittobehavelikeatinymagnet.Theprincipleidenticaltothatofanelectromagnet,whichconsistsofacoilofwire throughwhichelectricalchargespass;thecirclingmotionofthe chargesinthecoilofwiremakesitmagnetic,andinthesame way,thecirclingmotionoftheproton'schargemakesitmagnetic. Nowaprotoninoneofyourbody'shydrogenatomsndsitself surroundedbymanyotherwhirling,electricallychargedparticles: itsownelectron,plustheelectronsandnucleioftheothernearby atoms.Theseneighborsactlikemagnets,andexertmagnetic forcesontheproton,h/2.The k ofthevibratingprotonissimplya measureofthetotalstrengthofthesemagneticforces.Dependingonthestructureofthemoleculeinwhichthehydrogenatom ndsitself,therewillbeaparticularsetofmagneticforcesacting ontheprotonandaparticularvalueof k .TheNMRapparatus bombardsthesamplewithradiowaves,andifthefrequencyof theradiowavesmatchestheresonantfrequencyoftheproton, theprotonwillabsorbradio-waveenergystronglyandoscillate wildly.Itsvibrationsaredampednotbyfriction,becausethereis Section2.3PuttingEnergyIntoVibrations 37

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k / Drivingatafrequencyabove resonance. l / Drivingatresonance. m / Drivingatafrequency belowresonance. nofrictioninsideanatom,butbythereemissionofradiowaves. Byworkingbackwardthroughthischainofreasoning,onecandeterminethegeometricarrangementofthehydrogenatom'sneighboringatoms.Itisalsopossibletolocateatomsinspace,allowing medicalimagestobemade. Finally,itshouldbenotedthatthebehavioroftheprotoncannot bedescribedentirelycorrectlybyNewtonianphysics.Itsvibrationsareofthestrangeandspookykinddescribedbythelawsof quantummechanics.Itisimpressive,however,thatthefewsimpleideaswehavelearnedaboutresonancecanstillbeapplied successfullytodescribemanyaspectsofthisexoticsystem. DiscussionQuestion A NikolaTesla,oneoftheinventorsofradioandanarchetypicalmad scientist,toldacredulousreporterthefollowingstoryaboutanapplicationofresonance.Hebuiltanelectricvibratorthattinhispocket,and attachedittooneofthesteelbeamsofabuildingthatwasunderconstructioninNewYork.Althoughthearticleinwhichhewasquoteddidn'tsay so,hepresumablyclaimedtohavetunedittotheresonantfrequencyof thebuilding.Inafewminutes,Icouldfeelthebeamtrembling.Gradually thetremblingincreasedinintensityandextendedthroughoutthewhole greatmassofsteel.Finally,thestructurebegantocreakandweave,and thesteelworkerscametothegroundpanic-stricken,believingthatthere hadbeenanearthquake....[If]Ihadkeptontenminutesmore,Icould havelaidthatbuildingatinthestreet.Isthisphysicallyplausible? 2.4 ? Proofs Ourrstgoalistopredicttheamplitudeofthesteady-state vibrationsasafunctionofthefrequencyofthedrivingforceand theamplitudeofthedrivingforce.Withthatequationinhand,we willthenprovestatements2,3,and4fromtheprevioussection. Weassumewithoutproofstatement1,thatthesteady-statemotion occursatthesamefrequencyasthedrivingforce. Aswiththeproofinchapter1,wemakeuseofthefactthat asinusoidalvibrationisthesameastheprojectionofcircularmotionontoaline.Wevisualizethesystemshowninguresk-m, inwhichthemassswingsinacircleontheendofaspring.The springdoesnotactuallychangeitslengthatall,butitappearsto fromtheattenedperspectiveofapersonviewingthesystemedgeon.Theradiusofthecircleistheamplitude, A ,ofthevibrations asseenedge-on.Thedampingforcecanbeimaginedasabackwarddragforcesuppliedbysomeuidthroughwhichthemassis moving.Asusual,weassumethatthedampingisproportionalto velocity,andweusethesymbol b fortheproportionalityconstant, j F d j = bv .Thedrivingforce,representedbyahandtowingthemass withastring,hasatangentialcomponent j F t j whichcounteractsthe dampingforce, j F t j = j F d j ,andaradialcomponent F r whichworks 38 Chapter2Resonance

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eitherwithoragainstthespring'sforce,dependingonwhetherwe aredrivingthesystemaboveorbelowitsresonantfrequency. Thespeedoftherotatingmassisthecircumferenceofthecircle dividedbytheperiod, v =2 A=T ,itsaccelerationwhichisdirectly inwardis a = v 2 =r ,andNewton'ssecondlawgives a = F=m = kA + F r =m .Wewrite f res for 1 2 p k=m .Straightforwardalgebra yields [1] F r F t = 2 m bf )]TJ/F20 10.9091 Tf 5 -8.836 Td [(f 2 )]TJ/F20 10.9091 Tf 10.909 0 Td [(f 2 res Thisistheratioofthewastedforcetotheusefulforce,andwesee thatitbecomeszerowhenthesystemisdrivenatresonance. Theamplitudeofthevibrationscanbefoundbyattackingthe equation j F t j = bv =2 bAf ,whichgives [2] A = j F t j 2 bf However,wewishtoknowtheamplitudeintermsof| F |,not j F t j .Fromnowon,let'sdropthecumbersomemagnitudesymbols. WiththePythagoreantheorem,itiseasilyproventhat [3] F t = F r 1+ F r F t 2 andequations1-3canthenbecombinedtogivethenalresult [4] A = F 2 q 4 2 m 2 f 2 )]TJ/F20 10.9091 Tf 10.909 0 Td [(f 2 res 2 + b 2 f 2 Statement2:maximumAmplitudeatresonance Equation4showsdirectlythattheamplitudeismaximizedwhen thesystemisdrivenatitsresonantfrequency.Atresonance,therst terminsidethesquarerootvanishes,andthismakesthedenominatorassmallaspossible,causingtheamplitudetobeasbigas possible.Actuallythisisonlyapproximatelytrue,becauseitis possibletomake A alittlebiggerbydecreasing f alittlebelow f res ,whichmakesthesecondtermsmaller.Thistechnicalissueis addressedinhomeworkproblem3onpage43. Statement3:Amplitudeatresonanceproportionalto q Equation4showsthattheamplitudeatresonanceisproportionalto1 =b ,andthe Q ofthesystemisinverselyproportionalto b ,sotheamplitudeatresonanceisproportionalto Q Section2.4 ? Proofs 39

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Statement4:fwhmrelatedto q Wewillsatisfyourselvesbyprovingonlytheproportionality FWHM / f res =Q ,nottheactualequation FWHM = f res =Q Theenergyisproportionalto A 2 ,i.e.,totheinverseofthequantity insidethesquarerootinequation4.Atresonance,therstterm insidethesquarerootvanishes,andthehalf-maximumpointsoccur atfrequenciesforwhichthewholequantityinsidethesquareroot isdoubleitsvalueatresonance,i.e.,whenthetwotermsareequal. Atthehalf-maximumpoints,wehave f 2 )]TJ/F20 10.9091 Tf 10.909 0 Td [(f 2 res = f res FWHM 2 2 )]TJ/F20 10.9091 Tf 10.909 0 Td [(f 2 res = f res FWHM+ 1 4 FWHM 2 Ifweassumethatthewidthoftheresonanceissmallcomparedto theresonantfrequency,thentheFWHM 2 termisnegligiblecomparedtothe f res FWHMterm,andsettingthetermsinequation 4equaltoeachothergives 4 2 m 2 f res FWHM 2 = b 2 f 2 Weareassumingthatthewidthoftheresonanceissmallcompared totheresonantfrequency,so f and f res canbetakenassynonyms. Thus, FWHM= b 2 m Wewishtoconnectthisto Q ,whichcanbeinterpretedastheenergyofthefreeundrivenvibrationsdividedbytheworkdoneby dampinginonecycle.Theformerequals kA 2 = 2,andthelatteris proportionaltotheforce, bv / bAf res ,multipliedbythedistance traveled, A .Thisisonlyaproportionality,notanequation,since theforceisnotconstant.Wethereforendthat Q isproportional to k=bf res .TheequationfortheFWHMcanthenberestatedasa proportionalityFWHM / k=Qf res m / f res =Q 40 Chapter2Resonance

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Summary SelectedVocabulary damping.....thedissipationofavibration'senergyinto heatenergy,orthefrictionalforcethatcauses thelossofenergy qualityfactor..thenumberofoscillationsrequiredforasystem'senergytofallobyafactorof535due todamping drivingforce...anexternalforcethatpumpsenergyintoavibratingsystem resonance....thetendencyofavibratingsystemtorespond moststronglytoadrivingforcewhosefrequencyisclosetoitsownnaturalfrequency ofvibration steadystate...thebehaviorofavibratingsystemafterithas hadplentyoftimetosettleintoasteadyresponsetoadrivingforce Notation Q .........thequalityfactor f res ........thenaturalresonantfrequencyofavibrating system,i.e.,thefrequencyatwhichitwould vibrateifitwassimplykickedandleftalone f ..........thefrequencyatwhichthesystemactuallyvibrates,whichinthecaseofadrivensystemis equaltothefrequencyofthedrivingforce,not thenaturalfrequency Summary Theenergyofavibrationisalwaysproportionaltothesquareof theamplitude,assumingtheamplitudeissmall.Energyislostfrom avibratingsystemforvariousreasonssuchastheconversiontoheat viafrictionortheemissionofsound.Thiseect,calleddamping, willcausethevibrationstodecayexponentiallyunlessenergyis pumpedintothesystemtoreplacetheloss.Adrivingforcethat pumpsenergyintothesystemmaydrivethesystematitsown naturalfrequencyoratsomeotherfrequency.Whenavibrating systemisdrivenbyanexternalforce,weareusuallyinterestedin itssteady-statebehavior,i.e.,itsbehaviorafterithashadtimeto settleintoasteadyresponsetoadrivingforce.Inthesteadystate, thesameamountofenergyispumpedintothesystemduringeach cycleasislosttodampingduringthesameperiod. Thefollowingarefourimportantfactsaboutavibratingsystem beingdrivenbyanexternalforce: Thesteady-stateresponsetoasinusoidaldrivingforceoccursatthefrequencyoftheforce,notatthesystem'sownnatural frequencyofvibration. Summary 41

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Avibratingsystemresonatesatitsownnaturalfrequency. Thatis,theamplitudeofthesteady-stateresponseisgreatestin proportiontotheamountofdrivingforcewhenthedrivingforce matchesthenaturalfrequencyofvibration. Whenasystemisdrivenatresonance,thesteady-statevibrationshaveanamplitudethatisproportionalto Q TheFWHMofaresonanceisrelatedtoits Q anditsresonant frequency f res bytheequation FWHM= f res Q Thisequationisonlyagoodapproximationwhen Q islarge. 42 Chapter2Resonance

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Problems Key p Acomputerizedanswercheckisavailableonline. R Aproblemthatrequirescalculus. ? Adicultproblem. 1 Ifonestereosystemiscapableofproducing20wattsofsound powerandanothercanputout50watts,howmanytimesgreater istheamplitudeofthesoundwavethatcanbecreatedbythemore powerfulsystem?Assumetheyareplayingthesamemusic. 2 Manyshhaveanorganknownasaswimbladder,anair-lled cavitywhosemainpurposeistocontrolthesh'sbuoyancyanallow ittokeepfromrisingorsinkingwithouthavingtouseitsmuscles. Insomesh,however,theswimbladderorasmallextensionofit islinkedtotheearandservestheadditionalpurposeofamplifying soundwaves.Foratypicalshhavingsuchananatomy,thebladder hasaresonantfrequencyof300Hz,thebladder's Q is3,andthe maximumamplicationisaboutafactorof100inenergy.Overwhat rangeoffrequencieswouldtheamplicationbeatleastafactorof 50? 3 Asnotedinsection2.4,itisonlyapproximatelytruethatthe amplitudehasitsmaximumat f = = 2 p k=m .Beingmorecareful,weshouldactuallydenetwodierentsymbols, f 0 = = 2 p k=m and f res fortheslightlydierentfrequencyatwhichtheamplitude isamaximum,i.e.,theactualresonantfrequency.Inthisnotation, theamplitudeasafunctionoffrequencyis A = F 2 q 4 2 m 2 )]TJ/F20 10.9091 Tf 5 -8.837 Td [(f 2 )]TJ/F20 10.9091 Tf 10.909 0 Td [(f 2 0 2 + b 2 f 2 Showthatthemaximumoccursnotat f o butratheratthefrequency f res = r f 2 0 )]TJ/F20 10.9091 Tf 22.951 7.38 Td [(b 2 8 2 m 2 = r f 2 0 )]TJ/F15 10.9091 Tf 12.104 7.38 Td [(1 2 FWHM 2 Hint:Findingthefrequencythatminimizesthequantityinsidethe squarerootisequivalentto,butmucheasierthan,ndingthefrequencythatmaximizestheamplitude. R Problems 43

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Problem6. 4 aLet W betheamountofworkdonebyfrictionintherst cycleofoscillation,i.e.,theamountofenergylosttoheat.Find thefractionoftheoriginalenergy E thatremainsintheoscillations after n cyclesofmotion. bFromthis,provetheequation 1 )]TJ/F20 10.9091 Tf 12.105 7.38 Td [(W E Q = e )]TJ/F18 7.9701 Tf 6.586 0 Td [(2 recallingthatthenumber535inthedenitionof Q is e 2 cUsethistoprovetheapproximation1 =Q = 2 W=E .Hint: Usetheapproximationln+ x x ,whichisvalidforsmallvalues of x 5 Thegoalofthisproblemistorenetheproportionality FWHM / f res =Q intotheequationFWHM= f res =Q ,i.e.,toprove thattheconstantofproportionalityequals1. aShowthattheworkdonebyadampingforce F = )]TJ/F20 10.9091 Tf 8.485 0 Td [(bv overone cycleofsteady-statemotionequals W damp = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 2 bfA 2 .Hint:It islessconfusingtocalculatetheworkdoneoverhalfacycle,from x = )]TJ/F20 10.9091 Tf 8.485 0 Td [(A to x =+ A ,andthendoubleit. bShowthatthefractionoftheundrivenoscillator'senergylostto dampingoveronecycleis j W damp j =E =4 2 bf=k cUsethepreviousresult,combinedwiththeresultofproblem4, toprovethat Q equals k= 2 bf dCombinetheprecedingresultfor Q withtheequationFWHM= b= 2 m fromsection2.4toprovetheequationFWHM= f res =Q R ? 6 Thegureisfrom ShapememoryinSpiderdraglines ,Emile, LeFloch,andVollrath, Nature 440:621.Panel1showsan electronmicroscope'simageofathreadofspidersilk.In2,aspiderishangingfromsuchathread.Fromanevolutionarypointof view,it'sprobablyabadthingforthespiderifittwistsbackand forthwhilehanginglikethis.We'rereferringtoaback-and-forth rotationabouttheaxisofthethread,notaswingingmotionlikea pendulum.Theauthorsspeculatethatsuchavibrationcouldmake thespidereasierforpredatorstosee,anditalsoseemstomethat itwouldbeabadthingjustbecausethespiderwouldn'tbeable tocontrolitsorientationanddowhatitwastryingtodo.Panel3 showsagraphofsuchanoscillation,whichtheauthorsmeasured usingavideocameraandacomputer,witha0.1gmasshungfromit inplaceofaspider.Comparedtohuman-madeberssuchaskevlar orcopperwire,thespiderthreadhasanunusualsetofproperties: 1.Ithasalow Q ,sothevibrationsdampoutquickly. 44 Chapter2Resonance

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2.Itdoesn'tbecomebrittlewithrepeatedtwistingasacopper wirewould. 3.Whentwisted,ittendstosettleintoanewequilibriumangle, ratherthaninsistingonreturningtoitsoriginalangle.You canseethisinpanel2,becausealthoughtheexperimenters initiallytwistedthewireby33degrees,thethreadonlyperformedoscillationswithanamplitudemuchsmallerthan 35 degrees,settlingdowntoanewequilibriumat27degrees. 4.Overmuchlongertimescaleshours,thethreadeventually resetsitselftoitsoriginalequilbriumangleshownaszero degreesonthegraph.Thegraphreproducedhereonlyshows themotionoveramuchshortertimescale.Somehumanmadematerialshavethismemory"propertyaswell,butthey typicallyneedtobeheatedinordertomakethemgobackto theiroriginalshapes. Focusingonpropertynumber1,estimatethe Q ofspidersilkfrom thegraph. Problems 45

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46 Chapter2Resonance

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a / Dippingangerinsome water,1,causesadisturbance thatspreadsoutward,2. TheGreatWaveOffKanagawa,byKatsushikaHokusai-1849. Chapter3 FreeWaves Yourvocalcordsorasaxophonereedcanvibrate,butbeingable tovibratewouldn'tbeofmuchuseunlessthevibrationscouldbe transmittedtothelistener'searbysoundwaves.Whatarewaves andwhydotheyexist?Putyourngertipinthemiddleofacup ofwaterandthenremoveitsuddenly.Youwillhavenoticedtwo resultsthataresurprisingtomostpeople.First,theatsurface ofthewaterdoesnotsimplysinkuniformlytollinthevolume vacatedbyyournger.Instead,ripplesspreadout,andtheprocess ofatteningoutoccursoveralongperiodoftime,duringwhich thewateratthecentervibratesaboveandbelowthenormalwater level.Thistypeofwavemotionisthetopicofthepresentchapter. Second,youhavefoundthattheripplesbounceoofthewallsof thecup,inmuchthesamewaythataballwouldbounceoofa wall.Inthenextchapterwediscusswhathappenstowavesthat haveaboundaryaroundthem.Untilthen,weconneourselvesto wavephenomenathatcanbeanalyzedasifthemediume.g.,the waterwasinniteandthesameeverywhere. Itisn'thardtounderstandwhyremovingyourngertipcreates ripplesratherthansimplyallowingthewatertosinkbackdown uniformly.Theinitialcrater,a,leftbehindbyyourngerhas slopingsides,andthewaternexttothecraterowsdownhilltoll inthehole.Thewaterfaraway,ontheotherhand,initiallyhas 47

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nowayofknowingwhathashappened,becausethereisnoslope forittoowdown.Astheholellsup,therisingwateratthe centergainsupwardmomentum,andovershoots,creatingalittle hillwheretherehadbeenaholeoriginally.Theareajustoutsideof thisregionhasbeenrobbedofsomeofitswaterinordertobuild thehill,soadepressedmoat"isformed,b.Thiseectcascades outward,producingripples. 48 Chapter3FreeWaves

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b / Thetwocircularpatternsof ripplespassthrougheachother. Unlikematerialobjects,wavepatternscanoverlapinspace,and whenthishappenstheycombine byaddition. 3.1WaveMotion Therearethreemainwaysinwhichwavemotiondiersfromthe motionofobjectsmadeofmatter. 1.superposition Themostprofounddierenceisthatwavesdonotdisplayhave anythinganalogoustothenormalforcesbetweenobjectsthatcome incontact.Twowavepatternscanthereforeoverlapinthesame regionofspace,asshowningureb.Wherethetwowavescoincide, theyaddtogether.Forinstance,supposethatatacertainlocation inatacertainmomentintime,eachwavewouldhavehadacrest 3cmabovethenormalwaterlevel.Thewavescombineatthis pointtomakea6-cmcrest.Weusenegativenumberstorepresent depressionsinthewater.Ifbothwaveswouldhavehadatroughs measuring-3cm,thentheycombinetomakeanextra-deep-6cm trough.A+3cmcrestanda-3cmtroughresultinaheightofzero, i.e.,thewavesmomentarilycanceleachotheroutatthatpoint. Thisadditiveruleisreferredtoastheprincipleofsuperposition, superposition"beingmerelyafancywordforadding." Superpositioncanoccurnotjustwithsinusoidalwaveslikethe onesinthegureabovebutwithwavesofanyshape.Thegures onthefollowingpageshowsuperpositionofwave pulses .Apulseis simplyawaveofveryshortduration.Thesepulsesconsistonlyof asinglehumportrough.Ifyouhitaclotheslinesharply,youwill observepulsesheadingoinbothdirections.Thisisanalogousto Section3.1WaveMotion 49

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thewayripplesspreadoutinalldirectionswhenyoumakeadisturbanceatonepointonwater.Thesameoccurswhenthehammer onapianocomesupandhitsastring. Experimentstodatehavenotshownanydeviationfromthe principleofsuperpositioninthecaseoflightwaves.Forothertypes ofwaves,itistypicallyaverygoodapproximationforlow-energy waves. DiscussionQuestion A Ingurec/3,thefthframeshowsthespringjustaboutperfectly at.Ifthetwopulseshaveessentiallycanceledeachotheroutperfectly, thenwhydoesthemotionpickupagain?Whydoesn'tthespringjuststay at? c / Thesepicturesshowthemotionofwavepulsesalongaspring.Tomakeapulse,oneendofthe springwasshakenbyhand.Movieswerelmed,andaseriesofframechosentoshowthemotion.1.Apulse travelstotheleft.2.Superpositionoftwocollidingpositivepulses.3.Superpositionoftwocollidingpulses,one positiveandonenegative. 50 Chapter3FreeWaves

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e / Asthewavepulsegoes by,theribbontiedtothespring isnotcarriedalong.Themotion ofthewavepatternistothe right,butthemediumspringis movingupanddown,nottothe right. d / Asthewavepatternpassestherubberduck,theduckstays put.Thewaterisn'tmovingforwardwiththewave. 2.themediumisnottransportedwiththewave. Figuredshowsaseriesofwaterwavesbeforeithasreacheda rubberduckleft,havingjustpassedtheduckmiddleandhaving progressedaboutameterbeyondtheduckright.Theduckbobs arounditsinitialposition,butisnotcarriedalongwiththewave. Thisshowsthatthewateritselfdoesnotowoutwardwiththe wave.Ifitdid,wecouldemptyoneendofaswimmingpoolsimply bykickingupwaves!Wemustdistinguishbetweenthemotionof themediumwaterinthiscaseandthemotionofthewavepattern throughthemedium.Themediumvibrates;thewaveprogresses throughspace. self-checkA Inguree,youcandetecttheside-to-sidemotionofthespringbecause thespringappearsblurry.Atacertaininstant,representedbyasingle photo,howwouldyoudescribethemotionofthedifferentpartsofthe spring?Otherthantheatparts,doanypartsofthespringhavezero velocity? Answer,p.98 Awormexample1 Theworminthegureismovingtotheright.Thewavepattern, apulseconsistingofacompressedareaofitsbody,movesto theleft.Inotherwords,themotionofthewavepatternisinthe oppositedirectioncomparedtothemotionofthemedium. Section3.1WaveMotion 51

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f / Example2.Thesurferis dragginghishandinthewater. g / Example3:abreaking wave. h / Example4.Theboathas runupagainstalimitonitsspeed becauseitcan'tclimboverits ownwave.Dolphinsgetaround theproblembyleapingoutofthe water. Surngexample2 Theincorrectbeliefthatthemediummoveswiththewaveisoften reinforcedbygarbledsecondhandknowledgeofsurng.Anyone whohasactuallysurfedknowsthatthefrontoftheboardpushes thewatertothesides,creatingawakethesurfercaneven draghishandthroughthewater,asininguref.Ifthewaterwas movingalongwiththewaveandthesurfer,thiswouldn'thappen. Thesurferiscarriedforwardbecauseforwardisdownhill,notbecauseofanyforwardowofthewater.Ifthewaterwasowing forward,thenapersonoatinginthewateruptoherneckwould becarriedalongjustasquicklyassomeoneonasurfboard.In fact,itisevenpossibletosurfdownthebacksideofawave,althoughtheridewouldn'tlastverylongbecausethesurferandthe wavewouldquicklypartcompany. 3.awave'svelocitydependsonthemedium. Amaterialobjectcanmovewithanyvelocity,andcanbesped uporsloweddownbyaforcethatincreasesordecreasesitskinetic energy.Notsowithwaves.Themagnitudeofawave'svelocity dependsonthepropertiesofthemediumandperhapsalsoonthe shapeofthewave,forcertaintypesofwaves.Soundwavestravel atabout340m/sinair,1000m/sinhelium.Ifyoukickupwater wavesinapool,youwillndthatkickinghardermakeswavesthat aretallerandthereforecarrymoreenergy,notfaster.Thesound wavesfromanexplodingstickofdynamitecarryalotofenergy,but arenofasterthananyotherwaves.Inthefollowingsectionwewill giveanexampleofthephysicalrelationshipbetweenthewavespeed andthepropertiesofthemedium. Breakingwavesexample3 Thevelocityofwaterwavesincreaseswithdepth.Thecrestofa wavetravelsfasterthanthetrough,andthiscancausethewave tobreak. Onceawaveiscreated,theonlyreasonitsspeedwillchangeis ifitentersadierentmediumorifthepropertiesofthemedium change.Itisnotsosurprisingthatachangeinmediumcanslow downawave,butthereversecanalsohappen.Asoundwavetravelingthroughaheliumballoonwillslowdownwhenitemergesinto theair,butifitentersanotherballoonitwillspeedbackupagain! Similarly,waterwavestravelmorequicklyoverdeeperwater,soa wavewillslowdownasitpassesoveranunderwaterridge,butspeed upagainasitemergesintodeeperwater. Hullspeedexample4 Thespeedsofmostboats,andofsomesurface-swimmingani52 Chapter3FreeWaves

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i / Circularandlinearwave patterns. j / Planeandsphericalwave patterns. mals,arelimitedbythefactthattheymakeawaveduetotheir motionthroughthewater.Theboatingurehisgoingatthe samespeedasitsownwaves,andcan'tgoanyfaster.Nomatterhowhardtheboatpushesagainstthewater,itcan'tmake thewavemoveaheadfasterandgetoutoftheway.Thewave's speeddependsonlyonthemedium.Addingenergytothewave doesn'tspeeditup,itjustincreasesitsamplitude. Awaterwave,unlikemanyothertypesofwave,hasaspeedthat dependsonitsshape:abroaderwavemovesfaster.Theshape ofthewavemadebyaboattendstomolditselftotheshapeof theboat'shull,soaboatwithalongerhullmakesabroaderwave thatmovesfaster.Themaximumspeedofaboatwhosespeedis limitedbythiseffectisthereforecloselyrelatedtothelengthofits hull,andthemaximumspeediscalledthehullspeed.Sailboats designedforracingarenotjustlongandskinnytomakethem morestreamlinedtheyarealsolongsothattheirhullspeeds willbehigh. Wavepatterns Ifthemagnitudeofawave'svelocityvectorispreordained,what aboutitsdirection?Wavesspreadoutinalldirectionsfromevery pointonthedisturbancethatcreatedthem.Ifthedisturbanceis small,wemayconsideritasasinglepoint,andinthecaseofwater wavestheresultingwavepatternisthefamiliarcircularripple,i/1. If,ontheotherhand,welayapoleonthesurfaceofthewater andwiggleitupanddown,wecreatealinearwavepattern,i/2. Forathree-dimensionalwavesuchasasoundwave,theanalogous patternswouldbesphericalwavesandplanewaves,j. Innitelymanypatternsarepossible,butlinearorplanewaves areoftenthesimplesttoanalyze,becausethevelocityvectorisin thesamedirectionnomatterwhatpartofthewavewelookat.Since allthevelocityvectorsareparalleltooneanother,theproblemis eectivelyone-dimensional.Throughoutthischapterandthenext, wewillrestrictourselvesmainlytowavemotioninonedimension, whilenothesitatingtobroadenourhorizonswhenitcanbedone withouttoomuchcomplication. DiscussionQuestions A [seeabove] B Sketchtwopositivewavepulsesonastringthatareoverlappingbut notrightontopofeachother,anddrawtheirsuperposition.Dothesame forapositivepulserunningintoanegativepulse. C Atravelingwavepulseismovingtotherightonastring.Sketchthe velocityvectorsofthevariouspartsofthestring.Nowdothesamefora pulsemovingtotheleft. D Inasphericalsoundwavespreadingoutfromapoint,howwould theenergyofthewavefalloffwithdistance? Section3.1WaveMotion 53

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k / Hittingakeyonapiano causesahammertocomeup fromunderneathandhitastring actuallyasetofthreestrings. Theresultisapairofpulses movingawayfromthepointof impact. l / Astringisstruckwitha hammer,1,andtwopulsesyoff, 2. m / Acontinuousstringcan bemodeledasaseriesof discretemassesconnectedby springs. 3.2WavesonaString Sofaryouhavelearnedsomecounterintuitivethingsaboutthebehaviorofwaves,butintuitioncanbetrained.Thersthalfofthis sectionaimstobuildyourintuitionbyinvestigatingasimple,onedimensionaltypeofwave:awaveonastring.Ifyouhaveever stretchedastringbetweenthebottomsoftwoopen-mouthedcans totalktoafriend,youwereputtingthistypeofwavetowork. Stringedinstrumentsareanothergoodexample.Althoughweusuallythinkofapianowiresimplyasvibrating,thehammeractually strikesitquicklyandmakesadentinit,whichthenripplesoutin bothdirections.Sincethischapterisaboutfreewaves,notbounded ones,wepretendthatourstringisinnitelylong. Afterthequalitativediscussion,wewillusesimpleapproximationstoinvestigatethespeedofawavepulseonastring.Thisquick anddirtytreatmentisthenfollowedbyarigorousattackusingthe methodsofcalculus,whichmaybeskippedbythestudentwhohas notstudiedcalculus.Howfaryoupenetrateinthissectionisupto you,anddependsonyourmathematicalself-condence.Ifyouskip thelaterpartsandproceedtothenextsection,youshouldneverthelessbeawareoftheimportantresultthatthespeedatwhicha pulsemovesdoesnotdependonthesizeorshapeofthepulse.This isafactthatistrueformanyothertypesofwaves. Intuitiveideas Considerastringthathasbeenstruck,l/1,resultinginthecreationoftwowavepulses,2,onetravelingtotheleftandonetothe right.Thisisanalogoustothewayripplesspreadoutinalldirectionsfromasplashinwater,butonaone-dimensionalstring,all directions"becomesbothdirections." Wecangaininsightbymodelingthestringasaseriesofmasses connectedbysprings.Intheactualstringthemassandthespringinessarebothcontributedbythemoleculesthemselves.Ifwelook atvariousmicroscopicportionsofthestring,therewillbesomeareasthatareat,m/1,somethatareslopingbutnotcurved,2,and somethatarecurved,3and4.Inexample1itisclearthatboththe forcesonthecentralmasscancelout,soitwillnotaccelerate.The sameistrueof2,however.Onlyincurvedregionssuchas3and4 isanaccelerationproduced.Intheseexamples,thevectorsumof thetwoforcesactingonthecentralmassisnotzero.Theimportantconceptisthatcurvaturemakesforce:thecurvedareasofa wavetendtoexperienceforcesresultinginanaccelerationtoward themouthofthecurve.Note,however,thatanuncurvedportion ofthestringneednotremainmotionless.Itmaymoveatconstant velocitytoeitherside. 54 Chapter3FreeWaves

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n / Atriangularpulsespreadsout. Approximatetreatment Wenowcarryoutanapproximatetreatmentofthespeedat whichtwopulseswillspreadoutfromaninitialindentationona string.Forsimplicity,weimagineahammerblowthatcreatesatriangulardent,n/1.Wewillestimatetheamountoftime, t ,required untileachofthepulseshastraveledadistanceequaltothewidth ofthepulseitself.Thevelocityofthepulsesisthen w=t Asalways,thevelocityofawavedependsonthepropertiesof themedium,inthiscasethestring.Thepropertiesofthestringcan besummarizedbytwovariables:thetension, T ,andthemassper unitlength, Greeklettermu. Ifweconsiderthepartofthestringencompassedbytheinitial dentasasingleobject,thenthisobjecthasamassofapproximately w mass = length length=mass.Here,andthroughout thederivation,weassumethat h ismuchlessthan w ,sothatwecan ignorethefactthatthissegmentofthestringhasalengthslightly greaterthan w .Althoughthedownwardaccelerationofthissegmentofthestringwillbeneitherconstantovertimenoruniform acrossthestring,wewillpretendthatitisconstantforthesakeof oursimpleestimate.Roughlyspeaking,thetimeintervalbetween n/1and2istheamountoftimerequiredfortheinitialdenttoacceleratefromrestandreachitsnormal,attenedposition.Ofcourse thetipofthetrianglehasalongerdistancetotravelthantheedges, butagainweignorethecomplicationsandsimplyassumethatthe segmentasawholemusttraveladistance h .Indeed,itmightseem surprisingthatthetrianglewouldsoneatlyspringbacktoaperfectlyatshape.Itisanexperimentalfactthatitdoes,butour analysisistoocrudetoaddresssuchdetails. Thestringiskinked,i.e.,tightlycurved,attheedgesofthe triangle,soitisherethattherewillbelargeforcesthatdonot cancelouttozero.Therearetwoforcesactingonthetriangular hump,oneofmagnitude T actingdownandtotheright,andone ofthesamemagnitudeactingdownandtotheleft.Iftheangle oftheslopingsidesis ,thenthetotalforceonthesegmentequals 2 T sin .Dividingthetriangleintotworighttriangles,weseethat sin equals h dividedbythelengthofoneoftheslopingsides.Since h ismuchlessthan w ,thelengthoftheslopingsideisessentially thesameas w= 2,sowehavesin = h=w ,and F =4 Th=w .The accelerationofthesegmentactuallytheaccelerationofitscenter ofmassis a = F=m =4 Th=w 2 Thetimerequiredtomoveadistance h underconstantacceleration Section3.2WavesonaString 55

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a isfoundbysolving h = 1 2 at 2 toyield t = r 2 h a = w r 2 T Ournalresultforthevelocityofthepulsesis j v j = w t = s 2 T Theremarkablefeatureofthisresultisthatthevelocityofthe pulsesdoesnotdependatallon w or h ,i.e.,anytriangularpulse hasthesamespeed.Itisanexperimentalfactandwewillalso proverigorouslyinthefollowingsubsectionthatanypulseofany kind,triangularorotherwise,travelsalongthestringatthesame speed.Ofcourse,aftersomanyapproximationswecannotexpect tohavegottenallthenumericalfactorsright.Thecorrectresultfor thevelocityofthepulsesis v = s T Theimportanceoftheabovederivationliesintheinsightit brings|thatallpulsesmovewiththesamespeed|ratherthanin thedetailsofthenumericalresult.Thereasonforourtoo-highvalue forthevelocityisnothardtoguess.Itcomesfromtheassumption thattheaccelerationwasconstant,whenactuallythetotalforceon thesegmentwoulddiminishasitattenedout. Rigorousderivationusingcalculusoptional Afterexpendingconsiderableeortforanapproximatesolution, wenowdisplaythepowerofcalculuswitharigorousandcompletely generaltreatmentthatisneverthelessmuchshorterandeasier.Let theatpositionofthestringdenethe x axis,sothat y measures howfarapointonthestringisfromequilibrium.Themotionof thestringischaracterizedby y x t ,afunctionoftwovariables. Knowingthattheforceonanysmallsegmentofstringdepends onthecurvatureofthestringinthatarea,andthatthesecond derivativeisameasureofcurvature,itisnotsurprisingtondthat theinnitesimalforced F actingonaninnitesimalsegmentd x is givenby d F = T d 2 y d x 2 d x Thiscanbeprovedbyvectoradditionofthetwoinnitesimalforces actingoneitherside.Theaccelerationisthen a =d F= d m ,or, 56 Chapter3FreeWaves

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substitutingd m = dx d 2 y d t 2 = T d 2 y d x 2 Thesecondderivativewithrespecttotimeisrelatedtothesecond derivativewithrespecttoposition.Thisisnomorethanafancy mathematicalstatementoftheintuitivefactdevelopedabove,that thestringacceleratessoastoattenoutitscurves. Beforeevenbotheringtolookforsolutionstothisequation,we notethatitalreadyprovestheprincipleofsuperposition,because thederivativeofasumisthesumofthederivatives.Thereforethe sumofanytwosolutionswillalsobeasolution. Basedonexperiment,weexpectthatthisequationwillbesatisedbyanyfunction y x t thatdescribesapulseorwavepattern movingtotheleftorrightatthecorrectspeed v .Ingeneral,such afunctionwillbeoftheform y = f x )]TJ/F20 10.9091 Tf 10.406 0 Td [(vt or y = f x + vt ,where f isanyfunctionofonevariable.Becauseofthechainrule,each derivativewithrespecttotimebringsoutafactorof v .Evaluating thesecondderivativesonbothsidesoftheequationgives v 2 f 00 = T f 00 Squaringgetsridofthesign,andwendthatwehaveavalid solutionforanyfunction f ,providedthat v isgivenby v = s T 3.3SoundandLightWaves Soundwaves Thephenomenonofsoundiseasilyfoundtohaveallthecharacteristicsweexpectfromawavephenomenon: Soundwavesobeysuperposition.Soundsdonotknockother soundsoutofthewaywhentheycollide,andwecanhearmore thanonesoundatonceiftheybothreachourearsimultaneously. Themediumdoesnotmovewiththesound.Evenstanding infrontofatitanicspeakerplayingearsplittingmusic,wedo notfeeltheslightestbreeze. Thevelocityofsounddependsonthemedium.Soundtravels fasterinheliumthaninair,andfasterinwaterthaninhelium. Puttingmoreenergyintothewavemakesitmoreintense,not faster.Forexample,youcaneasilydetectanechowhenyou clapyourhandsashortdistancefromalarge,atwall,and thedelayoftheechoisnoshorterforalouderclap. Section3.3SoundandLightWaves 57

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Althoughnotallwaveshaveaspeedthatisindependentofthe shapeofthewave,andthispropertythereforeisirrelevanttoour collectionofevidencethatsoundisawavephenomenon,sounddoes neverthelesshavethisproperty.Forinstance,themusicinalarge concerthallorstadiummaytakeontheorderofasecondtoreach someoneseatedinthenosebleedsection,butwedonotnoticeor care,becausethedelayisthesameforeverysound.Bass,drums, andvocalsallheadoutwardfromthestageat340m/s,regardless oftheirdieringwaveshapes. Ifsoundhasallthepropertiesweexpectfromawave,thenwhat typeofwaveisit?Itmustbeavibrationofaphysicalmediumsuch asair,sincethespeedofsoundisdierentindierentmedia,such asheliumorwater.Furtherevidenceisthatwedon'treceivesound signalsthathavecometoourplanetthroughouterspace.Theroars andwhooshesofHollywood'sspaceshipsarefun,butscientically wrong. 1 Wecanalsotellthatsoundwavesconsistofcompressionsand expansions,ratherthansidewaysvibrationsliketheshimmyingofa snake.Onlycompressionalvibrationswouldbeabletocauseyour eardrumstovibrateinandout.Evenforaveryloudsound,the compressionisextremelyweak;theincreaseordecreasecompared tonormalatmosphericpressureisnomorethanapartpermillion. Ourearsareapparentlyverysensitivereceivers! Lightwaves Entirelysimilarobservationsleadustobelievethatlightisa wave,althoughtheconceptoflightasawavehadalongandtortuoushistory.ItisinterestingtonotethatIsaacNewtonveryinuentiallyadvocatedacontraryideaaboutlight.Thebeliefthatmatter wasmadeofatomswasstylishatthetimeamongradicalthinkers althoughtherewasnoexperimentalevidencefortheirexistence, anditseemedlogicaltoNewtonthatlightaswellshouldbemadeof tinyparticles,whichhecalledcorpusclesLatinforsmallobjects". Newton'striumphsinthescienceofmechanics,i.e.,thestudyof matter,broughthimsuchgreatprestigethatnobodybotheredto questionhisincorrecttheoryoflightfor150years.OnepersuasiveproofthatlightisawaveisthataccordingtoNewton'stheory, twointersectingbeamsoflightshouldexperienceatleastsomedisruptionbecauseofcollisionsbetweentheircorpuscles.Evenifthe 1 Outerspaceisnotaperfectvacuum,soitispossibleforsoundswavesto travelthroughit.However,ifwewanttocreateasoundwave,wetypicallydo itbycreatingvibrationsofaphysicalobject,suchasthesoundingboardofa guitar,thereedofasaxophone,oraspeakercone.Thelowerthedensityofthe surroundingmedium,thelessecientlytheenergycanbeconvertedintosound andcarriedaway.Anisolatedtuningfork,lefttovibrateininterstellarspace, woulddissipatetheenergyofitsvibrationintointernalheatataratemany ordersofmagnitudegreaterthantherateofsoundemissionintothenearly perfectvacuumaroundit. 58 Chapter3FreeWaves

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o / Agraphofpressureversustimeforaperiodicsound wave,thevowelah. p / Asimilargraphforanonperiodicwave,sh. corpuscleswereextremelysmall,andcollisionsthereforeveryinfrequent,atleastsomedimmingshouldhavebeenmeasurable.Infact, verydelicateexperimentshaveshownthatthereisnodimming. Thewavetheoryoflightwasentirelysuccessfulupuntilthe20th century,whenitwasdiscoveredthatnotallthephenomenaoflight couldbeexplainedwithapurewavetheory.Itisnowbelievedthat bothlightandmatteraremadeoutoftinychunkswhichhave both waveandparticleproperties.Fornow,wewillcontentourselves withthewavetheoryoflight,whichiscapableofexplainingagreat manythings,fromcamerastorainbows. Iflightisawave,whatiswaving?Whatisthemediumthat wiggleswhenalightwavegoesby?Itisn'tair.Avacuumisimpenetrabletosound,butlightfromthestarstravelshappilythrough zillionsofmilesofemptyspace.Lightbulbshavenoairinsidethem, butthatdoesn'tpreventthelightwavesfromleavingthelament. Foralongtime,physicistsassumedthattheremustbeamysterious mediumforlightwaves,andtheycalledittheaethernottobe confusedwiththechemical.Supposedlytheaetherexistedeverywhereinspace,andwasimmunetovacuumpumps.Thedetailsof thestoryaremorettinglyreservedforlaterinthiscourse,butthe endresultwasthatalongseriesofexperimentsfailedtodetectany evidencefortheaether,anditisnolongerbelievedtoexist.Instead, lightcanbeexplainedasawavepatternmadeupofelectricaland magneticelds. 3.4PeriodicWaves Periodandfrequencyofaperiodicwave Youchoosearadiostationbyselectingacertainfrequency.We havealreadydenedperiodandfrequencyforvibrations,butwhat dotheysignifyinthecaseofawave?Wecanrecycleourprevious denitionsimplybystatingitintermsofthevibrationsthatthe wavecausesasitpassesareceivinginstrumentatacertainpoint inspace.Forasoundwave,thisreceivercouldbeaneardrumor amicrophone.Ifthevibrationsoftheeardrumrepeatthemselves overandover,i.e.,areperiodic,thenwedescribethesoundwave thatcausedthemasperiodic.Likewisewecandenetheperiod andfrequencyofawaveintermsoftheperiodandfrequencyof thevibrationsitcauses.Asanotherexample,aperiodicwaterwave wouldbeonethatcausedarubberducktobobinaperiodicmanner astheypassedbyit. Theperiodofasoundwavecorrelateswithoursensoryimpressionofmusicalpitch.Ahighfrequencyshortperiodisahighnote. Thesoundsthatreallydenethemusicalnotesofasongareonly theonesthatareperiodic.Itisnotpossibletosinganon-periodic soundlikesh"withadenitepitch. Section3.4PeriodicWaves 59

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r / Awaterwaveprolecreatedbyaseriesofrepeating pulses. q / Astripchartrecorder. Thefrequencyofalightwavecorrespondstocolor.Violetisthe high-frequencyendoftherainbow,redthelow-frequencyend.A colorlikebrownthatdoesnotoccurinarainbowisnotaperiodic lightwave.Manyphenomenathatwedonotnormallythinkofas lightareactuallyjustformsoflightthatareinvisiblebecausethey falloutsidetherangeoffrequenciesoureyescandetect.Beyondthe redendofthevisiblerainbow,thereareinfraredandradiowaves. Pastthevioletend,wehaveultraviolet,x-rays,andgammarays. Graphsofwavesasafunctionofposition Somewaves,lightsoundwaves,areeasytostudybyplacinga detectoratacertainlocationinspaceandstudyingthemotionas afunctionoftime.Theresultisagraphwhosehorizontalaxisis time.Withawaterwave,ontheotherhand,itissimplerjustto lookatthewavedirectly.Thisvisualsnapshotamountstoagraph oftheheightofthewaterwaveasafunctionof position .Anywave canberepresentedineitherway. Aneasywaytovisualizethisisintermsofastripchartrecorder, anobsolescingdeviceconsistingofapenthatwigglesbackandforth asarollofpaperisfedunderit.Itcanbeusedtorecordaperson'selectrocardiogram,orseismicwavestoosmalltobefeltasa noticeableearthquakebutdetectablebyaseismometer.Takingthe seismometerasanexample,thechartisessentiallyarecordofthe ground'swavemotionasafunctionoftime,butifthepaperwasset tofeedatthesamevelocityasthemotionofanearthquakewave,it wouldalsobeafull-scalerepresentationoftheproleoftheactual wavepatternitself.Assuming,asisusuallythecase,thatthewave velocityisaconstantnumberregardlessofthewave'sshape,knowingthewavemotionasafunctionoftimeisequivalenttoknowing itasafunctionofposition. Wavelength Anywavethatisperiodicwillalsodisplayarepeatingpattern whengraphedasafunctionofposition.Thedistancespannedby onerepetitionisreferredtoasone wavelength .Theusualnotation forwavelengthis ,theGreekletterlambda.Wavelengthistospace asperiodistotime. Wavevelocityrelatedtofrequencyandwavelength Supposethatwecreatearepetitivedisturbancebykickingthe surfaceofaswimmingpool.Weareessentiallymakingaseriesof wavepulses.Thewavelengthissimplythedistanceapulseisableto travelbeforewemakethenextpulse.Thedistancebetweenpulses is ,andthetimebetweenpulsesistheperiod, T ,sothespeedof thewaveisthedistancedividedbythetime, v = =T 60 Chapter3FreeWaves

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u / Awaterwavetraveling intoaregionwithadifferent depthchangesitswavelength. s / Wavelengthsoflinearandcircularwaterwaves. Thisimportantandusefulrelationshipismorecommonlywrittenintermsofthefrequency, v = f Wavelengthofradiowavesexample5 Thespeedoflightis3.0 10 8 m/s.Whatisthewavelengthof theradiowavesemittedbyKKJZ,astationwhosefrequencyis 88.1MHz? Solvingforwavelength,wehave = v = f =.0 10 8 m = s = .1 10 6 s )]TJ/F39 7.9701 Tf 6.587 0 Td [(1 =3.4m Thesizeofaradioantennaiscloselyrelatedtothewavelengthof thewavesitisintendedtoreceive.Thematchneednotbeexact sinceafteralloneantennacanreceivemorethanonewavelength!,buttheordinarywhipantennasuchasacar'sis1/4 ofawavelength.AnantennaoptimizedtoreceiveKKJZ'ssignal wouldhavealengthof3.4m = 4=0.85m. Theequation v = f denesaxedrelationshipbetweenanytwo ofthevariablesiftheotherisheldxed.Thespeedofradiowaves inairisalmostexactlythesameforallwavelengthsandfrequencies itisexactlythesameiftheyareinavacuum,sothereisaxed relationshipbetweentheirfrequencyandwavelength.Thuswecan sayeitherAreweonthesamewavelength?"orAreweonthe samefrequency?" Adierentexampleisthebehaviorofawavethattravelsfrom aregionwherethemediumhasonesetofpropertiestoanarea Section3.4PeriodicWaves 61

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t / Ultrasound,i.e.,soundwithfrequencieshigherthantherange ofhumanhearing,wasusedto makethisimageofafetus.The resolutionoftheimageisrelatedtothewavelength,since detailssmallerthanaboutone wavelengthcannotberesolved. Highresolutionthereforerequires ashortwavelength,correspondingtoahighfrequency. wherethemediumbehavesdierently.Thefrequencyisnowxed, becauseotherwisethetwoportionsofthewavewouldotherwise getoutofstep,causingakinkordiscontinuityattheboundary, whichwouldbeunphysical.Amorecarefulargumentisthata kinkordiscontinuitywouldhaveinnitecurvature,andwavestend toattenouttheircurvature.Aninnitecurvaturewouldatten outinnitelyfast,i.e.,itcouldneveroccurintherstplace.Since thefrequencymuststaythesame,anychangeinthevelocitythat resultsfromthenewmediummustcauseachangeinwavelength. Thevelocityofwaterwavesdependsonthedepthofthewater, sobasedon = v=f ,weseethatwaterwavesthatmoveintoa regionofdierentdepthmustchangetheirwavelength,asshown inthegureontheleft.Thiseectcanbeobservedwhenocean wavescomeuptotheshore.Ifthedecelerationofthewavepattern issuddenenough,thetipofthewavecancurlover,resultingina breakingwave. Anoteondispersivewaves Thediscussionofwavevelocitygivenhereisactuallyanoversimplicationforawavewhosevelocitydependsonitsfrequencyandwavelength.Suchawaveiscalledadispersivewave.Nearlyallthewaves wedealwithinthiscoursearenon-dispersive,buttheissuebecomes importantinbook6ofthisseries,whereitisdiscussedinmoredetailin optionalsection4.2. Sinusoidalwaves Sinusoidalwavesarethemostimportantspecialcaseofperiodic waves.Infact,manyscientistsandengineerswouldbeuncomfortablewithdeningawaveformliketheah"vowelsoundashaving adenitefrequencyandwavelength,becausetheyconsideronly 62 Chapter3FreeWaves

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v / Thepatternofwavesmade byapointsourcemovingtothe rightacrossthewater.Note theshorterwavelengthofthe forward-emittedwavesand thelongerwavelengthofthe backward-goingones. sinewavestobepureexamplesofacertainfrequencyandwavelengths.Theirbiasisnotunreasonable,sincetheFrenchmathematicianFouriershowedthatanyperiodicwavewithfrequency f canbeconstructedasasuperpositionofsinewaveswithfrequencies f ,2 f ,3 f ,...Inthissense,sinewavesarethebasic,purebuilding blocksofallwaves.Fourier'sresultsosurprisedthemathematical communityofFrancethathewasridiculedthersttimehepublicly presentedhistheorem. However,whatdenitiontouseisamatterofutility.Oursense ofhearingperceivesanytwosoundshavingthesameperiodaspossessingthesamepitch,regardlessofwhethertheyaresinewaves ornot.Thisisundoubtedlybecauseourear-brainsystemevolved tobeabletointerprethumanspeechandanimalnoises,whichare periodicbutnotsinusoidal.Oureyes,ontheotherhand,judgea coloraspurebelongingtotherainbowsetofcolorsonlyifitisa sinewave. DiscussionQuestion A Supposewesuperimposetwosinewaveswithequalamplitudes butslightlydifferentfrequencies,asshowninthegure.Whatwillthe superpositionlooklike?Whatwouldthissoundlikeiftheyweresound waves? 3.5TheDopplerEffect Figurevshowsthewavepatternmadebythetipofavibrating rodwhichismovingacrossthewater.Iftherodhadbeenvibrating inoneplace,wewouldhaveseenthefamiliarpatternofconcentric circles,allcenteredonthesamepoint.Butsincethesourceof thewavesismoving,thewavelengthisshortenedononesideand lengthenedontheother.ThisisknownastheDopplereect. Notethatthevelocityofthewavesisaxedpropertyofthe medium,soforexampletheforward-goingwavesdonotgetanextra boostinspeedaswouldamaterialobjectlikeabulletbeingshot forwardfromanairplane. Wecanalsoinferachangeinfrequency.Sincethevelocityis constant,theequation v = f tellsusthatthechangeinwavelengthmustbematchedbyanoppositechangeinfrequency:higher frequencyforthewavesemittedforward,andlowerfortheones emittedbackward.ThefrequencyDopplereectisthereasonfor thefamiliardropping-pitchsoundofaracecargoingby.Asthecar Section3.5TheDopplerEffect 63

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approachesus,wehearahigherpitch,butafteritpassesuswehear afrequencythatislowerthannormal. TheDopplereectwillalsooccuriftheobserverismovingbut thesourceisstationary.Forinstance,anobservermovingtowarda stationarysourcewillperceiveonecrestofthewave,andwillthenbe surroundedbythenextcrestsoonerthansheotherwisewouldhave, becauseshehasmovedtowarditandhastenedherencounterwith it.Roughlyspeaking,theDopplereectdependsonlytherelative motionofthesourceandtheobserver,notontheirabsolutestate ofmotionwhichisnotawell-denednotioninphysicsorontheir velocityrelativetothemedium. Restrictingourselvestothecaseofamovingsource,andtowaves emittedeitherdirectlyalongordirectlyagainstthedirectionofmotion,wecaneasilycalculatethewavelength,orequivalentlythe frequency,oftheDoppler-shiftedwaves.Let v bethevelocityof thewaves,and v s thevelocityofthesource.Thewavelengthofthe forward-emittedwavesisshortenedbyanamount v s T equaltothe distancetraveledbythesourceoverthecourseofoneperiod.Using thedenition f =1 =T andtheequation v = f ,wendforthe wavelengthoftheDoppler-shiftedwavetheequation 0 = 1 )]TJ/F20 10.9091 Tf 12.105 7.38 Td [(v s v Asimilarequationcanbeusedforthebackward-emittedwaves,but withaplussignratherthanaminussign. Doppler-shiftedsoundfromaracecarexample6 Ifaracecarmovesatavelocityof50m/s,andthevelocityof soundis340m/s,bywhatpercentagearethewavelengthand frequencyofitssoundwavesshiftedforanobserverlyingalong itslineofmotion? Foranobserverwhomthecarisapproaching,wend 1 )]TJ/F102 10.9091 Tf 12.104 7.38 Td [(v s v =0.85, sotheshiftinwavelengthis15%.Sincethefrequencyisinversely proportionaltothewavelengthforaxedvalueofthespeedof sound,thefrequencyisshiftedupwardby 1 = 0.85=1.18, i.e.,achangeof18%.Forvelocitiesthataresmallcompared tothewavevelocities,theDopplershiftsofthewavelengthand frequencyareaboutthesame. Dopplershiftofthelightemittedbyaracecarexample7 Whatisthepercentshiftinthewavelengthofthelightwaves emittedbyaracecar'sheadlights? 64 Chapter3FreeWaves

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w / Example8.ADoppler radarimageofHurricaneKatrina, in2005. Lookingupthespeedoflightinthefrontofthebook, v =3.0 10 8 m/s,wend 1 )]TJ/F102 10.9091 Tf 12.104 7.38 Td [(v s v =0.99999983, i.e.,thepercentageshiftisonly0.000017%. Thesecondexampleshowsthatunderordinaryearthboundcircumstances,Dopplershiftsoflightarenegligiblebecauseordinary thingsgosomuchslowerthanthespeedoflight.It'sadierent story,however,whenitcomestostarsandgalaxies,andthisleads ustoastorythathasprofoundimplicationsforourunderstanding oftheoriginoftheuniverse. Dopplerradarexample8 TherstuseofradarwasbyBritainduringWorldWarII:antennasonthegroundsentradiowavesupintothesky,anddetected theechoeswhenthewaveswerereectedfromGermanplanes. Later,airforceswantedtomountradarantennasonairplanes, butthentherewasaproblem,becauseifanairplanewantedto detectanotherairplaneataloweraltitude,itwouldhavetoaim itsradiowavesdownward,andthenitwouldgetechoesfrom theground.ThesolutionwastheinventionofDopplerradar,in whichechoesfromthegroundweredifferentiatedfromechoes fromotheraircraftaccordingtotheirDopplershifts.Asimilar technologyisusedbymeteorologiststomapoutraincloudswithoutbeingswampedbyreectionsfromtheground,trees,and buildings. Optionaltopic:Dopplershiftsoflight IfDopplershiftsdependonlyontherelativemotionofthesourceand receiver,thenthereisnowayforapersonmovingwiththesourceand anotherpersonmovingwiththereceivertodeterminewhoismoving andwhoisn't.EithercanblametheDopplershiftentirelyontheother's motionandclaimtobeatrestherself.Thisisentirelyinagreementwith theprinciplestatedoriginallybyGalileothatallmotionisrelative. Ontheotherhand,acarefulanalysisoftheDopplershiftsofwater orsoundwavesshowsthatitisonlyapproximatelytrue,atlowspeeds, thattheshiftsjustdependontherelativemotionofthesourceandobserver.Forinstance,itispossibleforajetplanetokeepupwithitsown soundwaves,sothatthesoundwavesappeartostandstilltothepilot oftheplane.Thepilotthenknowssheismovingatexactlythespeed ofsound.Thereasonthisdoesn'tdisprovetherelativityofmotionis thatthepilotisnotreallydeterminingherabsolutemotionbutratherher motionrelativetotheair,whichisthemediumofthesoundwaves. Einsteinrealizedthatthissolvedtheproblemforsoundorwater waves,butwouldnotsalvagetheprincipleofrelativemotioninthecase oflightwaves,sincelightisnotavibrationofanyphysicalmediumsuch aswaterorair.Beginningbyimaginingwhatabeamoflightwould lookliketoapersonridingamotorcyclealongsideit,Einsteineventuallycameupwitharadicalnewwayofdescribingtheuniverse,in whichspaceandtimearedistortedasmeasuredbyobserversindifferentstatesofmotion.AsaconsequenceofthisTheoryofRelativity,he Section3.5TheDopplerEffect 65

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x / ThegalaxyM51.Under highmagnication,themilky cloudsrevealthemselvestobe composedoftrillionsofstars. showedthatlightwaveswouldhaveDopplershiftsthatwouldexactly, notjustapproximately,dependonlyontherelativemotionofthesource andreceiver. TheBigbang Assoonasastronomersbeganlookingattheskythroughtelescopes,theybegannoticingcertainobjectsthatlookedlikeclouds indeepspace.Thefactthattheylookedthesamenightafternight meantthattheywerebeyondtheearth'satmosphere.Notknowingwhattheyreallywere,butwantingtosoundocial,theycalled themnebulae,"aLatinwordmeaningclouds"butsoundingmore impressive.Intheearly20thcentury,astronomersrealizedthatalthoughsomereallywerecloudsofgase.g.,themiddlestar"of Orion'ssword,whichisvisiblyfuzzyeventothenakedeyewhen conditionsaregood,otherswerewhatwenowcallgalaxies:virtual islanduniversesconsistingoftrillionsofstarsforexampletheAndromedaGalaxy,whichisvisibleasafuzzypatchthroughbinoculars.ThreehundredyearsafterGalileohadresolvedtheMilky Wayintoindividualstarsthroughhistelescope,astronomersrealizedthattheuniverseismadeofgalaxiesofstars,andtheMilky Wayissimplythevisiblepartoftheatdiskofourowngalaxy, seenfrominside. Thisopenedupthescienticstudyofcosmology,thestructure andhistoryoftheuniverseasawhole,aeldthathadnotbeen seriouslyattackedsincethedaysofNewton.Newtonhadrealized thatifgravitywasalwaysattractive,neverrepulsive,theuniverse wouldhaveatendencytocollapse.Hissolutiontotheproblemwas topositauniversethatwasinniteanduniformlypopulatedwith matter,sothatitwouldhavenogeometricalcenter.Thegravitationalforcesinsuchauniversewouldalwaystendtocanceloutby symmetry,sotherewouldbenocollapse.Bythe20thcentury,the beliefinanunchangingandinniteuniversehadbecomeconventionalwisdominscience,partlyasareactionagainstthetimethat hadbeenwastedtryingtondexplanationsofancientgeological phenomenabasedoncatastrophessuggestedbybiblicaleventslike Noah'sood. Inthe1920'sastronomerEdwinHubblebeganstudyingthe Dopplershiftsofthelightemittedbygalaxies.Aformercollege footballplayerwithaseriousnicotineaddiction,Hubbledidnot setouttochangeourimageofthebeginningoftheuniverse.His autobiographyseldomevenmentionsthecosmologicaldiscoveryfor whichheisnowremembered.Whenastronomersbegantostudythe Dopplershiftsofgalaxies,theyexpectedthateachgalaxy'sdirection andvelocityofmotionwouldbeessentiallyrandom.Somewouldbe approachingus,andtheirlightwouldthereforebeDoppler-shifted totheblueendofthespectrum,whileanequalnumberwouldbe expectedtohaveredshifts.WhatHubblediscoveredinsteadwas 66 Chapter3FreeWaves

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y / Howdoastronomersknow whatmixtureofwavelengthsa staremittedoriginally,sothat theycantellhowmuchthe Dopplershiftwas?Thisimage obtainedbytheauthorwith equipmentcostingabout$5,and notelescopeshowsthemixture ofcolorsemittedbythestar Sirius.Ifyouhavethebookin blackandwhite,blueisontheleft andredontheright.Thestar appearswhiteorbluish-whiteto theeye,butanylightlookswhite ifitcontainsroughlyanequal mixtureoftherainbowcolors, i.e.,ofallthepuresinusoidal waveswithwavelengthslyingin thevisiblerange.Notetheblack gapteeth.Thesearethengerprintofhydrogenintheouter atmosphereofSirius.These wavelengthsareselectivelyabsorbedbyhydrogen.Siriusisin ourowngalaxy,butsimilarstars inothergalaxieswouldhave thewholepatternshiftedtoward theredend,indicatingtheyare movingawayfromus. z / ThetelescopeatMount WilsonusedbyHubble. thatexceptforafewverynearbyones,allthegalaxieshadred shifts,indicatingthattheywererecedingfromusataheftyfractionofthespeedoflight.Notonlythat,buttheonesfartheraway wererecedingmorequickly.Thespeedsweredirectlyproportional totheirdistancefromus. Didthismeanthattheearthoratleastourgalaxywasthe centeroftheuniverse?No,becauseDopplershiftsoflightonly dependontherelativemotionofthesourceandtheobserver.If weseeadistantgalaxymovingawayfromusat10%ofthespeed oflight,wecanbeassuredthattheastronomerswholiveinthat galaxywillseeoursrecedingfromthematthesamespeedinthe oppositedirection.Thewholeuniversecanbeenvisionedasarising loafofraisinbread.Asthebreadexpands,thereismoreandmore spacebetweentheraisins.Thefartheraparttworaisinsare,the greaterthespeedwithwhichtheymoveapart. Extrapolatingbackwardintimeusingtheknownlawsofphysics, theuniversemusthavebeendenseranddenseratearlierandearlier times.Atsomepoint,itmusthavebeenextremelydenseandhot, andwecanevendetecttheradiationfromthisearlyreball,inthe formofmicrowaveradiationthatpermeatesspace.ThephraseBig Bangwasoriginallycoinedbythedoubtersofthetheorytomakeit soundridiculous,butitstuck,andtodayessentiallyallastronomers accepttheBigBangtheorybasedontheverydirectevidenceofthe redshiftsandthecosmicmicrowavebackgroundradiation. WhattheBigbangisnot FinallyitshouldbenotedwhattheBigBangtheoryisnot.Itis notanexplanationof why theuniverseexists.Suchquestionsbelong totherealmofreligion,notscience.Sciencecanndeversimpler andevermorefundamentalexplanationsforavarietyofphenomena,butultimatelysciencetakestheuniverseasitisaccordingto observations. Furthermore,thereisanunfortunatetendency,evenamongmany scientists,tospeakoftheBigBangtheoryasadescriptionofthe veryrsteventintheuniverse,whichcausedeverythingafterit. AlthoughitistruethattimemayhavehadabeginningEinstein's theoryofgeneralrelativityadmitssuchapossibility,themethods ofsciencecanonlyworkwithinacertainrangeofconditionssuch astemperatureanddensity.Beyondatemperatureofabout10 9 degreesC,therandomthermalmotionofsubatomicparticlesbecomessorapidthatitsvelocityiscomparabletothespeedoflight. Earlyenoughinthehistoryoftheuniverse,whenthesetemperatures existed,Newtonianphysicsbecomeslessaccurate,andwemustdescribenatureusingthemoregeneraldescriptiongivenbyEinstein's theoryofrelativity,whichencompassesNewtonianphysicsasaspecialcase.Atevenhighertemperatures,beyondabout10 33 degrees, physicistsknowthatEinstein'stheoryaswellbeginstofallapart, Section3.5TheDopplerEffect 67

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aa / Shockwavesfromby theX-15rocketplane,yingat 3.5timesthespeedofsound. ab / Thisghterjethasjust acceleratedpastthespeedof sound.Thesuddendecompressionoftheaircauseswater dropletstocondense,forminga cloud. butwedon'tknowhowtoconstructtheevenmoregeneraltheory ofnaturethatwouldworkatthosetemperatures.Nomatterhow farphysicsprogresses,wewillneverbeabletodescribenatureat innitelyhightemperatures,sincethereisalimittothetemperatureswecanexplorebyexperimentandobservationinorderto guideustotherighttheory.Wearecondentthatweunderstand thebasicphysicsinvolvedintheevolutionoftheuniversestartinga fewminutesaftertheBigBang,andwemaybeabletopushbackto millisecondsormicrosecondsafterit,butwecannotusethemethods ofsciencetodealwiththebeginningoftimeitself. DiscussionQuestions A Ifanairplanetravelsatexactlythespeedofsound,whatwouldbe thewavelengthoftheforward-emittedpartofthesoundwavesitemitted? Howshouldthisbeinterpreted,andwhatwouldactuallyhappen?What happensifit'sgoingfasterthanthespeedofsound?Canyouusethisto explainwhatyouseeinguresaaandab? B Ifbulletsgoslowerthanthespeedofsound,whycanasupersonic ghterplanecatchuptoitsownsound,butnottoitsownbullets? C Ifsomeoneinsideaplaneistalkingtoyou,shouldtheirspeechbe Dopplershifted? 68 Chapter3FreeWaves

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Summary SelectedVocabulary superposition..theaddingtogetherofwavesthatoverlapwith eachother medium.....aphysicalsubstancewhosevibrationsconstituteawave wavelength....thedistanceinspacebetweenrepetitionsofa periodicwave Dopplereect..thechangeinawave'sfrequencyandwavelengthduetothemotionofthesourceorthe observerorboth Notation ..........wavelengthGreekletterlambda Summary Wavemotiondiersinthreeimportantwaysfromthemotionof materialobjects: Wavesobeytheprincipleofsuperposition.Whentwowaves collide,theysimplyaddtogether. Themediumisnottransportedalongwiththewave.The motionofanygivenpointinthemediumisavibrationaboutits equilibriumlocation,notasteadyforwardmotion. Thevelocityofawavedependsonthemedium,notonthe amountofenergyinthewave.Forsometypesofwaves,notably waterwaves,thevelocitymayalsodependontheshapeofthewave. Soundwavesconsistofincreasesanddecreasestypicallyvery smallonesinthedensityoftheair.Lightisawave,butitisa vibrationofelectricandmagneticelds,notofanyphysicalmedium. Lightcantravelthroughavacuum. Aperiodicwaveisonethatcreatesaperiodicmotioninareceiver asitpassesit.Suchawavehasawell-denedperiodandfrequency, anditwillalsohaveawavelength,whichisthedistanceinspace betweenrepetitionsofthewavepattern.Thevelocity,frequency, andwavelengthofaperiodicwavearerelatedbytheequation v = f Awaveemittedbyamovingsourcewillbeshiftedinwavelength andfrequency.Theshiftedwavelengthisgivenbytheequation 0 = 1 )]TJ/F20 10.9091 Tf 12.105 7.38 Td [(v s v where v isthevelocityofthewavesand v s isthevelocityofthe source,takentobepositiveornegativesoastoproduceaDopplerlengthenedwavelengthifthesourceisrecedingandaDopplershortenedoneifitapproaches.Asimilarshiftoccursiftheobserver Summary 69

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ismoving,andingeneraltheDopplershiftdependsapproximately onlyontherelativemotionofthesourceandobserveriftheirvelocitiesarebothsmallcomparedtothewaves'velocity.Thisisnot justapproximatelybutexactlytrueforlightwaves,andthisfact formsthebasisofEinstein'sTheoryofRelativity. 70 Chapter3FreeWaves

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Problem3. Problem2. Problems Key p Acomputerizedanswercheckisavailableonline. R Aproblemthatrequirescalculus. ? Adicultproblem. 1 Thefollowingisagraphoftheheightofawaterwaveasa functionof position ,atacertainmomentintime. Tracethisgraphontoanotherpieceofpaper,andthensketchbelow itthecorrespondinggraphsthatwouldbeobtainedif atheamplitudeandfrequencyweredoubledwhilethevelocity remainedthesame; bthefrequencyandvelocitywerebothdoubledwhiletheamplituderemainedunchanged; cthewavelengthandamplitudewerereducedbyafactorofthree whilethevelocitywasdoubled. [ProblembyArnoldArons.] 2 aThegraphshowstheheightofawaterwavepulseasa functionofposition.Drawagraphofheightasafunctionoftime foraspecicpointonthewater.Assumethepulseistravelingto theright. bRepeatparta,butassumethepulseistravelingtotheleft. cNowassumetheoriginalgraphwasofheightasafunctionof time,anddrawagraphofheightasafunctionofposition,assuming thepulseistravelingtotheright. dRepeatpartc,butassumethepulseistravelingtotheleft. [ProblembyArnoldArons.] 3 Thegureshowsonewavelengthofasteadysinusoidalwave travelingtotherightalongastring.Deneacoordinatesystem inwhichthepositive x axispointstotherightandthepositive y axisup,suchthattheattenedstringwouldhave y =0.Copy thegure,andlabelwith y =0alltheappropriatepartsofthe string.Similarly,labelwith v =0allpartsofthestringwhose velocitiesarezero,andwith a =0allpartswhoseaccelerations arezero.Thereismorethanonepointwhosevelocityisofthe greatestmagnitude.Pickoneofthese,andindicatethedirectionof itsvelocityvector.Dothesameforapointhavingthemaximum magnitudeofacceleration. Problems 71

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[ProblembyArnoldArons.] 4 FindanequationfortherelationshipbetweentheDopplershiftedfrequencyofawaveandthefrequencyoftheoriginalwave, forthecaseofastationaryobserverandasourcemovingdirectly towardorawayfromtheobserver. 5 Suggestaquantitativeexperimenttolookforanydeviation fromtheprincipleofsuperpositionforsurfacewavesinwater.Make itsimpleandpractical. 6 Themusicalnotemiddle C hasafrequencyof262Hz.What areitsperiodandwavelength? p 7 Singingthatiso-pitchbymorethanabout1%soundsbad. Howfastwouldasingerhavetobemovingrelativetoatherestof abandtomakethismuchofachangeinpitchduetotheDoppler eect? 8 Insection3.2,wesawthatthespeedofwavesonastring dependsontheratioof T= ,i.e.,thespeedofthewaveisgreaterif thestringisundermoretension,andlessifithasmoreinertia.This istrueingeneral:thespeedofamechanicalwavealwaysdepends onthemedium'sinertiainrelationtotherestoringforcetension, stiness,resistancetocompression,...Basedontheseideas,explain whythespeedofsoundinagasdependsstronglyontemperature, whilethespeedofsoundsinliquidsandsolidsdoesnot. 72 Chapter3FreeWaves

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Across-sectionalviewofahumanbody,showingthevocaltract. Chapter4 BoundedWaves Speechiswhatseparateshumansmostdecisivelyfromanimals.No otherspeciescanmastersyntax,andeventhoughchimpanzeescan learnavocabularyofhandsigns,thereisanunmistakabledierence betweenahumaninfantandababychimp:startingfrombirth,the humanexperimentswiththeproductionofcomplexspeechsounds. Sincespeechsoundsareinstinctiveforus,weseldomthinkabout themconsciously.Howdowedocontrolsoundwavessoskillfully? Mostlywedoitbychangingtheshapeofaconnectedsetofhollow cavitiesinourchest,throat,andhead.Somehowbymovingthe boundariesofthisspaceinandout,wecanproduceallthevowel sounds.Upuntilnow,wehavebeenstudyingonlythoseproperties ofwavesthatcanbeunderstoodasiftheyexistedinaninnite, openspace.Inthischapterweaddresswhathappenswhenawaveis connedwithinacertainspace,orwhenawavepatternencounters theboundarybetweentwodierentmedia,aswhenalightwave movingthroughairencountersaglasswindowpane. 73

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a / Adiverphotographedthissh, anditsreection,fromunderwater.Thereectionistheoneon top,andisformedbylightwaves thatwentuptothesurfaceof thewater,butwerethenreected backdownintothewater. 4.1Reection,Transmission,andAbsorption Reectionandtransmission Soundwavescanechobackfromacli,andlightwavesare reectedfromthesurfaceofapond.Weusethewordreection, normallyappliedonlytolightwavesinordinaryspeech,todescribe anysuchcaseofawavereboundingfromabarrier.Figurebshows acircularwaterwavebeingreectedfromastraightwall.Inthis chapter,wewillconcentratemainlyonreectionofwavesthatmove inonedimension,asingurec. Wavereectiondoesnotsurpriseus.Afterall,amaterialobject suchasarubberballwouldbouncebackinthesameway.Butwaves arenotobjects,andtherearesomesurprisesinstore. First,onlypartofthewaveisusuallyreected.Lookingout throughawindow,weseelightwavesthatpassedthroughit,buta personstandingoutsidewouldalsobeabletoseeherreectionin theglass.Alightwavethatstrikestheglassispartly reected and partly transmitted passedbytheglass.Theenergyoftheoriginal waveissplitbetweenthetwo.Thisisdierentfromthebehaviorof therubberball,whichmustgoonewayortheother,notboth. Second,considerwhatyouseeifyouareswimmingunderwater andyoulookupatthesurface.Youseeyourownreection.This isutterlycounterintuitive,sincewewouldexpectthelightwavesto burstforthtofreedominthewide-openair.Amaterialprojectile shotuptowardthesurfacewouldneverreboundfromthewater-air 74 Chapter4BoundedWaves

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b / Circularwaterwavesare reectedfromaboundaryonthe left. c / Awaveonaspring,initiallytravelingtotheleft,is reectedfromthexedend. boundary!Figureashowsasimilarexample. Whatisitaboutthedierencebetweentwomediathatcauses wavestobepartlyreectedattheboundarybetweenthem?Is ittheirdensity?Theirchemicalcomposition?Ultimatelyallthat mattersisthespeedofthewaveinthetwomedia. Awaveispartially reectedandpartiallytransmittedattheboundarybetweenmediain whichithasdierentspeeds. Forexample,thespeedoflightwaves inwindowglassisabout30%lessthaninair,whichexplainswhy windowsalwaysmakereections.Figuresd/1and2showexamples ofwavepulsesbeingreectedattheboundarybetweentwocoil springsofdierentweights,inwhichthewavespeedisdierent. Reectionssuchasbandc,whereawaveencountersamassive xedobject,canusuallybeunderstoodonthesamebasisascases liked/1and2laterinhissection,wheretwomediameet.Example c,forinstance,islikeamoreextremeversionofexampled/1.Ifthe heavycoilspringind/1wasmadeheavierandheavier,itwouldend upactinglikethexedwalltowhichthelightspringinchasbeen attached. self-checkA Ingurec,thereectedpulseisupside-down,butitsdepthisjustas bigastheoriginalpulse'sheight.Howdoestheenergyofthereected pulsecomparewiththatoftheoriginal? Answer,p.98 Fishhaveinternalears.example1 Whydon'tshhaveear-holes?Thespeedofsoundwavesin ash'sbodyisnotmuchdifferentfromtheirspeedinwater,so soundwavesarenotstronglyreectedfromash'sskin.They passrightthroughitsbody,soshcanhaveinternalears. Whalesongstravelinglongdistancesexample2 Soundwavestravelatdrasticallydifferentspeedsthroughrock, water,andair.Whalesongsarethusstronglyreectedatboth thebottomandthesurface.Thesoundwavescantravelhundredsofmiles,bouncingrepeatedlybetweenthebottomandthe surface,andstillbedetectable.Sadly,noisepollutionfromships hasnearlyshutdownthiscetaceanversionoftheinternet. Long-distanceradiocommunication.example3 Radiocommunicationcanoccurbetweenstationsonopposite sidesoftheplanet.Themechanismissimilartotheoneexplainedinexample2,butthethreemediainvolvedaretheearth, theatmosphere,andtheionosphere. self-checkB Sonarisamethodforshipsandsubmarinestodetecteachotherby producingsoundwavesandlisteningforechoes.Whatpropertieswould anunderwaterobjecthavetohaveinordertobeinvisibletosonar? Answer,p.98 Theuseofthewordreection"naturallybringstomindthecreSection4.1Reection,Transmission,andAbsorption 75

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ationofanimagebyamirror,butthismightbeconfusing,because wedonotnormallyrefertoreection"whenwelookatsurfaces thatarenotshiny.Nevertheless,reectionishowweseethesurfaces ofallobjects,notjustpolishedones.Whenwelookatasidewalk, forexample,weareactuallyseeingthereectingofthesunfrom theconcrete.Thereasonwedon'tseeanimageofthesunatour feetissimplythattheroughsurfaceblurstheimagesodrastically. d / 1.Awaveinthelighterspring,wherethewavespeedisgreater, travelstotheleftandisthenpartlyreectedandpartlytransmittedatthe boundarywiththeheaviercoilspring,whichhasalowerwavespeed. Thereectionisinverted.2.Awavemovingtotherightintheheavier springispartlyreectedattheboundarywiththelighterspring.The reectionisuninverted. 76 Chapter4BoundedWaves

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e / 1.Anuninvertedreection.Thereectedpulseis reversedfronttoback,butis notupside-down.2.Aninverted reection.Thereectedpulseis reversedbothfronttobackand toptobottom. f / Apulsetravelingthrough ahighlyabsorptivemedium. Invertedanduninvertedreections Noticehowthepulsereectedbacktotherightinexampled/1 comesbackupside-down,whereastheonereectedbacktotheleft in2returnsinitsoriginaluprightform.Thisistrueforotherwaves aswell.Ingeneral,therearetwopossibletypesofreections,a reectionbackintoafastermediumandareectionbackintoa slowermedium.Onetypewillalwaysbeaninvertingreectionand onenoninverting. It'simportanttorealizethatwhenwediscussinvertedanduninvertedreectionsonastring,wearetalkingaboutwhetherthe waveisippedacrossthedirectionofmotioni.e.,upside-downin thesedrawings.Thereectedpulsewillalwaysbereversedfront toback,asshowninguree.Thisisbecauseitistravelinginthe otherdirection.Theleadingedgeofthepulseiswhatgetsreected rst,soitisstillaheadwhenitstartsbacktotheleft|it'sjust thatahead"isnowintheoppositedirection. Absorption Sofarwehavetacitlyassumedthatwaveenergyremainsaswave energy,andisnotconvertedtoanyotherform.Ifthiswastrue,then theworldwouldbecomemoreandmorefullofsoundwaves,which couldneverescapeintothevacuumofouterspace.Inreality,any mechanicalwaveconsistsofatravelingpatternofvibrationsofsome physicalmedium,andvibrationsofmatteralwaysproduceheat,as whenyoubendacoat-hangarbackandforthanditbecomeshot. Wecanthusexpectthatinmechanicalwavessuchaswaterwaves, soundwaves,orwavesonastring,thewaveenergywillgradually beconvertedintoheat.Thisisreferredtoas absorption Thewavesuersadecreaseinamplitude,asshowninguref. Thedecreaseinamplitudeamountstothesamefractionalchange foreachunitofdistancecovered.Forexample,ifawavedecreases fromamplitude2toamplitude1overadistanceof1meter,then aftertravelinganothermeteritwillhaveanamplitudeof1/2.That is,thereductioninamplitudeisexponential.Thiscanbeproven asfollows.Bytheprincipleofsuperposition,weknowthatawave ofamplitude2mustbehavelikethesuperpositionoftwoidentical wavesofamplitude1.Ifasingleamplitude-1wavewoulddiedownto amplitude1/2overacertaindistance,thentwoamplitude-1waves superposedontopofoneanothertomakeamplitude1+1=2must diedowntoamplitude1 = 2+1 = 2=1overthesamedistance. self-checkC Asawaveundergoesabsorption,itlosesenergy.Doesthismeanthat itslowsdown? Answer,p.98 Inmanycases,thisfrictionalheatingeectisquiteweak.Sound wavesinair,forinstance,dissipateintoheatextremelyslowly,and thesoundofchurchmusicinacathedralmayreverberateforasmuch Section4.1Reection,Transmission,andAbsorption 77

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g / X-raysarelightwaveswitha veryhighfrequency.Theyare absorbedstronglybybones,but weaklybyesh. as3or4secondsbeforeitbecomesinaudible.Duringthistimeit hastraveledoverakilometer!Eventhisverygradualdissipation ofenergyoccursmostlyasheatingofthechurch'swallsandbythe leakingofsoundtotheoutsidewhereitwilleventuallyendupas heat.Undertherightconditionshumidairandlowfrequency,a soundwaveinastraightpipecouldtheoreticallytravelhundredsof kilometersbeforebeingnoticeablyattenuated. Ingeneral,theabsorptionofmechanicalwavesdependsagreat dealonthechemicalcompositionandmicroscopicstructureofthe medium.Ripplesonthesurfaceofantifreeze,forinstance,dieout extremelyrapidlycomparedtoripplesonwater.Forsoundwaves andsurfacewavesinliquidsandgases,whatmattersistheviscosity ofthesubstance,i.e.,whetheritowseasilylikewaterormercury ormoresluggishlylikemolassesorantifreeze.Thisexplainswhy ourintuitiveexpectationofstrongabsorptionofsoundinwateris incorrect.Waterisaveryweakabsorberofsoundviz.whalesongs andsonar,andourincorrectintuitionarisesfromfocusingonthe wrongpropertyofthesubstance:water'shighdensity,whichis irrelevant,ratherthanitslowviscosity,whichiswhatmatters. Lightisaninterestingcase,sincealthoughitcantravelthrough matter,itisnotitselfavibrationofanymaterialsubstance.Thus wecanlookatthestarSirius,10 14 kmawayfromus,andbeassuredthatnoneofitslightwasabsorbedinthevacuumofouter spaceduringits9-yearjourneytous.TheHubbleSpaceTelescope routinelyobserveslightthathasbeenonitswaytoussincethe earlyhistoryoftheuniverse,billionsofyearsago.Ofcoursethe energyoflightcanbedissipatedifitdoespassthroughmatterand thelightfromdistantgalaxiesisoftenabsorbediftherehappento becloudsofgasordustinbetween. Soundproongexample4 Typicalamateurmusicianssettingouttosoundprooftheirgarages tendtothinkthattheyshouldsimplycoverthewallswiththe densestpossiblesubstance.Infact,soundisnotabsorbedvery stronglyevenbypassingthroughseveralinchesofwood.Abetter strategyforsoundproongistocreateasandwichofalternating layersofmaterialsinwhichthespeedofsoundisverydifferent, toencouragereection. Theclassicdesignisalternatinglayersofberglassandplywood. Thespeedofsoundinplywoodisveryhigh,duetoitsstiffness, whileitsspeedinberglassisessentiallythesameasitsspeed inair.Bothmaterialsarefairlygoodsoundabsorbers,butsound wavespassingthroughafewinchesofthemarestillnotgoing tobeabsorbedsufciently.Thepointofcombiningthemisthat asoundwavethattriestogetoutwillbestronglyreectedat eachoftheberglass-plywoodboundaries,andwillbounceback andforthmanytimeslikeapingpongball.Duetoalltheback78 Chapter4BoundedWaves

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and-forthmotion,thesoundmayenduptravelingatotaldistance equaltotentimestheactualthicknessofthesoundproongbeforeitescapes.Thisistheequivalentofhavingtentimesthe thicknessofsound-absorbingmaterial. Theswimbladderexample5 Theswimbladderofash,whichwasrstdiscussedinhomeworkproblem2inchapter2,isoftenlocatedrightnexttothe sh'sear.Asdiscussedinexample1onpage75,thesh'sbody isnearlytransparenttosound,soit'sactuallydifculttogetany ofthesoundwaveenergytodeposititselfintheshsothatthe shcanhearit!Thephysicshereisalmostexactlythesameas thephysicsofexample4above,withthegas-lledswimbladder playingtheroleofthelow-densitymaterial. Radiotransmissionexample6 Aradiotransmittingstation,suchasacommercialstationoran amateurhamradiostation,musthavealengthofwireorcable connectingtheampliertotheantenna.Thecableandtheantennaactastwodifferentmediaforradiowaves,andtherewill thereforebepartialreectionofthewavesastheycomefromthe cabletotheantenna.Ifthewavesbouncebackandforthmany timesbetweentheamplierandtheantenna,agreatdealoftheir energywillbeabsorbed.Therearetwowaystoattacktheproblem.Onepossibilityistodesigntheantennasothatthespeedof thewavesinitisascloseaspossibletothespeedofthewaves inthecable;thisminimizestheamountofreection.Theother methodistoconnecttheampliertotheantennausingatype ofwireorcablethatdoesnotstronglyabsorbthewaves.Partial reectionthenbecomesirrelevant,sinceallthewaveenergywill eventuallyexitthroughtheantenna. DiscussionQuestion A Asoundwavethatunderwentapressure-invertingreectionwould haveitscompressionsconvertedtoexpansionsandviceversa.How woulditsenergyandfrequencycomparewiththoseoftheoriginalsound? Woulditsoundanydifferent?Whathappensifyouswapthetwowires wheretheyconnecttoastereospeaker,resultinginwavesthatvibratein theoppositeway? Section4.1Reection,Transmission,andAbsorption 79

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h / 1.Achangeinfrequency withoutachangeinwavelength wouldproduceadiscontinuityin thewave.2.Asimplechangein wavelengthwithoutareection wouldresultinasharpkinkinthe wave. 4.2 ? QuantitativeTreatmentofReection Inthisoptionalsectionweanalyzethereasonswhyreectionsoccur ataspeed-changingboundary,predictquantitativelytheintensities ofreectionandtransmission,anddiscusshowtopredictforany typeofwavewhichreectionsareinvertingandwhicharenoninverting.Thegorydetailsarelikelytobeofinterestmainlytostudentswithconcentrationsinthephysicalsciences,butallreaders areencouragedatleasttoskimthersttwosubsectionsforphysical insight. Whyreectionoccurs Tounderstandthefundamentalreasonsforwhatdoesoccurat theboundarybetweenmedia,let'srstdiscusswhatdoesn'thappen. Forthesakeofconcreteness,considerasinusoidalwaveonastring. Ifthewaveprogressesfromaheavierportionofthestring,inwhich itsvelocityislow,toalighter-weightpart,inwhichitishigh,then theequation v = f tellsusthatitmustchangeitsfrequency,or itswavelength,orboth.Ifonlythefrequencychanged,thenthe partsofthewaveinthetwodierentportionsofthestringwould quicklygetoutofstepwitheachother,producingadiscontinuityin thewave,h/1.Thisisunphysical,soweknowthatthewavelength mustchangewhilethefrequencyremainsconstant,2. Butthereisstillsomethingunphysicalaboutgure2.Thesuddenchangeintheshapeofthewavehasresultedinasharpkink attheboundary.Thiscan'treallyhappen,becausethemedium tendstoaccelerateinsuchawayastoeliminatecurvature.Asharp kinkcorrespondstoaninnitecurvatureatonepoint,whichwould produceaninniteacceleration,whichwouldnotbeconsistentwith thesmoothpatternofwavemotionenvisionedingure2.Waves canhavekinks,butnotstationarykinks. Weconcludethatwithoutpositingpartialreectionofthewave, wecannotsimultaneouslysatisfytherequirementsofcontinuity ofthewave,and2nosuddenchangesintheslopeofthewave. Thestudentwhohasstudiedcalculuswillrecognizethisasamountingtoanassumptionthatboththewaveanditsderivativearecontinuousfunctions. Doesthisamounttoaproofthatreectionoccurs?Notquite. Wehaveonlyproventhatcertaintypesofwavemotionarenot validsolutions.Inthefollowingsubsection,weprovethatavalid solutioncanalwaysbefoundinwhichareectionoccurs.Nowin physics,wenormallyassumebutseldomproveformallythatthe equationsofmotionhaveauniquesolution,sinceotherwiseagiven setofinitialconditionscouldleadtodierentbehaviorlateron, buttheNewtonianuniverseissupposedtobedeterministic.Since thesolutionmustbeunique,andwederivebelowavalidsolution involvingareectedpulse,wewillhaveendedupwithwhatamounts 80 Chapter4BoundedWaves

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i / Apulsebeingpartiallyreectedandpartiallytransmitted attheboundarybetweentwo stringsinwhichthespeedof wavesisdifferent.Thetop drawingshowsthepulseheading totheright,towardtheheavier string.Forclarity,allbuttherst andlastdrawingsareschematic. Oncethereectedpulsebegins toemergefromtheboundary, itaddstogetherwiththetrailing partsoftheincidentpulse.Their sum,shownasawiderline,is whatisactuallyobserved. toaproofofreection. Intensityofreection Wewillnowshow,inthecaseofwavesonastring,thatitispossibletosatisfythephysicalrequirementsgivenabovebyconstructingareectedwave,andasabonusthiswillproduceanequation fortheproportionsofreectionandtransmissionandaprediction astowhichconditionswillleadtoinvertedandwhichtouninverted reection.Weassumeonlythattheprincipleofsuperpositionholds, whichisagoodapproximationsforwavesonastringofsuciently smallamplitude. Lettheunknownamplitudesofthereectedandtransmitted wavesbe R and T ,respectively.Aninvertedreectionwouldbe representedbyanegativevalueof R .Wecanwithoutlossofgeneralitytaketheincidentoriginalwavetohaveunitamplitude. Superpositiontellsusthatif,forinstance,theincidentwavehad doublethisamplitude,wecouldimmediatelyndacorresponding solutionsimplybydoubling R and T Justtotheleftoftheboundary,theheightofthewaveisgiven bytheheight1oftheincidentwave,plustheheight R ofthepart ofthereectedwavethathasjustbeencreatedandbegunheading back,foratotalheightof1+ R .Ontherightsideimmediatelynext totheboundary,thetransmittedwavehasaheight T .Toavoida discontinuity,wemusthave 1+ R = T Nextweturntotherequirementofequalslopesonbothsidesof theboundary.Lettheslopeoftheincomingwavebe s immediately totheleftofthejunction.Ifthewavewas100%reected,and withoutinversion,thentheslopeofthereectedwavewouldbe )]TJ/F20 10.9091 Tf 8.485 0 Td [(s sincethewavehasbeenreversedindirection.Ingeneral,theslope ofthereectedwaveequals )]TJ/F20 10.9091 Tf 8.485 0 Td [(sR ,andtheslopesofthesuperposed wavesontheleftsideaddupto s )]TJ/F20 10.9091 Tf 11.534 0 Td [(sR .Ontheright,theslope dependsontheamplitude, T ,butisalsochangedbythestretching orcompressionofthewaveduetothechangeinspeed.If,for example,thewavespeedistwiceasgreatontherightside,then theslopeiscutinhalfbythiseect.Theslopeontherightis therefore s v 1 =v 2 T ,where v 1 isthevelocityintheoriginalmedium and v 2 thevelocityinthenewmedium.Equalityofslopesgives s )]TJ/F20 10.9091 Tf 10.909 0 Td [(sR = s v 1 =v 2 T ,or 1 )]TJ/F20 10.9091 Tf 10.909 0 Td [(R = v 1 v 2 T Solvingthetwoequationsfortheunknowns R and T gives R = v 2 )]TJ/F20 10.9091 Tf 10.909 0 Td [(v 1 v 2 + v 1 and T = 2 v 2 v 2 + v 1 Section4.2 ? QuantitativeTreatmentofReection 81

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j / Adisturbanceinfreeway trafc. k / Inthemirrorimage,the areasofpositiveexcesstrafc densityarestillpositive,but thevelocitiesofthecarshave allbeenreversed,soareasof positiveexcessvelocityhave beenturnedintonegativeones. Therstequationshowsthatthereisnoreectionunlessthetwo wavespeedsaredierent,andthatthereectionisinvertedinreectionbackintoafastmedium. Theenergiesofthetransmittedandreectedwaversalwaysadd uptothesameastheenergyoftheoriginalwave.Thereisnever anyabruptlossorgaininenergywhenawavecrossesaboundary.Conversionofwaveenergytoheatoccursformanytypesof waves,butitoccursthroughoutthemedium.Theequationfor T ,surprisingly,allowstheamplitudeofthetransmittedwavetobe greaterthan1,i.e.,greaterthanthatoftheincidentwave.This doesnotviolateconservationofenergy,becausethisoccurswhen thesecondstringislessmassive,reducingitskineticenergy,andthe transmittedpulseisbroaderandlessstronglycurved,whichlessens itspotentialenergy. Invertedanduninvertedreectionsingeneral Forwavesonastring,reectionsbackintoafastermediumare inverted,whilethosebackintoaslowermediumareuninverted.Is thistrueforalltypesofwaves?Therathersubtleansweristhatit dependsonwhatpropertyofthewaveyouarediscussing. Let'sstartbyconsideringwavedisturbancesoffreewaytrac. Anyonewhohasdrivenfrequentlyoncrowdedfreewayshasobserved thephenomenoninwhichonedrivertapsthebrakes,startingachain reactionthattravelsbackwarddownthefreewayaseachpersonin turnexercisescautioninordertoavoidrear-endinganyone.The reasonwhythistypeofwaveisrelevantisthatitgivesasimple, easilyvisualizedexampleofourdescriptionofawavedependson whichaspectofthewavewehaveinmind.Insteadilyowingfreewaytrac,boththedensityofcarsandtheirvelocityareconstant allalongtheroad.Sincethereisnodisturbanceinthispatternof constantvelocityanddensity,wesaythatthereisnowave.Nowif awaveistouchedobyapersontappingthebrakes,wecaneither describeitasaregionofhighdensityorasaregionofdecreasing velocity. Thefreewaytracwaveisinfactagoodmodelofasoundwave, andasoundwavecanlikewisebedescribedeitherbythedensity orpressureoftheairorbyitsspeed.Likewisemanyothertypes ofwavescanbedescribedbyeitheroftwofunctions,oneofwhich isoftenthederivativeoftheotherwithrespecttoposition. Nowlet'sconsiderreections.Ifweobservethefreewaywavein amirror,thehigh-densityareawillstillappearhighindensity,but velocityintheoppositedirectionwillnowbedescribedbyanegativenumber.Apersonobservingthemirrorimagewilldrawthe samedensitygraph,butthevelocitygraphwillbeippedacrossthe x axis,anditsoriginalregionofnegativeslopewillnowhavepositiveslope.AlthoughIdon'tknowanyphysicalsituationthatwould 82 Chapter4BoundedWaves

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l / Seenfromthisangle,the opticalcoatingonthelensesof thesebinocularsappearspurple andgreen.Thecolorvaries dependingontheanglefrom whichthecoatingisviewed,and theanglevariesacrossthefaces ofthelensesbecauseoftheir curvature. correspondtothereectionofatracwave,wecanimmediatelyapplythesamereasoningtosoundwaves,whichoftendogetreected, anddeterminethatareectioncaneitherbedensity-invertingand velocity-noninvertingordensity-noninvertingandvelocity-inverting. Thissametypeofsituationwilloccuroverandoverasoneencountersnewtypesofwaves,andtoapplytheanalogyweneed onlydeterminewhichquantities,likevelocity,becomenegatedina mirrorimageandwhich,likedensity,staythesame. Alightwave,forinstanceconsistsofatravelingpatternofelectricandmagneticelds.Allyouneedtoknowinordertoanalyzethe reectionoflightwavesishowelectricandmagneticeldsbehave underreection;youdon'tneedtoknowanyofthedetailedphysics ofelectricityandmagnetism.Anelectriceldcanbedetected,for example,bythewayone'shairstandsonend.Thedirectionof thehairindicatesthedirectionoftheelectriceld.Inamirrorimage,thehairpointstheotherway,sotheelectriceldisapparently reversedinamirrorimage.Thebehaviorofmagneticelds,however,isalittletricky.Themagneticpropertiesofabarmagnet, forinstance,arecausedbythealignedrotationoftheoutermost orbitingelectronsoftheatoms.Inamirrorimage,thedirectionof rotationisreversed,sayfromclockwisetocounterclockwise,andso themagneticeldisreversedtwice:oncesimplybecausethewhole pictureisippedandoncebecauseofthereversedrotationofthe electrons.Inotherwords,magneticeldsdonotreversethemselves inamirrorimage.Wecanthuspredictthattherewillbetwopossibletypesofreectionoflightwaves.Inone,theelectriceldis invertedandthemagneticelduninverted.Intheother,theelectric eldisuninvertedandthemagneticeldinverted. 4.3InterferenceEffects Ifyoulookatthefrontofapairofhigh-qualitybinoculars,you willnoticeagreenish-bluecoatingonthelenses.Thisisadvertised asacoatingtopreventreection.Nowreectionisclearlyundesirable|wewantthelighttogo in thebinoculars|butsofarI've describedreectionasanunalterablefactofnature,dependingonly onthepropertiesofthetwowavemedia.Thecoatingcan'tchange thespeedoflightinairoringlass,sohowcanitwork?Thekeyis thatthecoatingitselfisawavemedium.Inotherwords,wehave athree-layersandwichofmaterials:air,coating,andglass.Wewill analyzethewaythecoatingworks,notbecauseopticalcoatingsare animportantpartofyoureducationbutbecauseitprovidesagood exampleofthegeneralphenomenonofwaveinterferenceeects. Therearetwodierentinterfacesbetweenmedia:anair-coating boundaryandacoating-glassboundary.Partialreectionandpartialtransmissionwilloccurateachboundary.Foreaseofvisualizationlet'sstartbyconsideringanequivalentsystemconsistingof Section4.3InterferenceEffects 83

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m / Aropeconsistingofthree sections,themiddleonebeing lighter. n / Tworeections,aresuperimposed.Onereectionis inverted. o / Asoapbubbledisplays interferenceeffects. threedissimilarpiecesofstringtiedtogether,andawavepattern consistinginitiallyofasinglepulse.Figurem/1showstheincident pulsemovingthroughtheheavyrope,inwhichitsvelocityislow. Whenitencountersthelighter-weightropeinthemiddle,afaster medium,itispartiallyreectedandpartiallytransmitted.The transmittedpulseisbigger,butneverthelesshasonlypartofthe originalenergy.Thepulsetransmittedbytherstinterfaceisthen partiallyreectedandpartiallytransmittedbythesecondboundary,3.Ingure4,twopulsesareonthewaybackouttotheleft, andasinglepulseisheadingototheright.Thereisstillaweak pulsecaughtbetweenthetwoboundaries,andthiswillrattleback andforth,rapidlygettingtooweaktodetectasitleaksenergyto theoutsidewitheachpartialreection. Notehow,ofthetworeectedpulsesin4,oneisinvertedand oneuninverted.Oneunderwentreectionattherstboundarya reectionbackintoaslowermediumisuninverted,buttheother wasreectedatthesecondboundaryreectionbackintoafaster mediumisinverted. Nowlet'simaginewhatwouldhavehappenediftheincoming wavepatternhadbeenalongsinusoidalwavetraininsteadofa singlepulse.Thersttwowavestoreemergeontheleftcouldbe inphase,n/1,oroutofphase,2,oranywhereinbetween.The amountoflagbetweenthemdependsentirelyonthewidthofthe middlesegmentofstring.Ifwechoosethewidthofthemiddlestring segmentcorrectly,thenwecanarrangefordestructiveinterference tooccur,2,withcancellationresultinginaveryweakreectedwave. Thiswholeanalysisappliesdirectlytoouroriginalcaseofoptical coatings.Visiblelightfrommostsourcesdoesconsistofastreamof shortsinusoidalwave-trainssuchastheonesdrawnabove.Theonly realdierencebetweenthewaves-on-a-ropeexampleandthecaseof anopticalcoatingisthattherstandthirdmediaareairandglass, inwhichlightdoesnothavethesamespeed.However,thegeneral resultisthesameaslongastheairandtheglasshavelight-wave speedsthateitherbothgreaterthanthecoating'sorbothlessthan thecoating's. Thebusinessofopticalcoatingsturnsouttobeaveryarcane one,withaplethoraoftradesecretsandblackmagic"techniques handeddownfrommastertoapprentice.Nevertheless,theideas youhavelearnedaboutwavesingeneralaresucienttoallowyou tocometosomedeniteconclusionswithoutanyfurthertechnical knowledge.Theself-checkanddiscussionquestionswilldirectyou alongtheselinesofthought. Theexampleofanopticalcoatingwastypicalofawidevariety ofwaveinterferenceeects.Withalittleguidance,youarenow readytogureoutforyourselfotherexamplessuchastherainbow patternmadebyacompactdisc,alayerofoilonapuddle,ora 84 Chapter4BoundedWaves

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soapbubble. self-checkD 1.Colorcorrespondstowavelengthoflightwaves.Isitpossibleto chooseathicknessforanopticalcoatingthatwillproducedestructive interferenceforallcolorsoflight? 2.Howcanyouexplaintherainbowcolorsonthesoapbubbleingure o? Answer,p.98 DiscussionQuestions A Isitpossibletoget complete destructiveinterferenceinanoptical coating,atleastforlightofonespecicwavelength? B Sunlightconsistsofsinusoidalwave-trainscontainingontheorder ofahundredcyclesback-to-back,foralengthofsomethinglikeatenthof amillimeter.Whathappensifyoutrytomakeanopticalcoatingthicker thanthis? C Supposeyoutaketwomicroscopeslidesandlayoneontopofthe othersothatoneofitsedgesisrestingonthecorrespondingedgeofthe bottomone.Ifyouinsertasliverofpaperorahairattheoppositeend, awedge-shapedlayerofairwillexistinthemiddle,withathicknessthat changesgraduallyfromoneendtotheother.Whatwouldyouexpectto seeiftheslideswereilluminatedfromabovebylightofasinglecolor? Howwouldthischangeifyougraduallyliftedtheloweredgeofthetop slideuntilthetwoslideswerenallyparallel? D AnobservationliketheonedescribedindiscussionquestionCwas usedbyNewtonasevidence against thewavetheoryoflight!IfNewton didn'tknowaboutinvertingandnoninvertingreections,whatwouldhave seemedinexplicabletohimabouttheregionwheretheairlayerhadzero ornearlyzerothickness? Section4.3InterferenceEffects 85

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p / Amodelofaguitarstring. q / Themotionofapulseon thestring. r / Atrickywaytodoublethe frequency. 4.4WavesBoundedonBothSides Intheexamplesdiscussedinsection4.3,itwastheoreticallytrue thatapulsewouldbetrappedpermanentlyinthemiddlemedium, butthatpulsewasnotcentraltoourdiscussion,andinanycaseit wasweakeningseverelywitheachpartialreection.Nowconsider aguitarstring.Atitsendsitistiedtothebodyoftheinstrument itself,andsincethebodyisverymassive,thebehaviorofthewaves whentheyreachtheendofthestringcanbeunderstoodinthesame wayasiftheactualguitarstringwasattachedontheendstostrings thatwereextremelymassive,p.Reectionsaremostintensewhen thetwomediaareverydissimilar.Becausethewavespeedinthe bodyissoradicallydierentfromthespeedinthestring,weshould expectnearly100%reection. Althoughthismayseemlikearatherbizarrephysicalmodelof theactualguitarstring,italreadytellsussomethinginteresting aboutthebehaviorofaguitarthatwewouldnototherwisehave understood.Thebody,farfrombeingapassiveframeforattaching thestringsto,isactuallytheexitpathforthewaveenergyinthe strings.Witheveryreection,thewavepatternonthestringloses atinyfractionofitsenergy,whichisthenconductedthroughthe bodyandoutintotheair.Thestringhastoolittlecross-sectionto makesoundwavesecientlybyitself.Bychangingtheproperties ofthebody,moreover,weshouldexpecttohaveaneectonthe mannerinwhichsoundescapesfromtheinstrument.Thisisclearly demonstratedbytheelectricguitar,whichhasanextremelymassive, solidwoodenbody.Herethedissimilaritybetweenthetwowave mediaisevenmorepronounced,withtheresultthatwaveenergy leaksoutofthestringevenmoreslowly.Thisiswhyanelectric guitarwithnoelectricpickupcanhardlybeheardatall,anditis alsothereasonwhynotesonanelectricguitarcanbesustainedfor longerthannotesonanacousticguitar. Ifweinitiallycreateadisturbanceonaguitarstring,howwill thereectionsbehave?Inreality,thengerorpickwillgivethe stringatriangularshapebeforelettingitgo,andwemaythinkof thistriangularshapeasaverybroaddent"inthestringwhich willspreadoutinbothdirections.Forsimplicity,however,let'sjust imagineawavepatternthatinitiallyconsistsofasingle,narrow pulsetravelinguptheneck,q/1.Afterreectionfromthetopend, itisinverted,3.Nowsomethinginterestinghappens:gure5is identicaltogure1.Aftertworeections,thepulsehasbeeninvertedtwiceandhaschangeddirectiontwice.Itisnowbackwhere itstarted.Themotionisperiodic.Thisiswhyaguitarproduces soundsthathaveadenitesensationofpitch. self-checkE Noticethatfromq/1toq/5,thepulsehaspassedbyeverypointonthe stringexactlytwice.Thismeansthatthetotaldistanceithastraveled 86 Chapter4BoundedWaves

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s / Usingthesumoffoursine wavestoapproximatethetriangularinitialshapeofaplucked guitarstring. equals2 L ,where L isthelengthofthestring.Giventhisfact,whatare theperiodandfrequencyofthesounditproduces,expressedinterms of L and v ,thevelocityofthewave? Answer,p.99 Notethatifthewavesonthestringobeytheprincipleofsuperposition,thenthevelocitymustbeindependentofamplitude,and theguitarwillproducethesamepitchregardlessofwhetheritis playedloudlyorsoftly.Inreality,wavesonastringobeytheprincipleofsuperpositionapproximately,butnotexactly.Theguitar, likejustaboutanyacousticinstrument,isalittleoutoftunewhen playedloudly.Theeectismorepronouncedforwindinstruments thanforstrings,butwindplayersareabletocompensateforit. Nowthereisonlyoneholeinourreasoning.Supposewesomehowarrangetohaveaninitialsetupconsistingoftwoidenticalpulses headingtowardeachother,asingurer.Theywillpassthrough eachother,undergoasingleinvertingreection,andcomebackto acongurationinwhichtheirpositionshavebeenexactlyinterchanged.Thismeansthattheperiodofvibrationishalfaslong. Thefrequencyistwiceashigh. Thismightseemlikeapurelyacademicpossibility,sincenobody actuallyplaystheguitarwithtwopicksatonce!Butinfactitisan exampleofaverygeneralfactaboutwavesthatareboundedonboth sides.AmathematicaltheoremcalledFourier'stheoremstatesthat anywavecanbecreatedbysuperposingsinewaves.Figuresshows howevenbyusingonlyfoursinewaveswithappropriatelychosen amplitudes,wecanarriveatasumwhichisadecentapproximation totherealistictriangularshapeofaguitarstringbeingplucked. Theone-humpwave,inwhichhalfawavelengthtsonthestring, willbehavelikethesinglepulseweoriginallydiscussed.Wecall itsfrequency f o .Thetwo-humpwave,withonewholewavelength, isverymuchlikethetwo-pulseexample.Forthereasonsdiscussed above,itsfrequencyis2 f o .Similarly,thethree-humpandfour-hump waveshavefrequenciesof3 f o and4 f o Theoreticallywewouldneedtoaddtogetherinnitelymany suchwavepatternstodescribetheinitialtriangularshapeofthe stringexactly,althoughtheamplitudesrequiredfortheveryhigh frequencypartswouldbeverysmall,andanexcellentapproximation couldbeachievedwithasfewastenwaves. Wethusarriveatthefollowingverygeneralconclusion.Wheneverawavepatternexistsinamediumboundedonbothsidesby mediainwhichthewavespeedisverydierent,themotioncanbe brokendownintothemotionofatheoreticallyinniteseriesofsine waves,withfrequencies f o ,2 f o ,3 f o ,...Exceptforsometechnical details,tobediscussedbelow,thisanalysisappliestoavastrangeof sound-producingsystems,includingtheaircolumnwithinthehumanvocaltract.Becausesoundscomposedofthiskindofpattern offrequenciesaresocommon,ourear-brainsystemhasevolvedso Section4.4WavesBoundedonBothSides 87

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t / Graphsofloudnessversusfrequencyforthevowelah, sungasthreedifferentmusical notes.GisconsonantwithD, sinceeveryovertoneofGthatis closetoanovertoneofD*isat exactlythesamefrequency.G andC#aredissonanttogether, sincesomeoftheovertonesofG xarecloseto,butnotrighton topof,thoseofC#. astoperceivethemasasingle,fusedsensationoftone. Musicalapplications Manymusiciansclaimtobeabletopickoutbyearseveralofthe frequencies2 f o ,3 f o ,...,calledovertonesor harmonics ofthefundamental f o ,buttheyarekiddingthemselves.Inreality,theovertone serieshastwoimportantrolesinmusic,neitherofwhichdepends onthisctitiousabilitytohearout"theindividualovertones. First,therelativestrengthsoftheovertonesisanimportant partofthepersonalityofasound,calleditstimbrerhymeswith amber".Thecharacteristictoneofthebrassinstruments,forexample,isasoundthatstartsoutwithaverystrongharmonicseries extendinguptoveryhighfrequencies,butwhosehigherharmonics diedowndrasticallyastheattackchangestothesustainedportion ofthenote. Second,althoughtheearcannotseparatetheindividualharmonicsofasinglemusicaltone,itisverysensitivetoclashesbetween theovertonesofnotesplayedsimultaneously,i.e.,inharmony.We tendtoperceiveacombinationofnotesasbeingdissonantifthey haveovertonesthatareclosebutnotthesame.Roughlyspeaking, strongovertoneswhosefrequenciesdierbymorethan1%andless than10%causethenotestosounddissonant.Itisimportantto realizethatthetermdissonance"isnotanegativeoneinmusic. Nomatterhowlongyousearchtheradiodial,youwillneverhear morethanthreesecondsofmusicwithoutatleastonedissonant combinationofnotes.Dissonanceisanecessaryingredientinthe creationofamusicalcycleoftensionandrelease.Musicallyknowledgeablepeopledon'tusetheworddissonant"asacriticismof music,althoughdissonancecanbeusedinaclumsyway,orwithout providinganycontrastbetweendissonanceandconsonance. Standingwaves Figureushowssinusoidalwavepatternsmadebyshakingarope. Iusedtoenjoydoingthisatthebankwiththepensonchains,back inthedayswhenpeopleactuallywenttothebank.Youmightthink thatIandthepersoninthephotoshadtopracticeforalongtime inordertogetsuchnicesinewaves.Infact,asinewaveistheonly shapethatcancreatethiskindofwavepattern,calledastanding wave,whichsimplyvibratesbackandforthinoneplacewithout moving.Thesinewavejustcreatesitselfautomaticallywhenyou ndtherightfrequency,becausenoothershapeispossible. Ifyouthinkaboutit,it'snotevenobviousthatsinewavesshould beabletodothistrick.Afterall,wavesaresupposedtotravelata setspeed,aren'tthey?Thespeedisn'tsupposedtobezero!Well,we canactuallythinkofastandingwaveasasuperpositionofamoving 88 Chapter4BoundedWaves

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v / Sinewavesaddtomake sinewaves.Otherfunctionsdon't havethisproperty. w / Example7. u / Standingwavesonaspring. sinewavewithitsownreection,whichismovingtheoppositeway. Sinewaveshavetheuniquemathematicalproperty,v,thatthesum ofsinewavesofequalwavelengthissimplyanewsinewavewith thesamewavelength.Asthetwosinewavesgobackandforth,they alwayscancelperfectlyattheends,andtheirsumappearstostand still. Standingwavepatternsareratherimportant,sinceatomsare reallystanding-wavepatternsofelectronwaves.Youareastanding wave! Harmonicsonstringinstrumentsexample7 Figurewshowsaviolistplayingwhatstringplayersrefertoasa naturalharmonic.Thetermharmonicisusedhereinasomewhatdifferentsensethaninphysics.Themusician'spinkieis pressingverylightlyagainstthestringnothardenoughto makeittouchthengerboardatapointpreciselyatthecenter ofthestring'slength.Asshowninthediagram,thisallowsthe stringtovibrateatfrequencies2 f o ,4 f o ,6 f o ::: ,whichhavestationarypointsatthecenterofthestring,butnotattheoddmultiples f o ,3 f o ::: .Sincealltheovertonesaremultiplesof2 f o ,the earperceives2 f o asthebasicfrequencyofthenote.Inmusical terms,doublingthefrequencycorrespondstoraisingthepitchby anoctave.Thetechniquecanbeusedinordertomakeiteasier toplayhighnotesinrapidpassages,orforitsownsake,because ofthechangeintimbre. Section4.4WavesBoundedonBothSides 89

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x / Surprisingly,soundwaves undergopartialreectionatthe openendsoftubesaswellas closedones. y / Graphsofexcessdensity versuspositionforthelowestfrequencystandingwavesof threetypesofaircolumns.Points ontheaxishavenormalair density. Standing-wavepatternsofaircolumns Theaircolumninsideawindinstrumentbehavesverymuch likethewave-on-a-stringexamplewe'vebeenconcentratingonso far,themaindierencebeingthatwemayhaveeitherinvertingor noninvertingreectionsattheends. Someorganpipesareclosedatbothends.Thespeedofsound isdierentinmetalthaninair,sothereisastrongreectionat theclosedends,andwecanhavestandingwaves.Thesereections arebothdensity-noninverting,sowegetsymmetricstanding-wave patterns,suchastheoneshowningurey/1. FigurexshowsthesoundwavesinandaroundabambooJapanese utecalledashakuhachi,whichis open atbothendsoftheaircolumn.Wecanonlyhaveastandingwavepatterniftherearereectionsattheends,butthatisverycounterintuitive|whyis thereanyreectionatall,ifthesoundwaveisfreetoemergeinto openspace,andthereisnochangeinmedium?Recallthereason whywegotreectionsatachangeinmedium:becausethewavelengthchanges,sothewavehastoreadjustitselffromonepattern toanother,andtheonlywayitcandothatwithoutdevelopinga kinkisifthereisareection.Somethingsimilarishappeninghere. Theonlydierenceisthatthewaveisadjustingfrombeingaplane wavetobeingasphericalwave.Thereectionsattheopenends aredensity-inverting,y/2,sothewavepatternispinchedoatthe ends.Comparingpanels1and2ofthegure,weseethatalthough thewavepattensaredierent,inbothcasesthewavelengthisthe same:inthelowest-frequencystandingwave,halfawavelengthts insidethetube.Thus,itisn'tnecessarytomemorizewhichtypeof reectionisinvertingandwhichisinverting.It'sonlynecessaryto knowthatthetubesaresymmetric. Finally,wecanhaveanasymmetrictube:closedatoneendand openattheother.Acommonexampleisthepanpipes,z,whichare closedatthebottomandopenatthetop.Thestandingwavewith thelowestfrequencyisthereforeoneinwhich1/4ofawavelength tsalongthelengthofthetube,asshowningurey/3. Sometimesaninstrument'sphysicalappearancecanbemisleading.Aconcertute,aa,isclosedatthemouthendandopenat theother,sowewouldexpectittobehavelikeanasymmetricair column;inreality,itbehaveslikeasymmetricaircolumnopenat bothends,becausetheembouchureholetheholetheplayerblows overactslikeanopenend.Theclarinetandthesaxophonelook similar,havingamoutpieceandreedatoneendandanopenend attheother,buttheyactdierent.Infacttheclarinet'saircolumnhaspatternsofvibrationthatareasymmetric,thesaxophone symmetric.Thediscrepancycomesfromthedierencebetweenthe conicaltubeofthesaxandthecylindricaltubeoftheclarinet.The adjustmentofthewavepatternfromaplanewavetoaspherical 90 Chapter4BoundedWaves

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z / Apanpipeisanasymmetricaircolumn,openatthetop andclosedatthebottom. aa / Aconcertutelookslike anasymmetricaircolumn,open atthemouthendandclosedat theother.However,itspatternsof vibrationaresymmetric,because theembouchureholeactslikean openend. waveismoregradualatthearingbellofthesaxophone. self-checkF Drawagraphofpressureversuspositionfortherstovertoneoftheair columninatubeopenatoneendandclosedattheother.Thiswillbe thenext-to-longestpossiblewavelengththatallowsforapointofmaximumvibrationatoneendandapointofnovibrationattheother.How manytimesshorterwillitswavelengthbecomparedtothewavelength ofthelowest-frequencystandingwave,showninthegure?Basedon this,howmanytimesgreaterwillitsfrequencybe? Answer,p.99 Section4.4WavesBoundedonBothSides 91

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Summary SelectedVocabulary reection.....thebouncingbackofpartofawavefroma boundary transmission...thecontinuationofpartofawavethrougha boundary absorption....thegradualconversionofwaveenergyinto heatingofthemedium standingwave..awavepatternthatstaysinoneplace Notation ..........wavelengthGreekletterlambda Summary Wheneverawaveencounterstheboundarybetweentwomedia inwhichitsspeedsaredierent,partofthewaveisreectedand partistransmitted.Thereectionisalwaysreversedfront-to-back, butmayalsobeinvertedinamplitude.Whetherthereectionis inverteddependsonhowthewavespeedsinthetwomediacompare, e.g.,awaveonastringisuninvertedwhenitisreectedbackintoa segmentofstringwhereitsspeedislower.Thegreaterthedierence inwavespeedbetweenthetwomedia,thegreaterthefractionof thewaveenergythatisreected.Surprisingly,awaveinadense materiallikewoodwillbestronglyreectedbackintothewoodat awood-airboundary. Aone-dimensionalwaveconnedbyhighlyreectiveboundaries ontwosideswilldisplaymotionwhichisperiodic.Forexample,if bothreectionsareinverting,thewavewillhaveaperiodequal totwicethetimerequiredtotraversetheregion,ortothattime dividedbyaninteger.Animportantspecialcaseisasinusoidal wave;inthiscase,thewaveformsastationarypatterncomposedof asuperpositionofsinewavesmovinginoppositedirection. 92 Chapter4BoundedWaves

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C261.6Hz D293.7 E329.6 F349.2 G392.0 A440.0 B [ 466.2 Problem5. Problems Key p Acomputerizedanswercheckisavailableonline. R Aproblemthatrequirescalculus. ? Adicultproblem. 1 Lighttravelsfasterinwarmerair.Usethisfacttoexplainthe formationofamirageappearingliketheshinysurfaceofapoolof waterwhenthereisalayerofhotairabovearoad.Forsimplicity, pretendthatthereisactuallyasharpboundarybetweenthehot layerandthecoolerlayeraboveit. 2 aUsingtheequationsfromoptionalsection4.2,compute theamplitudeoflightthatisreectedbackintoairatanair-water interface,relativetotheamplitudeoftheincidentwave.Thespeeds oflightinairandwaterare3.0 10 8 and2.2 10 8 m/s,respectively. bFindtheenergyofthereectedwaveasafractionoftheincident energy.[Hint:Theanswerstothetwopartsarenotthesame.] p 3 Aconcertuteproducesitslowestnote,atabout262Hz, whenhalfofawavelengthtsinsideitstube.Computethelength oftheute. Answer,p.99 4 aAgoodtenorsaxophoneplayercanplayallofthefollowingnoteswithoutchangingherngering,simplybyalteringthe tightnessofherlips:E [ Hz,E [ Hz,B [ Hz,and E [ Hz.Howisthispossible?I'mnotaskingyoutoanalyze thecouplingbetweenthelips,thereed,themouthpiece,andtheair column,whichisverycomplicated. bSomesaxophoneplayersareknownfortheirabilitytousethis techniquetoplayfreaknotes,"i.e.,notesabovethenormalrange oftheinstrument.Whyisn'titpossibletoplaynotesbelowthe normalrangeusingthistechnique? 5 Thetablegivesthefrequenciesofthenotesthatmakeup thekeyofFmajor,startingfrommiddleCandgoingupthrough allsevennotes.aCalculatetherstfourorveharmonicsofC andG,anddeterminewhetherthesetwonoteswillbeconsonantor dissonant.bDothesameforCandB [ .[Hint:Rememberthat harmonicsthatdierbyabout1-10%causedissonance.] 6 Brassandwindinstrumentsgoupinpitchasthemusician warmsup.Supposeaparticulartrumpet'sfrequencygoesupby 1.2%.Let'sconsiderpossiblephysicalreasonsforthechangein pitch.aSolidsgenerallyexpandwithincreasingtemperature,becausethestrongerrandommotionoftheatomstendstobumpthem apart.Brassexpandsby1.88 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(5 perdegreeC.Wouldthistend toraisethepitch,orlowerit?Estimatethesizeoftheeectin comparisonwiththeobservedchangeinfrequency.bThespeed ofsoundinagasisproportionaltothesquarerootoftheabsolute Problems 93

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temperature,wherezeroabsolutetemperatureis-273degreesC.As inparta,analyzethesizeanddirectionoftheeect.cDetermine thechangeintemperature,inunitsofdegreesC,thatwouldaccount fortheobservedeect. 7 Yourexhaledbreathcontainsabout4.5%carbondioxide,and isthereforemoredensethanfreshairbyabout2.3%.Byanalogy withthetreatmentofwavesonastringinsection3.2,weexpect thatthespeedofsoundwillbeinverselyproportionaltothesquare rootofthedensityofthegas.Calculatetheeectonthefrequency producedbyawindinstrument. 94 Chapter4BoundedWaves

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Appendix1:Exercises Exercise1A:Vibrations Equipment: airtrackandcartsoftwodierentmasses springs springscales Placethecartontheairtrackandattachspringssothatitcanvibrate. 1.Testwhethertheperiodofvibrationdependsonamplitude.Tryatleastonemoderate amplitude,forwhichthespringsdonotgoslack,atleastoneamplitudethatislargeenoughso thattheydogoslack,andoneamplitudethat'stheverysmallestyoucanpossiblyobserve. 2.Tryacartwithadierentmass.Doestheperiodchangebytheexpectedfactor,basedon theequation T =2 p m=k ? 3.Useaspringscaletopullthecartawayfromequilibrium,andmakeagraphofforceversus position.Isitlinear?Ifso,whatisitsslope? 4.Testtheequation T =2 p m=k numerically.

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Exercise2A:WorksheetonResonance 1.Comparetheoscillator'senergiesatA,B,C,andD. 2.ComparetheQvaluesofthetwooscillators. 3.Matchthex-tgraphsin#2withtheamplitude-frequencygraphsbelow. 96 Appendix1:Exercises

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Appendix2:PhotoCredits Exceptasspecicallynotedbeloworinaparentheticalcreditinthecaptionofagure,alltheillustrationsin thisbookareundermyowncopyright,andarecopyleftlicensedunderthesamelicenseastherestofthebook. Insomecasesit'sclearfromthedatethatthegureispublicdomain,butIdon'tknowthenameoftheartist orphotographer;Iwouldbegratefultoanyonewhocouldhelpmetogivepropercredit.Ihaveassumedthat imagesthatcomefromU.S.governmentwebpagesarecopyright-free,sinceproductsoffederalagenciesfallinto thepublicdomain.I'veincludedsomepublic-domainpaintings;photographicreproductionsofthemarenot copyrightableintheU.S.BridgemanArtLibrary,Ltd.v.CorelCorp.,36F.Supp.2d191,S.D.N.Y.1999. WhenPSSCPhysics"isgivenasacredit,itindicatesthatthegureisfromthersteditionofthetextbook entitledPhysics,bythePhysicalScienceStudyCommittee.Theearlyeditionsofthesebooksneverhadtheir copyrightsrenewed,andarenowthereforeinthepublicdomain.Thereisalsoablanketpermissiongivenin thelaterPSSCCollegePhysicsedition,whichstatesonthecopyrightpagethatThematerialstakenfromthe originalandsecondeditionsandtheAdvancedTopicsofPSSCPHYSICSincludedinthistextwillbeavailable toallpublishersforuseinEnglishafterDecember31,1970,andintranslationsafterDecember31,1975." CreditstoMillikanandGalerefertothetextbooksPracticalPhysicsandElementsofPhysics. Botharepublicdomain.The1927versiondidnothaveitscopyrightrenewed.Sinceispossiblethatsomeof theillustrationsinthe1927versionhadtheircopyrightsrenewedandarestillundercopyright,Ihaveonlyused themwhenitwasclearthattheywereoriginallytakenfrompublicdomainsources. Inafewcases,Ihavemadeuseofimagesunderthefairusedoctrine.However,Iamnotalawyer,andthelaws onfairusearevague,soyoushouldnotassumethatit'slegalforyoutousetheseimages.Inparticular,fairuse lawmaygiveyoulessleewaythanitgivesme,becauseI'musingtheimagesforeducationalpurposes,andgiving thebookawayforfree.Likewise,ifthephotocreditsayscourtesyof...,"thatmeansthecopyrightownergave mepermissiontouseit,butthatdoesn'tmeanyouhavepermissiontouseit. Contents Bridge,MRI,surfer,x-ray,galaxy: seebelow. 13 Electricbass: BrynjarVik,CC-BYlicense. 20 Jupiter: UncopyrightedimagefromtheVoyagerprobe.Lineartbytheauthor. 25 TacomaNarrowsBridge: Publicdomain,fromStillmanFiresCollection:TacomaFireDept,www.archive.org. 33 NimitzFreeway: Unknownphotographer,courtesyoftheUCBerkeleyEarthSciencesandMapLibrary. 37 Two-dimensional MRI: Imageoftheauthor'swife. 37 Three-dimensionalbrain: R.Malladi,LBNL. 44 Spideroscillations: Emile,LeFloch,andVollrath, Nature 440:621. 47 Paintingofwaves: KatsushikaHokusai-1849, publicdomain. 50 Superpositionofpulses: PhotofromPSSCPhysics. 51 Markeronspringaspulsepasses by: PSSCPhysics. 52 Surnghanddrag: StanShebs,GFDLlicensedWikimediaCommons. 62 Fetus: Imageoftheauthor'sdaughter. 52 Breakingwave: OleKils,olekilsatweb.de,GFDLlicensedWikipedia. 61 Wavelengthsofcircularandlinearwaves: PSSCPhysics. 61 Changingwavelength: PSSCPhysics. 63 Dopplereectforwaterwaves: PSSCPhysics. 65 Dopplerradar: PublicdomainimagebyNOAA,anagencyof theU.S.federalgovernment. 66 M51galaxy: publicdomainHubbleSpaceTelescopeimage,courtesyofNASA, ESA,S.BeckwithSTScI,andTheHubbleHeritageTeamSTScI/AURA. 67 MountWilson: AndrewDunn, cc-by-salicensed. 68 X15: NASA,publicdomain. 68 Jetbreakingthesoundbarrier: Publicdomainproduct oftheU.S.government,U.S.NavyphotobyEnsignJohnGay. 73 Humancross-section: CourtesyoftheVisible HumanProject,NationalLibraryofMedicine,USNIH. 74 Reectionofsh: JanDerk,Wikipediauserjanderk, publicdomain. 75 Reectionofcircularwaves: PSSCPhysics. 75 Reectionofpulses: PSSCPhysics. 76 Reectionofpulses: PhotofromPSSCPhysics. 78 X-rayimageofhand: 1896imageproducedbyRoentgen. 84 Soapbubble: WikimediaCommons,GFDL/CC-BY-SA,userTagishsimon. 86 Photoofguitar: Wikimedia Commons,dedicatedtothepublicdomainbyuserTsca. 89 Standingwaves: PSSCPhysics. 82 Trac: WikipediauserDili,CC-BYlicensed. 91 Panpipes: WikipediauserAndrewDunn,CC-BY-SAlicensed. 91 Flute: WikipediauserGrendelkhan,GFDLlicensed.

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Appendix3:HintsandSolutions AnswerstoSelf-Checks AnswerstoSelf-ChecksforChapter2 Page28,self-checkA: Thehorizontalaxisisatimeaxis,andtheperiodofthevibrationsis independentofamplitude.Shrinkingtheamplitudedoesnotmakethecylesandfaster. Page29,self-checkB: Energyisproportionaltothesquareoftheamplitude,soitsenergyis fourtimessmalleraftereverycycle.Itlosesthreequartersofitsenergywitheachcycle. Page35,self-checkC: Sheshouldtapthewineglassesshendsinthestoreandlookforone withahigh Q ,i.e.,onewhosevibrationsdieoutveryslowly.Theonewiththehighest Q will havethehighest-amplituderesponsetoherdrivingforce,makingitmorelikelytobreak. AnswerstoSelf-ChecksforChapter3 Page51,self-checkA: Theleadingedgeismovingup,thetrailingedgeismovingdown,and thetopofthehumpismotionlessforoneinstant. AnswerstoSelf-ChecksforChapter4 Page75,self-checkA: Theenergyofawaveisusuallyproportionaltothesquareofits amplitude.Squaringanegativenumbergivesapositiveresult,sotheenergyisthesame. Page75,self-checkB: Asubstanceisinvisibletosonarifthespeedofsoundwavesinitis thesameasinwater.Reectionsonlyoccuratboundariesbetweenmediainwhichthewave speedisdierent. Page77,self-checkC: No.Amaterialobjectthatloseskineticenergyslowsdown,buta waveisnotamaterialobject.Thevelocityofawaveordinarilyonlydependsonthemedium, nottheamplitude.Thespeedofasoftsound,forexample,isthesameasthespeedofaloud sound. Page85,self-checkD: 1.No.Togetthebestpossibleinterference,thethicknessofthe coatingmustbesuchthatthesecondreectedwavetrainlagsbehindtherstbyaninteger numberofwavelengths.Optimalperformancecanthereforeonlybeproducedforonespecic coloroflight.Thetypicalgreenishcolorofthecoatingsshowsthattheydotheworstjobfor greenlight. 2.Lightcanbereectedeitherfromtheoutersurfaceofthelmorfromtheinnersurface,and therecanbeeitherconstructiveordestructiveinterferencebetweenthetworeections.Wesee apatternthatvariesacrossthesurfacebecauseitsthicknessisn'tconstant.Weseerainbow colorsbecausetheconditionfordestructiveorconstructiveinterferencedependsonwavelength.

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Whitelightisamixtureofallthecolorsoftherainbow,andataparticularplaceonthesoap bubble,partofthatmixture,sayred,maybereectedstrongly,whileanotherpart,bluefor example,isalmostentirelytransmitted. Page86,self-checkE: Theperiodisthetimerequiredtotraveladistance2 L atspeed v T =2 L=v .Thefrequencyis f =1 =T = v= 2 L Page91,self-checkF: Thewavepatternwilllooklikethis: .Threequartersofa wavelengthtinthetube,sothewavelengthisthreetimesshorterthanthatofthelowestfrequencymode,inwhichonequarterofawavets.Sincethewavelengthissmallerbyafactor ofthree,thefrequencyisthreetimeshigher.Insteadof f o ,2 f o ,3 f o ,4 f o ::: ,thepatternofwave frequenciesofthisaircolumngoes f o ,3 f o ,5 f o ,7 f o ::: AnswerstoSelectedHomeworkProblems SolutionsforChapter4 Page93,problem3: Check:Theactuallengthofauteisabout66cmfromthetip ofthemouthpiecetotheendofthebell. 99

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Index absorptionofwaves,77 amplitude dened,16 peak-to-peak,16 relatedtoenergy,27 comet,13 damping dened,28 decibelscale,28 Dopplereect,63 drivingforce,31 eardrum,31 Einstein,Albert,14 energy relatedtoamplitude,27 exponentialdecay dened,29 Fourier'stheorem,87 frequency dened,15 fundamental,88 Galileo,19 Halley'sComet,13 harmonics,88 Hooke'slaw,17 interferenceeects,83 light,57 motion periodic,15 overtones,88 period dened,15 pitch,13 principleofsuperposition,49 pulse dened,49 qualityfactor dened,29 reectionofwaves,74 resonance dened,33 simpleharmonicmotion dened,18 periodof,18 sound,57 speedof,52 standingwave,88 steady-statebehavior,31 swing,30 timbre,88 tuningfork,17 work donebyavaryingforce,14,17,19

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Index 101

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102 Index

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Index 103

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104 Index

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Index 105

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UsefulData MetricPrexes M-mega-10 6 k-kilo-10 3 m-milli-10 )]TJ/F19 5.9776 Tf 5.756 0 Td [(3 -Greekmumicro-10 )]TJ/F19 5.9776 Tf 5.756 0 Td [(6 n-nano-10 )]TJ/F19 5.9776 Tf 5.756 0 Td [(9 p-pico-10 )]TJ/F19 5.9776 Tf 5.756 0 Td [(12 f-femto-10 )]TJ/F19 5.9776 Tf 5.756 0 Td [(15 Centi-,10 )]TJ/F19 5.9776 Tf 5.756 0 Td [(2 ,isusedonlyinthecentimeter. TheGreekAlphabet Aalpha Nnu Bbeta xi )-1633(gamma oOomicron delta pi Eepsilon Prho Zzeta sigma Heta Ttau theta Yupsilon Iiota phi Kkappa Xchi lambda psi Mmu omega SpeedsofLightand Sound speedoflight c =3.00 10 8 m/s speedofsound c =340m/s SubatomicParticles particlemasskgradiusfm electron9.109 10 )]TJ/F19 5.9776 Tf 5.756 0 Td [(31 0.01 proton1.673 10 )]TJ/F19 5.9776 Tf 5.756 0 Td [(27 1.1 neutron1.675 10 )]TJ/F19 5.9776 Tf 5.756 0 Td [(27 1.1 Theradiiofprotonsandneutronscanonlybegivenapproximately,sincetheyhavefuzzysurfaces.Forcomparison,a typicalatomisaboutamillionfminradius. NotationandUnits quantityunitsymbol distancemeter,m x x timesecond,s t t masskilogram,kg m densitykg = m 3 velocitym/s v accelerationm = s 2 a gravitationaleldJ = kg morm = s 2 g forcenewton,1N=1kg m = s 2 F pressure1Pa=1N = m 2 P energyjoule,J E powerwatt,1W=1J/s P amplitudevaries A periods T frequencyHz f wavelengthm qualityfactorunitless Q FWHMHzFWHM Conversions Nonmetricunitsintermsofmetricones: 1inch=25.4mmbydenition 1pound-force=4.5newtonsofforce kg g =2.2pounds-force 1scienticcalorie=4.18J 1kcal=4.18 10 3 J 1gallon=3.78 10 3 cm 3 1horsepower=746W Whenspeakingoffoodenergy,thewordCalorie"isused tomean1kcal,i.e.,1000calories.Inwriting,thecapitalC maybeusedtoindicate1Calorie=1000calories. RelationshipsamongU.S.units: 1footft=12inches 1yardyd=3feet 1milemi=5280feet Earth,Moon,andSun bodymasskgradiuskmradiusoforbitkm earth5.97 10 24 6.4 10 3 1.49 10 8 moon7.35 10 22 1.7 10 3 3.84 10 5 sun1.99 10 30 7.0 10 5 | 106 Index

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Index 107