《物理双语教学课件》Chapter 10 Waves 波动.docx
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《物理双语教学课件》Chapter 10 Waves 波动.docx
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《物理双语教学课件》Chapter10Waves波动
Chapter10Waves
10.1TypesofWaves
1.Mechanicalwaves:
Thesewavesaremostfamiliarbecauseweencounterthemalmostconstantly;commonexamplesincludewaterwaves,soundwaves,andseismicwaves.Allthesewaveshavecertaincentralfeatures:
theyaregovernedbyNewton’slaws,andtheycanexistonlywithinamaterialmedium,suchaswater,air,androck.
2.Electromagneticwaves:
Thesewavesarelessfamiliar,butyouusethemconstantly;commonexamplesincludevisibleandultravioletlight,radioandtelevisionwaves,microwaves,x-rays,andradarwaves.Thesewavesrequirenomaterialmediumtoexist.Lightwavesfromstars,forexample,travelthroughthevacuumofspacetoreachus.Allelectromagneticwavestravelthroughavacuumatthesamespeedc,givenbyc=299,792,458m/s.
3.Matterwaves:
Althoughthesewavesarecommonlyusedinmoderntechnology,theirtypeisprobablyveryunfamiliartoyou.Electrons,protons,andotherfundamentalparticles,andevenatomsandmolecules,travelaswaves.Becausewecommonlythinkofthesethingsasconstitutingmatter,thesewavesarecalledmatterwaves.
4.Muchofwhatwediscussinthischapterappliestowavesofallkinds.However,forspecificexamplesweshallrefertomechanicalwaves.
10.2TransverseandLongitudinalWaves
1.Transversewave
(1).Awavesentalongastretched,tautstringisthesimplestmechanicalwave.Ifyougiveoneendofastretchedstringasingleup-and-downjerk,awaveintheformofasinglepulsetravelsalongthestring,asinthefigure.Thispulseanditsmotioncanoccurbecausethestringisundertension.Whenyoupullyourendofthestringupward,itbeginstopullupwardontheadjacentsectionofthestringviatensionbetweenthetwosections.Astheadjacentsectionmovesupward,itbeginstopullthenextsectionupward,andsoon.Meanwhile,youhavepulleddownonyourendofthestring.So,aseachsectionmovesupwardinturn,itbeginstobepulledbackdownwardbyneighboringsectionsthatalreadyonthewaydown.Thenetresultisthatadistortioninthestring’sshape(thepulse)movesalongthestringatsomevelocityv.
(2).Ifyoumoveyourhandupanddownincontinuoussimpleharmonicmotion,acontinuouswavetravelsalongthestringatvelocityv.Becausethemotionofyourhandisasinusoidalfunctionoftime,thewavehasasinusoidalshapeatanygiveninstant,asinthefigure(b).Thatis,thewavehastheshapeofasinecurveoracosinecurve.
(3).Weconsiderhereonlyan“ideal”string,inwhichnofriction-likeforceswithincausethewavetodieoutasittravelsalongthestring.Inaddition,weassumethatthestringissolongthatweneednotconsiderawavereboundingfromthefarend.
(4).Onewaytostudythewavesofthefigureistomonitorthewave’sform(shapeofwave)asitmovestotheright.Alternatively,wecanmonitorthemotionofanelementofthestringastheelementoscillatesupanddownwhilethewavepassesthroughit.Wewouldfindthatthedisplacementofeverysuchoscillatingstringelementisperpendiculartothedirectionoftravelofthewave,asindicatedinthefigure.Thismotionissaidtobetransverse,andthewaveissaidtobeatransversewave.
2.Longitudinalwave:
(1).Therightfigureshowshowasoundwavecanbeproducedbyapistoninalong,airfilledpipe.Ifyousuddenlymovethepistonrightwardandthenleftward,youcansendapulseofsoundalongthepipe.Therightwardmotionofthepistonmovestheelementsofairnexttoitrightward,changingtheairpressurethere.Theincreasedairpressurethenpushesrightwardontheelementofairsomewhatfartheralongthepipe.Oncetheyhavemovedrightward,theelementsmovebackleftward.Thusthemotionoftheairandthechangeinairpressuretravelrightwardalongthepipeasapulse.
(2).Ifyoupushandpullonthepistoninsimpleharmonicmotion,asisbeingdoneinthefigure,asinusoidalwavetravelsalongthepipe.Becausethemotionoftheelementsofairisparalleltothedirectionofthewave’stravel.Themotionissaidtobelongitudinalwave.
3.Bothatransversewaveandalongitudinalwavearesaidtobetravelingwavesbecausethewavetravelsfromonepointtoanother,asfromoneendofthestringtotheotherendorfromoneendofthepipetotheotherend.Notethatitisthewavethatmovesbetweenthetwopointsandnotthematerial(stringorair)throughwhichthewavemoves.
10.3WavelengthandFrequency
1.Introduction
(1).Tocompletelydescribeawaveonastring,weneedafunctionthatgivestheshapeofthewave.Thismeansthatweneedarelationintheform
inwhichyisthetransversedisplacementofanystringelementasafunctionhofthetimetandthepositionxoftheelementalongthestring.Ingeneral,asinusoidalshapelikethewavecanbedescribedwithhbeingeitherasinefunctionoracosinefunction;bothgivethesamegeneralshapeforthewave.Inthischapterweusethesinefunction.
(2).Forasinusoidalwave,travelingtowardincreasingvaluesofx,thetransversedisplacementyofastringelementatpositionxattimeisgivenby
here
istheamplitudeofthewave;thesubscriptmstandsformaximum,becausetheamplitudeisthemagnitudeofthemaximumdisplacementofthestringelementineitherdirectionparalleltotheyaxis.Thequantitieskand
areconstantswhosemeaningsareabouttodiscuss.Thequantity
iscalledthephaseofthewave.
2.
WavelengthandangularWaveNumber
(1).Thefigureshowshowthetransversedisplacementyvarieswithpositionxataninstant,arbitrarilycalledt=0.Thatis,thefigureisa“snapshot”ofthewaveatthatinstant.Witht=0,thewaveequationbecomes
.TheFigure(a)isaplotofthisequation;itshowstheshapeoftheactualwaveattimet=0.
(2).Thewavelength
ofawaveisthedistancebetweenrepetitionsofthewaveshape.Atypicalwavelengthismarkedinfigure(a).Bydefinition,thedisplacementyisthesameatbothendsofthiswavelength,thatis,at
and
.Thus
.
(3).Asinefunctionbeginstorepeatitselfwhenitsangleisincreasedby
rad;sowehave
.Wecallktheangularwavenumberofthewave;itsSIunitistheradianpermeter.
3.Period,angularfrequency,andfrequency:
(1).Thefigure(b)showshowthedisplacementyvarieswithtimetatafixedposition,takentobex=0.Ifyouweretomonitorthestring,youwouldseethatthesingleelementofthestringatthatpositionmovesupanddowninsimpleharmonicmotionwithx=0:
.Thefigure(b)isaplotofthisequation;itdoesnotshowtheshapeofthewave.
(2).WedefinetheperiodofoscillationsTofawavetobethetimeintervalbetweenrepetitionsofthemotionofanoscillatingstringelement.Atypicalperiodismarkedinthefigure(b).Wehave
.
(3).Thiscanbetrueonlyif
.Wecall
theangularfrequencyofthewave;itsSIunitistheradianpersecond.
(4).Thefrequencyfofthewaveisdefinedas1/Tandisrelatedtotheangularfrequencyby
.Thisfrequencyfisanumberofoscillationsperunittime-madebyastringelementasthewavemovesthroughit,andfisusuallymeasuredinhertzoritsmultiples.
10.4TheSpeedofaTravelingwave
1.Thefigureshowstwosnapshotsofthewavetakenasmalltimeinterval
apart.Thewaveistravelinginthedirectionofincreasingx,theentirewavepatternmovingadistance
inthatdirectionduringtheinterval
.Theratio
(or,inthedifferentiallimit,dx/dt)isthewavespeedv.Howcanwefinditsvalue?
2.Asthewavemoves,eachpointofthemovingwaveformretainsitsdisplacementy.Foreachsuchpoint,theargumentofthesinefunctionmustbeaconstant:
.
3.Tofindwavespeedv,wetakethederivativeoftheequation,get
.Theequationtellsusthatthewavespeedisonewavelengthperperiod.
4.Thewaveequation
describesawavemovinginthedirectionofincreasingx.
(1).Wecanfindtheequationofawavetravelingintheoppositedirectionbyreplacingtwith–t.
(2).Thiscorrespondstothecondition
.(3).Thusawavetravelingtowarddecreasingxisdescribedbytheequation
.(4).Itsvelocityis
.
5.Considernowawaveofgeneralizedshape,givenby
wherehrepresentsanyfunction,thesinefunctionbeingonepossibility.Ouranalysisaboveshowsthatallwavesinwhichthevariablesxandtenterinthecombination
aretravelingwaves.Furthermore,alltravelingwavesmustbetheformabove.Thus
representsapossibletravelingwave.Thefunction
ontheotherhand,doesnotrepresentatravelingwave.
6.WaveSpeedonaStretchedString
(1).Thespeedofawaveisrelatedtothewave’swavelengthandfrequency,butitissetbythemedium.Ifawaveisthroughamediumsuchaswater,air,steel,orastretchedstring,itmustcausetheparticlesofthatmediumtooscillateasitpasses.Forthathappen,themediummustpossessbothinertiaandelasticity.Thesetwopropertiesdeterminehowfastthewavecantravelinthemedium.Andconversely,itshouldbepossibletocalculatethespeedofthewavethroughthemediumintermsoftheseproperties.
(2).WecanderivethespeedfromNewton’ssecondlawas
where
isthelineardensityofthestring,and
thetensioninthestring.
(3).Theequationtellsusthatthespeedofawavealongastretchedidealstringdependsonlyonthecharacteristicsofthestringandnotonthefrequencyofthewave.
10.5EnergyandPowerofaTravelingStringWave
Whenwesetupawaveonastretchedstring,weprovideenergyforthemotionofthestring.Asthewavemovesawayfromus,ittransportsthatenergyasbothkineticenergyandelasticpotentialenergy.Letusconsidereachformin
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