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November 1, 2015 CH 16 Waves‐I
I.
TypesofWaves
A. Mechanicalwaves.Thesewaveshavetwocentralfeatures:They
aregovernedbyNewton’slaws,andtheycanexistonlywithina
materialmedium,suchaswater,air,androck.Commonexamples
includewaterwaves,soundwaves,andseismicwaves.
B. Electromagneticwaves.Thesewavesrequirenomaterialmedium
toexist.Allelectromagneticwavestravelthroughavacuumatthe
sameexactspeedc=299,792,458m/s.Commonexamplesinclude
visibleandultravioletlight,radioandtelevisionwaves,microwaves,x
rays,andradar(WewillcovertheseinPhysicsIIinmoredetail.)
C. Matterwaves.Thesewavesareassociatedwithelectrons,protons,
andotherfundamentalparticles,andevenatomsandmolecules.
Thesewavesarealsocalledmatterwaves(studyinQuantum
Mechanics.)
II.
TransverseandLongitudinalWaves
A. Inatransversewave,thedisplacementofeverysuchoscillating
elementalongthewaveisperpendiculartothedirectionoftravelof
thewave,asindicatedinFig.below.
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November 1, 2015 B. Inalongitudinalwavethemotionoftheoscillatingparticlesis
paralleltothedirectionofthewave’stravel,asshowninFig.below.
III.
Wavevariables
A. TransverseWaveEquation
1.
Theamplitudeymofawaveisthemagnitudeofthemaximum
displacementoftheelementsfromtheirequilibriumpositionsasthewave
passesthroughthem.
2.
Thephaseofthewaveistheargument(kx–wt)ofthesinefunction;as
thewavesweepsthroughastringelementataparticularpositionx,thephase
changeslinearlywithtimet.
3.
Thewavelengthofawaveisthedistanceparalleltothedirectionof
thewave’stravelbetweenrepetitionsoftheshapeofthewave(orwave
shape).Itisrelatedtotheangularwavenumber,k,by
4.
TheperiodofoscillationTofawaveisthetimeforanelementtomove
throughonefulloscillation.Itisrelatedtotheangularfrequency,w,by
5.
Thefrequencyfofawaveisdefinedas1/Tandisrelatedtothe
angularfrequencywby
6.
Aphaseconstantfinthewavefunction:y=Ymsin(kx–t+f).The
valueoffcanbechosensothatthefunctiongivessomeotherdisplacement
andslopeatx=0whent=0.
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November 1, 2015 IV.
TheSpeedofaTravelingWave
A. AsthewaveinFig.16‐7abovemoves,eachpointofthemoving
waveform,suchaspointAmarkedonapeak,retainsitsdisplacement
y.(Pointsonthestringdonotretaintheirdisplacement,butpointson
thewaveformdo.)IfpointAretainsitsdisplacementasitmoves,the
phasegivingitthatdisplacementmustremainaconstant:
1.
2.
Takingthederivativeweget
3.
Thus
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November 1, 2015 B. Sampleproblem:TransverseWave
1.
Awavetravelingalongastringisdescribedby
y(x,t) = 0.00327 sin (72.1x – 2.72t) inwhichthenumericalconstantsareinSIunits(0.00327m,72.1rad/m,and
2.72rad/s).
a)
Whatistheamplitudeofthiswave?
(1)
b)
ym=
Whatisthewavelength,period,andfrequencyofthiswave?
c)
Whatisthevelocityofthiswave?
d)
Whatisthedisplacementofthestringatx=22.5cm,andt=18.9s?
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November 1, 2015 V.
WaveSpeedonaStretchedString
A. Thespeedofawavealongastretchedidealstringdependsonlyon
thetensionandlineardensityofthestringandnotonthefrequency
ofthewave.
1.
Asmallstringelementoflengthlwithinthepulseisanarcofacircle
ofradiusRandsubtendinganangle2atthecenterofthatcircle.Aforcewith
amagnitudeequaltothetensioninthestring,,pullstangentiallyonthis
elementateachend.Thehorizontalcomponentsoftheseforcescancel,but
theverticalcomponentsaddtoformaradialrestoringforce.Forsmallangles,
2.
Ifmisthelinearmassdensityofthestring,andmthemassofthe
smallelement,
3.
Theelementhasacceleration:
4.
Therefore,
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November 1, 2015 VI.
EnergyandPowerofaWaveTravelingalongaString
A. Transversespeed
B. Kineticenergy
1.
2.
Thus
3.
Finally
C. Theaveragepower,whichistheaveragerateatwhichenergyof
bothkinds(kineticenergyandelasticpotentialenergy)istransmitted
bythewave,is:
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November 1, 2015 D. Example,TransverseWave:
1.
Astringalongwhichwavescantravelis2.70mlongandhasamassof
260g.Thetensioninthestringis36.0N.Whatmustbethefrequencyofthe
travelingwavesofamplitude7.70mmfortheaveragepowertobe85.0W?
a)
Solution:
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November 1, 2015 VII.
TheWaveEquation
A. Derivation
1.
Letusstartwithadrawing:
2.
Ifdisplacementinthey‐directionisnotabsurdlyhigh,thentensionis
equalonbothsidesofthestringsegment.Thus,
3.
Then,thenetforceinthey‐directionisFy=
4.
ApplyingNSL:
5.
Deltamass:
6.
Usingtheslopeofthestringsegment:
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November 1, 2015 7.
Puttingittogether:
8.
Solutionofthisdifferentialequationtakesform:
9.
UnitsoftheConstant?
B. Using__________________wefinallygetthewaveequation:Thegeneral
differentialequationthatgovernsthetravelofwavesofalltypes
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November 1, 2015 VIII. TheSuperpositionofWaves
A. Thedisplacementofthestringwhenwavesoverlapisthenthe
algebraicsum
1.
Overlappingwavesalgebraicallyaddtoproducearesultantwave(or
netwave).
2.
Overlappingwavesdonotinanywayalterthetravelofeachother.
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November 1, 2015 IX.
InterferenceofWaves
A. Iftwosinusoidalwavesofthesameamplitudeandwavelength
travelinthesamedirectionalongastretchedstring,theyinterfereto
producearesultantsinusoidalwavetravelinginthatdirection.
1.
Letusstartwithtwowaves:
2.
Theiralgebraicsum:
3.
UsingAppendixe,thesumofthesinesoftwoangles:
4.
Thustheirdisplacementis:
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November 1, 2015 5.
Graphicalrepresentation
6.
Table:
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November 1, 2015 B. Sampleproblem:
1.
Twoidenticaltravelingwaves,movinginthesamedirection,areoutof
phasebyπ/2rad.Whatistheamplitudeoftheresultantwaveintermsofthe
commonamplitudeymofthetwocombiningwaves?
a)
Solution:
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November 1, 2015 X.
StandingWaves
A. Diagram
B. Iftwosinusoidalwavesofthesameamplitudeandwavelength
travelinoppositedirectionsalongastretchedstring,their
interferencewitheachotherproducesastandingwave.
1.
Letusstartwithourtwowaves
2.
Theiralgebraicsum:
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November 1, 2015 3.
UsingAppendixe,thesumofthesinesoftwoangles:
4.
Thustheirdisplacementis:
5.
Inthestandingwaveequation,theamplitudeiszeroforvaluesofkx
thatgive
a)
Thosevaluesare
b)
Since
, weget
, for n =0,1,2, . . . (nodes),
asthepositionsofzeroamplitudeorthenodes.
Theadjacentnodesarethereforeseparatedby,
halfawavelength.
6.
Theamplitudeofthestandingwavehasamaximumvalueof
, whichoccursforvaluesofkxthatgive
a)
.
Thosevaluesare
.
b)
Thatis,
,
asthepositionsofmaximumamplitudeortheantinodes.The
antinodesareseparatedby
betweenpairsofnodes.
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November 1, 2015 XI.
StandingWaves,ReflectionsataBoundary
A. Diagram
B. Rememberinordertotrulysolveadifferentialequationyouneed
toapplyyourboundaryconditions.
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November 1, 2015 XII.
StandingWavesandResonance
A. Forcertainfrequencies,theinterferenceproducesastanding
wavepattern(oroscillationmode)withnodesandlargeantinodes
likethoseinFig.16‐19.
1.
Fig.16‐19Stroboscopicphotographsreveal(imperfect)standingwave
patternsonastringbeingmadetooscillatebyanoscillatorattheleftend.The
patternsoccuratcertainfrequenciesofoscillation.(Richard
Megna/FundamentalPhotographs)
B. Suchastandingwaveissaidtobeproducedatresonance,andthe
stringissaidtoresonateatthesecertainfrequencies,calledresonant
frequencies.
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November 1, 2015 C. Thefrequenciesassociatedwiththesemodesareoftenlabeledf1,
f2,f3,andsoon.Thecollectionofallpossibleoscillationmodesis
calledtheharmonicseries,andniscalledtheharmonicnumberofthe
nthharmonic.
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November 1, 2015 D. Sampleproblems:
1.
Astringstretchedbetweentwoclampsismadetooscillateinstanding
wavepatterns.Whatisthewavelengthforeachofthestandingpatterns
shownbelowifL=100cm?Whatistheharmonicineachcase?
λ=________m,_____harmonic
λ=________m,_____harmonic
2.
StringsAandBhaveidenticallengthsandlineardensities,butstringB
isundergreatertensionthanstringA.Figurebelowshowsfoursituations,(a)
through(d),inwhichstandingwavepatternsexistonthetwostrings.Inwhich
situationsistherethepossibilitythatstringsAandBareoscillatingatthe
sameresonantfrequency?
a)
Answer:
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November 1, 2015 3.
Anylonguitarstringhasalineardensityof7.20g/mandisundera
tensionof150N.ThefixedsupportsaredistanceD=90.0cmapart.Thestring
isoscillatinginthestandingwavepatternshowninFig.below.Calculatethe(a)
speed,(b)wavelength,and(c)frequencyofthetravelingwaveswhose
superpositiongivesthisstandingwave.
a)
Solution
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