CH 16 Structured Notes with blanks frequencies associated with these modes are often labeled f 1, f 2, f 3, and so on. The collection of all possible oscillation modes is called the

[SHIVOK SP211] November 1, 2015

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CH 16

Waves‐I

I. TypesofWaves

A. Mechanicalwaves.Thesewaveshavetwocentralfeatures:TheyaregovernedbyNewton’slaws,andtheycanexistonlywithinamaterialmedium,suchaswater,air,androck.Commonexamplesincludewaterwaves,soundwaves,andseismicwaves.

B. Electromagneticwaves.Thesewavesrequirenomaterialmediumtoexist.Allelectromagneticwavestravelthroughavacuumatthesameexactspeedc=299,792,458m/s.Commonexamplesincludevisibleandultravioletlight,radioandtelevisionwaves,microwaves,xrays,andradar(WewillcovertheseinPhysicsIIinmoredetail.)

C. Matterwaves.Thesewavesareassociatedwithelectrons,protons,andotherfundamentalparticles,andevenatomsandmolecules.Thesewavesarealsocalledmatterwaves(studyinQuantumMechanics.)

II. TransverseandLongitudinalWaves

A. Inatransversewave,thedisplacementofeverysuchoscillatingelementalongthewaveisperpendiculartothedirectionoftravelofthewave,asindicatedinFig.below.


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B. Inalongitudinalwavethemotionoftheoscillatingparticlesisparalleltothedirectionofthewave’stravel,asshowninFig.below.

III. Wavevariables

A. TransverseWaveEquation

1. Theamplitudeymofawaveisthemagnitudeofthemaximumdisplacementoftheelementsfromtheirequilibriumpositionsasthewavepassesthroughthem.

2. Thephaseofthewaveistheargument(kx–wt)ofthesinefunction;asthewavesweepsthroughastringelementataparticularpositionx,thephasechangeslinearlywithtimet.

3. Thewavelengthofawaveisthedistanceparalleltothedirectionofthewave’stravelbetweenrepetitionsoftheshapeofthewave(orwaveshape).Itisrelatedtotheangularwavenumber,k,by

4. TheperiodofoscillationTofawaveisthetimeforanelementtomovethroughonefulloscillation.Itisrelatedtotheangularfrequency,w,by

5. Thefrequencyfofawaveisdefinedas1/Tandisrelatedtotheangularfrequencywby

6. Aphaseconstantfinthewavefunction:y=Ymsin(kx–t+f).Thevalueoffcanbechosensothatthefunctiongivessomeotherdisplacementandslopeatx=0whent=0.


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IV. TheSpeedofaTravelingWave

A. AsthewaveinFig.16‐7abovemoves,eachpointofthemovingwaveform,suchaspointAmarkedonapeak,retainsitsdisplacementy.(Pointsonthestringdonotretaintheirdisplacement,butpointsonthewaveformdo.)IfpointAretainsitsdisplacementasitmoves,thephasegivingitthatdisplacementmustremainaconstant:

1.

2. Takingthederivativeweget

3. Thus


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B. Sampleproblem:TransverseWave

1. Awavetravelingalongastringisdescribedbyy(x,t) = 0.00327 sin (72.1x – 2.72t)

inwhichthenumericalconstantsareinSIunits(0.00327m,72.1rad/m,and2.72rad/s).

a) Whatistheamplitudeofthiswave?

(1) ym=

b) Whatisthewavelength,period,andfrequencyofthiswave?

c) Whatisthevelocityofthiswave?

d) Whatisthedisplacementofthestringatx=22.5cm,andt=18.9s?


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V. WaveSpeedonaStretchedString

A. Thespeedofawavealongastretchedidealstringdependsonlyonthetensionandlineardensityofthestringandnotonthefrequencyofthewave.

1. AsmallstringelementoflengthlwithinthepulseisanarcofacircleofradiusRandsubtendinganangle2atthecenterofthatcircle.Aforcewithamagnitudeequaltothetensioninthestring,,pullstangentiallyonthiselementateachend.Thehorizontalcomponentsoftheseforcescancel,buttheverticalcomponentsaddtoformaradialrestoringforce.Forsmallangles,

2. Ifmisthelinearmassdensityofthestring,andmthemassofthesmallelement,

3. Theelementhasacceleration:

4. Therefore,


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VI. EnergyandPowerofaWaveTravelingalongaString

A. Transversespeed

B. Kineticenergy

1.

2. Thus

3. Finally

C. Theaveragepower,whichistheaveragerateatwhichenergyofbothkinds(kineticenergyandelasticpotentialenergy)istransmittedbythewave,is:


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D. Example,TransverseWave:

1. Astringalongwhichwavescantravelis2.70mlongandhasamassof260g.Thetensioninthestringis36.0N.Whatmustbethefrequencyofthetravelingwavesofamplitude7.70mmfortheaveragepowertobe85.0W?

a) Solution:

.


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VII. TheWaveEquation

A. Derivation

1. Letusstartwithadrawing:

2. Ifdisplacementinthey‐directionisnotabsurdlyhigh,thentensionisequalonbothsidesofthestringsegment.Thus,

3. Then,thenetforceinthey‐directionisFy=

4. ApplyingNSL:

5. Deltamass:

6. Usingtheslopeofthestringsegment:


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7. Puttingittogether:

8. Solutionofthisdifferentialequationtakesform:

9. UnitsoftheConstant?

B. Using__________________wefinallygetthewaveequation:Thegeneraldifferentialequationthatgovernsthetravelofwavesofalltypes


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VIII. TheSuperpositionofWaves

A. Thedisplacementofthestringwhenwavesoverlapisthenthealgebraicsum

1. Overlappingwavesalgebraicallyaddtoproducearesultantwave(ornetwave).

2. Overlappingwavesdonotinanywayalterthetravelofeachother.


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IX. InterferenceofWaves

A. Iftwosinusoidalwavesofthesameamplitudeandwavelengthtravelinthesamedirectionalongastretchedstring,theyinterferetoproducearesultantsinusoidalwavetravelinginthatdirection.

1. Letusstartwithtwowaves:

2. Theiralgebraicsum:

3. UsingAppendixe,thesumofthesinesoftwoangles:

4. Thustheirdisplacementis:


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5. Graphicalrepresentation

6. Table:


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B. Sampleproblem:

1. Twoidenticaltravelingwaves,movinginthesamedirection,areoutofphasebyπ/2rad.Whatistheamplitudeoftheresultantwaveintermsofthecommonamplitudeymofthetwocombiningwaves?

a) Solution:


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X. StandingWaves

A. Diagram

B. Iftwosinusoidalwavesofthesameamplitudeandwavelengthtravelinoppositedirectionsalongastretchedstring,theirinterferencewitheachotherproducesastandingwave.

1. Letusstartwithourtwowaves

2. Theiralgebraicsum:


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3. UsingAppendixe,thesumofthesinesoftwoangles:

4. Thustheirdisplacementis:

5. Inthestandingwaveequation,theamplitudeiszeroforvaluesofkxthatgive

a) Thosevaluesare

b) Since

, weget , for n =0,1,2, . . . (nodes), asthepositionsofzeroamplitudeorthenodes.

Theadjacentnodesarethereforeseparatedby, halfawavelength.

6. Theamplitudeofthestandingwavehasamaximumvalueof

, whichoccursforvaluesofkxthatgive . a) Thosevaluesare

. b) Thatis,

, asthepositionsofmaximumamplitudeortheantinodes.The

antinodesareseparatedby andarelocatedhalfwaybetweenpairsofnodes.


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XI. StandingWaves,ReflectionsataBoundary

A. Diagram

B. Rememberinordertotrulysolveadifferentialequationyouneedtoapplyyourboundaryconditions.


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XII. StandingWavesandResonance

A. Forcertainfrequencies,theinterferenceproducesastandingwavepattern(oroscillationmode)withnodesandlargeantinodeslikethoseinFig.16‐19.

1. Fig.16‐19Stroboscopicphotographsreveal(imperfect)standingwavepatternsonastringbeingmadetooscillatebyanoscillatorattheleftend.Thepatternsoccuratcertainfrequenciesofoscillation.(RichardMegna/FundamentalPhotographs)

B. Suchastandingwaveissaidtobeproducedatresonance,andthestringissaidtoresonateatthesecertainfrequencies,calledresonantfrequencies.


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C. Thefrequenciesassociatedwiththesemodesareoftenlabeledf1,

f2,f3,andsoon.Thecollectionofallpossibleoscillationmodesis

calledtheharmonicseries,andniscalledtheharmonicnumberofthenthharmonic.


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D. Sampleproblems:

1. Astringstretchedbetweentwoclampsismadetooscillateinstandingwavepatterns.WhatisthewavelengthforeachofthestandingpatternsshownbelowifL=100cm?Whatistheharmonicineachcase?

λ=________m,_____harmonic λ=________m,_____harmonic

2. StringsAandBhaveidenticallengthsandlineardensities,butstringBisundergreatertensionthanstringA.Figurebelowshowsfoursituations,(a)through(d),inwhichstandingwavepatternsexistonthetwostrings.InwhichsituationsistherethepossibilitythatstringsAandBareoscillatingatthesameresonantfrequency?

a) Answer:


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3. Anylonguitarstringhasalineardensityof7.20g/mandisunderatensionof150N.ThefixedsupportsaredistanceD=90.0cmapart.ThestringisoscillatinginthestandingwavepatternshowninFig.below.Calculatethe(a)speed,(b)wavelength,and(c)frequencyofthetravelingwaveswhosesuperpositiongivesthisstandingwave.

a) Solution

CH 16 Structured Notes with blanks frequencies associated with these modes are often labeled f 1, f 2, f 3, and so on. The collection of all possible oscillation modes is called the

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