[SHIVOK SP211] November 1, 2015 

 

 Page1

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.

 

 

 [SHIVOK SP211] November 1, 2015 

 

 Page2

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.

 

 [SHIVOK SP211] November 1, 2015 

 

 Page3

IV. TheSpeedofaTravelingWave 

 

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

1.  

2. Takingthederivativeweget

 

 

 

 

3. Thus

 

 

 

 

 [SHIVOK SP211] November 1, 2015 

 

 Page4

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? 

 

 

 [SHIVOK SP211] November 1, 2015 

 

 Page5

V. WaveSpeedonaStretchedString

A. Thespeedofawavealongastretchedidealstringdependsonlyonthetensionandlineardensityofthestringandnotonthefrequencyofthewave.

 

 

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

 

 

2. Ifmisthelinearmassdensityofthestring,andmthemassofthesmallelement,

 

 

3. Theelementhasacceleration: 

 

4. Therefore, 

 [SHIVOK SP211] November 1, 2015 

 

 Page6

VI. EnergyandPowerofaWaveTravelingalongaString

 

A. Transversespeed 

 

B. Kineticenergy

1.  

2. Thus

 

3. Finally

 

 

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

 

 [SHIVOK SP211] November 1, 2015 

 

 Page7

D. Example,TransverseWave:

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

a) Solution: 

 

 

 

 

. 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 [SHIVOK SP211] November 1, 2015 

 

 Page8

VII. TheWaveEquation

A. Derivation

1. Letusstartwithadrawing: 

 

 

 

   

 

 

2. Ifdisplacementinthey‐directionisnotabsurdlyhigh,thentensionisequalonbothsidesofthestringsegment.Thus,

 

 

3. Then,thenetforceinthey‐directionisFy= 

 

4. ApplyingNSL: 

 

5. Deltamass: 

 

6. Usingtheslopeofthestringsegment: 

 

 

 

 [SHIVOK SP211] November 1, 2015 

 

 Page9

7. Puttingittogether: 

 

 

8. Solutionofthisdifferentialequationtakesform: 

 

 

9. UnitsoftheConstant? 

 

 

B. Using__________________wefinallygetthewaveequation:Thegeneraldifferentialequationthatgovernsthetravelofwavesofalltypes

 

 

 

 

 

 

 

 

 

 

 

 

 [SHIVOK SP211] November 1, 2015 

 

  Page10

VIII. TheSuperpositionofWaves 

A. Thedisplacementofthestringwhenwavesoverlapisthenthealgebraicsum

 

1. Overlappingwavesalgebraicallyaddtoproducearesultantwave(ornetwave).

2. Overlappingwavesdonotinanywayalterthetravelofeachother. 

 

 

 

 

 [SHIVOK SP211] November 1, 2015 

 

  Page11

IX. InterferenceofWaves 

A. Iftwosinusoidalwavesofthesameamplitudeandwavelengthtravelinthesamedirectionalongastretchedstring,theyinterferetoproducearesultantsinusoidalwavetravelinginthatdirection.

 

1. Letusstartwithtwowaves: 

 

 

 

2. Theiralgebraicsum:

 

 

 

3. UsingAppendixe,thesumofthesinesoftwoangles:

4. Thustheirdisplacementis: 

 

 

 

 

 

 [SHIVOK SP211] November 1, 2015 

 

  Page12

5. Graphicalrepresentation

 

 

6. Table:

 

 

 

 

 [SHIVOK SP211] November 1, 2015 

 

  Page13

B. Sampleproblem:

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

 

a) Solution: 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 [SHIVOK SP211] November 1, 2015 

 

  Page14

X. StandingWaves

A. Diagram

 

B. Iftwosinusoidalwavesofthesameamplitudeandwavelengthtravelinoppositedirectionsalongastretchedstring,theirinterferencewitheachotherproducesastandingwave.

1. Letusstartwithourtwowaves

 

 

2. Theiralgebraicsum: 

 

 

 [SHIVOK SP211] November 1, 2015 

 

  Page15

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.

 [SHIVOK SP211] November 1, 2015 

 

  Page16

XI. StandingWaves,ReflectionsataBoundary

A. Diagram

 

 

 

B. Rememberinordertotrulysolveadifferentialequationyouneedtoapplyyourboundaryconditions.

 

 [SHIVOK SP211] November 1, 2015 

 

  Page17

XII. StandingWavesandResonance

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

 

 

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

B. Suchastandingwaveissaidtobeproducedatresonance,andthestringissaidtoresonateatthesecertainfrequencies,calledresonantfrequencies.

 

 [SHIVOK SP211] November 1, 2015 

 

  Page18

C. Thefrequenciesassociatedwiththesemodesareoftenlabeledf1,

f2,f3,andsoon.Thecollectionofallpossibleoscillationmodesis

calledtheharmonicseries,andniscalledtheharmonicnumberofthenthharmonic.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 [SHIVOK SP211] November 1, 2015 

 

  Page19

D. Sampleproblems:

1. Astringstretchedbetweentwoclampsismadetooscillateinstandingwavepatterns.WhatisthewavelengthforeachofthestandingpatternsshownbelowifL=100cm?Whatistheharmonicineachcase?

  

     

λ=________m,_____harmonic λ=________m,_____harmonic

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

 

 

  

 

a) Answer: 

 

 

 

 [SHIVOK SP211] November 1, 2015 

 

  Page20

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

 

 

  

 

a) Solution