v

v v

v

2

2

2

2

2

2

2

1

1

1

1

1

1

1

Superposition of Waves

Whathappenswhentwowavestouch.

Principle of Superposition

Whentwomarblescollide,theybounceback.That'showmaterialobjectsbehave.Twodifferentobjectscan'toccupythesamespaceatthesametime.

Waves,ontheotherhand,canoccupythesamespace.Thesetwowavepulsesaremovingtowardseachother,thenstarttooverlap,thencontinueon.

Principle of Superposition

Thisisoneofthemajorconceptsofphysics-superposition.

Theconceptofsuperpositionisaveryfundamentalpartofphysics.Itistheunderlyingcauseformanyphenomenainoptics,acoustics,quantummechanics,andothersubfields.Superpositionalsoplaysalargeroleinengineeringfields,especiallyelectrical.Ifyouwanttosendseveralchunksofinformationdownthesameelectricalwire,you'llrelyonaspectsofsuperposition.We'lllookathowitoccursinsimplesystemslikeawaveonastringaswellasmorecomplicatedsituationslike2dimensionalacousticinterference.Later,you'llusethesameconceptsandtoolstotreatmorecomplicatedsituations.

Whateverthetwowavesare,allweneedtodoisaddthemupinordertofindthenewwave.

Thisanimationshowstwodifferentwaveapproaching,interacting,thenreceding.

(x, t) = (x, t) + (x, t)y′ y1 y2

PHY 208 - superposition

updated on 2018-02-01 J. Hedberg | © 2018 Page 1

0 1 2 3 4 5 6 7 8 9 10

2 3 4 5 6 7 8

1 m/s 1 m/s

2 3 4 5 6 7 82 3 4 5 6 7 8

2 3 4 5 6 7 8

A B

DC

Thesetwowaves(theredandtheblue)areaddedtogethertogetthepurple

Redcurve:wave1:

Bluecurve:

Quick Question 1

Thesetwowavesareapproachingeachotheratt=0.Whatwillthesumlooklikeatt=2s?

Math of Superposition

Let'saddtwowavestravelinginthesamedirectiononthesamestring.( , ,and arethesame)

wave1:

wave2:

Since,

withatrigidentity(below)

Thisisaveryusefultrigidentity:

Theonlydifferencebetweenthesetwowavesisthephasefactorthatappearsinthesecondone.Thisjustindicatesthatthewavesmighthavedifferentamplitudesatt=0.

The sum of two waves

k ω A

(x, t) = A sin(kx − ωt)y1

(x, t) = A sin(kx − ωt+ ϕ)y2

(x, t)y′ ==

(x, t) + (x, t)y1 y2A sin(kx − ωt) + A sin(kx − ωt+ ϕ)

(1)(2)

(x, t) = [2Acos ] sin(kx − ωt+ )y′ϕ

2ϕ

2

sinα + sinβ = 2 sin (α + β) cos (α − β)12

12

ϕ

(x, t) = A sin(kx − ωt)y1

(x, t) = A sin(kx − ωt+ ϕ)2

PHY 208 - superposition

updated on 2018-02-01 J. Hedberg | © 2018 Page 2

wave. wave2:

PurpleCurve:

Summary of interference types

Interferenceisusedtodescribethisphenomenon.

PhasedifferenceinAmplitude InterferenceType

Degrees Radians wavelength0º 0 0 FullyConstructive120º .33 Intermediate

180º .5 0 Fullydestructive240º .67 Intermediate

360º 1 FullyConstructive

(Applicableforwaveswiththesameamplitudeandwavelengthtravelinginthesamedirection.)

Waves traveling in opposite directions.

Herearetwowaveswithequalamplitudesandfrequenciestravelinginoppositedirectionsonastring.(Thebluewaveisthesumofthetworedwaves)

Asyouwatchtheanimation,keepaneyeonthedarkblueline.Thisisthesumofthetworedlines.You'llnotethatitdoesn'tappeartomovinginthexdirection,onlytheydirection.Therearealsosomepointsthatnevermoveintheydirection(i.e.havenodisplacment,

(x, t) = A sin(kx − ωt+ ϕ)y2

(x, t) = [2Acos ] sin(kx − ωt+ )y′ϕ

2ϕ

2

2A2π3 A

π4π3 A

2π 2A

PHY 208 - superposition

updated on 2018-02-01 J. Hedberg | © 2018 Page 3

ever),andotherpointsthatoscillatebetween+Aand-A(maximumdisplacement).

Math of Superposition for opposite direction

Let'saddtwowavestravelinginoppositedirectiononthesamestring.( , ,and arethesame)

wave1:

wave2:

Since,

withatrigindentity(below)

Thisisnotofthestandardtravelingwaveformat!Indeed,thisequationdescribesastandingwave.

Standing waves

Fortravelingwaves,theamplitudeofdisplacementofeachelementwasthesame.Theywouldallgetdisplacedtoamaximum .Inastandingwave,theamplitudeswillbepositiondependent.

The argumentofthesinefunctionleadstothisphenomenon.

Nodes

Since ,wecandeterminewhereexactlytheamplitudeswillbezero.

Whenever , ,we'llobtainazeroforthedisplacement.

Rearranging:

.

Anti-Nodes

Likewise,when =1,theropewillundergoamaximumdisplacement.

k ω A

(x, t) = A sin(kx − ωt)y1

(x, t) = A sin(kx + ωt)y2

(x, t)y′ ==

(x, t) + (x, t)y1 y2A sin(kx − ωt) + A sin(kx + ωt)

(3)(4)

(x, t) = [2A sinkx] cos(ωt)y′

sinα + sinβ = 2 sin (α + β) cos (α − β)12

12

A

(x, t) = [2A sin ] cos(ωt)y′ (kx) positiondependent

kx

sin(nπ) = 0

kx = nπ n = 0,1,2,3...

x = nλ

2

sin(kx)

, , , . . .

PHY 208 - superposition

updated on 2018-02-01 J. Hedberg | © 2018 Page 4

Ifweoscillateafixedstringinsuchawaythatthereisanodeateachendpoint,wehaveeffectivelysetupastandingwave.Thiswillhappenwhenthefrequencyofoscillationisinresonancewiththestringcharacteristics.

Thus,onlywavelengthswhich'fit'inthestringwillcreateresonantoscillations.

Thesearecalled'harmonicmodes'(shownare1,2,and3).

L

Nodes and Antinodes

Hereisourstandingwave.

Onenodeandoneantinodearepointedout.(Althoughtherearemanymore)

Inabstractmath-land,allwehavetodotocreateastandingwaveisadduptwowavefunctions.Inthephysicaluniversehowever,wehavetodoalittlebitmore.Tounderstandhowastandingwaveiscreatedinaphysicalsystem,we'llneedtoseewhathappenswhenwavesbounce.

Boundary (hard)

Ifwesendawavepulsedownastring,wherethestringisfixedatthefarend,weseethatthewaveformflips.

Anotherwayofphrasingitistosay,thewaveformisinverteduponreflection,orundergoesaphaseflipof180°

Boundary (soft)

Ifwesendawavepulsedownastring,wherethestringislooseatthefarend,weseethatthewaveformdoesnotflip.

Creation of a standing wave

Standing waves and resonance

Theimagehereshowsthefirst3harmonicmodesofthestring,modes1,2and3.Thefirstmodeisoftencalledthefundamental,orlowestharmonic.

Resonance frequencies

Whatdeterminestheresonancefrequencies?

kx =

=

, , , . . .π

23π

25π

2

(n+ )π for n = 0,1,2...12

(5)

(6)

PHY 208 - superposition

updated on 2018-02-01 J. Hedberg | © 2018 Page 5

Rememberthat .Now,thevelocityofthewavewavegivenbythephysicalcharacteristicsofthestring:thelinearmassdensity, ,andthetension, .

Thereforewecanwrite:

Quick Question 2

Whenawireundertensionoscillatesinitsthirdharmonicmode,howmanywavelengthsareobserved?

1. 1/32. 2/33. 1/24. 3/25. 2

Quick Question 3

Whichofthesecouldbethefrequencyofastandingwavewithawavespeedof12m/sasitoscillatesona4.0-mstringfixedatbothends?(itmightnotbethelowestharmonic)

A. 2.5HzB. 5.0HzC. 10HzD. 15HzE. 20Hz

Example Problem: Piano String

ThelowestnoteonmostpianosisA0.Ithasafrequencyof27.5Hz.Thevibratingsectionofthiswireonagrandpianois1.9meterslong.[1meterofpianowirehasamassof200g]Whatisthetensioninthestring?

Interference

Consideraspeakerplayingaprettysinusoidalwavewithawavelength .

Then,anotherspeakerplayingtheexactsamepitchisplacedinfrontofSpeaker1,sothatthetwospeakersareexactlyonewavelengthapart.

Thetwosignals,orwaves,willconstructivelyinterferecreatingasignaloflargeramplitude.

v = λf

μ τ

f = = n = nv

λ

v

2L

τ

μ

−−√ 12L

Example Problem #1:

λ

PHY 208 - superposition

updated on 2018-02-01 J. Hedberg | © 2018 Page 6

spkr 1 spkr 2

sum of 1+ 2

Now,let’simaginethesecondspeakerwasproducingawaveexactlyoppositetothespeaker1wave.

Thesumofthesetwowaveswillnowbeequaltozero,sincetheyarealwayscreatingdestructiveinterference.

Thesetwowavesarecalled“outofphase”.

spkr 1 spkr 2

sum of 1+ 2

Math of 1-D audio interference

IfwanttoknowwhatthesoundislikeatpointA,we’llneedtoknowhowfaritisfrombothsources:

spkr 1 spkr 2

A

If isequaltowavelengthtimesawholenumber,thentheamplitudeoftheoscillationsisincreased:

If isequaltowavelengthtimeshalfanintegernumber,thentheamplitudeoftheoscillationsisdecreased:

Two speakers

Δd

Δd = nλ

Δd

Δd = nλ

2

PHY 208 - superposition

updated on 2018-02-01 J. Hedberg | © 2018 Page 7

Math of 1d interference

Thephasedifferencebetweentwowaveswilldeterminewhethertheinterferenceisconstructiveordestructive.Forconstructive:

andfordestructive:

Theequationsoftwotravelingwaves(travelinginthesamedirectionwiththesamefrequencyandwavelength)aregivenby:

and

Forthesewaves, meanstheintitialphaseconstantofwave1.

Thus,thephases,ortheargumentsofthesinetermsaregivenby:

and

Ifwesubtractthesetwophases,thatisfind (akathephasedifference):

Andso,forthecaseoftwosoundsourcesinonedimension,ifwewanttofigureoutwhenconstructiveinterferencewilloccur,basedoneithertheinitialphaseconstantsofthetwowaves,ortheseparationinspace:

Orfordestructiveinterference:

1d-interference

Hereisthephasedifferenceequation:

Wecanseethattherearetwocontributions:

1. Thepath-lengthdifference, ,inproportiontothewavelength.

Δϕ = 0, 2π, 4π…

Δϕ = π, 3π, 5π…

= A sin(k − ωt+ )y1 x1 ϕ10

= A sin(k − ωt+ )y2 x2 ϕ20

ϕ10

= k − ωt+ϕ1 x1 ϕ10

= k − ωt+ϕ2 x2 ϕ20

Δϕ

Δϕ = − = k( − ) + ( − ) = ( − ) + Δϕ2 ϕ1 x2 x1 ϕ20 ϕ102π

λx2 x1 ϕ0

Δϕ = ( − ) + Δϕ = 0, 2π, 4π…2π

λx2 x1

Δϕ = ( − ) + Δϕ = π, 3π, 5π…2π

λx2 x1

Δϕ = ( − ) + Δ2π

λx2 x1 ϕ0

−x2 x1

PHY 208 - superposition

updated on 2018-02-01 J. Hedberg | © 2018 Page 8

Consider2points.Weneedtofigureouthowfartheyarefromeachsoundsource.

point 1

point 2

crest

trough

L1L2

L2L1

2. Theinherentphasedifferencebetweenthetwooscillators.

Ifwehavetwoidenticalsourcesthatareinphase(i.e. ),thenonlythepathlengthdifferencewilldetermineifthewavesconstructivelyordestructivelyinterfere.

where isanintegerwillcreatemaximumconstructiveinterference.Thismakessense.Ifthewavesareseparatedbyawholenumberofwavelengths,thenit'sessentiallythesameasiftheyarenotseparatedatall.

Sounds waves in 2-dimensions (or 3)

Hereisasinglesource.Imaginejustonespeaker,creatingpressurewavesintheair.

Nowweaddasecondsource.

We'llneedtobeverycarefulwithinterpretingthecontrast(i.ecolorscheme)ofthisplot.Ifapointisredorblue,thatmeansthemedium'sdisplacementislarge.Ifapointiscoloredwithwhite,thatmeansthemedium'sdisplacementiszero.Mostpositionswillalternatebetweenred,white,blue,white,redastimeadvances,andthewavepropagates.However,someregions,asyoucanseeintheanimatedversion,remainwhiteatalltimes.Thesearetheregionsofdestructiveinterferences,wherethewavesfromthetwosorcesinterferedestructively.Attheselocations,thereisnodisplacementofthemedium,andthusnosoundisheard.

Constructive and Destructive Interference

Wecanfindsomegeneralguidelinestodetermineifwe'llhaveconstructiveordestructiveinterferencebasedonthepositionofthelistener.

1. Case1:Thepathlengthdifferenceisequaltoawholenumberofwavelengths:

Thisyieldsconstructiveinterference2. Case2:Thepathlengthdifferenceisequaltoahalfnumberofwavelengths:

Thisyieldsdestructiveinterference

Rephrased in terms of phase.

Δ = 0ϕ0

Δx = mλ

m

ΔL = nλ

ΔL = (n+ ) λ12

PHY 208 - superposition

updated on 2018-02-01 J. Hedberg | © 2018 Page 9

Mathematicallyspeaking,interferenceisdeterminedbythephasedifferencebetweentwowaves.

If is0, ,oranymultipleof ,constructiveinterferenceoccurs

i.e. where

Thisiscontrarytodestructiveinterferencewhichoccursatoddmultiplesof

i.e. where

ϕ 2π 2π

ϕ = m× 2π m = 0,1,2,…

2π

ϕ = (2m+ 1)π m = 0,1,2,…

PHY 208 - superposition

updated on 2018-02-01 J. Hedberg | © 2018 Page 10

Quick Question 4

crest trough trough crest

12

3

Herearetwosoundsourceemittingwoundwavesinphase.Thesolidlinesarethemaxiumumpressureregions,thedashedlinesshowthelocationoftheminimumpressureregions,at

Atpoint1,theinterferenceis

1. MaximumConstructive2. Constructive,butlessthanmaximum3. PerfectlyDestructive4. Destructive,butonlypartially5. Nointerference

Quick Question 5

Atpoint2,theinterferenceis

1. MaximumConstructive2. Constructive,butlessthanmaximum3. PerfectlyDestructive4. Destructive,butonlypartially5. Nointerference

Quick Question 6

Atpoint3,theinterferenceis

1. MaximumConstructive2. Constructive,butlessthanmaximum3. PerfectlyDestructive4. Destructive,butonlypartially5. Nointerference

t = t0

Example Problem #2:

PHY 208 - superposition

updated on 2018-02-01 J. Hedberg | © 2018 Page 11

Twospeakersinaplaneare2.0mapartandinphasewitheachother.Bothemit700HzSoundwavesintoaroomwherethespeedofsoundis341m/s.Alistenerstands5.0minfrontofthespeakersand2.0mtoonesideofthecenter.Describetheinterferenceatthispointinspace.

Reflecting sound waves

Justlikewithawaveonastring,iftheconditionsareright,we'llobtainastandingwaveinsidethetube.

And,justlikewithwavesonastring,onlycertainwavelengthswill'fit'.

Pressure and Displacement in tubes

closed-closed open-open∆(pressure)

displacement

∆(pressure)

displacement

m = 1

m = 2

m = 3

m = 1

m = 2

m = 3

Thisfigureshowsthefirstthreemodesofoscillationforstandingwavesintubes.Ontheleft,weseeatubethatisclosedonboth

Example Problem #2:PHY 208 - superposition

updated on 2018-02-01 J. Hedberg | © 2018 Page 12

closed-open∆(pressure)

displacement

m = 1

m = 3

m = 5

…arejusttubeswithairflowing.Thelengthofthetubeisoneimportantaspect.Differentnotesarecreatedbycoveringholes,whicheffectivelychangethelengthofthetube.

ends.Sincethetubeisclosed,particlesofaircannotmovepasttheboundary.Thus,thedisplacementgraphsshowanodeattheendsofthetube.However,inthepressuregraph,theendsareoccupiedbyanti-nodes.Ontheright,thesameisillustratedbutfortubesthatareopenonbothends.Now,theparticlesarefreetomoveinandoutofthetubeattheend.However,sincethepressureattheendofthetubeissetbyatmosphericpressure,thisvaluecannotchange.Therefore,weseenodesinthepressuregraphsattheends,andanti-nodesinthedisplacementgraphs.

Asyoucanseeinthetopleft,thefundamentalmodeforaclosed-closedtubehasafrequencywhosewavelengthisequaltotwicethelengthofthetube.Thesecondharmonicfrequencycontains1fullwavelength.

Pressure and Displacement in tubes

Theseplotsshowtubesthatareclosedononeend,andopenontheother.Thefundamentalmodeforaclosed-opentubewillhaveawavelengthequalto4timesthelengthofthetube.

Wind Instruments...

PHY 208 - superposition

updated on 2018-02-01 J. Hedberg | © 2018 Page 13

Tuningforksareverycleanoscillators.

Theends(calledtines)movebackandforthinaverysinusoidalmotion.

Thismotioncreatesthepressurewavesthatwehearasapuresinewave.

Thewavelengthandthusthefrequencyaregivenbythegeometryandmaterialsofthetuningfork.

disp

lace

men

t

time

Tuning Forks

PHY 208 - superposition

updated on 2018-02-01 J. Hedberg | © 2018 Page 14

Hereisatuningfork

Andhereisthefrequencyspectrumofthesoundthetuningforkmakes.Thespectrumisagraphofthesignalstrengthversesfrequency.Wecanseethatonefrequencyhasaveryhighvalue.

Sound, in general

Rarelydowehearperfectsinusoidaloscillations.Mostinstruments,noises,vocalizationsaremixturesofmanydifferentoscillations.

ThisismiddleConthepiano.Youcanseethefundamentalat261Hertz.Buttherearealotofhighertonesalsopresent.Thezoominshowsevenmorelittlepeaks.

Beats

Sofar,allthistalkofinterferencehasbeenabouttwosourceswiththesamefrequency.Inreallife,that’susuallynotthecase.

Herearetwoplotsofsinewavescomingfromtwospeakerslocatedatthesameplace.

Thetopsoundhasafrequencyof1Hertzwhilethebottomhasafrequencyof1.1Hertz.

It’shardtotelltheyaredifferent.

PHY 208 - superposition

updated on 2018-02-01 J. Hedberg | © 2018 Page 15

5 10 15 20

2

1

1

2

region of constructive interference

region of destructive interference

5 10 15 20

2

1

1

2

overlap

out of phase

Leteachofthetwosourcewavesbegivenby:

Theresultantdisplacementisthen:

= cos t and = cos ts1 sm ω1 s2 sm ω2

s = + = (cos t+ cos t)s1 s2 sm ω1 ω2 (7)

PHY 208 - superposition

updated on 2018-02-01 J. Hedberg | © 2018 Page 16

5 10 15 20

2

1

1

2

region of constructive interference

region of destructive interference

Since ,theterminbracketscanbeconsideredamodulatedenvelopeoscillatingatangularfrequency .

Itsminandmaxvaluesoccurtwiceinagivencycle.

Thisoscillationbetweenaminandmaxiswhatwehearasbeats.

or,intermsoffrequency, :

M

L

Mass(kg) (Hz)2.00 684.00 976.00 1178.00 13510.00 152

Usingatrigidentity,thisbecomes:

if and ,thentheaboveequationbecomesmoretidy:

Aheavymassissuspendedfroma1.65mlongsteelwire.(Thewirehasamassof5.85g)Thefrequenciesofthe3rdharmonicoscillationofthewireasafunctionofmassaregivenbelowinthetable.Usethisdatatodetermineavalueof.

Enter:Fit{{0,0},{2,68^2},{4,97^2},{6,117^2},{8,135^2},{10,152^2}}atwolframalpha.

Link

s = 2 cos[ ( − ) t] cos[ ( + ) t]sm

12

ω1 ω212

ω1 ω2 (8)

= ( − )ω′ 12

ω1 ω2 ω = ( + )12

ω1 ω2

s(t) = [2 cos t] cosωtsm ω′

s(t) = [2 cos t] cosωtsm ω′

ω≫ ω′

ω′

= 2 = 2( )( − ) = −ωbeat ω′ 12

ω1 ω2 ω1 ω2

f

= −fbeat f1 f2

f3

g

PHY 208 - superposition

updated on 2018-02-01 J. Hedberg | © 2018 Page 17