PHYC10001Physics1AdvancedChapter16:WavesI

ProfRobertScholtenSchoolofPhysics

UniversityofMelbourne

29

A/Prof RE Scholten (2011)ProfREScholten(2015) 2

WhereweareLastlecture Doppler Soundwaves,supersonictransport

Thislecture Standingwaves Optics:diffraction

Nextlecture Optics:interference Optics:reflectionandrefraction

A/Prof RE Scholten (2011)ProfREScholten(2015) 3

Recall I A2 andinversesquarelaw Hearingislogarithmic sotakelogofpower decibel scale

Forwavesthatpropagateinamaterialmedium(e.g.sound):Dopplershift

Forlight

Sonicboom

S

D

vvvvff

'

cvcv

S

S

/1/1'

SS vv

tvvt sin

Logarithmic scale: decibels

010log10 I

IdB

A/Prof RE Scholten (2011)ProfREScholten(2015)

Interference:whyisitso important? Ifyouhavepoormobilephonereception,whydoesmovingafewcmmake

adifference,ifthebasestationis10kmaway? Whyareorganpipesspecificlengths? Howisitthatalaserhassuchpurecolour?

4

A/Prof RE Scholten (2011)ProfREScholten(2015) 5

ConcepTest

Astringisclampedatbothendsandpluckedsoitvibratesinastandingmodebetweentwoextremepositionsa andb.Letupwardmotioncorrespondtopositivevelocities.Whenthestringisinpositionc,theinstantaneousvelocityofpointsalongthestring:

A.iszeroeverywhere.B.ispositiveeverywhere.C.isnegativeeverywhere.D.dependsonlocation.

From Peer Instruction by E. Mazur

A/Prof RE Scholten (2011)ProfREScholten(2015) 6

16.7Superposition:standingwaves Inconsideringsuperposition,wecanconsider

Copropagating,counterpropagating Sameordifferentfrequency,and/orwavelength,and/orvelocity Sameordifferentamplitudes

Startsimply: same frequency,wavelength,velocity,amplitude different direction:counterpropagating

Twoidenticalcounterpropagatingtravellingwaves

DR 2/ 2DL

2/ 2

2121 cossin2sinsin

Travellingtotheright:Forconstantphase,

as increases, mustincrease

sin sin

A/Prof RE Scholten (2011)ProfREScholten(2015) 7

16.7Superposition:standingwavesTwoidentical counterpropagatingtravellingwaves

Write , as , cos

DR 2/ 2DL

2/ 2

2121 cossin2sinsin

Not atravellingwave

sin sin

, 2 sin cos

A/Prof RE Scholten (2011)ProfREScholten(2015) 8

16.7Superposition:standingwavesTwoidentical counterpropagatingtravellingwaves

, 2 sin cos cos

Mobilephone

deadzones!

D(x,t)

x2/ 2

,...2,0 tD(x,t)

x2/ 2

,..., 23

2 t

A/Prof RE Scholten (2011)ProfREScholten(2015) 9

16.7Superposition:standingwavesTwoidentical counterpropagatingtravellingwaves Sometimes zeroeverywhere:

cost=0 when t = T/4, 3T/4,

Someplaces zeroalways Nodes (zeros)andantinodes Nodes(zeros)at Antinodes

Separationofnodesis/2 Separationofantinodesis/2 Note:standingwavesdonottransmitenergy

Node

Antinode2,1,0 nnkx 2,1,0 21 nnkx

, 2 sin cos

A/Prof RE Scholten (2011)ProfREScholten(2015) 10

Superposition:standingwaves Boundaryconditionslimittheallowed

wavelengths MUSThavenodesateachend!

Frequencies

Lowestfrequencyf1 isthefundamentalm isthemodenumber:thenumberofantinodes

Separationofnodes/2Node Node

...,3,2,12

20sin2

mmL

mLkL

kLa

m

12mf

Lvmfm

Thisisasimpleexampleofresonance:frequenciesallowedarethosewithintegerhalfwavelengths.Importantforelectronwavefunction conceptslater

, 2 sin cos

A/Prof RE Scholten (2011)ProfREScholten(2015) 11

21.4Standingsoundwaves Narrowtubee.g.organpipe standingsoundwave Soundwave

Longitudinaldisplacement 90 outofphasetopressure Displacementzeroatpressurepeaksandtroughs

P

xD

x

A/Prof RE Scholten (2011)ProfREScholten(2015) 12

21.4Standingsoundwaves CLOSEDend:nodisplacementpossible displacementnode OPENend:atmosphericpressure pressurenode CLOSEDCLOSED

Displacementnodesatbothends;similartostring

OPENOPENPressurenodesatbothends;alsosimilartostring

P

xD

x...,3,2,12 m

Lvmfm

A/Prof RE Scholten (2011)ProfREScholten(2015) 13

21.4Standingsoundwaves Organpipes=openopen Windinstrumentsclosedatoneend CLOSEDOPEN

(exercise!E.g.inexam?)Closed(node)

...,7,5,3,14

mLvmfm

A/Prof RE Scholten (2011)ProfREScholten(2015) 14

21.4Examples:lengthoforganpipe;opticalmodesOrganpipe,secondharmonicat523Hz(oneoctaveabovemiddleC261.626Hz) Whatisthelengthofthepipe?

Optical laser cavity;lightwavesbetweenmirrors;electricfieldzeroatends(conductor) Howmanyantinodesinatypicallaser

cavity,L = 3.0cm?

Notelaseronlyoscillatesondiscretemodes,separatedinfrequencybyfreespectralrange(FSR),f

Originoflaserspectralpurity!

m656.0523343

22

2

2

fvL

Lvmfm

L

/2

Lcmfm 2

fm1 fm fm+1

Lc

2FSR

Lcfff mm 21

000,7710780

06.02

2

9

m

m

Lm

mL

A/Prof RE Scholten (2011)ProfREScholten(2015) 15

Standingwaves:summary Twoidentical counterpropagatingtravellingwaves

, 2 sin cos cos

Nodes andantinodes Separationofnodesis/2 Separationofantinodesis/2

Cavity nodeateachend discreteresonantfrequenciesNodes

Antinodes

12mf

Lvmfm

A/Prof RE Scholten (2011)ProfREScholten(2015)

Thatsalotoffussaboutstandingwaves Basisofchurchorgans,windinstrumentsingeneral,

stringinstruments, Basisoflasers

Homework:IfIhavealaseroperatingat ,cavitylength3.0cm,howmuchdoesthefrequencyoflightemittedbythelaserchangeifIchangethecavitylengthby1.0nm?

Basisofatomicstructure

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A/Prof RE Scholten (2011)ProfREScholten(2015)

Electronwavesandatoms Onlycertainmomentap allowed(N.Bohr) 4

A/Prof RE Scholten (2011)ProfREScholten(2015) 18

16.5Interferenceinonedimension Superposition interference Standingwave=interferencepattern

fortwocounterpropagatingwavesofequalfrequencyandamplitude Nowconsidercopropagating

Anewtravellingwave Amplitudedependsonthephasedifference

50/50 beamsplitter(semi-silvered mirror)

tkxytxytkxytxy

m

m

sin,sin,

2

1

2121 sincos2,' tkxytxy m

Dependsonrelativephasedifference(whereeachstartsfrom),noton or

Timevariation

A/Prof RE Scholten (2011)ProfREScholten(2015)

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16.5Interference Phasedifference 2

where isequivalentpathlengthdifference betweentwowaves

Considerthreesituations:1. Constructive, = 0,2,

2 cos 2Pathlengthdifference=integermultipleof2. Destructive, = ,3,

2 cos 0Pathlengthdifference=halfintegermultipleof

3. Partial

y'(x,t)

kx2/ 2

y'(x,t)

kx2/ 2

y'(x,t)

kx2/ 2

Huh?Wheredidtheenergy

go???

2121 sincos2,' tkxytxy m

A/Prof RE Scholten (2011)ProfREScholten(2015) 20

Interferenceintwoandthreedimensions Circularorsphericalwave

(assumeym constantoverregionofinterest) Lineardistancex r,theradialdistancefromsource

0sin, tkrytry m

PS2

S1

A/Prof RE Scholten (2011)ProfREScholten(2015) 21

17.6Beats Whatifwaveshavedifferentfrequencies? Considercopropagating,equalamplitudebutdifferent frequency

cos cos (Considertimedependenceonly(nox dependence);putdetectoratx = 0 )

Sousingtrigidentitycos cos 2 cos cos 2 cos cos 2 cos cos

; No dependenceonlybecausethisisatonefixedposition, 0

A/Prof RE Scholten (2011)ProfREScholten(2015) 22

21.8Beats Iffrequenciesaresimilar,1 2 Sumisrapidlyvaryingcosinewithfrequencywithslowlyoscillatingcosineamplitudeenvelope

2121mod

2121avg tsm modcos2

Destructive Constructive Destructive

21modbeat 2 ffff Twobeatspermod

wavelength

t

, 2 cos cos

, 2 cos cos

A/Prof RE Scholten (2011)ProfREScholten(2015) 23

Summaryandhomework Superpositionoftwoidenticalcounterpropagatingtravellingwaves

standingwave Copropagating,equalfrequency interference

Constructive,destructive,partial:dependsonphasedifference Copropagating, differentfrequency beats

Readingfornextlecture: Chapters35,36Waveoptics Recommendedexercises

Chapter 16:Q9,Q10,P33,P40,P51 Chapter 17:Q6,Q7,Q9,Q11,P21,P23,P51,P53

position

time

21mod

, 2 sin cos cos

2121 sincos2,' tkxytxy m

, 2 cos cos