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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
16
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