Classical Mechanics Lecture 23 - Simon Fraser University · Types of Waves Movie (lwave) Movie...
Transcript of Classical Mechanics Lecture 23 - Simon Fraser University · Types of Waves Movie (lwave) Movie...
Classical MechanicsLecture 23
Today’sConcept:
HarmonicWaves
MechanicsLecture23,Slide1
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So,thetensioninaropeisrepresentedwiththesymbol"T",insteadofaforcewithasubscriptlike"F_t".Wellthatmakessense,becausethere'snootherquanEtysymbolizedwith"T"thatcomesupalltheEmewhendealingwithwaves...OHWAIT,YESTHEREIS,IT'SPERIOD!!!Seriously,whoismakingthesedecisions?!IsitthesameguywhodecidedtouseomegaforangularfrequencyANDangularvelocity?Thatguyisoverdueforagoodlynching...
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What is a Wave?
AwaveisatravelingdisturbancethattransportsenergybutnotmaXer.
Examples:!Soundwaves(airmovesback&forth)!Stadiumwaves(peoplemoveup&down)!Waterwaves(watermovesup&down)!Lightwaves(whatmoves?)
MechanicsLecture23,Slide2
Types of Waves
Movie(lwave)
Movie(twave)
Longitudinal:ThemediumoscillatesinthesamedirecEonasthewaveismoving.
MechanicsLecture23,Slide3
Transverse:ThemediumoscillatesperpendiculartothedirecEonthewaveismoving.ThisiswhatwestudyinPhysics140.
MechanicsLecture23,Slide4
Amplitude:ThemaximumdisplacementA ofapointonthewave.
A
Wavelength:Thedistanceλ betweenidenCcalpointsonthewave. Movie(tspeed)
Period:TheCmeP ittakesforanelementofthemediumtomakeonecompleteoscillaCon.
Wave Properties
MechanicsLecture23,Slide5
Wavelengthλ
String:
WavesAandBshownabovearepropagaCnginthesamemedium.Howdotheirfrequenciescompare?
A)ωA > ωB
B)ωA < ωB
C)ωA = ωB
A
B
x
y
x
y
CheckPoint
MechanicsLecture23,Slide6
WavesAandBshownabovearepropagaCnginthesamemedium.Howdotheirfrequenciescompare?B)ωA < ωB C)ωA = ωB A)ωA > ωB
B)AstheybothhavethesamevelociCes,andthewavelengthofAislarger,thefrequencymustbesmallerthanthatofB.C)Thetensionandmassdensityarethesameinbothstringssotheyhavethesamefrequency.
A)It'speriodisshorter.
A
B
x
y
x
y
MechanicsLecture23,Slide7
Thespeedofsoundinairisabitover300 m/s,andthespeedoflightinairisabout300,000,000 m/s.
Supposewemakeasoundwaveandalightwavethatbothhaveawavelengthof3 meters.WhatistheraEoofthefrequencyofthelightwavetothatofthesoundwave?
A)About1,000,000B)About0.000001C)About1000
Clicker Question
MechanicsLecture23,Slide8
How to make a Function Move
SupposewehavesomefuncEony = f (x):
x
y
0
MechanicsLecture23,Slide9
f (x − a) isjustthesameshapemovedadistanceatotheright:
x = ax
y
0
Leta = vt Then
f (x − vt) willdescribethesameshapemovingtotherightwithspeedv.
x = vt
v
x
y
0
y(x) = A cos
2⇡
�x
!
A
λ
x
y
Harmonic Wave
Considerawavethatisharmonicinxandhasawavelengthofλ.
MechanicsLecture23,Slide10
Iftheamplitudeismaximumatx = 0 thishasthefuncEonalform:
y(x) = A cos
2⇡
�(x � vt)
!
Now,ifthisismovingtotherightwithspeedvitwillbedescribedby:
x
y
v
k =2⇡�
! =2⇡v�
“wavenumber”
1 cycle per second
2 cycles per second
IfafuncEonmovingtotherightwithspeedvisdescribedbyf (x − vt) thenwhatdescribesthesamefuncEonmovingtothelehwithspeedv?
v
x
y
0
y = f (x − vt)
v A) y = − f (x − vt)
B)y = f (x + vt)
C)y = f (−x + vt)
Clicker Question
MechanicsLecture23,Slide11
x
y
0 x
y
0 x
y
SupposethefuncConhasitsmaximumatf (0).
A)y = − f (x − vt)
B)y = f (x + vt)
C)y = f (−x + vt)
x – vt = 0
x = vty = f (x − vt)
x – vt = 0
x = vt
x + vt = 0
x = −vt
−x + vt = 0
x = vt
v
v
v
v
MovestotherightwhenthesignsinfrontofthexandttermsaredifferentMovestotheleYwhenthesignsinfrontofthexandttermsarethesame
MechanicsLecture23,Slide12
v
x
y
0 x
y
0 x
y
v
0 x
y
x
y
0
WehaveshownthatthefuncConalformy(x,t) = Acos(kx − ωt) representsawavemovinginthe+x direcCon.
Whichofthefollowingrepresentsawavemovinginthe–xdirecCon?
A)y(x,t) = Acos(ωt − kx)
B)y(x,t) = Asin(kx - ωt)
C)y(x,t) = Acos(kx + ωt)
CheckPoint
MechanicsLecture23,Slide13
Ax
y
MechanicsLecture23,Slide14
WehaveshownthatthefuncConalformy(x,t) = Acos(kx − ωt) representsawavemovinginthe+x direcCon.
Ax
y
CheckPoint
Whichofthefollowingrepresentsawavemovinginthe –x direcCon?
A)y(x,t) = Acos(ωt − kx)CosineisanoddfuncCon,soreversingkxandwtwillreversethesignofthefuncConfromposiCvetonegaCve
C)y(x,t) = Acos(kx + ωt)Withparabolas,(x-a),whereaisanynumber,resultsinagraphshiYedtotheright/orposiCvexdirecCon.Therefore,theopposit[e](x+a)shi[f]tsittotheleY.
Energy
MechanicsLecture23,Slide15
WavesAandB shownabovearepropagaEnginthesamemediumwiththesameamplitude.Whichonecarriesthemostenergyperunitlength?
A)A
B)B
C)Theycarrythesameenergyperunitlength
CheckPoint
MechanicsLecture23,Slide16
A
B
x
y
x
y
A
B
x
y
x
y
Whichonecarriesthemostenergyperunitlength?
A)AB)B C)Same
A)SinceAhasalargerwavelength,itcarriesmoreenergyperunit.
B)kineCcenergyisdirectlyproporConalto(ωA2),sincetheiramplitudearethesameandBhasalargerω,Bhasmoreenergy
C)AssumingthatCwassupposedtobeA=B,theirenergiesareequalsincetheiramplitudesareequal.
MechanicsLecture23,Slide17
CheckPoint
MechanicsLecture23,Slide18
Homework Problem
MechanicsLecture23,Slide19
Homework Problem
MechanicsLecture23,Slide20
Homework Problem
Distancetraveledbyapieceofropeduringoneperiodis4A
Averagespeed=distancetraveled/Cmetaken
MechanicsLecture23,Slide21
Homework Problem