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