PHY2048, Lecture 23 – Wave MotionPHY2048, Lecture 23 – Wave MotionChapter 14 – Tuesday November 16thChapter 14 – Tuesday November 16th

•Fifth and final mini exam on Thu. (Chs. 13-15)Fifth and final mini exam on Thu. (Chs. 13-15)•Make up labs Nov. 29 at 12:30pm and 3:30pm in Make up labs Nov. 29 at 12:30pm and 3:30pm in UPL107UPL107•Final exam – Final exam – Wed. Dec. 8Wed. Dec. 8thth, 10am to noon, in MOR , 10am to noon, in MOR 104104

(Moore (Moore Auditorium)Auditorium)

•Review of wave solutionsReview of wave solutions•The wave equationThe wave equation•Wave superposition and interferenceWave superposition and interference

•Spatial interference and standing Spatial interference and standing waveswaves•Temporal beatingTemporal beating•Sources of musical soundSources of musical sound

•Doppler effect (if time)Doppler effect (if time)

Stephen Hill (covering for Prof. Wiedenhover)

Review - wavelength and Review - wavelength and frequencyfrequency

2k

kk is the is the angular wavenumberangular wavenumber..

Tra

nsv

ers

e w

ave

Tra

nsv

ers

e w

ave

2

T

is the is the angular frequencyangular frequency..

1

2f

T

v fk T

frequencyfrequency

velocityvelocity

cos(( ) ), k ty At xx

AmplitudeDisplacement Phase}

Phaseshift

angular wavenumberangular frequency

Traveling waves on a stretched Traveling waves on a stretched stringstring

is the string's linear density, or mass per unit length.is the string's linear density, or mass per unit length.

•Tension Tension FFTT provides the restoring force (kg.m.s provides the restoring force (kg.m.s-2-2) in the string. ) in the string. Without tension, the wave could not propagate.Without tension, the wave could not propagate.

•The mass per unit length The mass per unit length (kg.m (kg.m-1-1) determines the response ) determines the response of the string to the restoring force (tension), through Newtorn's of the string to the restoring force (tension), through Newtorn's 2nd law.2nd law.

•Look for combinations of Look for combinations of FFTT and and that give dimensions of that give dimensions of speed (m.sspeed (m.s-1-1).).

TFv

FTFT Fneta

a

a

+ve curve+ve curve +ve curve+ve curve

ve curveve curve

Zero curveZero curve

Traveling waves on a stretched Traveling waves on a stretched stringstring

is the string's linear density, or mass per unit length.is the string's linear density, or mass per unit length.

The Wave EquationThe Wave Equation

FTFT Fneta

a

a

+ve curve+ve curve +ve curve+ve curve

ve curveve curve

Zero curveZero curve

2 2

2 2net T y

y yF F l l ma

x t

l

masstransverseacceleration

Dimensionless parameter proportional to curvature

2 2

2 2T

y yF

x t

The wave equationThe wave equation2 2

2 2TF y y

x t

•General solution:General solution:

( , ) sin or ( , )m my x t y kx t y x t y f kx t

2 2

2 22 2

, ,y y

k y x t y x tx t

2

2 2 22

orT TF Fk v

k

TFv

vv

The principle of superposition for The principle of superposition for waveswaves•It often happens that waves travel simultaneously through the It often happens that waves travel simultaneously through the

same region, same region, e.g.e.g.

Radio waves from many broadcastersRadio waves from many broadcastersSound waves from many musical instrumentsSound waves from many musical instrumentsDifferent colored light from many locations from your TVDifferent colored light from many locations from your TV

2 22

2 2

y yv

x t

•And have solutions:And have solutions: ( , ) or sinm my x t y f kx t y kx t

•This is a result of the This is a result of the principle of superpositionprinciple of superposition, which applies , which applies to all to all harmonic wavesharmonic waves, , i.e.i.e., waves that obey the linear wave , waves that obey the linear wave equationequation

•Nature is such that all of these waves can exist without Nature is such that all of these waves can exist without altering each others' motionaltering each others' motion

•Their effects simply addTheir effects simply add

The principle of superposition for The principle of superposition for waveswaves•If two waves travel simultaneously along the same stretched If two waves travel simultaneously along the same stretched

string, the resultant displacement string, the resultant displacement y'y' of the string is simply of the string is simply given by the summationgiven by the summation

1 2' , , ,y x t y x t y x t

where where yy11 andand yy22 would have been the displacements had the would have been the displacements had the waves traveled alone.waves traveled alone.

Overlapping waves algebraically add to produce a Overlapping waves algebraically add to produce a resultant wave resultant wave (or (or net wavenet wave).).

Overlapping waves do not in any way alter the travel Overlapping waves do not in any way alter the travel of each otherof each other

•This is the This is the principle of superpositionprinciple of superposition..

Interference of wavesInterference of waves•Suppose two sinusoidal waves with the same frequency and Suppose two sinusoidal waves with the same frequency and amplitude travel in the same direction along a string, such thatamplitude travel in the same direction along a string, such that

1

2

sin

sin

m

m

y y kx t

y y kx t

•The waves will add.The waves will add.

Interference of wavesInterference of waves

Noise canceling Noise canceling headphonesheadphones

Interference of wavesInterference of waves•Suppose two sinusoidal waves with the same frequency and Suppose two sinusoidal waves with the same frequency and amplitude travel in the same direction along a string, such thatamplitude travel in the same direction along a string, such that

1

2

sin

sin

m

m

y y kx t

y y kx t

•The waves will add.The waves will add.

•If they are in phase (If they are in phase (i.e. i.e. = 0 = 0), they combine to double the ), they combine to double the displacement of either wave acting alone.displacement of either wave acting alone.

•If they are out of phase (If they are out of phase (i.e. i.e. = = ), they combine to cancel ), they combine to cancel everywhere, since everywhere, since sin(sin() = ) = sin(sin())..

•This phenomenon is called This phenomenon is called interferenceinterference..

Interference of wavesInterference of waves•Mathematical proof:Mathematical proof:

1

2

sin

sin

m

m

y y kx t

y y kx t

1 2' , , ,

sin sinm m

y x t y x t y x t

y kx t y kx t

Then:Then:

1 12 2sin sin 2sin cos But:But:

1 12 2' , 2 cos sinmy x t y kx t So:So:

Amplitude Wave part

Phaseshift

Interference of wavesInterference of waves

1 12 2' , 2 cos sinmy x t y kx t

If two sinusoidal waves of the same amplitude and If two sinusoidal waves of the same amplitude and frequency travel in the same direction along a frequency travel in the same direction along a stretched string, they interfere to produce a resultant stretched string, they interfere to produce a resultant sinusoidal wave traveling in the same direction.sinusoidal wave traveling in the same direction.

•If If = 0 = 0, the waves interfere , the waves interfere constructivelyconstructively, , cos½cos½ = 1 = 1 and the and the wave amplitude is wave amplitude is 22yymm..

•If If = = , the waves interfere , the waves interfere destructivelydestructively, , cos(cos(/2) = 0/2) = 0 and the and the wave amplitude is wave amplitude is 00, , i.e.i.e. no wave at all. no wave at all.

•All other cases are intermediate between an amplitude of All other cases are intermediate between an amplitude of 00 and and 22yymm..

•Note that the phase of the resultant wave also depends on the Note that the phase of the resultant wave also depends on the phase difference.phase difference.Adding waves as vectors (phasors) described by amplitude and phaseAdding waves as vectors (phasors) described by amplitude and phase

Interference - Standing WavesInterference - Standing WavesIf two sinusoidal waves of the same amplitude and If two sinusoidal waves of the same amplitude and wavelength travel in opposite directions along a wavelength travel in opposite directions along a stretched string, their interference with each other stretched string, their interference with each other produces a produces a standing wavestanding wave..

1 2

1 12 2

' , , ,

sin sin

2 sin cos

m m

m

y x t y x t y x t

y kx t y kx t

y kx t

•This is clearly not a traveling wave, because it does not have This is clearly not a traveling wave, because it does not have the form the form ff((kx kx tt))..

•In fact, it is a stationary wave, with a sinusoidal varying In fact, it is a stationary wave, with a sinusoidal varying amplitude amplitude 22yymmcos(cos(tt))..

t dependencex dependence

Link Link

Reflections at a boundaryReflections at a boundary•Waves reflect from boundaries.Waves reflect from boundaries.

•This is the reason for echoes - you This is the reason for echoes - you hear sound reflecting back to you.hear sound reflecting back to you.

•However, the nature of the However, the nature of the reflection depends on the boundary reflection depends on the boundary condition.condition.

•For the two examples on the left, For the two examples on the left, the nature of the reflection depends the nature of the reflection depends on whether the end of the string is on whether the end of the string is fixed or loose.fixed or loose.

Standing waves Standing waves and resonanceand resonance•At ordinary frequencies, At ordinary frequencies, waves travel backwards and waves travel backwards and forwards along the string.forwards along the string.

•Each new reflected wave Each new reflected wave has a new phase.has a new phase.

•The interference is The interference is basically a mess, and no basically a mess, and no significant oscillations build significant oscillations build up.up.

Standing waves and resonanceStanding waves and resonance•However, at certain special However, at certain special frequencies, the interference frequencies, the interference produces strong standing wave produces strong standing wave patterns.patterns.

•Such a standing wave is said to Such a standing wave is said to be produced at be produced at resonanceresonance..

•These certain frequencies are These certain frequencies are called called resonant frequenciesresonant frequencies..

Standing waves and resonanceStanding waves and resonance•Standing waves occur whenever Standing waves occur whenever the phase of the wave returning to the phase of the wave returning to the oscillating end of the string is the oscillating end of the string is precisely in phase with the forced precisely in phase with the forced oscillations.oscillations.•Thus, the trip along the string Thus, the trip along the string and back should be equal to an and back should be equal to an integral number of wavelengths, integral number of wavelengths, i.ei.e.. 22 or for 1,2,3...

LL n n

n

, for 1,2,3...2

v vf n n

L

•Each of the frequencies Each of the frequencies ff11, , ff11, , ff11, , etcetc, are called , are called harmonicsharmonics, or a , or a harmonic seriesharmonic series; ; nn is the is the harmonic harmonic numbernumber..

determined by geometrydetermined by geometry

Standing waves and resonanceStanding waves and resonance

•Here is an example of a two-dimensional vibrating diaphragm.Here is an example of a two-dimensional vibrating diaphragm.

•The dark powder shows the positions of the nodes in the The dark powder shows the positions of the nodes in the vibration.vibration.

Standing waves in air columnsStanding waves in air columns•Simplest case:Simplest case:

- 2 open ends- 2 open ends

- Antinode at each end- Antinode at each end

- 1 node in the middle- 1 node in the middle

•Although the wave is Although the wave is longitudinal, we can represent it longitudinal, we can represent it schematically by the solid and schematically by the solid and dashed green curves.dashed green curves.

1 2 2 /1L L

Standing waves in air columnsStanding waves in air columnsA harmonic series

2

3

4

1 2 2 /1L L

2, for 1,2,3,....

Ln

n , for 1,2,3,...

2

v vf n n

L

Standing waves in air columnsStanding waves in air columns

4, for 1,3,5,....

Ln

n , for 1,3,5,...

4

v vf n n

L

A different harmonic series

1

3

5

7

Musical instrumentsMusical instruments

Flute

Oboe

Saxophone

Link

Musical instrumentsMusical instruments

4

vf n

L

•n = even if B.C. same at both ends of pipe/string•n = odd if B.C. different at the two ends

Wave interference - spatialWave interference - spatial

Interference - temporal (or beats)Interference - temporal (or beats)( , ) cos( )ms x t s kx t

•In order to obtain a spatial interference pattern, we placed In order to obtain a spatial interference pattern, we placed two sources at different locations, two sources at different locations, i.e.i.e. we varied the first term we varied the first term in the phase of the waves.in the phase of the waves.

•We can do the same in the time domain whereby, instead of We can do the same in the time domain whereby, instead of placing sources at different locations, we give them different placing sources at different locations, we give them different angular frequencies angular frequencies and and . For simplicity, we analyze the . For simplicity, we analyze the sound at sound at xx = 0 = 0.. 1 2 1 2cos cosms s s s t t

1 12 2cos cos 2cos cos

1 11 2 1 22 22 cos cos

2 cos ' cos

m

m

s s t t

s t t

11 22

11 22

'

Interference - temporal (or beats)Interference - temporal (or beats)

•A maximum amplitude occurs whenever A maximum amplitude occurs whenever ''tt has the value has the value or or ..

•This happens twice in each time period of the cosine function.This happens twice in each time period of the cosine function.

•Therefore, the Therefore, the beat frequencybeat frequency is twice the frequency is twice the frequency '', , i.e.i.e.

2 cos ' cosms s t t

1 2

1 2

2 '

2 '

beat

beatf f f f

Link Link

Doppler effectDoppler effect•Consider a source of sound at the ‘proper frequency’, Consider a source of sound at the ‘proper frequency’, f 'f ', , moving relative to a stationary observer.moving relative to a stationary observer.

•The observer will hear the sound with an apparent frequency, The observer will hear the sound with an apparent frequency, ff, which is shifted from the proper frequency according to the , which is shifted from the proper frequency according to the following equation:following equation:

1 'u

f fv

•Here, Here, vv is the sound velocity (~330 m/s in air), and is the sound velocity (~330 m/s in air), and uu is the is the relative speed between the source and detector.relative speed between the source and detector.

When the motion of the detector or source is towards When the motion of the detector or source is towards the other, use the plus (+) sign so that the formula the other, use the plus (+) sign so that the formula gives an upward shift in frequency. When the motion gives an upward shift in frequency. When the motion of the detector or source is away from the other, use of the detector or source is away from the other, use the minus (the minus () sign so that the formula gives a ) sign so that the formula gives a downward shift. downward shift.

Mach cone angle: sinS

v

v

Kinetic energy: dK = 1/2 dm vy2

( , ) sinmy x t y kx t

cosy m

yv y kx t

t

2 212 cos ( )mdK dx y kx t

Divide both sides by dt, where dx/dt = vx

2 2 212 cos ( )x m

dKv y kx t

dt

Energy in traveling wavesEnergy in traveling waves

Similar expression for elastic potential energy

2 2 212 cos ( )x m

dUv y kx t

dt

2 2 2 2 2 2 21 1 1 12 2 2 22 cos ( ) 2avg m m mP v y kx t v y v y

Energy is pumped in an oscillatory fashion down the stringNote: I dropped the subscript on v since it represents the wave speed

dm = dx