sy29_may05_09

Physics 207 Lecture 29

Lecture 29Goals: Chapter 20 Work with a few important characteristics of sound waves.(e.g., Doppler effect) Chapter 21 Recognize standing waves are the superposition of twotraveling waves of same frequency Study the basic properties of standing waves Model interference occurs in one and two dimensions Understand beats as the superposition of two waves ofunequal frequency.

Assignment HW12, Due Friday, May 8th Thursday, Finish up, begin review for final, evaluationsPhysics 207: Lecture 29, Pg 1

Doppler effect, moving sources/receivers

Physics 207: Lecture 29, Pg 2

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Doppler effect, moving sources/receivers

If the source of sound is moving Toward the observer f observer seems smaller Away from observer seems larger

=

f sourcev1 vs

f observer =

f sourcev1 + vs

If the observer is moving v Toward the source f observer = 1 + o f sourcev seems smaller

Away from source seems larger

v f observer = 1 o fsourcev Doppler Example AudioDoppler Example VisualPhysics 207: Lecture 29, Pg 3

Doppler Example

A speaker sits on a small moving cart and emits a short 1Watt sine wave pulse at 340 Hz (the speed of sound in air is340 m/s, so = 1m ). The cart is 30 meters away from thewall and moving towards it at 20 m/s.The sound reflects perfectly from the wall. To an observeron the cart, what is the Doppler shifted frequency of thedirectly reflected sound?Considering only the position of the cart, what is theintensity of the reflected sound? (In principle on would haveto look at the energy per unit time in the moving frame.)t030 mA


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Doppler Example

The sound reflects perfectly from the wall. To an observer onthe cart, what is the Doppler shifted frequency of the directlyreflected sound?f

f observer =

At the wall: fwall = 340 / (1-20/340) = 361 Hz

source

v

1 vs

Wall becomes source for the subsequent part v f observer = 1 + o f sourcevAt the speaker f = fwall (1+ 20/340) = 382 Hz

t0

t130 m


Example Interference

Considering only the position of the cart, what is the intensity ofthe reflected sound to this observer? (In principle one wouldhave to look at the energy per unit time in the moving frame.)vcart t + vsound t = 2 x 30 m = 60 mt = 60 / (340+20) = 0.17 s dsound = 340 * 0.17 m = 58 mI = 1 / (4 582) = 2.4 x 10-5 W/m2 or 74 dBst0

t130 m


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Doppler effect, moving sources/receivers Three key pieces of information

Time of echo Intensity of echo Frequency of echoPlus prior knowledge of object being studied With modern technology (analog and digital) this can be done in real time.


Superposition

Q: What happens when two waves collide ?

A: They ADD together! We say the waves are superimposed.


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Interference of Waves

2D Surface Waves on Water

In phase sources separatedby a distance d

d


Principle of superposition

The superposition of 2 or more waves is called interferenceConstructive interference:These two waves are in phase.Their crests are aligned.

Destructive interference:These two waves are out ofphase.The crests of one are alignedwith the troughs of the other.

Their superposition produces awave with amplitude 2a

Their superposition produces awave with zero amplitude


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Interference: space and time

Is this a point of constructiveor destructive interference?What do we need to do tomake the sound from thesetwo speakers interfereconstructively?


Interference of SoundSound waves interfere, just like transverse waves do. Theresulting wave (displacement, pressure) is the sum of the two (ormore) waves you started with.AD ( r2 , t ) = cos[ 2 ( r2 / t / T ) + 2 ]r2rr

r =| r1 | | r2 |

D ( r1 , t ) =

Maximum constructive interference

Acos[ 2 ( r1 / t / T ) + 1 ]r1

= 2 r + 1 2 = 2 m

= r +(1 2 ) = m22Maximum destructive interference = 2 r + 1 2 = 2 ( m + 1 )2m = 0,1,2,...

rPhysics 207: Lecture 29, Pg 12

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A speaker sits on a pedestal 2 m tall and emits a sine waveat 343 Hz (the speed of sound in air is 343 m/s, so = 1m ).Only the direct sound wave and that which reflects off theground at a position half-way between the speaker and theperson (also 2 m tall) makes it to the persons ear.How close to the speaker can the person stand (A to D) sothey hear a maximum sound intensity assuming there is nophase change at the ground (this is a bad assumption)?t1

t0dA

t0B

D

h

C

The distances AD and BCD have equal transit times so thesound waves will be in phase. The only need is for AB = Physics 207: Lecture 29, Pg 13


The geometry dictates everything else.AB = AD = BC+CD = BC + (h2 + (d/2)2) = dAC = AB+BC = +BC = (h2 + d/22)Eliminating BC gives+d = 2 (h2 + d2/4) + 2d + d2 = 4 h2 + d21 + 2d = 4 h2 / d = 2 h2 / = 7.5 mt1t07.5tD0A4.253.25BCBecause the ground is more dense than air there will be a phasechange of and so we really should set AB to /2 or 0.5 m.Physics 207: Lecture 29, Pg 14

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Exercise Superposition

Two continuous harmonic waves with the same frequencyand amplitude but, at a certain time, have a phasedifference of 170are superimposed. Which of the follo wingbest represents the resultant wave at this moment?Original wave(the other has a different phase)(A)

(B)(D)

(C)

(E)


Wave motion at interfacesReflection of a Wave, Fixed End

When the pulse reaches the support,the pulse moves back along thestring in the opposite direction

This is the reflection of the pulse

The pulse is inverted


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Animation

Reflection of a Wave, Fixed End


Reflection of a Wave, Free EndAnimation


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Transmission of a Wave, Case 1When the boundary is intermediate between the last twoextremes ( The right hand rope is massive or massless.)then part of the energy in the incident pulse is reflected andpart is transmitted Some energy passesthrough the boundary Here rhs > lhs

Animation


Transmission of a Wave, Case 2Now assume a heavier string is attached to a lightstring Part of the pulse is reflected and part is transmitted The reflected part is not inverted

Animation


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Standing wavesTwo waves traveling in opposite direction interfere with eachother.If the conditions are right, same k & , their interferencegenerates a standing wave:DRight(x,t)= a sin(kx-t) DLeft(x,t)= a sin(kx+t)A standing wave does not propagate in space, it stands in place.A standing wave has nodes and antinodesAnti-nodesD(x,t)= DL(x,t) + DR(x,t)D(x,t)= 2a sin(kx) cos(t)The outer curve is theamplitude functionA(x) = 2a sin(kx)when t = 2n n = 0,1,2,

k = wave number = 2 /NodesPhysics 207:Lecture 29, Pg 21

Standing waves on a stringLongest wavelength allowed isone half of a waveFundamental: /2 = L = 2 L

m = 2 L = vm

fm

m = 1, 2 ,3,...Recall v = f

fm = m v2LOvertones m > 1Physics 207: Lecture 29, Pg 22

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Vibrating Strings- Superposition Principle

Antinode D(0,t)

D(x,0)

Violin, viola, cello, string bassGuitarsUkulelesMandolinsBanjos


Standing waves in a pipeOpen end: Must be a displacement antinode (pressure minimum)Closed end: Must be a displacement node (pressure maximum)Blue curves are displacement oscillations. Red curves, pressure.Fundamental:

/2

/2

/4


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Standing waves in a pipem = 2L

m = 2L

m = 4L

v2Lm = 1 , 2 , 3 ,...

v2Lm = 1 , 2 , 3 ,...

v4Lm = 1 , 3 , 5 ,...

m

fm = m

m

m

fm = m

fm = m


Combining Waves

Fourier SynthesisPhysics 207: Lecture 29, Pg 26

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Lecture 29

Assignment HW12, Due Friday, May 8th


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