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UW-Madison PHYSICS 207 - PHYSICS 207 Lecture Notes

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Lecture 29Doppler effect, moving sources/receiversDoppler effect, moving sources/receiversDoppler ExampleDoppler ExampleExample Sound IntensitySlide 7Ch. 21: Wave SuperpositionInterference of WavesPrinciple of superpositionInterference: space and timeInterference of SoundSlide 13Example InterferenceExample InterferenceExercise SuperpositionWave motion at interfaces Reflection of a Wave, Fixed EndReflection of a Wave, Fixed EndReflection of a Wave, Free EndStanding wavesStanding waves on a stringVibrating Strings- Superposition PrincipleStanding waves in a pipeSlide 26Combining WavesOrgan Pipe ExampleStanding WavesSlide 30Physics 207: Lecture 29, Pg 1Lecture 29Goals:Goals:•Chapter 20Chapter 20 Work with a few important characteristics of sound waves. (e.g., Doppler effect)•Chapter 21Chapter 21 Recognize standing waves are the superposition of two traveling 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 of unequal frequency.•AssignmentAssignment HW13, Due Friday, May 7h  Thursday, Finish up, begin review for finalPhysics 207: Lecture 29, Pg 2Doppler effect, moving sources/receiversPhysics 207: Lecture 29, Pg 3Doppler effect, moving sources/receivers If the source of sound is moving Toward the observer   seems smaller  Away from observer   seems largerIf the observer is moving Toward the source   seems smaller Away from source   seems largervvsff1sourceobserversourceoobservervv1 ffDoppler Example AudioDoppler Example Visualvvsff1sourceobserversourceoobservervv1 ffPhysics 207: Lecture 29, Pg 4Doppler Example A speaker sits on a small moving cart and emits a short 1 Watt sine wave pulse at 340 Hz (the speed of sound in air is ~340 m/s, so  = 1m ). The cart is 30 meters away from the wall and moving towards it at 20 m/s. The sound reflects perfectly from the wall. To an observer on the cart, what is the Doppler shifted frequency of the directly reflected sound? Considering only the position of the cart, what is the intensity of the reflected sound? (In principle on would have to look at the energy per unit time in the moving frame.)t0A30 mPhysics 207: Lecture 29, Pg 5Doppler ExampleThe sound reflects perfectly from the wall. To an observer on the cart, what is the Doppler shifted frequency of the directly reflected sound? At the wall: fwall = 340 / (1-20/340) = 361 Hz Wall becomes “source” for the subsequent part At the speaker f ’ = fwall (1+ 20/340) = 382 Hzt030 mt1vvsff1sourceobserversourceoobservervv1 ffPhysics 207: Lecture 29, Pg 6Example Sound IntensityConsidering only the position of the cart, what is the intensity of the reflected sound to this observer? (In principle one would have to look at the energy per unit time in the moving frame.) vcart t + vsound t = 2 x 30 m = 60 m t = 60 / (340+20) = 0.17 s  dsound = 340 * 0.17 m = 58 m I = 1 / (4 582) = 2.4 x 10-5 W/m2 or 74 dBst030 mt1Physics 207: Lecture 29, Pg 7Doppler 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.Physics 207: Lecture 29, Pg 8Ch. 21: Wave SuperpositionQ: What happens when two waves “collide” ?A: They ADD together! We say the waves are “superimposed”.Physics 207: Lecture 29, Pg 9Interference of Waves2D Surface Waves on WaterIn phase sources separated by a distance ddPhysics 207: Lecture 29, Pg 10Principle of superpositionThe superposition of 2 or more waves is called interferenceConstructive interference:These two waves are in phase.Their crests are aligned.Their superposition produces a wave with amplitude 2aDestructive interference:These two waves are out of phase.The crests of one are aligned with the troughs of the other.Their superposition produces a wave with zero amplitudePhysics 207: Lecture 29, Pg 11Interference: space and timeIs this a point of constructiveor destructive interference?What do we need to do to make the sound from these two speakers interfere constructively?Physics 207: Lecture 29, Pg 12Interference of SoundSound waves interfere, just like transverse waves do. The resulting wave (displacement, pressure) is the sum of the two (or more) waves you started with.||||21rrr])//(2cos[),(2222 TtrrAtrD])//(2cos[),(1111 TtrrAtrDrDifferencePathPhysics 207: Lecture 29, Pg 13Interference of SoundSound waves interfere, just like transverse waves do. The resulting wave (displacement, pressure) is the sum of the two (or more) waves you started with.||||21rrr,...2,1,0)21( 22ceinterferen edestructiv Maximum)(22 22ceinterferen veconstructi Maximum212121mmrmrmr])//(2cos[),(2222 TtrrAtrD])//(2cos[),(1111 TtrrAtrDrPhysics 207: Lecture 29, Pg 14Example InterferenceA speaker sits on a pedestal 2 m tall and emits a sine wave at 340 Hz (the speed of sound in air is 340 m/s, so  = 1m ). Only the direct sound wave and that which reflects off the ground at a position half-way between the speaker and the person (also 2 m tall) makes it to the persons ear.How close to the speaker can the person stand (A to D) so they hear a maximum sound intensity assuming there is no “phase change” at the ground (this is a bad assumption)?The distances AD and BCD have equal transit times so the sound waves will be in phase. The only need is for AB = t1t0t0ABADCdhPhysics 207: Lecture 29, Pg 15Example InterferenceThe geometry dictates everything else.AB = AD = BC+CD = BC + (h2 + (d/2)2)½ = dAC = AB+BC =   +BC = (h2 + d/22)½ Eliminating BC gives  +d = 2 (h2 + d2/4)½ + 2d + d2 = 4 h2 + d2 1 + 2d = 4 h2 /   d = 2 h2 /  – ½ = 7.5 mt1t0t0ABADC7.54.253.25Because the ground is more dense than air there will be a phase change of  and so we really should set AB to/2 or 0.5 m.Physics 207: Lecture 29, Pg 16Exercise SuperpositionTwo continuous


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UW-Madison PHYSICS 207 - PHYSICS 207 Lecture Notes

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