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UW-Madison PHYSICS 207 - Lecture 29

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Page 1Physics 207 – Lecture 29Physics 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/receiversPage 2Physics 207 – Lecture 29Physics 207: Lecture 29, Pg 3Doppler effect, moving sources/receivers If the source of sound is moving Toward the observer ⇒λ seems smaller Away from observer ⇒λ seems larger If the observer is moving Toward the source ⇒λ seems smaller Away from source ⇒λ 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 λ = 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 mPage 3Physics 207 – Lecture 29Physics 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 partAt 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 mI = 1 / (4π 582) = 2.4 x 10-5W/m2or 74 dBst030 mt1Page 4Physics 207 – Lecture 29Physics 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”.Page 5Physics 207 – Lecture 29Physics 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 awave 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 awave with zero amplitudePage 6Physics 207 – Lecture 29Physics 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.||||21rrrrr−=∆])//(2cos[),(2222φλπ+−= TtrrAtrD])//(2cos[),(1111φλπ+−= TtrrAtrDr∆DifferencePathPage 7Physics 207 – Lecture 29Physics 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.||||21rrrrr−=∆,...2,1,0)21( 22ceinterferen edestructiv Maximum)(22 22ceinterferen veconstructi Maximum212121=+=−+∆≡∆=−+∆=∆=−+∆=∆mmrmrmrπφφλπφλφφπλφπλπφφλπφ])//(2cos[),(2222φλπ+−= TtrrAtrD])//(2cos[),(1111φλπ+−= 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 λ = 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 = λt1t0t0ABADCdhPage 8Physics 207 – Lecture 29Physics 207: Lecture 29, Pg 15Example Interference The geometry dictates everything else.AB = λ AD = BC+CD = BC + (h2+ (d/2)2)½= dAC = AB+BC = λ +BC = (h2+ d/22)½Eliminating BC gives λ+d = 2 (h2+ d2/4)½λ + 2λd + d2= 4 h2+ d21 + 2d = 4 h2 / λ d = 2 h2 / λ – ½= 7.5 mt1t0t0ABADC7.54.253.25Because the ground is more dense than air there will be a phase change of π and so we really should set AB to λ/2 or 0.5 m.Physics 207: Lecture 29, Pg 16Exercise Superposition Two continuous harmonic waves with the same frequency and amplitude but, at a certain time, have a phase difference of 170°are superimposed. Which of the follo wing best represents the resultant wave at this moment?(A)(E)(D)(C)(B)Original wave (the other has a different phase)Page 9Physics 207 – Lecture 29Physics 207: Lecture 29, Pg 17Wave motion at interfacesReflection of a Wave, Fixed End When the pulse reaches the support, the pulse moves back along the string in the opposite direction This is the reflection of the pulse The pulse is invertedPhysics


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UW-Madison PHYSICS 207 - Lecture 29

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