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Lecture 29

Lecture 29. Goals:. Chapter 20 Work with a few important characteristics of sound waves. (e.g., Doppler effect) Chapter 21 Recognize standing waves are the superposition of two traveling waves of same frequency Study the basic properties of standing waves

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Lecture 29

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  1. Lecture 29 Goals: • Chapter 20 • Work with a few important characteristics of sound waves. (e.g., Doppler effect) • Chapter 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. • Assignment • HW13, Due Friday, May 7h • Thursday, Finish up, begin review for final

  2. Doppler effect, moving sources/receivers

  3. Doppler 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 larger Doppler Example Audio Doppler Example Visual

  4. Doppler 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 l = 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.) t0 30 m A

  5. t0 t1 30 m Doppler 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 Hz

  6. t0 t1 30 m Example 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.) vcartDt + vsoundDt = 2 x 30 m = 60 m Dt = 60 / (340+20) = 0.17 s  dsound = 340 * 0.17 m = 58 m I = 1 / (4p 582) = 2.4 x 10-5 W/m2 or 74 dBs

  7. Doppler effect, moving sources/receivers • Three key pieces of information • Time of echo • Intensity of echo • Frequency of echo Plus prior knowledge of object being studied • With modern technology (analog and digital) this can be done in real time.

  8. Ch. 21: Wave Superposition • Q: What happens when two waves “collide” ? • A: They ADD together! • We say the waves are “superimposed”.

  9. Interference of Waves • 2D Surface Waves on Water In phase sources separated by a distance d d

  10. Principle of superposition Destructive interference: These two waves are out of phase. The crests of one are aligned with the troughs of the other. • The superposition of 2 or more waves is called interference Constructive interference: These two waves are in phase. Their crests are aligned. Their superposition produces a wave with amplitude 2a Their superposition produces a wave with zero amplitude

  11. Interference: space and time • Is this a point of constructive or destructive interference? What do we need to do to make the sound from these two speakers interfere constructively?

  12. Interference of Sound Sound waves interfere, just like transverse waves do. The resulting wave (displacement, pressure) is the sum of the two (or more) waves you started with.

  13. Interference of Sound Sound waves interfere, just like transverse waves do. The resulting wave (displacement, pressure) is the sum of the two (or more) waves you started with.

  14. t1 t0 d t0 D h A A B C Example 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 l = 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 = l

  15. Example Interference • The geometry dictates everything else. AB = lAD = BC+CD = BC + (h2 + (d/2)2)½ = d AC = AB+BC = l +BC = (h2 + d/22)½ Eliminating BC gives l+d = 2 (h2 + d2/4)½ l + 2ld + d2 = 4 h2 + d2 1 + 2d = 4 h2 / l d = 2 h2 / l – ½ = 7.5 m t1 t0 7.5 t0 D A A 4.25 3.25 B C Because the ground is more dense than air there will be a phase change of p and so we really should set AB to l/2 or 0.5 m.

  16. Exercise Superposition • Two continuous harmonic waves with the samefrequency and amplitude but, at a certain time, have a phase difference of 170° are superimposed. Which of the following best represents the resultant wave at this moment? Original wave (the other has a different phase) (A) (B) (D) (C) (E)

  17. Wave 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 reflectionof the pulse • The pulse is inverted

  18. Reflection of a Wave, Fixed End Animation

  19. Reflection of a Wave, Free End Animation

  20. Standing waves • Two waves traveling in opposite direction interfere with each other. If the conditions are right, same k & w, their interference generates a standing wave: DRight(x,t)= a sin(kx-wt) DLeft(x,t)= a sin(kx+wt) A standing wave does not propagate in space, it “stands” in place. A standing wave has nodes and antinodes Anti-nodes D(x,t)= DL(x,t) + DR(x,t) D(x,t)= 2a sin(kx) cos(wt) The outer curve is the amplitude function A(x) = 2a sin(kx) when wt = 2pn n = 0,1,2,… k = wave number = 2π/λ Nodes

  21. Standing waves on a string • Longest wavelength allowed is one half of a wave Fundamental: l/2 = L  l = 2 L Recall v = fl Overtones m > 1

  22. Violin, viola, cello, string bass Guitars Ukuleles Mandolins Banjos Vibrating Strings- Superposition Principle D(x,0) Antinode D(0,t)

  23. Standing waves in a pipe Open end: Mustbe a displacement antinode (pressure minimum) Closed end: Must be a displacement node (pressure maximum) Blue curves are displacement oscillations. Red curves, pressure. Fundamental: l/2l/2 l/4

  24. Standing waves in a pipe

  25. Combining Waves Fourier Synthesis

  26. Organ Pipe Example A 0.9 m organ pipe (open at both ends) is measured to have it’s first harmonic (i.e., its fundamental) at a frequency of 382 Hz. What is the speed of sound (refers to energy transfer) in this pipe? L=0.9 m f = 382 Hzandf l = vwith l = 2 L / m(m = 1) v = 382 x 2(0.9) m  v = 687 m/s

  27. Standing Waves • What happens to the fundamental frequency of a pipe, if the air (v =300 m/s) is replaced by helium (v = 900 m/s)? Recall: f l = v (A) Increases (B) Same (C) Decreases

  28. Lecture 29 • Assignment • HW13, Due Friday, May 7th

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