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Lecture 28, Dec. 8

Lecture 28, Dec. 8. 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 28, Dec. 8

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  1. Lecture 28, Dec. 8 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 • HW12, Due Friday, Dec. 12th • For Wednesday, Review for final, Evaluations

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

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

  6. 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

  7. 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?

  8. 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.

  9. 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 343 Hz (the speed of sound in air is 343 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

  10. 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.

  11. 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)

  12. 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

  13. Reflection of a Wave, Fixed End Animation

  14. Reflection of a Wave, Free End Animation

  15. Transmission of a Wave, Case 1 • When the boundary is intermediate between the last two extremes ( The right hand rope is massive or massless.) then part of the energy in the incident pulse is reflected and part is transmitted • Some energy passes through the boundary • Here mrhs > mlhs Animation

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

  17. 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

  18. 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

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

  20. 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

  21. Standing waves in a pipe

  22. Combining Waves Fourier Synthesis

  23. 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

  24. 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

  25. DESTRUCTIVEINTERFERENCE CONSTRUCTIVEINTERFERENCE Superposition & Interference • Consider two harmonic waves A and B meet at t=0. • They have same amplitudes and phase, but 2 = 1.15 x 1. • The displacement versus time for each is shown below: Beat Superposition A(1t) B(2t) C(t) =A(t)+B(t)

  26. Superposition & Interference • Consider A + B, yA(x,t)=A cos(k1x–w1t)yB(x,t)=A cos(k2x–w2t) And let x=0, y=yA+yB = 2A cos[2p (f1 – f2)t/2] cos[2p (f1 + f2)t/2] and |f1 – f2| ≡ fbeat = = 1 / Tbeat A(1t) B(2t) t Tbeat C(t)=A(t)+B(t)

  27. Exercise Superposition • The traces below show beats that occur when two different pairs of waves are added (the time axes are the same). • For which of the two is the difference in frequency of the original waves greater? Pair 1 Pair 2 The frequency difference was the samefor both pairs of waves. Need more information.

  28. Lecture 28, Dec. 8 • Assignment • HW12, Due Friday, Dec. 12th

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