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Unformatted text preview: 16 Wave Motion CHAPTER OUTLINE 16.1 Propagation of a Disturbance 16.2 The Traveling Wave Model 16.3 The Speed of Waves on Strings 16.5 Rate of Energy Transfer by Sinusoidal Waves on Strings 16.6 The Linear Wave Equation ANSWERS TO QUESTIONS Q16.1 As the pulse moves down the string, the particles of the string itself move side to side. Since the mediumhere, the stringmoves perpendicular to the direction of wave propagation, the wave is transverse by defi nition. Q16.2 To use a slinky to create a longitudinal wave, pull a few coils back and release. For a transverse wave, jostle the end coil side to side. *Q16.3 (i) Look at the coeffi cients of the sine and cosine functions: 2, 4, 6, 8, 8, 7. The ranking is d = e > f > c > b > a. (ii) Look at the coeffi cients of x . Each is the wave number, 2 , so the smallest k goes with the largest wavelength. The ranking is d > a = b = c > e > f. (iii) Look at the coeffi cients of t . The absolute value of each is the angular frequency = 2 f . The ranking is f > e > a = b = c = d. (iv) Period is the reciprocal of frequency, so the ranking is the reverse of that in part iii: d = c = b = a > e > f. (v) From v = f = k , we compute the absolute value of the ratio of the coeffi cient of t to the coeffi cient of x in each case. From a to f respectively the numerical speeds are 5, 5, 5, 7.5, 5, 4. The ranking is d > a = b = c = e > f. *Q16.4 From v = T , we must increase the tension by a factor of 4 to make v double. Answer (b). *Q16.5 Answer (b). Wave speed is inversely proportional to the square root of linear density. *Q16.6 (i) Answer (a). Higher tension makes wave speed higher. (ii) Answer (b). Greater linear density makes the wave move more slowly. Q16.7 It depends on from what the wave refl ects. If refl ecting from a less dense string, the refl ected part of the wave will be right side up. Q16.8 Yes, among other things it depends on. The particle speed is described by v y ,max v = = = A f A A 2 2 . Here v is the speed of the wave. 427 13794_16_ch16_p427-448.indd 427 13794_16_ch16_p427-448.indd 427 12/11/06 5:03:19 PM 12/11/06 5:03:19 PM 428 Chapter 16 *Q16.9 (a) through (d): Yes to all. The maximum particle speed and the wave speed are related by v y ,max v = = = A f A A 2 2 . Thus the amplitude or the wavelength of the wave can be adjusted to make either v y ,max or v larger. Q16.10 Since the frequency is 3 cycles per second, the period is 1 3 second = 333 ms. Q16.11 Each element of the rope must support the weight of the rope below it. The tension increases with height. (It increases linearly, if the rope does not stretch.) Then the wave speed v = T increases with height....
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