SM_chapter18

# SM_chapter18 - 18 Superposition and Standing Waves CHAPTER...

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18 Superposition and Standing Waves CHAPTER OUTLINE 18.1 Superposition and Interference 18.2 Standing Waves 18.3 Standing Waves in a String Fixed at Both Ends 18.4 Resonance 18.5 Standing Waves in Air Columns 18.6 Standing Waves in Rod and Membranes 18.7 Beats: Interference in Time 18.8 Nonsinusoidal Wave Patterns ANSWERS TO QUESTIONS Q18.1 No. Waves with all waveforms interfere. Waves with other wave shapes are also trains of disturbance that add together when waves from different sources move through the same medium at the same time. Q18.2 No. The total energy of the pair of waves remains the same. Energy missing from zones of destructive interfer- ence appears in zones of constructive interference. *Q18.3 In the starting situation, the waves interfere constructively. When the sliding section is moved out by 0.1 m, the wave going through it has an extra path length of 0.2 m = λ / 4, to show partial interference. When the slide has come out 0.2 m from the starting conF guration, the extra path length is 0.4 m = λ / 2, for destructive interference. Another 0.1 m and we are at r 2 r 1 = 3 / 4 for partial interference as before. At last, another equal step of sliding and one wave travels one wavelength farther to interfere constructively. The ranking is then d > a = c > b. *Q18.4 (i) If the end is F xed, there is inversion of the pulse upon reﬂ ection. Thus, when they meet, they cancel and the amplitude is zero. Answer (d). (ii) If the end is free, there is no inversion on reﬂ ection. When they meet, the amplitude is 2 2 01 02 A = ( ) = .. m m . Answer (b). *Q18.5 The strings have different linear densities and are stretched to different tensions, so they carry string waves with different speeds and vibrate with different fundamental frequencies. They are all equally long, so the string waves have equal wavelengths. They all radiate sound into air, where the sound moves with the same speed for different sound wavelengths. The answer is (b) and (e). *Q18.6 The fundamental frequency is described by f L 1 2 = v , where v = T μ 12 (i) If L is doubled, then fL 1 1 ~ will be reduced by a factor 1 2 . Answer (f ). (ii) If is doubled, then f 1 ~ will be reduced by a factor 1 2 . Answer (e). (iii) If T is doubled, then fT 1 ~ will increase by a factor of 2. Answer (c). Q18.7 What is needed is a tuning fork—or other pure-tone generator—of the desired frequency. Strike the tuning fork and pluck the corresponding string on the piano at the same time. If they are pre- cisely in tune, you will hear a single pitch with no amplitude modulation. If the two pitches are a bit off, you will hear beats. As they vibrate, retune the piano string until the beat frequency goes to zero. 425

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426 Chapter 18 *Q18.8 The bow string is pulled away from equilibrium and released, similar to the way that a guitar string is pulled and released when it is plucked. Thus, standing waves will be excited in the bow string. If the arrow leaves from the exact center of the string, then a series of odd harmonics will be excited. Even harmonies will not be excited because they have a node at the point where the string exhibits its maximum displacement. Answer (c).
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SM_chapter18 - 18 Superposition and Standing Waves CHAPTER...

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