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Unformatted text preview: 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 (i) If the end is fi 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 0 1 0 2 A = ( ) = . . m m . Answer (b). *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 confi 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 λ / 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 No. The total energy of the pair of waves remains the same. Energy missing from zones of destructive interference appears in zones of constructive interference. *Q18.5 Answer (c). The two waves must have slightly different amplitudes at P because of their different distances, so they cannot cancel each other exactly. Q18.6 Damping, and non–linear effects in the vibration turn the energy of vibration into internal energy. *Q18.7 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). *Q18.8 The fundamental frequency is described by f L 1 2 = v , where v = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ T μ 1 2 (i) If L is doubled, then f L 1 1 ~ − will be reduced by a factor 1 2 . Answer (f ). (ii) If μ is doubled, then f 1 1 2 ~ μ − will be reduced by a factor 1 2 . Answer (e). (iii) If T is doubled, then f T 1 ~ will increase by a factor of 2 . Answer (c). 473 13794_18_ch18_p473496.indd 473 13794_18_ch18_p473496.indd 473 1/3/07 8:14:55 PM 1/3/07 8:14:55 PM *Q18.9 Answer (d). The energy has not disappeared, but is still carried by the wave pulses. Each par ticle of the string still has kinetic energy. This is similar to the motion of a simple pendulum....
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 Spring '10
 Smith
 Physics, Wavelength, Standing wave, Hz

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