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18
Superposition and Standing Waves
CHAPTER OUTLINE
18.1 18.2 18.3 18.4 18.5 18.6 18.7 18.8 Superposition and Interference Standing Waves Standing Waves in a String Fixed at Both Ends Resonance Standing Waves in Air Columns Standing Waves in Rod and Membranes Beats: Interference in Time Nonsinusoidal Wave Patterns

ANSWERS TO QUESTIONS
Q18.1 No. Waves with all waveforms interfere. Waveswith 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. If the end is fixed, there is inversion of the pulse upon reflection. Thus, when they meet, they cancel and the amplitude is zero. Answer (d). If the end is free, there is no inversion on reflection. When they meet, the amplitude is 2 A = 2 ( 0.1 m ) =0.2 m. Answer (b).

*Q18.2 (i)

(ii)

*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 configuration, the extra path length is 0.4 m = λ 2, for destructive interference.Another 0.1 m and we are at r2 − r1 = 3λ 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. 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.

Q18.4*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 differentspeeds 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 f1 = (i) (ii) If L is doubled, then f1 If μ is doubled, then f1 ⎛T ⎞ v , where v = ⎜ ⎟ ⎝μ⎠ 2L 1 L−1 will be reduced by a factor . Answer (f ). 2 1 μ −1 2 will be reduced by a factor . Answer (e). 2 T will increase by a factor of 2. Answer (c).
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(iii) If T is doubled, then f1

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*Q18.9

Answer (d). The energy has not disappeared, but is still carried by the wave pulses. Each particle of the stringstill has kinetic energy. This is similar to the motion of a simple pendulum. The pendulum does not stop at its equilibrium position during oscillation—likewise the particles of the string do not stop at the equilibrium position of the string when these two waves superimpose.

*Q18.10 The resultant amplitude is greater than either individual amplitude, wherever the two waves are nearly enoughin phase that 2Acos(φ 2) is greater than A. This condition is satisfied whenever the absolute value of the phase difference φ between the two waves is less than 120°. Answer (d). Q18.11 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 precisely in tune, you willhear 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.

*Q18.12 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....