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Chapter 18

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

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18 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 Plates 18.7 Beats: Interference in Time 18.8 Non-Sinusoidal Wave Patterns Superposition and Standing Waves ANSWERS TO QUESTIONS Q18.1 No. Waves with other waveforms are also trains of disturbance that add together when waves from different sources move through the same medium at the same time. Q18.2 The energy has not disappeared, but is still carried by the wave pulses. Each particle of the string still 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. Q18.3 No. A wave is not a solid object, but a chain of disturbance. As described by the principle of superposition, the waves move through each other. Q18.4 They can, wherever the two waves are nearly enough in phase that their displacements will add to create a total displacement greater than the amplitude of either of the two original waves. When two one-dimensional sinusoidal waves of the same amplitude interfere, this condition is satisfied whenever the absolute value of the phase difference between the two waves is less than 120°. Q18.5 When the two tubes together are not an efficient transmitter of sound from source to receiver, they are an efficient reflector. The incoming sound is reflected back to the source. The waves reflected by the two tubes separately at the junction interfere constructively. Q18.6 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. Q18.7 Each of these standing wave patterns is made of two superimposed waves of identical frequencies traveling, and hence transferring energy, in opposite directions. Since the energy transfer of the waves are equal, then no net transfer of energy occurs. Q18.8 Damping, and non–linear effects in the vibration turn the energy of vibration into internal energy. Q18.9 The air in the shower stall can vibrate in standing wave patterns to intensify those frequencies in your voice which correspond to its free vibrations. The hard walls of the bathroom reflect sound very well to make your voice more intense at all frequencies, giving the room a longer reverberation time. The reverberant sound may help you to stay on key. 523

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524 Superposition and Standing Waves Q18.10 The trombone slide and trumpet valves change the length of the air column inside the instrument, to change its resonant frequencies. Q18.11 The vibration of the air must have zero amplitude at the closed end. For air in a pipe closed at one end, the diagrams show how resonance vibrations have NA distances that are odd integer submultiples of the NA distance in the fundamental vibration. If the pipe is open, resonance
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