/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 166 A sound wave of frequency \(v\) ... [FREE SOLUTION] | 91影视

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Chapter 9: Problem 166

A sound wave of frequency \(v\) travels horizontally to the right. It is reflected from a large vertical plane surface moving to the left with a speed \(V\). The speed of sound in the medium is \(c\). (A) The number of wave pulse striking the surface per second is \(\frac{v(c+V)}{c}\) (B) The wavelength of the reflected wave is \(\frac{c(c-V)}{v(c+V)}\) (C) The frequency of the reflected wave is \(\frac{v(c+V)}{(c-V)}\) (D) The number of beats heard by a stationary listener to the left of the reflecting surface is \(\frac{v V}{c-V}\)

Short Answer

Expert verified
The number of wave pulses striking the surface per second is \(v\frac{c + V}{c}\), the wavelength of the reflected wave is \(\frac{c(c - V)}{v(c + V)}\), the frequency of the reflected wave is \(\frac{v(c + V)}{c - V}\), and the number of beats heard by a stationary listener to the left of the reflecting surface is \(\frac{vV}{c - V}\)

Step by step solution

01

Number of waves striking the surface per second

When a wave hits a moving object, the frequency of the wave striking the object is different from the source frequency. This can be given by \(f' = f \frac{c + V}{c}\), where \(f'\) is the effective frequency, \(f\) is the original frequency, \(c\) is the speed of sound and \(V\) is the speed of the reflecting surface. Substituting the given values, we find \(f' = v\frac{c + V}{c}\)
02

Wavelength of the reflected wave

The wavelength of the sound after reflection can be given by the formula \(位' = \frac{c - V}{f'}\), where \(位'\) is the wavelength after reflection and \(f'\) is the effective frequency. Substituting the values, we get \(位' = \frac{c - V}{v\frac{c + V}{c}} = \frac{c(c - V)}{v(c + V)}\)
03

Frequency of the reflected wave

The frequency of the reflected wave can be given by the formula \(f'' = f' \frac{c}{c - V}\), where \(f''\) is the frequency of the reflected wave. Substituting the values, we get \(f'' = v\frac{c + V}{c} \frac{c}{c - V} = \frac{v(c + V)}{c - V}\)
04

Number of beats heard by a stationary listener

The number of beats heard by a stationary listener can be determined by the beat frequency equation which is the absolute difference in frequencies, which in this case would be the original frequency and the frequency of the reflected wave. The beat frequency \(b = f'' - v\). Substituting the values, we get \(b = \frac{v(c + V)}{c - V} - v = \frac{vV}{c - V}\)

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