Question

(II) As a bat flies toward a wall at a speed of 6.0 m/s, the bat emits an ultrasonic sound wave with frequency 30.0 kHz. What frequency does the bat hear in the reflected wave?

Short answer

The frequency the bat hear in the reflected wave is \(3.1 \times {10^4}\;{\rm{Hz}}\).

Step-by-step solution

Understanding the frequency the bat hears

In this problem, for the calculation of the frequency consider the wall towards which the bat flies as observer.

Given Data

The sound wave frequency is\(f = 30\;{\rm{kHz}}\).

The speed of the bat is \(v = 6\;{\rm{m/s}}\).

Evaluating the frequency does the bat hear in the reflected wave

The relation of frequency is given by,

\({f_{\rm{w}}} = f\left( {\frac{1}{{1 - \frac{v}{{{v_{\rm{s}}}}}}}} \right)\)

Here, \({v_{\rm{s}}}\) is the speed of the sound.

The relation to calculate the frequency of the bat is given by,

\(\begin{aligned}{c}{f_{\rm{b}}} &= {f_{\rm{w}}}\left( {1 + \frac{v}{{{v_{\rm{s}}}}}} \right)\\{f_{\rm{b}}} &= f\left( {\frac{1}{{1 - \frac{v}{{{v_{\rm{s}}}}}}}} \right)\left( {1 + \frac{v}{{{v_{\rm{s}}}}}} \right)\\{f_{\rm{b}}} &= f\left( {\frac{{{v_{\rm{s}}} + v}}{{{v_{\rm{s}}} - v}}} \right)\end{aligned}\)

Here, \({v_{\rm{s}}}\) is the speed of the sound.

On plugging the values in the above relation.

\(\begin{aligned}{l}{f_{\rm{b}}} &= \left( {30\;{\rm{Hz}} \times \frac{{{{10}^3}\;{\rm{kHz}}}}{{1\;{\rm{Hz}}}}} \right)\left( {\frac{{343\;{\rm{m/s}} + 6\;{\rm{m/s}}}}{{343\;{\rm{m/s}} - 6\;{\rm{m/s}}}}} \right)\\{f_{\rm{b}}} &= 3.1 \times {10^4}\;{\rm{Hz}}\end{aligned}\)

Thus, \({f_{\rm{b}}} = 3.1 \times {10^4}\;{\rm{Hz}}\) is the required frequency.