Question

A sound source emits a frequency of 180 Hz when moving towards a rigid wall with speed 5 m/s and an observer is moving away from the wall with speed 5 m/s. Both source and observer moves on a straight line which is perpendicular to the wall. The number of beats per second heard by the observer will be [speed of sound=355 m/s]
A. 5 beats/s
B. 6 beats/s
C. 10 beats/s
D. 8 beats/s

Answer 1

Hint:When the source of the sound and the observer are in relative motion then due to Doppler’s effect there is apparent change in the frequency. The beat frequency is the difference of the frequencies of the two waves.

Formula used:
\[{f_{ap}} = {f_o}\left( {\dfrac{{v \pm {v_o}}}{{v \pm {v_s}}}} \right)\]
where \[{f_{ap}}\] is the apparent frequency heard by the listener moving with speed \[{v_o}\] with respect to the source which is moving with speed \[{v_s}\], \[{f_o}\] is the original frequency and v is the speed of sound in air.
\[{f_{beat}} = \left| {{f_1} - {f_2}} \right|\]
Here \[{f_{beat}}\] is the beat frequency, \[{f_1}\] and \[{f_2}\] are the frequencies of two sound waves.

Complete step by step solution:
The frequency of the reflected wave is the same as the incident wave. So, when the sound wave is reflected from the wall then the stationary wall acts as the sound source and the observer is moving away from the wall. When the sound source is stationary wall and the observer is moving away from the source, then the apparent frequency reaching the observer will be,
\[{f_{ap2}} = \left( {{f_o}} \right)\left( {\dfrac{{355 - 5}}{{355}}} \right) \\ \]
\[\Rightarrow {f_{ap2}} = \left( {180Hz} \right)\left( {\dfrac{{350}}{{355}}} \right) \\ \]
\[\Rightarrow {f_{ap2}} = 177.46\,Hz \\ \]
The apparent frequency of the sound when the observer is moving towards the approaching sound source is,
\[{f_{ap3}} = {f_o}\left( {\dfrac{{355 + 5}}{{355 - 5}}} \right) \\ \]
\[\Rightarrow {f_{ap3}} = \left( {180Hz} \right)\left( {\dfrac{{360}}{{350}}} \right) \\ \]
\[\Rightarrow {f_{ap3}} = 185.14\,Hz \\ \]
The beat will be formed by the wave coming from the source and the wave coming from the wall. So, the beat frequency is,
\[{f_{beat}} = \left( {185.14 - 177.45} \right)\,beats/s \\ \]
\[\therefore {f_{beat}} \approx 8\,beats/s\]
Hence, the number of beats heard by the observer is 8 beats per seconds.

Therefore, the correct option is D.

Note: While taking the wall as the source of the reflected sound wave, we should take the frequency of the reflected sound wave as original. We don’t have to apply Doppler’s effect for the sound source and the stationary wall.