Question

Does the change in frequency due to Doppler effect depend on
(i) distance between source and observer?
(ii) The fact that the source is moving towards the observer or observer is moving towards the source?

Answer 1

Hint:When the sounding source is in motion with respect to the listener then we use the Doppler’s frequency shift formula for longitudinal waves to determine the relation between the actual frequency of the wave and the apparent frequency of the wave.

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 which is moving with speed \[{v_o}\], the source which is moving with speed \[{v_s}\], \[{f_o}\] is the original frequency and v is the speed of sound in air.

Complete step by step solution:
From the Doppler’s effect formula,
\[{f_{ap}} = {f_o}\left( {\dfrac{{v \pm {v_o}}}{{v \pm {v_s}}}} \right)\]
Here, the distance between the source and the observer has no effect on the apparent frequency of the sound. So, there is no change in frequency due to Doppler’s effect.

When the source is moving towards an stationary observer then the apparent frequency heard by the observer is,
\[{f_{ap}} = {f_o}\left( {\dfrac{v}{{v - {v_s}}}} \right)\]
As \[{v_s} \ne 0\], so the apparent frequency is greater than the original frequency. Hence, there is change in the frequency due to the Doppler Effect when the source is moving towards the observer.

When the observer is moving towards an stationary sound source then the apparent frequency heard by the observer is,
\[{f_{ap}} = {f_o}\left( {\dfrac{{v + {v_o}}}{v}} \right)\]
As \[{v_o} \ne 0\], so the apparent frequency is greater than the original frequency. Hence, there is change in the frequency due to the Doppler Effect when the observer is moving towards the source.

Hence,
(i) There is no effect on the frequency due to distance between the source and the observer.
(ii) There is change in frequency when either the source is moving towards the observer or the observer is moving towards the source due to Doppler’s effect.

Note: We should not be confused with the relative motion between the source and the observer with the rate of change of distance between them as the distance causes change in frequency due to Doppler’s effect.