Question

If wind blows from a stationary sounding object to a stationary listener, then the apparent frequency n' and actual frequency n are related as :
A. \[n' \ge n\]
B. \[n' < n\]
C. \[n' = n\]
D. \[n' > n\]

Answer 1

Hint:When there is relative motion between the sounding source and 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 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.

Complete step by step solution:
It is given that the sounding source is at stationary,
\[{v_s} = 0\,m/s\]
The listener is also at rest,
\[{v_o} = 0\,m/s\]
Using Doppler’s effect formula, the apparent frequency heard by the listener will be,
\[{f_{ap}} = {f_o}\left( {\dfrac{{v \pm {v_o}}}{{v \pm {v_s}}}} \right) \\ \]
Where \[{f_{ap}}\]the apparent frequency is heard by the listener \[ = n'\], and \[{f_o}\] is the original frequency of the sound\[ = n\]

Putting the values in the Doppler’s effect formula, we get
\[n' = n\left( {\dfrac{{v \pm 0}}{{v \pm 0}}} \right) \\ \]
\[\Rightarrow n' = n\left( {\dfrac{v}{v}} \right) \\ \]
\[\therefore n' = n\]
So, the apparent frequency heard by the listener is equal to the original frequency of the sound originated from the sounding source.

Therefore, the correct option is C.

Note: The speed of air affects the speed of air. When the air is flowing in the direction of the sound then the speed of the sound is more than the speed of sound in still air. When the air is flowing in the direction opposite to the direction of the sound then the speed of the sound is less than the speed of sound in still air. But the frequency remains unchanged unless there is non-zero relative motion between the sounding source and the listener.