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  4. A train sounds its whistle as it approaches an observer standing at a point near the track. The observer measures a frequency of 216 Hz as the train approaches and a frequency of 184 Hz as the train leaves. What is the frequency of its whistle?

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Q. A train sounds its whistle as it approaches an observer standing at a point near the track. The observer measures a frequency of $216\, Hz$ as the train approaches and a frequency of $184\, Hz$ as the train leaves. What is the frequency of its whistle?

KEAMKEAM 2019Waves

Solution:

Case I When train is moving towards a stationary observer.
According to Doppler's effect of sound,
apparent frequency heard, $n'=\frac{n v}{v-v_{s}}$
where, $n=$ frequency of source, $v=$ speed of sound and $v_{s}=$ speed of source.
$\Rightarrow v_{s}=v\left[1-\frac{n}{n'}\right]$ ...(i)
Case II When train is leaving the standing observer.
Again, according to the Doppler's effect of sound,
Apparent frequency heard,
$n''=\frac{n v}{v+ v_{s}} \Rightarrow v_{s}=v\left[\frac{n}{n'}-1\right]$ ...(ii)
From Eqs. (i) and (ii), we get
$\frac{n}{n'}+\frac{n}{n''}=2$
Given, $n'=216\, Hz$ and $n''=184\, Hz$
Substituting these values in the above equation, we get
$n=\frac{2 \times 216 \times 184}{(216+184)} \approx 200\, Hz$