An engine whistling at a constant frequency $n_0$ and moving with a constant velocity goes past a stationary observer. As the engine crosses him, the frequency of the sound heard by him changes by a factor $f$. The actual difference in the frequencies of the sound heard by him before and after the engine crosses him is
Diffcult
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Say $n_{1}$ is the freq observed by the observer before source cross the observer, and $n_{2}$ after source crossed the observer. and we are given that,
$n_{2}=f n_{1}$$\ldots \ldots .(1)$
Also the natural freq of the source is $n_{0}$
Then we can write,
Q$n_{1}=\frac{V}{V-V_{s}} n_{0}$
$n_{2}=\frac{V}{V+V_{s}} n_{0}$
Substitute these values in equation $(1),$
And apply componendo-dividendo,
We will get, $\frac{V_{s}}{V}=\frac{1-f}{1+f}, \ldots \ldots \ldots \ldots \ldots .(2)$
we are asked, the difference of
$n_{1}-n_{2}=\left(\frac{1}{1-\frac{V_{s}}{V}}-\frac{1}{1+\frac{V_{s}}{V}}\right) n_{0}$
Substitute the value of $\frac{V_{s}}{V}$ from equation $(2)$
$n_{1}-n_{2}=\frac{1}{2} n_{0} \frac{1-f^{2}}{f}$
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