Question

An engine whistling at a constant frequency $n_o$ 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 $f(f_{after}=f\times f_{before})$. The actual difference in the frequencies of the sound heard by him before and after the engine crosses him is :

• A. $\frac{1}{2}{n}_o(1-f^2)$
• B. $\frac{1}{2}{n}_o\left(\frac{1-f^2}{f}\right)$
• C. $n_o\left(\frac{1-f}{1+f}\right)$
• D. $\frac{1}{2}{n}_o\left(\frac{1-f}{1+f}\right)$

Answer 1

The correct option is B $\frac{1}{2}{n}_o\left(\frac{1-f^2}{f}\right)$

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,

$n_2=f{n}_1$, $............................\left(1\right)$

Also the natural freq of the source is $n_0$.
Then we can write,

$n_1=\frac{V}{V-V_s}{n}_0$

$n_2=\frac{V}{V+V_s}{n}_0$

Substitute these values in equation $\left(1\right)$,

And apply componendo-dividendo,

We will get, $\frac{V_s}{V}=\frac{1-f}{1+f}$, $.....................\left(2\right)$,

We are asked, the difference of

$n_1-n_2=(\frac{1}{1-\frac{V_s}{V}}-\frac{1}{1+\frac{V_s}{V}})n_0$

Substitute the value of $\frac{V_s}{V}$ from equation $\left(2\right),$

$n_1-n_2=\frac{1}{2}{n}_0\frac{1-f^2}{f}$,

Option "B" is correct.