Question

When a source of sound of frequency $f$ crosses a stationary observer with a speed $v_s$($<<$ speed of sound $v$), the apparent change in frequency $\Delta f$ is given by

• A. $\frac{2f{v}_s}{v}$
• B. $2fv{v}_s$
• C. $\frac{2fv}{v_s}$
• D. $\frac{f{v}_s}{v}$

Answer 1

The correct option is A $\frac{2f{v}_s}{v}$

Let $f_1$ be apparent frequency when source is moving towards observer
$f_2$ be apparent frequency when source is moving away from observer
Then change in frequency
$\Delta f=f_1-f_2$

$=f\left(\frac{v}{v-v_s}\right)-f\left(\frac{v}{v+v_s}\right)$

$=f[{(1-\frac{v_s}{v})}^{-1}-{(1+\frac{v_s}{v})}^{-1}]$
Using Binomial Theorem

$\approx f[(1+\frac{v_s}{v})-(1-\frac{v_s}{v})]=\frac{2f{v}_s}{v}$