Question

When a source of sound of frequency ‘n’ crosses a stationary observer with a speed V_s (<< speed of sound V), the apparent change in frequency is given by________;

• A. $\frac{n{V}_s}{v}$
• B. $2nV{V}_s$
• C. $\frac{2n{V}_s}{V}$
• D. $\frac{2n{V}_{}}{V_s}$

Answer 1

The correct option is C

$\frac{2n{V}_s}{V}$

Option C is correct: $\frac{2n{V}_s}{V}$

Explanation:

1. Let $n_1$ be the apparent frequency when the source is moving towards the observer and $n_2$ be the apparent frequency when the source is moving away from the observer.
2. Then change in frequency $∆n=n_1-n_2$
3. Apparent frequency when source is moving towards observer $n_1=n\left(\frac{v}{v-v_s}\right)$
4. Apparent frequency when source is moving away from observer $n_2=n\left(\frac{v}{v+v_s}\right)$

Substituting values in equation,

$$\begin{gathered} ∆n=n\left(\frac{v}{v-v_s}\right)-n\left(\frac{v}{v+v_s}\right) \\ ∆n=n[{(1-\frac{v_s}{v})}^{-1}-{(1+\frac{v_s}{v})}^{-1}] \\ U\sin gbinomial, \\ ∆n=n\left[\right(1+\frac{v_s}{v})-(1-\frac{v_s}{v}\left)\right] \\ ∆n=\frac{2n{v}_s}{v} \end{gathered}$$