Question

A source emitting a sound of frequency f is placed at a large distance from an observer. The source starts moving towards the observer with a uniform acceleration ‘a’. Find frequency heard by the observer corresponding to the wave emitted just after the source starts. The speed of sound in medium is v.

• A. $\frac{v{f}^2}{2vf-a}$
  
• B. $\frac{2v{f}^2}{2vf+a}$
  
• C. $\frac{2v{f}^2}{3vf-a}$
• D. $\frac{2v{f}^2}{2vf-a}$

Answer 1

The correct option is D $\frac{2v{f}^2}{2vf-a}$

Suppose at t = 0, distance between source and observer is l. First, wave pulse (say $p_1$) is emitted at this instant. This pulse will reach the observer after a time

$t_1=\frac{l}{v}$
Source will emit the next pulse (say $p_2$) after a time $T(=\frac{1}{f})$.
During this time the source will move a distance $\left(\frac{1}{2}\right)a{T}^2$ towards the observer. This pulse $p_2$ will reach the observer in a time
$t_2=T+\frac{1-\frac{1}{2}a{T}^2}{v}$
The changed time period as observed by the observer is
$T^1=t_2-t_1=T+\frac{1}{v}-\frac{1}{2}\frac{a{T}^2}{v}-\frac{1}{v}$
Substituting $T^1=\frac{1}{f^{\prime }}andt=\frac{1}{f}$ in the above equation, we get
$f^{\prime }=\frac{2v{f}^2}{2vf-a}$