Question

A fixed source of sound emitting a certain frequency appears as $f_a$ when the observer is approaching the source with speed $v_0$ and $f_r$ when the observer recedes from the source with the same speed. The frequency of the source is

• A. $\frac{f_r+f_a}{2}$
• B. $\frac{f_r-f_a}{2}$
  
• C. $\sqrt{f_a{f}_r}$
  
• D. $\frac{2f_r{f}_a}{f_r+f_a}$

Answer 1

The correct option is A $\frac{f_r+f_a}{2}$

$f_a=\frac{v+v_0}{v}f=(1+\frac{v_0}{v})f$ and

$f_r=(1-\frac{v_0}{v})f$

$\frac{f_a}{f_r}=\frac{v+v_0}{v-v_0}$

$\Rightarrow (f_a-f_r)v=(f_a+f_r)v_0$

$\Rightarrow \frac{v}{v_0}=\frac{f_a+f_r}{f_a-f_r}$
and

$f_a-f_r=\frac{2v_0}{v}f=2\left(\frac{f_a-f_r}{f_a+f_r}\right)f$

$\Rightarrow f=\frac{f_a+f_r}{2}$