Question

A siren of frequency n approaches a stationary observer and then recedes from the observer. If the velocity of source (V) << the velocity of sound(C), the apparent change in frequency is :

• A. 2nV/C
• B. 2nC/V
• C. n/V
• D. 2VC/n

Answer 1

The correct option is A 2nV/C

Given : $V_s=Vm/s$ (source moving forwards observes )
then $V_s=-Vm/s$ (source moving away form observer)
$V=Cm/s$
$V_0=0m/s$
$\delta =nHz$

By doppler effect formula: ${\delta }^1=\delta \left(\frac{V-V_0}{V-V_s}\right)$

For case when source is moving towards observer:
${\delta }_1^1=n\left(\frac{C-0}{C-V}\right)=\frac{nC}{C-V}$

For case when source is moving away from observer:
${\delta }_2^1=n\left(\frac{C-0}{C-(-V)}\right)=\frac{nC}{C+V}$

${\delta }_1^1-{\delta }_2^1=\frac{nC}{C-V}-\frac{nC}{C+V}$

$=nC\left(\frac{c+v-(c-v)}{C^2-V^2}\right)$

$=nC\left(\frac{2V}{C^2-V^2}\right)$

$=nC\left(\frac{2V}{C^2}\right)(\therefore V<<C)$

$=\frac{2nV}{C}$