Question

What is Doppler effect? Obtain an expression for the apparent frequency of sound heard when the source is in motion with respect to an observer at rest.

Answer 1

The Apparent change in the frequency of sound when the source, the observer and the medium are in a relative motion is called Doppler Effect.
Let S be source of sound moving with velocity $v_s$ towards stationary listener O.
Distance travelled in time time period, $T=v_s.T$
Therefore apparent wavelength ${\lambda }^{\prime }=\lambda -v_s.T$
If frequency is $\nu$ then ,
$\lambda ={\lambda }^{\prime }-v_{s}T$
$\lambda ={\lambda }^{\prime }-\frac{v_s}{\nu }=\frac{v-v_s}{\nu }$; $v=\nu \lambda$
If apparent frequency heard by listener is ${\nu }^{\prime }$
${\nu }^{\prime }=\frac{v}{{\lambda }^{\prime }}=\frac{\nu v}{v-v_s}$
Thus apparent frequency is greater than the actual frequency as source moves towards listener.
Similarly if source moves away from listener.
${\nu }^{\prime }=\frac{v}{{\lambda }^{\prime }}=\frac{\nu v}{v+v_s}$