Question

What is the Doppler effect? Find an expression for the apparent frequency heard when the source is in motion and the listener is at rest. The third overtone produced by a vibrating string 2m long is 1200 Hz. What is the velocity of propagation of the wave?

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.
${\nu }^{\prime }=\frac{v}{{\lambda }^{\prime }}=\frac{\nu v}{v+v_s}$
Third overtone or 4th harmonic produced by vibrating string of $l=2m$ is $1200Hz$.
Frequency, $F_1=$ $\frac{4v}{2l}$

$\Rightarrow 1200=\frac{4\times v}{2\times 2}$
$\Rightarrow v=1200m/s$