Doppler's effect

Question

A band playing music at a frequency f is moving towards a wall at a speed u. A motorist is following the band with the same speed u. If v is the speed of sound, the beat frequency heard by the motorist is

Moderate

Solution

Let u_{m} = speed of motorist and u_{b} = Speed of the band. The motorist will hear two sounds-one coming directly from the band and the other reflected from the wall. The apparent frequency of the direct sound is given by f.

The reflected sound can be regarded as coming from the mirror image which is approaching the motorist with a speed v_{b}. Hence, the apparent frequency of the reflected sound is given by (the observer and the source of sound are both approaching each other)

${\mathrm{\Delta f}}_{1}=\mathrm{f}\left(\frac{\mathrm{v}+{\mathrm{u}}_{\mathrm{m}}}{\mathrm{v}-{\mathrm{u}}_{\mathrm{b}}}\right)-\mathrm{f}=\mathrm{f}\left(\frac{\mathrm{v}+\mathrm{u}-\mathrm{v}+{\mathrm{u}}_{}}{\mathrm{v}-\mathrm{u}}\right)=\frac{2\mathrm{uf}}{\mathrm{v}-\mathrm{u}}\phantom{\rule{1em}{0ex}}\left(\because {\mathrm{u}}_{\mathrm{m}}={\mathrm{u}}_{\mathrm{b}}=\mathrm{u}\right)$

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