A band playing music at frequency f is moving towards a wall at a speed $v_{b}$ . A motorist is following the band with a speed $v_{m}$ . If v is the speed of sound, the expression for the beat frequency heard by the motorist is

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Solution

The correct option is C
The motorist receives two sound waves: direct one and that reflected from the wall.
$f^{'} = \frac{v + v_{m}}{v + v_{b}} f$
For reflected sound waves:
Frequency of sound wave reflected from the wall is
$f "= \frac{v}{v - v_{b}} \times f$

Frequency of the reflected waves as received by the moving motorist is
$f^{'} "= \frac{v + v_{b}}{v - v_{b}} \times f^{'} - \frac{v + v_{m}}{v + v_{b}} f$
Therefore, the beat frequency is
$f^{'} "= \frac{v + v_{m}}{v - v_{b}} \times f - \frac{v + v_{m}}{v + v_{b}} f$
$= \frac{2 v_{b} (v + v_{m})}{v^{2} - v_{b}^{2}} f$

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