A band playing music at frequency f is moving towards a wall at a speed vb. A mo

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Question -

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

Options:

A .

\frac{v + v_{m}}{v + v_{b}} f

B .

\frac{v + v_{m}}{v - v_{b}} f

C .

\frac{2 v_{b} (v + v_{m})}{v^{2} - v_{b}^{2}} f

D .

\frac{2 v_{m} (v + v_{b})}{v^{2} - v_{m}^{2}} f

Answer: Option C
:
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

A Band Playing Music At Frequency F Is Moving Towards A Wall...

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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