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

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 . v+vmv+vbf

B . v+vmv−vbf

C . 2vb(v+vm)v2−v2bf

D . 2vm(v+vb)v2−v2mf

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

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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