A band playing music at a 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, obtain an expression for the beat frequency heard by the motorist.

Options

A. `(v+v_m)/(v+v_b)f`
B. `(v+v_m)/(v-v_b)f`
C. `(2v_b(v+v_m))/(v^2-v_b^2)f`
D. `(2v_m(v+v_b))/(v^2-v_m^2)f`

Answer

C

Explanation

The motorist receives two sound waves: direct one and that reflected from the wall.
`f'=(v+v_m)/(v+v_b)f`
For reflected from the wall is
`f''=(v)/(v-v_b)xxf`
Frequency of the reflected waves as received by the moving motorist is
`f''=(v+v_m)/(v)xxf''=(v+v_m)/(v-v_b)xxf`
Therefore the beat frequency is
`f''-f'=(v+v_m)/(v-v_b)xxf-(v+v_m)/(v+v_b)f`
`=(2v_b(v+v_m))/(v^2-v_b^2)f`