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

Frequency received by motorist (directly)
`f_(1) = f((v+v_(o))/(v+v_(s))) = f((v+v_(m))/(v+v_(b)))`
Frequency received by wall
`f' = f((v)/(v-v_(b)))`
Now wall will act as source of frequency `f'` and motorist as observer.
Second frequency received by motorist
`f_(2) = f'((v+v_(m))/(v)) = f((v+v_(m))/(v-v_(b)))`
Beat frequency `= f_(2) - f_(1) = (f(v+v_(m)) (2v_(b)))/((v^(2) - v_(b)^(2)))`