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 the speed of sound, obtain an expression for the beat frequency heard by the motorist.

Answer

B

Explanation

KEY CONCEPT : Motorist will listen two sound waves. One directly from the sound source and other reflected from the fixed wall. Let the apparent frequencies of these two waves as received by motorist are `f' and f"` respectively. For direct sound : `V_(m)` will be position as it moves towards the source and tries to increase the apparent frequency. `V_(b)` hence tries to decrease the apparent frequency value.
`f' = (v + v_(m))/(v + v_(b))f` ....(1)

For reflected sound : For sound waves moving towards stantionary obsrever (i.e wall), frequency of sound as heard by wall
`f_(1) = (v)/(v - v_(b))f`
After reflection of sound waves having frequency `f_(1)` for the moving obsrever i.e. motorist. As direction of motion of motorist is opposite to direction of sound waves, hence frequency `f"` of reflected sound waves as received by the motorist is
`f" = (v + v_(m))/(v)f_(1) = (v + v_(m))/(v - v_(b))f` ...(2)
Hence, beat frequency as heard by the motorist
`Deltaf = f" - f' = ((v + v_(m))/(v - v_(b)) - (v + v_(m))/(v + v_(b)))f`
or, `Deltaf = (2v_(b)(v + v_(m)))/(v^(2) - v_(b)^(2))`