# A band playing music it a frequency f is moving towards a wall at a speed vb. A motorist .is following the band with a speed vm, If v is the speed of sound, obtain an expression for the beat frequency heard by the motorist.

9 years ago
Hello Student,
KEY CONCEPT : Motorist will listen two sound waves. One directly 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 positive as it moves towards the source and tries to increase the apparent frequency. V b will be taken positives as it move away from the observer and 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 stationary observer (i.e. wall ), frequency of sound as heard by wall
f1 = v/v – v b f
After reflection of sound waves having frequency f1 fixed wall acts as a stationary source of frequency f1 for the moving observer i.e. motorist. As direction motion of motorist is of opposite to direction of sound waves, hence frequency f” of reflected sound waves as received by the motorist is
f” = v + v m/v f1 = v + v m /v – v b f . . . . . . . . . . . . . . . . . . . . . . . (2)
Hence, beat frequency as heard by the motorist
∆f = f” – f’ = (v + vm/v – vb – v + vm/v + vb) f
Or, ∆f = 2vb (v + vm) f / v2 – vb2

Thanks