Question 6 solutionA band playing music at frequency f is moving towards a wall with speed Vb . A momotorist is following the band with speed vmObtain a expression for the beat frequency heard by the motorist.

 

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 + v_m/v – v_b – v + v_m/v + v_b) f

Or, ∆f = 2v_b (v + v_m) f / v₂ – v_b²