First slide

Doppler

Question

A band playing music at a frequency n 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, the expression for the beat frequency heard by the motorist is

Difficult

A

$(\frac{v + v_{m}}{v + v_{b}}) n$
B

$(\frac{v + v_{m}}{v - v_{b}}) n$
C

$[\frac{2 v_{b} (v + v_{m})}{v^{2} - v_{b}^{2}}] n$
D

$[\frac{2 v_{m} (v + v_{b})}{v^{2} - v_{b}^{2}}] n$

Solution

The motorist receives two sounds : direct one and that reflected from the wall' For direct sound,
$n^{'} = [\frac{v + v_{m}}{v + v_{b}}] n For reflected sound, the reflected frequency of sound will be n^{''} = (\frac{V}{V - v_{b}}) n This frequency is received by the moving motorist . The apparent frequency received by him is given by n^{'''} = (\frac{v + v_{m}}{v}) n^{''} = (\frac{v + v_{m}}{v}) (\frac{v}{v - v_{b}}) n = (\frac{v + v_{m}}{v - v_{b}}) n ∴ Beat frequency = n^{"'} - n^{'} = (\frac{v + v_{m}}{v - v_{b}}) n - (\frac{v + v_{m}}{v + v_{b}}) n = [\frac{(v + v_{m}) 2 v_{b}}{v^{2} - v_{b}^{2}}] n$

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