Question

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

• A. $\frac{\nu + \nu_{m}}{\nu + \nu_{b}}\upsilon$
• B. $\frac{\nu + \nu_{m}}{\nu - \nu_{b}}\upsilon$
• C. $\frac{2\nu_{b}(\nu + \nu_{m})}{\nu^{2} - \nu_{b}^{2}}\upsilon$
• D. $\frac{2\nu_{m}(\nu + \nu_{b})}{\nu^{2} - \nu_{m}^{2}}\upsilon$

Answer 1

Answer: (C)

$\frac{2\nu_{b}(\nu + \nu_{m})}{\nu^{2} - \nu_{b}^{2}}\upsilon$

Answer 2

The motorist receives two sound waves : direct one and that reflected from the wall.

For direction sound waves,

$\upsilon' = \frac{v + v_{m}}{v + v_{b}}\upsilon$

For reflected sound waves,

Frequency of sound wave reflected from the wall

$\upsilon' = \frac{v}{v - v_{b}} \times \upsilon$

Frequency of the reflected waves as received by the moving motorist

$\upsilon''' = \frac{v + v_{m}}{v} \times \upsilon'' = \frac{v + v_{m}}{v - v_{b}} \times \upsilon$

$\therefore$Beat frequency $= \upsilon''' - \upsilon'$

$= \frac{v + v_{m}}{v - v_{b}} \times \frac{v + v_{m}}{v + v_{b}}\upsilon = \frac{2v_{b}(v + v_{m})}{v^{2} - v_{b}^{2}}\upsilon$