Question

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 $v$ is the speed of sound, the expression for the beat frequency heard by the motorist is

• A. $\frac{\left(\right. v + v_{m} \left.\right) f}{v + v_{b}}$
• B. $\frac{\left(v + v_{m}\right) f}{v - v_{b}}$
• C. $\frac{2 v_{b} \left(\right. v + v_{m} \left.\right) f}{v^{2} - v_{b}^{2}}$
• D. $\frac{2 v_{m} \left(\right. v + v_{b} \left.\right) f}{v^{2} - v_{b}^{2}}$

Answer 1

Answer: (C)

The motorist receives two sound waves, direct one from the band and that reflected from the wall, figure. For direct sound waves, apparent frequency
$f^{′}=\frac{\left(\right. v + v_{m} \left.\right) f}{v + v_{b}}$

For reflected sound waves.
Frequency of sound wave reflected from the wall
$f^{′ ′}=\frac{v \times f}{v - v_{b}}$
Frequency of reflected waves as received by the moving motorist,
$f^{′}=\frac{\left(v + v_{m}\right) f^{′ ′}}{v}=\frac{\left(\right. v + v_{m} \left.\right) f}{v - v_{b}}$
$\therefore $ Beat frequency $=f^{′ ′}-f^{′}$
$=\frac{\left(\right. v + v_{m} \left.\right) f}{v - v_{b}}-\frac{\left(\right. v + v_{m} \left.\right) f}{v + v_{b}}=\frac{2 v_{b} \left(\right. v + v_{m} \left.\right) f}{v^{2} - v_{b}^{2}}$