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. $\frac{(v+v_m)f}{v+v_b}$
• B. $\frac{\left(v+v_m\right)f}{v-v_b}$
• C. $\frac{2v_b(v+v_m)f}{v^2-v_b^2}$
• D. $\frac{2v_m(v+v_b)f}{v^2-v_b^2}$

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
$f^{\prime }=\frac{(v+v_m)f}{v+v_b}$

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