Q. A band playing music at 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
Solution:
The motorist receives two sound waves are direct one and that reflected from the wall.
$f '=\left(\frac{v+v_{m}}{v+v_{b}}\right)\,f$
For reflected sound waves:
Frequency of sound wave reflected from the wall is,
$f''=\frac{v}{v-v_{b}}\times f$
Frequency of the reflected waves as received by the moving motorist is
$f'''=\frac{v+v_{m}}{v}\times f''=\frac{v+v_{m}}{v-v_{b}}\times f$
Therefore, the beat frequency is
$f'''-f'=\frac{v+v_{m}}{v-v_{b}}\times f -\frac{v+v_{m}}{v+v_{b}}\times f$
$=\frac{2v_{b}\left(v+v_{m}\right)}{\left(v^{2}-v^{2}_{b}\right)} f$
