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

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

Answer 1

Answer: (C)

$\frac{2 v_{b}\left(v+v_{m}\right)}{v^{2}-v_{b}^{2}} f$

Answer 2

According to Doppler's effect When observer is moving behind the source (band),apparent frequency heard $n'=n\left[\frac{v+v_{o}}{v+v_{s}}\right]$ Here, $v_{o}=v_{m}$ $v_{s}=v_{b}$ $\therefore n_{1}=\left[\frac{v+v_{m}}{v+v_{b}}\right] f$ The other sound is echo, reaching the observer from the wall and can be regarded as coming from the image of source formed by reflection at the wall. This image is approaching the observer in the direction of sound. Hence, for reflected sound, frequency heard by motorist $n_{2}=n\left[\frac{v+v_{o}}{v-v_{s}}\right]$ or $n_{2}=f\left[\frac{v+v_{m}}{v-v_{b}}\right]$ Then, number of beat frequency heard by motorist $=n_{2}-n_{1} .$ $n_{2}-n_{1}=\left(\frac{v+v_{m}}{v-v_{b}}\right) f-\left(\frac{v+v_{m}}{v+v_{b}}\right) f$ $=\frac{2 v_{b}\left(v+v_{m}\right)}{v^{2}-v_{b}^{2}} f$