Q. 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 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) $\frac{2v_{m}(v + v_{b})f}{v^{2} - v_{b}^{2}}$
Explanation
(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
$f^{''} = \frac{(v + v_{m})f}{v + v_{b}}$
[image:15_image71.emf]
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,
$f^{''} = \frac{\left( v + v_{m} \right)f^{''''}}{v} = \frac{(v + v_{m})f}{v - v_{b}}$
$\therefore$ Beat frequency $= f^{''''} - f^{''}$
$= \frac{(v + v_{m})f}{v - v_{b}} - \frac{(v + v_{m})f}{v + v_{b}} = \frac{2v_{b}(v + v_{m})f}{v^{2} - v_{b}^{2}}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.