Question

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

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

Answer 1

The correct option is C $\frac{2v_b(v+v_m)}{v^2-v_b^2}f$

The motorist receives two sound waves: direct one and that reflected from the wall.
$f^{\prime }=\frac{v+v_m}{v+v_b}f$
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
$f^{\prime }"=\frac{v+v_b}{v-v_b}\times f^{\prime }-\frac{v+v_m}{v+v_b}f$
Therefore, the beat frequency is
$f^{\prime }"=\frac{v+v_m}{v-v_b}\times f-\frac{v+v_m}{v+v_b}f$
$=\frac{2v_b(v+v_m)}{v^2-v_b^2}f$