Question

Two sources $S_1$ and $S_2$ of same frequency $f$ emits sound. The sources are moving as shown with speed $u$ each. A stationary observer hears that sound. The beat frequency is ($v=0$ velocity of sound)

• A. $\frac{2v^{2}f}{v^2-u^2}$
• B. $\frac{2u}{v}f$
• C. $\frac{2u^{2}f}{v^2-u^2}$
• D. $\frac{2uvf}{v^2-u^2}$

Answer 1

The correct option is D $\frac{2uvf}{v^2-u^2}$

Two sources $S_1$ and $S_2$ of same frequency $f$ emits sound.

Velocty of source $=u$

and velocity of sound is $v$,

Apparent frequency of $S_1$,

$f_1=\frac{v}{v-u}f$

Apparent frequency of $S_2$,

$f_2=\frac{v}{v+u}f$

Beat can be represented as,
Beat $=f_1-f_2$

$f_1-f_2=\frac{v}{v-u}f-\frac{v}{v+u}f$

$f_1-f_2=\left[\frac{v(v+u)-v(v-u)}{(v-u)(v+v)}\right]f$

$f_1-f_2=\left[\frac{v^2+uv-v^2+uv}{v^2-u^2}\right]f$

$f_1-f_2=\left[\frac{2uv}{v^2-u^2}\right]f$

Final answer: (c)