Question

A sound wave of frequency $f$ travels horizontal to the right. It is reflected from a large vertical plane surface moving to left with a speed $v$. If the speed of sound in medium is $c$ then

• A. The number of the beats heard by a stationary listner to the left of the reflecting surface is $\frac{vf}{c-v}$
• B. The wavelength of reflected wave is $\frac{c}{f}\left(\frac{c-v}{c+v}\right)$
• C. The number of waves striking the plane surface per second is $f\left(\frac{c+v}{c}\right)$
• D. The frequency of the reflected wave is $f\left(\frac{c+v}{c-v}\right)$

Answer 1

Answer: (B), (C), (D)

The frequency of the reflected wave is $f\left(\frac{c+v}{c-v}\right)$

Moving plane is like a moving observer. Therefore, number of waves encounted by moving plane,
$f_1=f\left(\frac{v+v_o}{v}\right)$

where, $v$ is velocity of sound, i.e., $v=c$
$v_o$ is velocity of plane surface, i.e., $v_o=v$

So, $f_1=f\left(\frac{c+v}{c}\right)$

For reflected wave, velocity of source $v_s$ will be equal to the velocity of plane surface $v$.
So, frequency of reflected wave,
$f_2=f_1\left(\frac{v}{v-v_s}\right)$

$\Rightarrow f_2=f\left(\frac{c+v}{c}\right)\left(\frac{c}{c-v}\right)$

$\Rightarrow f_2=f\left(\frac{c+v}{c-v}\right)$

Wavelength of reflected wave,
${\lambda }_2=\frac{c}{f_2}=\frac{c}{f}\left(\frac{c-v}{c+v}\right)$

Beat frequency, $f_b=f_2-f=f\left(\frac{c+v}{c-v}\right)-f=\frac{2fv}{c-v}$

Therefore, the correct option are (a), (b) and (c).