Question

A sound wave of frequency f propagating through air with a velocity C, is reflected from a surface which moving away from the source with a constant speed V. Find the frequency of the reflected wave, measure by the observer at the position of the source.

Answer 1

Let the frequency of the sound wave in air$=\frac{C}{\lambda }$; where C=velocity of sound and $\lambda$=wavelength of the sound wave

and $v$ is the velocity of the reflected surface which is moving away from the source.

From Doppler effect we know that if the source or reflected surface is moving towards or away from each other the frequency of the reflected sound so produced will be lower.

$f=\frac{C}{C+V}{f}_0$ where $f_0$=original frequency

Now frequency $f{}^{\prime }$ observed by the observer when the reflected surface is moving away.

$f{}^{\prime }=\frac{C-V}{C}f$

Now comparing both the equation we get

$f^{{}^{\prime }}=\frac{C-V}{C}\left(\frac{C}{C+V}\right)f_0{f}^{{}^{\prime }}=\left(\frac{C^2-CV}{C^2+CV}\right)f_0{f}^{{}^{\prime }}=f_0\frac{C-V}{C+V}$