Will the frequency of the reflected wave of a moving source be shifted by Doppler effect? I want to know if there is a moving source that emits wave to a reflecting object like a wall for example, then, will the frequency of the reflected wave be shifted because of the Doppler effect. As the source moves, will waves incidents on the wall have its frequency shifted, but: how about the reflected wave?
 A: In the rest frame of the wall/mirror, the light emitted from the moving source will be blue-shifted (if moving towards the mirror). The reflected light (in the mirror rest frame) will have the same blue-shifted frequency.  Then, in the rest frame of the moving source, the reflected light will be blue-shifted again. i.e. The Doppler shift formula must be applied twice.
$$ f^{\prime} = (1-v^2/c^2)^{-1/2} (1+v/c)\times (1-v^2/c^2)^{-1/2} (1+v/c)f$$
$$ f^{\prime} = \left( \frac{c+v}{c-v}\right) f $$
Conversely, the reflected light received by a source of radiation moving away from the mirror will be redshifted twice.
A: Short answer: yes.
The object moving towards you may emit a sound that is Doppler shifted to a higher frequency - but once the sound is "in the air" you don't know or care why it had the frequency that it had. So after reflection from the wall it will still be shifted.
Now if the object observed the sound it made (reflected and traveling back towards it) that would indeed be shifted to even higher frequency - if you imagine the object observed its "mirror image" that mirror image would be moving toward it twice as fast as the wall is moving toward it.
