Is the frequency of the echo (resulted from the reflected sound wave) equal to the frequency of the original sound wave?
[Physics] Frequency of Sound Waves
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Is the frequency of the echo (resulted from the reflected sound wave) equal to the frequency of the original sound wave?
Best Answer
Yes, the frequency is normally the same.
If the sound is reflected from a moving object then the frequency will be shifted higher or lower depending if the object is moving towards or away from the observer. This is the Doppler Effect.
I have assumed that the observer emits the sound and is not moving. If the observer is moving towards the reflector or away the frequency will also be shifted higher or lower.
If for the example the observer is emitting sound and moving away from the reflector then the frequency of the sound reflected back will be lower. To calculate this there are two steps. First, the frequency, $f$ received by the stationary reflector needs to be calculated with the doppler formula
$$f={c \over c+ v_s} f_0$$
where $f_0$ is the original frequency, $c$ is the speed of sound in air and $v_s$ is the speed of the source moving away from the reflector.
Now the frequency, $f'$, observed by the moving observer is given by
$$f'={c - v_o\over c } f$$
where $v_o$ is the speed the observer is moving away from the reflector.
Now putting the two equations together and remembering that the observer is the source of the sound (so that $v_s=v_o$) we get
$$f'={c - v_s\over c } f = {c \over c+ v_s} f_0$$
$$f'={c - v_s \over c+ v_s} f_0$$