|MadSci Network: Physics|
The doppler effect is an apparent change in frequency of a sound if the sound's source is moving toward or away from the receiver, or the receiver is moving toward or away from the source.
Let c be the speed of sound. Let there be a sound source of frequency f_s moving toward a stationary receiver at speed v_s. In a time delta-t, the source will emit (f delta-t) wavecrests. The first wavecrest will travel a distance c * delta-t from the start-point of the source, but the last wavecrest will be at the position of the source, (v_s*delta-t). So there are (f_s* delta-t) waves spread out over a distance (c*delta-t - v_s delta-t), which means the wavelength is distance / (# waves), which is (c*delta-t - v_s*delta-t)/(f_s*dela-t) = (c - v_r)/f_s. Since frequency = wavespeed / wavelength, the frequncy of the waves heard by a stationary observer is
c*f_s f_s f = ------- = --------- (1) c - v_s 1 - (v_s/c)The frequency is higher (higher pitch) when the source moves towards the receiver. When v_s = c, the frequency becomes infinite: this makes sense or a sound source moving at the speed of sound.
When the receiver is moving at speed v_r, the effect is similar. The receiver is moving away from the source, the wavecrests overtake it more slowly: they seem to be traveling at a speed of (c - vr). Wavecrests pass it at a rate (apparent speed)/(wavelength), which is f_r = (c - v_r)/(c/f), or
f_r = f (1 - v_r/c) (2)The frequency heard by the receiver is lower when it moves away from the source When v_r = c, the frequency is zero because no waves are able to pass the receiver.
We can combine (1) and (2) to get a unified doppler-shift frequency measured by a moving receiver f_r after being transmitted by a moving source at ource frequency f_s:
(1 - v_r/c) f_r = f_s ----------- (3) (1 - v_s/c)I've given this example for sound waves, but it works for any sort of wave which travels at constant speed.
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