The Doppler effect (or Doppler shift), named after Austrian physicist Christian Doppler who proposed it in 1842, is the change in frequency of a wave for an observer moving relative to the source of the wave. It is commonly heard when a vehicle sounding a siren or horn approaches, passes, and recedes from an observer. The received frequency is higher (compared to the emitted frequency) during the approach, it is identical at the instant of passing by, and it is lower during the recession. If the source moving away from the observer is emitting waves through a medium with an actual frequency f0, then an observer stationary relative to the medium detects waves with a frequency f given by
where vs is positive if the source is moving away from the observer, and negative if the source is moving towards the observer. A similar analysis for a moving observer and a stationary source yields the observed frequency (the receiver's velocity being represented asvr):
where the similar convention applies: vr is positive if the observer is moving towards the source, and negative if the observer is moving away from the source. These can be generalized into a single equation with both the source and receiver moving.
With a relatively slow moving source, vs,r is small in comparison to v and the equation approximates to
However the limitations mentioned above still apply. When the more complicated exact equation is derived without using any approximations (just assuming that source, receiver, and wave or signal are moving linearly relatively to each other) several interesting and perhaps surprising results are found. For example, as Lord Rayleigh noted in his classic book on sound, by properly moving it would be possible to hear a symphony being played backwards. This is the so-called "time reversal effect" of the Doppler effect. Other interesting conclusions are that the Doppler effect is time-dependent in general (thus we need to know not...