= Doppler effect
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{title2=$\Delta f=f_o-f_s$}
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= Doppler shift
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The <Doppler effect> is a change between emitted and received <frequency> caused by motion of the source or receiver. An approaching siren can send more <wave crests> per second to a <microphone> than the source emits per second; a receding siren sends fewer. The source need not change its own <oscillation>. The <Doppler shift> is $\Delta f=f_o-f_s$, where $f_s>0$ is emitted <frequency> and $f_o>0$ is received <frequency>, both measured in <hertz>.
For <acoustic waves>, motion must be specified relative to the material carrying the <wave>. The <speed of sound> is measured in that material's rest <reference frame>. Moving the source changes the spacing of successive <wavefronts>; moving the receiver changes how quickly it crosses those <wavefronts>. These two mechanisms give different exact factors in the <classical Doppler effect for sound>, even for the same closing <relative velocity>.
This makes a <microphone> a motion probe: a stable tone supplies a repeating clock, and changes in arrival spacing reveal the source's motion toward or away from the receiver. An <acoustic Doppler measurement> can test the prediction directly. A passing source also illustrates why geometry matters: its speed can stay constant while the received <frequency> changes continuously, as in <Doppler frequency along a trajectory>.
The following sound model assumes <linear acoustics> in uniform stationary material with <isotropy>, constant <speed of sound> and negligible <wave dispersion>. Source and receiver speeds are below the <speed of sound>, and there is a single direct propagation path. Air motion requires transforming source and receiver <velocities> into the air's rest <reference frame>; a nonuniform flow requires a more general propagation model. At the sonic threshold the simple periodic-arrival picture fails; <moving-point acoustic wavefronts> and the <Mach cone> describe the relevant geometry, while finite <shock waves> require further physics. <Light> in <vacuum> instead requires the <relativistic Doppler effect>: the acoustic formula cannot be converted into its exact optical counterpart just by replacing the <speed of sound> with the <speed of light>.
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