Result

Doppler shift

f^{'} = f (\frac{v \pm v _{o}}{v \mp v _{s}})

Statement

The observed frequency f′ when source and observer move relative to the medium at speed v_s and v_o respectively.

Symbols and units

$f^{'}$	observed frequency	Hz
$f$	source frequency	Hz
$v$	wave speed in the medium	m/s
$v_{o}, v_{s}$	observer, source speed	m/s

When it applies

Source and observer speeds much less than the wave speed in the medium (for sound) — otherwise relativistic corrections apply.

Don't use when

Assumes source and observer speeds much less than the wave speed in the medium — near or above the wave speed (e.g. supersonic sources) requires relativistic or shock-wave treatment instead.

Derivation

A moving observer encounters wavefronts at a different rate than a stationary one; a moving source compresses wavefronts ahead of it and stretches them behind — combining both effects gives the general shift.

Worked examples

The source-speed limit

As the source speed v_s approaches the wave speed v, the denominator approaches zero:

f^{'} = f (\frac{v}{v - v _{s}}) v_{s} \to v \infty

This divergence marks the onset of a shock wave — the source is about to outrun its own wavefronts entirely.

Watch out for

Sign convention: use + in the numerator when the observer moves TOWARD the source, and − in the denominator when the source moves TOWARD the observer (opposite signs for motion away). Getting a sign wrong shifts the frequency the wrong way.