A source that emits light while approaching moves forward itself between one wavefront and the next. The wavefronts ahead close up and the wavelength shortens.
Moving away, the spacing opens and the wavelength stretches. It is the same picture as the Doppler effect for sound, where an ambulance sounds higher approaching and lower receding.
In visible light the short-wavelength side is blue and the long side red. An approaching source looks bluer, a receding one redder.
For sound, the frequency is when the source approaches at speed and when the listener approaches, two different formulas ( is the speed of sound). There is a medium, the air, and the answer depends on which one moves relative to it.
Light has no medium, and its speed is for everyone. There is no way to decide whether the source or the observer is moving, so the formula must be written with the relative speed of the two frames alone.
The upper row of the figure is sound, the lower row light. Sound waves travel at a speed set against the air; light waves travel at for everyone.
In a time on its own clock, the source emits waves. In the observer's frame, time dilation stretches this time to .
During the source approaches by , so the train of waves is long. It passes the observer in .
The frequency is the number of waves divided by this time.
Putting in gives
For a receding source, replace by . This formula is the geometric mean of the two sound factors and , lying exactly between them.
For a receding source is smaller and the wavelength longer. The colour shifts toward the red, which is why this is called redshift.
The figure takes a source emitting light at nm, moves it toward the observer at and , then away at and . The wavelengths seen are , , and nm.
The wavelengths atoms emit are fixed by the kind of atom. If hydrogen's nm line is seen at nm, the source is receding at about km/s.
In Hubble showed that more distant galaxies have larger redshifts, a clue that the universe is expanding. For distant galaxies, though, the redshift is calculated not with this Doppler formula but as wavelengths stretched by the expansion of the universe.
As a source passes an observer, the light emitted at the point of closest approach arrives, in the observer's frame, at right angles to the source's motion. That direction is neither approaching nor receding, so for sound the frequency would not change.
For light, only the slowing of the source's clock remains, and . This is called the transverse Doppler effect.
In Ives and Stilwell compared the wavelengths of light emitted forward and backward by fast hydrogen atoms, and confirmed this factor coming from time dilation.