The Doppler shift equation for sound waves
where
is positive if the observer is moving towards the source and
is positive if the source is moving towards the observer.
The same equation does not hold for electromagnetic waves. Suppose a light flashes once a second as observed by someone stationary relative to the light and moving at a speed v relative to some other observer O'. The light flashes at the instant it is at the position of O'. A clock moving with the light is running slow as observe by O', and will take
seconds before O' observes the light to flash again. The light is also moving away from O' so he won’t see the next flash until the light has travelled back to O' over the distance covered by the spaceship between flashes. Since the spaceship is travelling at speed v , and the time between flashes as measured in O's frame is
The light source will have travelled a distance
in this time and the light from the flash will take a time
to reach O'.
Thus the total time between O' observing the first flash as the spaceship passes close by us and the second flash emitted one second later by the spaceship clock is
We write
or
Expanding to first order in v/c gives
so that
for a source receding.