http://rockpile.phys.virginia.edu/mod04/mod34.pdf
Paul Fendley: "Now let's see what this does to the frequency of the light. We know that even without special relativity, observers moving at different velocities measure different frequencies. (This is the reason the pitch of an ambulance changes as it passes you it doesn't change if you're on the ambulance). This is called the Doppler shift, and for small relative velocity v it is easy to show that the frequency shifts from f to f(1+v/c) (it goes up heading toward you, down away from you). There are relativistic corrections, but these are negligible here."

That is, if the frequency measured by the stationary observer is f=c/L (L is the wavelength), the frequency measured by an observer moving towards the light source with speed v is:

f' = f(1+v/c) = (c+v)/L = c'/L

where c'=c+v has a definite physical meaning: it is the speed of the light waves relative to the moving observer. The formula f'=f(1+v/c) is an approximation (the relativistic corrections are not taken into account) and so is c'=c+v. However Paul Fendley explains that the relativistic corrections "are negligible here", which means that both f'=f(1+v/c) and c'=c+v are virtually exact. Special relativity is violated.

Let us still add the relativistic corrections (time dilation is taken into account):

f' = f(1+v/c)(gamma) = (c+v)(gamma)/L = c'/L

where c'=(c+v)(gamma) is the speed of the light waves relative to the moving observer. Clearly both the non-relativistic and relativistic Doppler effect violate special relativity. If v is small enough, we have c'=c+v in both cases, which means that the speed of light relative to the observer varies with the speed of the observer as predicted by both Newton's emission theory of light and Maxwell's 19th century electromagnetic theory.

Pentcho Valev