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."
Yes it is easy to show that the frequency shifts from f to f'=f(1+v/c) but Einsteinians are horrified since the demonstration implies that the speed of light as measured by the observer shifts from c to c'=c+v:
http://physics-animations.com/Physics/English/dopp_txt.htm Next, we shall consider the case when observer moves and the source of the wave is still. In this case the wavelength is not changed and Doppler frequency shift appears because the velocity w of the wave relatively the observer is changed:
w = u + v (observer is moving toward the stationary source)
w = u - v (observer is moving away from the stationary source)
Because f_dop = w/lambda, initial f = u/lambda_0 and lambda = lambda_0 we find that
f_dop = f(1+v/u) (observer moves toward the stationary source)
f_dop = f(1-v/u) (observer is moving away stationary source) ______________________________________________ [end of quotation]