http://www.hep.man.ac.uk/u/roger/PHY.../lecture18.pdf
Roger Barlow, Professor of Particle Physics: "The Doppler effect - changes in frequencies when sources or observers are in motion - is familiar to anyone who has stood at the roadside and watched (and listened) to the cars go by. It applies to all types of wave, not just sound. (...) Moving Observer. Now suppose the source is fixed but the observer is moving towards the source, with speed v. In time t, ct/lambda waves pass a fixed point. A moving point adds another vt/lambda. So f'=(c+v)/lambda. (...) Relativistic Doppler Effect (...) If the source is regarded as fixed and the observer is moving, then the observer's clock runs slow. They will measure time intervals as being shorter than they are in the rest frame of the source, and so they will measure frequencies as being higher, again by a gamma factor: f'=(1+v/c)(gamma)f..."

If "in time t, ct/lambda waves pass a fixed point", and if "a moving point adds another vt/lambda", then the speed of the light waves relative to the moving observer is:

c' = (lambda)(ct/lambda + vt/lambda)/t = c + v

in violation of special relativity. This shift in the speed of the light waves relative to the observer (from c to c'=c+v) causes the frequency he measures to shift from f=c/lambda to:

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

If v is smaller than (1/3)c, the relativistic corrections are negligible (for v=(1/3)c gamma is 1.05) and both c'=c+v and f'=c'/lambda are virtually exact formulas no matter whether the classical or the relativistic Doppler effect is calculated.

Pentcho Valev