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)."

"In time t, ct/(lambda) waves pass a fixed point." That is, the speed of the waves relative to the fixed observer is c.

"A moving point adds another vt/(lambda)." That is, the speed of the waves relative to the moving observer becomes c'=c+v. The new frequency, f'=c'/(lambda)=(c+v)/(lambda), is consistent with c'=c+v.

Pentcho Valev