Einsteinians readily admit that, for waves other than light waves, the speed of the waves (relative to the observer) varies with the speed of the observer and this explains the Doppler frequency shift:
http://faculty.washington.edu/wilkes/116/slides/Physics116_L08-interference.pdf "Sound waves have speed c, and f and L are related by c=Lf. For an observer moving relative to medium with speed u, apparent propagation speed c' will be different: c'=c±u. Wavelength cannot change - it's a constant length in the medium, and same length in moving coordinate system (motion does not change lengths). Observed frequency has to change, to match apparent speed and fixed wavelength: f'=c'/L."
Einsteinians use the same argument and deduce the same frequency shift (f'=c'/L=(c+v)/L) for light waves as well:
http://www.hep.man.ac.uk/u/roger/PHYS10302/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."
The only difference between Doppler in sound (moving observer) and Doppler in light (moving observer) is that in the former case Einsteinians explicitly refer to the variable speed of the waves (relative to the moving observer) while in the case of light the variation is implicit. This guarantees the safety of special relativity because Einsteinians know: if something is implicit, then it is dangerous and should never become explicit, Divine Einstein, yes we all believe in relativity, relativity, relativity. Yet even Divine Albert's world is not eternal and sooner or later Einsteinians will be faced with an embarrassing question:
http://www.informaworld.com/smpp/content~content=a909857880 Peter Hayes "The Ideology of Relativity: The Case of the Clock Paradox" : Social Epistemology, Volume 23, Issue 1 January 2009, pages 57-78: "The gatekeepers of professional physics in the universities and research institutes are disinclined to support or employ anyone who raises problems over the elementary inconsistencies of relativity. A winnowing out process has made it very difficult for critics of Einstein to achieve or maintain professional status. Relativists are then able to use the argument of authority to discredit these critics. Were relativists to admit that Einstein may have made a series of elementary logical errors, they would be faced with the embarrassing question of why this had not been noticed earlier. Under these circumstances the marginalisation of antirelativists, unjustified on scientific grounds, is eminently justifiable on grounds of realpolitik. Supporters of relativity theory have protected both the theory and their own reputations by shutting their opponents out of professional discourse."