![speed of sound vs speed of light speed of sound vs speed of light](https://i.ytimg.com/vi/_aaFGm9uqqo/maxresdefault.jpg)
But in case of light in a physical medium, slowed down to significantly below c, I can't see why the same formulas as for sound waves wouldn't apply. In the case of light in a vacuum, c survives any addition unharmed, so the result relative to any observer is always c.
![speed of sound vs speed of light speed of sound vs speed of light](https://vignette.wikia.nocookie.net/vsbattles/images/8/8b/Cebdaf11276eabe2f595d6da9e582a04.jpg)
I think the simplest way to view it is in the medium rest frame, then transform to whatever frame we are interested in. the receiver has no impact either (though his perceived frequency is affected), except that we may care to inquire about the wave velocity relative to him, which is again obtained by addition. Typically, we don't care about it (except for instance for supersonic emitters travelling faster than the wave the emit). the emitter movement wrt the medium has no impact on that (it does yield a shift in frequency but this is another matter - let's assume for simplicity that the wave velocity is independent of frequency here), but the wave speed relative to the emitter is impacted and can be calculated by addition. This brings an issue for light in vacuum, but it's ok since c is invariant anayway, so light propagates at c wrt the vacuum, whatever velocity we may arbitrarily assign to that vacuum. the speed of the wave relative to its medium of propagation is constant. I think I agree with that other poster, but actually I am not sure we can make general statements (speed relative to what ? Is there a medium of progagation ? )