The speed of light is constant no matter if there be motion in whatever is emitting it (the source) or motion in whatever is receiving it (the observer). Does this principle apply to other kinds of waves like sound or water waves?
The invariance of the speed of light is a result of a symmetry called Lorentz covariance. Any massless particle behaves the same way as light, though with the possible exception of gravitons the photon is the only known massless particle that propagates over long distances. Gluons are also massless but they are confined to within baryons.
Sound and water waves are oscillations in a medium not due to elementary particles. In both cases the propagation speed is a function of the properties of the medium (and in the case of gravity waves like water waves the gravitational force).