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The invariance of the phase of waves among inertial frames is questionable

Two types of relativistic transformation for the four-vector (ω/c, k) of waves (including light waves in media, or acoustic waves) are obtained based on the differential Lorentz transformation. One of them is just the usual Lorentz transformation, whereas the second one is not. The second one is required to secure the frequency of waves from being negative, in the case that the speed of waves in a medium is less than the speed of that medium moving in the direction opposite to the propagation direction of waves. Therefore, the four-vector (ω/c, k) of waves is in general not Lorentz-covariant. The invariance of the phase of waves among inertial frames is questionable.