Is the phase of plane waves an invariant?

Based on the invariance of the phase of waves, plane waves was shown to propagate with negative frequencies in a medium which moves at superluminal speeds opposite to the propagation direction of plane waves. The validity of the invariance of the phase of plane waves was then called into question. A radical change of the conventional concept of plane waves is recently proposed to solve the problem of negative frequency of waves. Here, we point out flaws in that proposal. Thus, the validity of the invariance of the phase of plane waves remains questionable.