There's no problem with the non-invariance of D'Alembert equation for mechanical waves, if I understand what you mean, because mechanical waves *do have* a preferred inertial frame, an "aether". 

For example, a sound wave in a fluid satisfies the wave equation with speed: $$c^2=(\frac{\partial p}{\partial \rho})_s$$ in the rest frame of the fluid. 

The point is that Maxwell's equations are supposed to be valid in *every* inertial frame of reference. Since, in vacuum, they lead to the wave equation, the wave equation must be valid in *every* inertial frame, that's the problem.