Why invariance holds for the electromagnetic waves but not for the mechanical waves? Why the electromagnetic wave equation provides a wave with constant speed in all reference frames, but the mechanical elastic wave equation (from linear elasticity) does not?
The electromagnetic wave travels in a constant speed independently of the observer's reference frame but the mechanical waves' speed do depend on the relative motion of the observer. What causes this if both are described by identical equation - the wave equation?
The equations in question are:
1) Electromagnetic wave equations, describing wave with invariant velocity
$$
 c^2\nabla^2E=\frac{\partial^2 E}{\partial^2 t}
$$
$$
 c^2\nabla^2B=\frac{\partial^2 B}{\partial^2 t}
$$
2) Acoustic wave equation, describing mechanical wave with frame dependent velocity
$$
 c^2\nabla^2p=\frac{\partial^2 p}{\partial^2 t}
$$
 A: From your question, I believe you are talking about mechanical waves in a non relativistic setting and electromagnetic waves in the conventional relativistic setting. The key to resolving your problem is that the appropriate transformation from one frame to another in relativistic and non-relativistic settings are basically different.
So if you make a transformation on the wave equation in the non-relativistic setting, since implicitly you are making a Galilean transformation,  you are bound to get the speed of the wave as frame dependent. However a Lorentz transformation on the electromagnetic waves ensures that the speed of the electromagnetic waves remains the same in all frames, i.e, it propagates at $c$. The wave-equation is the same for both, but the frame transformation is different for both.
A: I think it has been mentioned in bits and pieces but I would like to kind of sum it up along with adding my point: Lorentz transformation is the general transformation law that should be used for switching between any two inertial reference frames. For low relative speeds, Galilean transformation approximately resembles the Lorentz one and that is why we use it at low speeds. Of course, the derivation of Lorentz transformation uses an empirical result, the invariance of the speed of light. But it is not like defining something and then saying everything is consistent. The more general truth depicted via Lorentz transformations is that 'anything' moving at a particular speed, namely $c$, will travel at the same speed in each frame. Light is not 'special' in that way. Bring any other wave or particle which moves at that particular speed of $c$ and its speed will be invariant. So in a way, the reason a mechanical wave doesn't have the property of invariance associated with it is because its speed is not $c$.
