Isn't D'Alembert's wave equation enough to see that Galilean transformations are wrong? The D'Alembert equation for mechanical waves was written in 1750:
$$\frac{\partial^2u}{\partial x^2}=\dfrac{1}{v^2}\dfrac{\partial^2u}{\partial t^2}$$
(in 1D, $v$ being the propagation speed of the wave)
It is not invariant under a Galilean transformation. 
Why was nobody shocked about this at the time? Why did we have to wait more than a hundred years (Maxwell's equations) to discover that Galilean transformations are wrong? Couldn't we see that they are wrong already by looking at the D'Alembert equation for mechanical waves?
Am I missing something?
 A: Your reasoning is spot on and the D'Alembert equation is indeed not Galilean-invariant: you are not missing anything aside from some historical knowledge; this is not my specialty either, but I think I can answer. 
This non-Galilean-invariance was simply taken to be evidence for the existence of a luminiferous aether. The D'Alembert wave equation also perfectly well describes sound, and there is of course no problem with its non-Galilean-invariance here: this is exactly what you expect when there is a "privileged" frame defined by a wave's medium. When Maxwell's equations were discovered, the physics community simple assumed that they were correct only for the frame at rest relative to the luminiferous aether. In the middle of the 19th century, most researchers had abandoned Galileo's relativity postulate, at least for light. This was not an unreasonable stance until invalidated by experiments such as the Michelson-Morely experiment: Galileo knew nothing of electromagnetism or any details of physics at relative speeds comparable to the speed of light.
A: 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: $$c^2=\left(\frac{\partial p}{\partial \rho}\right)_s$$ 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.
