# 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?

• Could you explain why you expect the equation for mechanical waves to be Galilei invariant? Jan 30 '17 at 15:10
• @ACuriousMind Because I was actually missing something! Now I am starting to understand the issue here. Let's say I'm on a train moving with uniform velocity on a linear rail, and that on this train I have a big airtight container full of air. Now, if I do any sound-related experiment inside this container, I will not be able to determine if my laboratory is in motion or not, because the medium in which the mechanical wave is propagating is moving (or not) with the reference frame of the lab. The issue is that his can't be done with light because there isn't a medium! Am I correct? Jan 30 '17 at 15:39
• Do you mind explicitly showing how it's not invariant to clarify the question? Jan 31 '17 at 15:25
• consider how complex the concept of "invariance under galilean transformations" is. was there even much similar concept of a "transformation" in the 1700s? its a thoroughly modern concept. or perhaps it was pondered in some other historical context. one would have to research physics history very carefully to be able to answer correctly. the wave equation was indeed pondered at the time...
– vzn
Jan 31 '17 at 18:47
• @vzn Well, Galileo Galilei wrote his transformations in a book published in 1638, the "Discourses and Mathematical Demonstrations Relating to Two New Sciences", so yes, I believe there was the concept of transformation in the 1700s. The problem for me is more conceptual, rather than simpy historical. Feb 1 '17 at 0:27

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.