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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?

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    $\begingroup$ Could you explain why you expect the equation for mechanical waves to be Galilei invariant? $\endgroup$
    – ACuriousMind
    Jan 30, 2017 at 15:10
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    $\begingroup$ @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? $\endgroup$
    – Tropilio
    Jan 30, 2017 at 15:39
  • $\begingroup$ Do you mind explicitly showing how it's not invariant to clarify the question? $\endgroup$ Jan 31, 2017 at 15:25
  • $\begingroup$ 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... $\endgroup$
    – vzn
    Jan 31, 2017 at 18:47
  • $\begingroup$ @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. $\endgroup$
    – Tropilio
    Feb 1, 2017 at 0:27

2 Answers 2

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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.

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  • $\begingroup$ Researchers did not abandon the principle of relativity. They equated the principle of relativity with invariance under Galilean transformations, and they were firmly convinced that the true laws of physics were Galilean invariant. So the fact that Maxwell’s equations are not Galilean invariant just implies that they aren’t true laws of physics, just like the fact that the sound wave equation is not Galilean invariant implies that it is not a true law of physics. To obtain Galilean invariant laws you need to add terms depending on aether speed: physics.stackexchange.com/q/378861/27396 $\endgroup$ Jul 20, 2018 at 15:43
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    $\begingroup$ It’s precisely because of their faith in Galilean invariance that they thought that speed of light would not be the same in all frames, and that the Michelson-Morley experiment would be able to determine the velocity of the aether. And that’s why it came as a shock when the Michelson-Morley experiment yielded a null result, because that seemingly invalidated the principle of relativity (which again they equated with Galilean invariance). But then Einstein showed that it doesn’t invalidate the principle of relativity at all, you just need to modify your notions of space and time. $\endgroup$ Jul 20, 2018 at 15:47
  • $\begingroup$ @KeshavSrinivasan What you seem to be saying, however (which is correct) is that the 19th century physicists assumed that the Galilean Transformation of co-ordinates was correct. Indeed they did, but this violates Galilean invariance. The proof you reference applies the Galilean transformation in the way assumed in the 19th century, giving rise to the extra "aether wind" terms in Maxwell's equations and, importantly, a way to measure motion relative to the aether, by measuring the change in group velocity. The violation of Galilean invariance that is seen in the application of the .... $\endgroup$ Apr 7, 2023 at 7:25
  • $\begingroup$ @KeshavSrinivasan ... Galilean co-ordinate transformation to Maxwell's equations is a total invalidation of what Galileo was driving at in his “Allegory of Salviati’s Ship”, where he defined his notion of invariance (look this up). It is precisely the disproof of this predicted violation that the Michelson Morley experiment showed up, thus restoring a generalized assumption of Galilean invariance to light. To do this, we must assume frame dependence of time co-ordinates, thus turning the Galilean into the Lorentz transformation. This restores the validity of the Salviati Thought Experiment.🌺 $\endgroup$ Apr 7, 2023 at 7:31
  • $\begingroup$ I just completely disagree with you. Adding velocity dependent terms to Maxwell’s equations does not result in something that violates Galilean invariance, anymore than adding velocity dependent terms to the sound wave equation violates Galilean invariance. To this day everyone uses velocity dependent terms in the sound wave equation when working in reference frames moving with respect to air. Before the Michelson-Morley experiment, everyone thought the true laws of electromagnetism obeyed Galilean invariance. $\endgroup$ Apr 7, 2023 at 21:03
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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.

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