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I D'Alembert equation and Galilean transformation

  1. Jan 30, 2017 #1
    The D'Alembert equation for the mechanical waves was written in 1750. It is not invariant under a Galilean transformation.
    Why nobody was shocked about this at the time? Why we had to wait more than a hundred years (Maxwell's equations) to discover that Galilean transformations are wrong? Couldn't we see that they wrong already by looking at the D'Alembert equation for the mechanical waves?
    Am I missig something?

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  3. Jan 30, 2017 #2


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    D'Alembert's equation for sound waves through air is only valid in the rest frame of the air. If you are moving relative to the air, then the equation as described in your rest frame is modified. It was assumed that Maxwell's equations were similarly modified if you are moving relative to the rest frame of whatever medium electromagnetic waves propagate through.
  4. Jan 30, 2017 #3


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    If you consider mechanical waves in a medium, there is nothing strange with the wave equation not satisfying Galilei invariance because the presence of the medium itself violates it. There is really nothing to be upset about in that respect, e.g., sound waves carried by air travel at different velocities to you depending on your motion relative to the medium and this can be confirmed by experiment.

    The new thing in relativity is that there seemingly was no medium and the speed of light turned out to be invariant and the same regardless of the state of motion of the observer. There is now no medium to break Galilei invariance.
  5. Jan 31, 2017 #4
    Maybe, it is worth to mention that exact dynamical equations for media, say, continuity and Euler ones, are invariant under Galilean transformations. D'Alembert equation for sound waves is just approximation for small perturbations of homogeneous non-moving background. It is background, not physical laws themselves, which makes wave equation non-invariant.

    The same is true for solids.
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