Hi! I was reading some notes on relativity (Special relativity) (http://teoria-de-la-relatividad.blogspot.com/2009/03/3-la-fisica-es-parada-de-cabeza.html) and it says that the classical wave equation is not Galilean Invariant. I tried to show it by myself, but I think there is some point that I'm missing.(adsbygoogle = window.adsbygoogle || []).push({});

Given two coordinate frames, if [tex] x=\bar{x}+vt [/tex] the classical wave equation transforms to:

[tex]\frac{\partial^2 \phi}{\partial \bar{x}^2} + \frac{\partial^2 \phi}{\partial \bar{y}^2} + \frac{\partial^2 \phi}{\partial \bar{z}^2} - \frac{1}{c^2} \frac{\partial^2 \phi}{\partial \bar{t}^2} + \frac{1}{c^2} \left( 2v \frac{\partial^2 \phi}{\partial \bar{x} \partial \bar{t}}- v^2 \frac{\partial^2 \phi}{\partial \bar{x}^2} \right) = 0[/tex]

But i can really get that answer. Supose I want to compute

[tex] \frac{\partial \phi}{\partial \bar{x}} = \frac{\partial \phi}{\partial x} \frac{\partial x }{\partial \bar{x}} + \frac{\partial \phi}{\partial y } \frac{\partial y }{\partial \bar{x}} + \frac{\partial \phi }{\partial z } \frac{\partial z }{\partial \bar{x}} + \frac{\partial \phi }{\partial t } \frac{\partial t }{\partial \bar{x}} [/tex]

And

[tex] t = \bar{t} = \frac{x-\bar{x}}{v} [/tex]

should be my hints to get to that result?

Thanks!

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# Wave equation and Galilean Transformation

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