Covariance of the Wave Equation in Modern Physics: A Proof

[mike217]

Question: Show that the wave equation is in not covariant with respect to Gallilean transformations, given the function y=Asin(2pi(x/lambda - ft))

My main question is inorder to show the covariance of a law, should I apply the law on the primed variables and show that it is satisfied by applying a transformation, or should I make the substitution, apply the law, and then show it is satisfied.

Thank you.

 

[dextercioby]

Interesting problem.Reminds me of the days when "Lorentz covariance" was unknown to me.
Do you know the form of a Galilei transformation??If so,write y(x,t) in the 'primed' system.

Daniel.

 

[mike217]

Thanks for your reply Daniel.

I have worked on this problem, but the result that I am getting is that the wave equation under the Gallilean transformation is satisfied. Please view key steps of my solution in the attachement.

 

[dextercioby]

I'm sorry,but i couldn't understand your formulas.
To show that the wave function (and hence the wave equation) is not invariant under the Group of Galilean transformations (is not Galilei covariant) means to see whether the transformed wavefunction:

WVFCT--------->GT (WVFCT)'

satisfies or not the transformed wave equation:

WVEQ--------->GT (WVEQ)'

Daniel.