Electromagnetic field in a medium

[Bob_for_short]

An electromagnetic field, as a plane wave, has a known space-time dependence:

 

E = E_0⋅cos(ωt-kx).

 

In a transparent medium it is the same except for involving the refraction indices n.

 

Now, let us look at the field in a moving reference frame - that with v = c/n. What solution is for the wave in such a frame? Isn't it a

 

E' = E'_0⋅cos(ω't') ?

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I meant E' = E'_0⋅cos(k'x').