Does the field of a unifirmly moving charge satisfy Maxwell's wave equation?

Question

The so-called "electromagnetic wave equation" is so general that it should obviously hold for every point in space and for all field configurations (if there are no charges).

Here it is: $$ \left(c^2 \nabla^2 - \dfrac{\partial^2}{\partial t^2}\right)\mathbf E = 0$$

Now, for a uniformly moving (not accelerating) charge, the field configuration "moves" with it, with velocity $v$ (let's ignore relativistic effects).

Although this is no "wave", any function that "moves" should satisfy its own "wave equation", but with $v^2$ instead of $c^2$.

(is that right?)

But now we have a contradiction, because in my example the electic field is supposed to satisfy two different equations at once.

Where is the mistake? I'll appreciate any input.

Answer 1

The speed of light in a vacuum, $c$, refers to the phase velocity of an EM wave. It is the speed of a pure sine wave.

What you are talking about for the speed of the disturbance that propagates with a moving charge is the group velocity; the velocity of the interference pattern of many individual sine waves with different frequencies. This is not constrained to move at $c$.