What is the solution of wave equation for a moving charge?

The electric field of a point charge at rest in an inertial frame is static in the frame. Any location outside the charge follows the Maxwell equations in the vacuum, and as a consequence the wave equation. It is easy to see that both the Laplacian of the field and its time second derivative are zero. The wave equation is fulfilled trivially. If this charge is moving with constant velocity in a lab, the electric (and magnetic) field are calculated by a Lorenz boost. In the lab's frame, the E-field is not linear with time for a given point, due to the Coulomb law, so the second derivative with respect to time is not zero. It seems that the wave equation is fulfilled not trivially by a function $f(\mathbf r, t)$ . If it fulfills the wave equation it is a EM wave, or not? If not, what is the form of this function?