Charges affect each other at a distance after a time delay, whether they are moving or accelerating or not.
When both are stationary we consider the force an electric field.
When the target is stationary and the source is moving directly toward or away from the target at constant velocity, it's still an electric field but the intensity of the force is different.
If the target is stationary and the source is moving sideways, it's still an electric field and the intensity is something else.
If the source and the target are both moving, and the source has some motion sideways, while the target has some motion in the same plane as the direction between them and the source velocity, then there is also a magnetic force. In this particular case, the equations for the electric force do not add up in different frames. In a frame where one charge is stationary magnetic force on the target is zero. In a frame where they are both moving, the equations give a different wrong result and magnetic force is needed to make up the error.
You get radiation only when the source charge is accelerating sideways.
The radiation will be a wave pattern when the motion of the source charge fits a wave.