But the wave seems to be both generated by the charged particle at the origin and and it also appears to be expanding at the head of the wave. If you were to take the charged particle out of the simulation at any time after t=0, it looks like the wave would still grow larger because the front of the wave is generating it's own oscillations and is moving faster than the rest of the profile. If you look at the simulation with the charged particle in place: after one period T, the wave is growing by 4pi as the both the front end and the back end of the wave seem to expand. This is where I am confused, if the front end were moving at the same speed as the rest of the profile then it would never have the speed to separate from the rest of the profile and expand the wave further. The front end of the wave would always be at the same phase. Thanks for your help!