Liénard-Wiechert Scalar Potential of a Relativistic Oscillating Particle (Classical Electrodynamics)
This is an animated plot of the Liénard-Wiechert potential due to a relativistic particle that is moving upward with Vy = 0.8333c, and which is also oscillating in the horizontal direction with frequency 3E9 Hz, amplitude 8 cm.
The Liénard-Wiechert potential can be thought of as the relativistic version of the Coulomb potential; it reduces to the Coulomb form when the particle is stationary, but takes into account that - when the particle moves - changes in the potential field propagate at a finite speed.
Two physical effects visible in this video are:
1) Doppler-shifting of the radiation toward higher frequencies in front of the particle, and toward lower frequencies behind it;
2) Relativistic beaming, which causes most of the radiation to be emitted in the direction of motion of the particle (compare to the video of the stationary oscillator). Note that no special care had to be taken to take SR into account for this video; classical electrodynamics is relativistically correct by nature.
The Liénard-Wiechert potentials can be found here:
http://en.wikipedia.org/wiki/Li%C3%A9...
I have written a C++ program that calculates these potentials for user-supplied particle motion. Computing the so-called "retarded time" was done using a binary search algorithm with accuracy on the order of the machine precision.
I created a series of bitmap images using my C++ program; I then used the excellent VirtualDub to create the video (virtualdub.sourceforge.net).