Time Evolution of the Wave Equation (pictures)

At this stage, with the initial conditions set, all of the coefficients $c_{ij}$ are determined ($= c_i d_j$), so it is a straightforward matter to evolve the magnetic field by using equation (23), for each point $(r,z,t)$ desired.

I chose a grid of $100 \times 100$ points in the $rz$ plane, which covered 0-20 light seconds in each direction. The equation was evolved for 40 seconds, which is long enough time for all waves to propagate across the grid. The initial condition is shown in this density plot. It shows a concentration of magnetic flux near the ring $(r,z) = (3.5,0)$.

(The box on the left shows the initial position of the toroid, as does the dashed ring on the right. The outer ring is the outer extent of the computation.)

After about 5 seconds of evolution, the magnetic field has expanded from its initial configuration, and has begun to interact near the origin. This is probably constructive interference due to the wave converging at the $z$-axis, similar to focusing of light.

Finally, after 25 seconds of evolution, the waves have moved well away from the original position of the toroid, and show no signs of returning. Indeed, after 30 seconds, the waves have completely left the grid.

Finally, CLICK HERE FOR THE FULL MOVIE, which is an animated GIF of about 400 kilobytes. It shows the time evolution in both $rz$ and $ry$ planes, up to to 40 seconds. The magnetic field is always in the azimuthal direction ($\hat{\phi}$), and is of arbitrary strength. Only the magnitude is shown, not the direction. A wave is seen to form, which indeed propagates outward at the speed of light. The final magnetic field intensity at the original position of the flux loop is less than 0.01% of the starting value.