Figure 3 shows the results of seeding the wave simulations with, clockwise from upper left, a single tent-peak, a sine wave, a Fourier sum, and random values. The sine wave seed yields a space-time pattern that looks a bit like a box of sushi. The random seed does not settle down for the linear wave scheme. The wave equation seeded by a spike, Wave, Fourier Sum, and Random noise. We discussed the issue that the space step must be greater than or equal to the time step.
Experiment shows the simulation to be more robust if it is strictly greater.
In Figure 4 we show how this rule can lead to an alternating pattern. More precisely, we patch in a small interval of values taken from a sine wave of a different phase. When the perturbation first happens, we see a pattern like in the middle of the spacetime picture, with the disturbance moving inward from both the left and the right ends. As the CA continues to evolve, the two perturbation waves repeatedly pass through each other at the center of the CA, proceed out to the ends, wrap around to move back towards the center and so on. Figure 4 shows the CA after it has already undergone several of these cycles since the initial perturbation event.