Waves

The two-dimensional wave equation describes drumheads, ponds, and ripple tanks alike. Discretized onto a grid it becomes a simple local rule: every point accelerates toward the average height of its four neighbors, so disturbances propagate outward at a fixed speed without any agent ever knowing about the wave it is part of.

Here every patch stores a height and a velocity. The center patch is a driver that oscillates sinusoidally with the tick counter; each staged tick pulls every other patch toward its neighborhood average, applies damping, and integrates velocity into height. The bounded box reflects wavefronts back inward, so interference rings and standing-wave patterns build up over time.