Pendulum waves: A lesson in aliasing

A set of uncoupled pendula may be used to exhibit "pendulum waves," patterns that alternately look like traveling waves, standing waves, and chaos. The pendulum patterns cycle spectacularly in a time that is large compared to the oscillation period of the individual pendula. In this article we derive a continuous function to explain the pendulum patterns using a simple extension to the equation for traveling waves in one dimension. We show that the cycling of the pendulum patterns arises from aliasing of this underlying continuous function, a function that does not cycle in time.