SVG animation of a pendulum wave with 12 pendulums, the lowest pendulum making 60 oscillations in one minute, the next 61, and so forth – in the animations, tap or hover over a pendulum to pause
In 2001, two University of Minnesota Morris researchers have derived a continuous function explaining the patterns in the pendulums using an extension to the equation for traveling waves in one dimension, and showed that their cycling arises from aliasing of the underlying continuous function.[4]
The lengths of the pendulums are set such that in a given time t, the first pendulum completes n oscillations, and each subsequent one completes one more oscillation than the previous. As all pendulums are started together, their relative phases change continuously, but after time t, they come back in sync and the sequence repeats.[1]
For small perturbations, the period of a pendulum is given by