Pendulum wave

The math

Each of the 24 pendulums is slightly shorter than the one before it, tuned so that in one minute the longest completes exactly 51 swings, the next 52, and so on up to 74. Because every count is a whole number, all 24 arrive back at their starting point together at the 60-second mark — perfectly in step. In between, the row drifts apart by fixed fractions of a swing: at 30 s neighbours differ by half a cycle, so odd and even pendulums are exactly opposite (that is the moment the colours split into two groups); at 20 s they form three groups, at 15 s four, and so on. The travelling "snake" you see is nothing more than those small fractions adding up along the row. The lengths come from the exact period of a real pendulum, T = 4·√(L/g)·K(sin²(θ₀/2)), with K the complete elliptic integral, and the animation integrates the full non-linear equation of motion rather than the small-angle approximation — so the 60-second return is exact, not approximate. Colour is position: teal on one side of centre, pink on the other, white as a bob crosses the middle at full speed. The second minute shows the very same motion while the camera rises to a bird's-eye view.

From the description of 24 Pendulums Fall Out of Sync.