15 pendulums, one pattern
Each pendulum swings a little faster than the last. Every 24 seconds they line up again.
x = cos(2π·(15 + k)·t / 24)Fifteen pendulums hang in a row, seen from above. Each one is a little shorter than the one before, so it swings a little faster. Pendulum k (counting from 0) swings exactly 15 + k times in 24 seconds: the first 15 times, the last 29 times.
Snakes, groups and a straight line
At the start they all swing together. Then the faster ones pull ahead and the row bends into a travelling wave, then into snakes that seem to move on their own.
At certain moments the chaos sorts itself out:
| Time | What lines up |
|---|---|
| 6 s | every 4th pendulum is in step |
| 8 s | every 3rd pendulum |
| 12 s | every 2nd: two clean rows swinging opposite ways |
| 24 s | all 15 are back in a straight line |
Why
By time t, neighbouring pendulums differ by t / 24 of a swing. At 12 seconds that is half a swing, so neighbours are opposite and every second one matches. At 8 seconds it is a third, at 6 seconds a quarter. At 24 seconds it is one whole swing, so everyone is back in line.
A pendulum's period is T = 2π√(L / g), so the lengths are chosen to give exactly these swing counts. This toy is called a pendulum wave.
