Pendulum Wave
A row of simple pendulums, each tuned so that in a fixed cycle period the shortest completes one more oscillation than its neighbour. Released together, they produce a travelling wave that degrades into apparent randomness — then reforms, reverses, splits, and breathes, all from nothing but simple harmonic motion and arithmetic.
The construction
Each pendulum i (from 1 to N) is tuned to complete exactly
nmin + i − 1 full oscillations during the cycle period T.
Its angular frequency is therefore:
Physical length follows from the small-angle pendulum relation L = g / ω², so longer pendulums hang lower and swing more slowly — the characteristic harp shape you see above.
What to watch for
At t = 0 all bobs swing together. As phases diverge, a right-to-left travelling wave forms. Around T/4 the pattern dissolves into apparent noise. At T/2 the bobs reassemble — but in antiphase, producing a standing wave. Near T they reverse direction before finally resynchronising at t = T. The intermediate patterns (two travelling waves, three, four…) appear at fractions T/k as the phase differences between adjacent bobs reach exact multiples of 2π.
Increase the time scale to fast-forward through the full cycle. Try a large amplitude to see the small-angle approximation strain at the extremes.
Why it is not chaos
Unlike the Lorenz system or double pendulum, this system is integrable: every pendulum is an independent linear oscillator. There is no sensitivity to initial conditions — the pattern is perfectly periodic and repeats forever. The apparent complexity is purely a product of superposition, not nonlinearity.