Pendulum waves
A row of uncoupled pendulums, each slightly longer than the last, released simultaneously from the same angle. Small differences in period compound into mesmerizing patterns — apparent traveling waves, standing waves, and total chaos — before the pendulums eventually realign and the cycle begins again.
Phase relationships and visual waves
Each pendulum in the row completes a slightly different number of oscillations during the full cycle time. The longest pendulum might complete 51 oscillations while the shortest completes 65. Because their frequencies form an arithmetic sequence, the phase differences between adjacent pendulums increase linearly with time. This linear phase gradient is exactly the condition that produces a traveling wave — even though no energy passes between the pendulums at all.
From order to chaos and back
At the start, all pendulums swing in unison. As time progresses, the slight frequency differences cause them to drift apart. The pattern passes through apparent traveling waves, standing waves, and moments of seeming disorder. But because every frequency is a rational multiple of a common base frequency (determined by the cycle time), all pendulums must eventually return to their starting phase simultaneously. The cycle time is the least common period — the moment when every pendulum has completed an integer number of swings.
Beat frequencies
When two pendulums with slightly different frequencies swing side by side, they produce a beat pattern — slowly alternating between constructive alignment (swinging together) and destructive cancellation (swinging opposite). The beat frequency equals the difference between the two oscillation frequencies. With many pendulums, these pairwise beats combine to create the rich visual texture you see. The same mathematics governs acoustic beats between tuning forks, the wobble of AM radio signals, and the shimmering of moiré patterns.
The physics of pendulum length
For a simple pendulum with small-angle oscillations, the period is T = 2π√(L/g), depending only on length and gravity — not on mass or amplitude. This means the period scales as the square root of length. To get an arithmetic sequence of frequencies (equal spacing), the lengths must follow a specific non-linear pattern. The Harvard Natural Sciences Lecture Demonstrations team popularized this apparatus, and it has become one of the most beloved physics demonstrations in the world.