A row of pendulums with carefully chosen lengths creates mesmerising patterns — all start together, then ripple and weave before reuniting. The beauty emerges purely from arithmetic: each pendulum has a period that is a ratio of the others.
Pendulum n has length Lₙ chosen so it completes n+1 swings in the time pendulum 1 completes 2 swings. As periods diverge, phase relationships produce apparent waves, spirals and braids. After one full period all N pendulums re-synchronise simultaneously.
Press Launch to release all pendulums from the same angle. Watch for the transient patterns at ¼, ½, ¾ of the full period. Adjust N (number of pendulums) and Amplitude. Hit Reset to restart from perfect alignment.
A viral 2010 Harvard Natural Sciences Demonstration video of a physical pendulum wave has over 60 million views. The same principle — closely spaced resonant frequencies beating against each other — explains the overtone shimmer of a piano string.
This simulation models a row of simple pendulums hanging from a shared bar, each tuned to a slightly different length. Within one fixed cycle time, pendulum k completes exactly (base + k) full swings, so each obeys the small-angle period law T = 2π√(L/g) with length L = g·T²/(4π²). Released in unison, the bobs gradually drift out of phase, sweeping through travelling waves, snaking ripples and seemingly chaotic patterns before snapping back into perfect alignment.
The effect is a visible form of beating, or aliasing, between closely spaced oscillation frequencies. The same physics of harmonic oscillation and resonance underlies the overtone shimmer of musical instruments, the interference fringes in optics and the Moiré patterns seen through overlapping grids. Pendulum-wave machines are a classic demonstration in physics teaching because they make abstract phase relationships directly observable.
What is a pendulum wave?
A pendulum wave is a set of uncoupled pendulums of progressively different lengths, hung in a row. Because each has a slightly different period, releasing them together produces a shifting sequence of wave, spiral and braid patterns that eventually realigns into a single swing.
Why do the pendulums realign after one full cycle?
The lengths are chosen so that, over one chosen cycle time, each pendulum completes a whole number of oscillations. When that cycle ends every pendulum has finished an integer number of swings, so they all return to their starting phase simultaneously.
What determines a pendulum's period?
For small swings the period depends only on the pendulum's length and the local gravitational acceleration, following T = 2π√(L/g). Mass and release angle have essentially no effect at small amplitudes, which is why the patterns are so reproducible.
No. In the small-angle approximation the period is independent of mass, so heavier or lighter bobs swing with the same timing. The patterns depend purely on the lengths.
For angles below roughly 15 degrees, sin θ is very close to θ, which makes the pendulum behave as a simple harmonic oscillator with a constant period. The simulation uses this approximation to keep each period exact.
In a real pendulum, large amplitudes slightly lengthen the period and break the perfect realignment. This simulation models the idealised small-angle case, so the amplitude slider changes how dramatic the sweep looks without spoiling the resync.
Two notes of nearly equal frequency produce a slow throb called beating as they drift in and out of phase. The pendulum wave is the spatial, visible equivalent: many oscillators with closely spaced frequencies beating against each other.
Count sets how many pendulums are in the row, and Cycle sets the full-cycle time after which they all realign. More pendulums give richer wave detail; a longer cycle slows the whole sequence down.
Yes. Physical pendulum-wave machines are widely used in physics classrooms and museums. A viral Harvard Natural Sciences demonstration of one has tens of millions of views online.
Midway through the cycle the pendulums are maximally out of phase, so no simple pattern is visible and the motion appears random. It is fully deterministic, though, and order re-emerges as the cycle completes.