Simple pendulums tuned to different frequencies spontaneously produce travelling waves, standing waves, and synchrony.
A pendulum wave apparatus consists of N pendulums with carefully chosen lengths. The lengths are not arbitrary — each pendulum k is designed to complete exactly (nmin + k) oscillations in one cycle time T. Because each pendulum executes simple harmonic motion (SHM) independently, their collective motion sweeps through a sequence of beautiful, predictable patterns over time.
nMin + k oscillations in T seconds.T_k = T / (nMin + k)L_k = g · T_k² / (4π²) = g·T² / (4π²·(nMin+k)²)x_k(t) = A · sin(2π·(nMin+k)·t / T)
All pendulums start in phase. Reoccurs exactly at every multiple of T — typically 60 seconds.
Phase differences grow linearly across pendulums, producing an apparent wave sweeping left to right.
At the half-cycle, even and odd pendulums are in antiphase, producing a symmetric nodal pattern.
Intermediate times reveal complex multi-node shapes — the number of apparent groups equals the time fraction denominator.
| Time | Pattern | Description |
|---|---|---|
| 0 s | Synchrony | All 15 bobs aligned — a single cluster |
| ~7.5 s | 8 groups | Bobs arrange into 8 visible clusters |
| ~10 s | 6 groups | 6 distinct groups visible |
| ~15 s | Travelling wave | Smooth phase ramp — apparent rightward motion |
| ~20 s | 3 groups | Three clusters oscillate together |
| ~30 s | Standing wave | Two antiphase halves — 7–8 node pattern |
| ~45 s | Travelling wave | Reversed — apparent leftward motion |
| 60 s | Synchrony | All pendulums return to exact alignment |
The key insight is that adjacent pendulums differ by exactly one oscillation per cycle. The relative frequency between pendulum k and k+1 is Δf = 1/T — meaning they drift apart at a constant rate and re-align after exactly T seconds. This predictability is why the patterns are so strikingly regular and reproducible.
The phenomenon is related to quantum revivals in physics: a wavepacket spreads over time due to dispersion, then re-forms (revives) when the phases realign — a direct analogue of all pendulums returning to synchrony at t = T. Similarly, the intermediate patterns correspond to fractional revivals where subsets of the pendulums phase-align.
gcd(nMin, floor(t·N/T)) approximately — the visible grouping is determined by
how many pendulums have completed an integer number of half-cycles relative to each other.
The most famous physical implementation was built by the Harvard Natural Sciences Lecture Demonstrations group. Their apparatus uses 15 pendulums with lengths calculated to complete 51 through 65 oscillations in 60 seconds — the same parameters used in the Classic 15 preset here. The video of this demonstration has been viewed millions of times online and introduced wave physics to a generation of students. The apparatus requires precise machining: the longest pendulum is about 2.4 m and the shortest about 1.1 m.