Pendulum WaveBeta

About Pendulum Wave

A pendulum wave is the opposite of chaos. The pendulums are not connected to each other; each simply swings at its own fixed period. Even so, they produce a sequence of travelling snakes, two-row and three-row patterns, and moments of apparent disorder, and then snap back into a single line at an exactly predictable time. The appeal of the device is that every one of those patterns can be calculated in advance.

This tool integrates each pendulum separately with θ'' = −(g/L)·sin θ − c·θ' (RK4 in steps of 2 ms or less). It does not replace sin θ by θ, so a wide swing lengthens the period as it does in a real pendulum. According to Harvard's lecture-demonstration notes, their apparatus follows a design published by Richard Berg of the University of Maryland, and the idea goes back to a device built by Ernst Mach around 1867.

How the lengths are chosen

During the realignment time T, the longest pendulum makes k complete swings, the next k + 1, and so on. Harvard's rig has 15 pendulums, T = 60 s and k = 51, so the shortest makes 65 swings. Pendulum i therefore has period T/(k + i), and inverting the simple-pendulum formula P = 2π√(L/g) gives L = g·(P/2π)².

With g = 9.81, the 51-swing pendulum has P = 1.176 s and L = 34.39 cm; the 65-swing pendulum has P = 0.923 s and L = 21.17 cm. All fifteen fit into a span of about 13 cm. Turning on amplitude compensation recomputes the lengths from the exact period at the working amplitude; at 10° the longest pendulum shrinks to 34.26 cm, 1.3 mm shorter.

When patterns appear: T·p/q

At time t, the phase difference between any two neighbouring pendulums is 2π·t/T, the same for every pair. At t = T/2 that is half a cycle, so alternate pendulums sit at opposite ends: two rows. At T/3 and 2T/3 you get three rows. In general, when p/q is a fraction in lowest terms, at t = T·p/q every pendulum sits at one of q phases and you see q rows. For the Harvard settings that means two rows at 30 s, three at 20 s and 40 s, and four at 15 s and 45 s.

The snake-like wave between patterns shows up when the phase difference is small, near t = 0 or t = T: phase then grows gently along the row, which looks like a long-wavelength sine curve. At times where p/q needs a large q there are too many rows to see, and the display looks random. The tool's pattern timeline and phase clock show exactly this calculation.

What goes wrong when you pull them wide

At amplitude θ₀ the true period is T₀ / AGM(1, cos(θ₀/2)), where AGM is the arithmetic–geometric mean. That is 0.19% longer than the small-angle value at 10°, 2.38% longer at 35° and 7.3% longer at 60°. Running the Pull it wide scenario (35°, no compensation) through the same equations, at 60 s the fifteen bobs are spread from 14° to −35°.

The interesting part is that, if all the pendulums are released at the same amplitude, they all slow down by the same fraction, 2.38%. The pattern is not destroyed, only delayed: at 60 × 1.0238 = 61.43 s all fifteen are back together at exactly 35°. The line looks broken at 60 s because the same fractional delay means more lost swings for a pendulum that makes more swings. The 51-swing pendulum is 1.2 swings behind and the 65-swing one is 1.5 swings behind, and that difference is what reshapes the picture at the 60-second mark. With compensated lengths the realignment happens right at 60 s.

The tuner: why a millimetre matters

Period goes as the square root of length, so a relative length error becomes half as large in the period. A 1 mm error on the middle pendulum (58 swings, L = 26.59 cm) shifts its period by 0.19%. Over 58 swings in 60 s that adds up to 0.11 of a swing, a phase error of about 39°, which is easy to see in the line-up. The tuner scenario detunes three randomly chosen pendulums so that each makes 0.25 to 0.45 swings more or fewer in 60 s. When hunting by ear, listen for the tick at the centre crossing that drifts out of rhythm with the others.

On another planet

Every period scales as 1/√g by the same factor, so a rig tuned on Earth shows the same patterns in the same order anywhere; only the clock stretches or shrinks. The realignment time becomes 60 × √(9.81/g): 147.6 s on the Moon (g = 1.62), 97.6 s on Mars (3.71) and 37.7 s on Jupiter (24.79).

What this simulation leaves out

The pendulums are perfectly independent. In a real rig the shared frame flexes slightly and can pass energy between pendulums. The default air drag is a single linear term with the same coefficient (0.004 per second by default) for every pendulum. In that case each amplitude decays as e^{−ct/2}, identically for all of them (10° falls to about 8.9° after 60 s), so the ratios survive. In reality drag depends on each bob's speed and each string's length, amplitudes decay at different rates, and because of the amplitude-dependent period above the pattern gradually blurs over long runs. Turn on "Light bobs: speed² air drag" in Free lab to see this: it adds −κ·L·|θ'|·θ' (κ = 0.15 per metre) to the angular acceleration, so the longer pendulums, whose bobs move faster, lose amplitude sooner. Released at 12°, after 60 s the longest pendulum is down to about 4.5° and the shortest to about 5.1°, so their amplitude ratio has drifted from 1 to about 0.87. The bobs are point masses and the strings do not stretch.

Worked examples

How to use it

1. Pick a scenario: Harvard demo shoot, Pull it wide, The tuner, On another planet or Free lab.
2. Check the pattern timeline for the next two-row or three-row moment and watch for it.
3. In Free lab, change pendulum count, T, swing count, amplitude, air drag and planet, and compare with amplitude compensation on and off.

Frequently asked questions