Pendulum Wave (3D)

What this page shows

A row of simple pendulums is released together from the same angle. Each string is a little shorter than its neighbour, so each swings a little faster. At first they move in step; then they drift out of phase and the row traces a travelling wave, then a spiral, then apparent chaos, and after the realignment time they all line up again. Every pendulum here is integrated on its own with the full nonlinear equation (no small-angle shortcut), so what you see is what the physics gives.

The design rule

Choose a realignment time Trev and a number N0. Pendulum i (0 = longest) is built to make exactly N0 + i swings in Trev, so its period is Ti = Trev / (N0 + i). For small swings T = 2π √(L/g), hence Li = g (Ti / 2π)2 with g = 9.80665 m/s². Because the swing counts differ by whole numbers, after Trev every pendulum has completed a whole number of cycles and is back where it started.

Equation of motion

θ'' = -(g/L) sin θ - γ θ', with air damping rate γ (zero by default). The page uses a fourth-order Runge-Kutta step of 2 ms. The energy per kilogram ½ L² θ'² + g L (1 - cos θ) is conserved when γ = 0.

Why big swings break the pattern

The period of a real pendulum grows with amplitude: T = 4 √(L/g) K(sin(θ0/2)), where K is the complete elliptic integral of the first kind (the page evaluates it with the arithmetic-geometric mean). At 14 degrees the correction is only 0.4 %, but at 75 degrees it is about 11 %, which is more than the spacing between neighbouring periods, so the carefully chosen lengths no longer give the right swing counts and the row never realigns. The option compensate the lengths redesigns the strings so that the exact period equals Ti and the pattern comes back. Because the correction factor is the same for every pendulum released from the same angle, the compensated row realigns exactly even though no single pendulum is small-angle.

Phase coherence

The phase of each pendulum is read from its phase-plane position, φ = atan2(-ω/ω0, θ) with ω0 = √(g/L), and the order parameter R = |Σ eiφ| / N measures how aligned they are: R = 1 when all are in step, near 0 when the phases are spread evenly. The fourth graph shows R dropping and returning; it also touches high values at Trev/2 and other simple fractions, where the pendulums form regular patterns.

Checks run in the page

1. The integrated small-angle period equals 2π√(L/g). 2. At 92 degrees the integrated period equals the elliptic-integral formula. 3. Energy is conserved over 200 s. 4. A 15-pendulum row released at 0.02 rad returns to its start after 60 s with phase coherence above 0.999.

Honest simplifications

The bobs are point masses on massless rigid strings and the pivots are frictionless; the damping is a simple linear term, whereas real air drag on a bob is partly quadratic; a real pendulum also has a finite bob size, which shifts the period slightly (the physical-pendulum correction), and strings stretch and twist. The 3D view uses the same numbers, only drawn with a real-looking rack.