2D Pendulum-Wave Resonance Dashboard: Exact Return-Time Math

Return-Time Parameters

Pendulum count (N) 15

Return time T 60.0 s

Start-count offset N0 40

Release amplitude 18°

Live Readout

Elapsed time

0.0 s

Cycle progress

Coherence r(t)

1.000

Next resync in

60.0 s

How it works

Each of the N pendulums swings completely independently, integrated from the real nonlinear equation of motion with 4th-order Runge–Kutta at a fixed 4 ms sub-step:

theta_n'' = -(g / L_n) * sin(theta_n)

Lengths are not chosen arbitrarily — they are solved backwards from the return time T so that pendulum n completes exactly N0+n whole oscillations in one cycle:

L_n = g * (T / (N0 + n))^2 / (4 * pi^2)

Because every L_n corresponds to an integer oscillation count over the same T, every pendulum's small-amplitude phase satisfies theta_n(T) = theta_n(0): the whole row must return in step at t = kT, for every integer k, purely from that arithmetic — nothing is nudged into sync by hand.

Coherence r(t) — a Kuramoto-style order parameter computed from each pendulum's phase-plane angle atan2(-theta_dot_n/omega_n, theta_n); r=1 means every pendulum is at the same point in its own cycle, r near 0 means the phases have scattered around the circle. Watch it collapse from 1 toward 0 mid-cycle, then climb back to 1 exactly at t=T.
Polar phase plot — the same phases plotted as N dots on a unit circle; a tight cluster is a coherent "wave" moment, a scattered ring is the chaotic-looking middle of the cycle.
Click any bob to inspect its exact solved length, period and running oscillation count (time elapsed / its own period), so the T = (N0+n)-cycles relationship in the formula is directly checkable per pendulum.

Pendulum row (RK4, real lengths) Coherence r(t) strip Polar phase plot