Each of the N pendulums is an independent small-angle simple harmonic oscillator. Pendulum i is given exactly (N_base + i) full swings during the re-sync period T, so its angular frequency is fixed at:
ω_i = 2π·(N_base + i) / T
θ_i(t) = A·sin(ω_i·t) (small-angle SHM)
L_i = g / ω_i² (pendulum length that gives ω_i)
Because every ω_i is an exact integer multiple of 2π/T, all N pendulums return to θ=0 moving the same way at every multiple of T — the row "explodes" into a traveling-wave pattern in between and snaps back into a single swinging line exactly on schedule. No two lengths are equal, so no two pendulums ever move in lockstep except at those sync instants.
- Coherence R — a Kuramoto-style order parameter, R(t) = |(1/N)·Σ e^{iω_i t}|. R = 1 means every pendulum shares the same phase (a single moving line); R → 0 means phases are scattered evenly around the circle (the row looks scrambled). Watch the strip below the pendulums.
- Length range — each visible rod is drawn to scale from its real physical length L_i = g/ω_i², so a longer rod always swings slower, exactly as gravity dictates.
- Next full sync — countdown to the next t = k·T instant, when R returns to 1 and the row realigns.