Pendulum Phase Array
\(N\) independent (uncoupled) pendulums with lengths tuned so pendulum \(k\) completes exactly \((n_0+k)\) oscillations in time \(T\) — producing snake, butterfly, and standing-wave patterns from differential phase alone.
The Phase Array Principle
The pendulums are not coupled — there are no springs or mechanical links between them. Each pendulum \(k = 0, 1, \ldots, N-1\) swings independently with length \(L_k = \dfrac{g\,T^2}{4\pi^2(n_0+k)^2}\), chosen so it completes exactly \((n_0+k)\) oscillations in the pattern period \(T\). The differential phases between adjacent pendulums create the apparent collective wave patterns, which repeat perfectly at \(t = T\).
Governing Equation
Each pendulum satisfies the exact nonlinear equation \(\ddot\theta_k = -(g/L_k)\sin\theta_k - d\,\dot\theta_k\), integrated with RK4 (step \(\Delta t = 0.005\,\text{s}\)). No small-angle approximation is used; the pattern periods are therefore slightly different from the ideal SHM prediction for large \(\theta_0\).
Reading the Displays
- Pendulum canvas: physical view showing rods, bobs, and the rainbow Catmull-Rom envelope through all bobs. Fading trail shows recent envelope history.
- Wave strip: 1D projection showing the horizontal displacement \(x_k = L_k\sin\theta_k\) of each bob, making the wave pattern instantly legible.
- Phase bar: progress within the pattern period \(T\), with key pattern times marked.
Parameters
Presets
Pattern Jump
Jump to key moments within the current cycle. The pattern repeats every \(T\) seconds. Key times: \(t=0\) (in-phase), \(t=\tfrac{T}{N}\) (snake), \(t=\tfrac{T}{2}\) (butterfly).
Cite this tool
Kapita, S. (2026). Pendulum Phase Array. Math Tools. https://shelvean.github.io/math-tools/pendulum_wave.html
Kapita, Shelvean. "Pendulum Phase Array." Math Tools, 2026, shelvean.github.io/math-tools/pendulum_wave.html.
@online{kapita2026pendulumwave,
author = {Shelvean Kapita},
title = {{Pendulum Phase Array}},
year = {2026},
organization = {Math Tools},
url = {https://shelvean.github.io/math-tools/pendulum_wave.html}
}