How are the pendulum lengths chosen?

Pendulum k must make exactly N₀+k swings in the time Γ, so its period is Γ/(N₀+k). Inverting T = 2π√(L/g) gives L_k = g(Γ/(2π(N₀+k)))². The classic 15-pendulum build uses Γ = 60 s and N₀ = 51, so the longest is about 34.4 cm and the shortest about 21.2 cm.

Why do two and three rows appear?

The pendulums are not coupled. Neighbours differ in frequency by 1/Γ, so at time t their phase difference is 2πt/Γ. At t = Γ/2 neighbours are in antiphase and the odd and even pendulums form two rows; at Γ/3 they group by k mod 3; at Γ the phase step is a whole multiple of 2π and everything lines up again. At times that are not simple fractions of Γ the phases look scrambled.

Why does a large release angle spoil the realignment?

A pendulum's period grows with amplitude, roughly T₀(1 + θ₀²/16). If every pendulum starts at the same angle, all periods stretch by the same factor and realignment is only delayed. Real builds usually push the bobs with a board to the same horizontal offset, so short pendulums start at larger angles and slow down more. The period ratios are then no longer exact and the line does not reform at Γ. Below about 10° the effect is negligible.

How accurate must a home-made pendulum wave be?

A length error δL gives a phase error of about (N₀+k)·δL/(2L) cycles after Γ. With the classic parameters, keeping that below 0.1 cycle needs lengths accurate to about 1 mm. Use a V-shaped bifilar suspension to stop the swing plane from wandering, measure the vertical distance from the bar to the bob centre, and tune each pendulum against the table.

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〰️Pendulum Wave Simulator - length calculator, travelling waves, DIY build sheet

A pendulum wave is a row of pendulums of decreasing length chosen so that pendulum k swings exactly N₀+k times in a cycle period Γ. Released together, they show a travelling snake, two and three interleaved rows, apparent chaos, and finally line up again at Γ. Adjust N, Γ, N₀, amplitude and damping, switch between perspective, top and end views, scrub the timeline or jump to Γ/2 and Γ/3. The exact large-angle mode shows why real builds lose alignment, and a build sheet gives the lengths for making one.

Free · No sign-up · Runs in your browserLast updated: 2026-10-29

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How to use Pendulum wave

1Pick a preset (classic 15 pendulums / 60 s) or set N, the cycle period Γ and N₀
2Press Play, or drag the timeline and use the Γ/2, Γ/3 buttons to jump to pattern moments
3Drag the perspective view to rotate it, or switch to the top view to see the snake
4Tick exact large-angle and choose board release to see whether the line reforms at Γ
5Enter the bob diameter in the build sheet and copy the lengths, string lengths and tolerances

FAQ

How are the pendulum lengths chosen?: Pendulum k must make exactly N₀+k swings in the time Γ, so its period is Γ/(N₀+k). Inverting T = 2π√(L/g) gives L_k = g(Γ/(2π(N₀+k)))². The classic 15-pendulum build uses Γ = 60 s and N₀ = 51, so the longest is about 34.4 cm and the shortest about 21.2 cm.
Why do two and three rows appear?: The pendulums are not coupled. Neighbours differ in frequency by 1/Γ, so at time t their phase difference is 2πt/Γ. At t = Γ/2 neighbours are in antiphase and the odd and even pendulums form two rows; at Γ/3 they group by k mod 3; at Γ the phase step is a whole multiple of 2π and everything lines up again. At times that are not simple fractions of Γ the phases look scrambled.
Why does a large release angle spoil the realignment?: A pendulum's period grows with amplitude, roughly T₀(1 + θ₀²/16). If every pendulum starts at the same angle, all periods stretch by the same factor and realignment is only delayed. Real builds usually push the bobs with a board to the same horizontal offset, so short pendulums start at larger angles and slow down more. The period ratios are then no longer exact and the line does not reform at Γ. Below about 10° the effect is negligible.
How accurate must a home-made pendulum wave be?: A length error δL gives a phase error of about (N₀+k)·δL/(2L) cycles after Γ. With the classic parameters, keeping that below 0.1 cycle needs lengths accurate to about 1 mm. Use a V-shaped bifilar suspension to stop the swing plane from wandering, measure the vertical distance from the bar to the bob centre, and tune each pendulum against the table.

Features

Lengths and periods from L_k = g(Γ/(2π(N₀+k)))²
Smooth animation in perspective (drag to rotate), top and end-on views
Draggable timeline with Γ/4, Γ/3, Γ/2 pattern marks and live annotation
Closed-form small-angle solution and exact sin θ RK4 integration, with damping
Same-angle or board release, phase-spread measure and realignment time
DIY build sheet in cm with string length, large-angle correction and tolerance, copyable as CSV

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