OneTaskBy Tim King · Updated
Pendulum Wave
Watch a Row of Pendulums Drift In and Out of Step
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How the Pendulum Wave works
This pendulum wave demo shows patterns from pendulums of different lengths. Press Release and the row of pendulums swings together, then slowly fans out into a travelling wave, breaks into two groups swinging against each other, then three, dissolves into apparent chaos and, exactly one cycle later, lines up again. Change the number of pendulums, the cycle time and the release angle, and turn the connecting line on or off to see the wave shape more clearly.
Pendulum wave: what it does and why
The pendulum wave is a classic physics demonstration usually built from a dozen or so steel balls on strings of graduated length. This version uses a 2D physics engine with real string constraints and gravity, so the pendulums really are swinging, not animated.
How the lengths are chosen. A pendulum's period depends only on its length: it is proportional to the square root of the length, so a pendulum four times as long swings half as often. The trick is to give the longest pendulum some whole number of swings per cycle — say 20 in 30 seconds — the next one 21, the next 22, and so on. Every pendulum then completes a whole number of swings in exactly one cycle, so they all arrive back at the release point together, however scrambled they look in between. The lengths that produce those periods are shown in the stats.
What you see. Early in the cycle neighbouring pendulums are only slightly out of phase, which shows as a smooth travelling wave. At a quarter of the cycle every second pendulum is a half-swing out of step, so the row splits into two groups moving opposite ways; at a third, three groups; near the middle the pattern looks random. The second half of the cycle runs the same sequence backwards until they align.
Why real ones drift. A physical pendulum wave slowly loses the pattern because the period of a real pendulum grows slightly with amplitude, and air drag shrinks the amplitude unevenly. This simulation has no air resistance, but the amplitude effect is still there, so wide release angles make the realignment less crisp than small ones — try 15 degrees against 60 to see the difference. Everything runs in the browser; Pause, Fullscreen and Save PNG work as in the other physics toys.
FAQ
How are the string lengths chosen?
The longest pendulum is given a whole number of swings per cycle, the next one more, and so on. Because period depends only on length, each pendulum completes a whole number of swings in exactly one cycle, so they all arrive back at the release point together. The lengths are listed in the stats.
Why does the row split into two groups partway through?
At a quarter of the cycle every second pendulum is half a swing out of step with its neighbours, so alternate pendulums move opposite ways. At a third of the cycle the same thing happens with three groups, and near the middle the pattern looks random.
Why is the realignment less crisp at wide release angles?
A pendulum's period grows slightly with amplitude, so at 60 degrees the pendulums are no longer in an exact integer ratio and drift out of step. Try 15 degrees for a sharp realignment and compare.
Does the cycle time change the wave?
The cycle time sets how long the whole pattern takes to repeat, which sets the string lengths. A longer cycle means longer strings and a slower, more spread-out wave; the sequence of patterns is the same.
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