The Swing That Hands It Over
Two pendulums joined by a weak spring. Start one and it stops dead a minute later with the other one swinging, then takes it all back. The clock is the difference between two frequencies a hundredth of a hertz apart.
What it is
Two pendulums hanging from a beam with a light spring between them. Headphones help, because each one is panned hard to its own ear.
Start one and leave the other still. Wait. The still one begins to move, the moving one begins to stop, and a minute later they have swapped completely. Then it comes back. The sound walks from one ear to the other and back for as long as you leave it.
Start them together and it never happens. Start them opposed and it never happens either. Only the lopsided beginning does this.
Everything runs nine times over, because a hand over takes a minute and nobody waits a minute.
How it works
The pair has two ways of moving that stay put. Both swinging together, which never stretches the spring, so it might as well not be there. And both swinging exactly against each other, where the spring is stretched the whole time and pulls harder, so it goes faster.
Those are the normal modes, and one pendulum swinging on its own is simply those two added together. They start in step and drift apart, and where they cancel one pendulum is still and where they add the other is at full swing.
So the hand over isn’t a transfer. Nothing decides to move the energy. Two frequencies drift past each other and the appearance of transfer is what that looks like.
Which makes the clock the difference between them, and nothing else:
| spring | together | opposed | apart | predicted | measured |
|---|---|---|---|---|---|
| 0.08 | 0.5255 Hz | 0.5293 Hz | 0.0038 Hz | 260.2 s | 260.0 s |
| 0.15 | 0.5255 Hz | 0.5326 Hz | 0.0072 Hz | 139.2 s | 139.2 s |
| 0.35 | 0.5255 Hz | 0.5421 Hz | 0.0166 Hz | 60.2 s | 60.2 s |
| 0.80 | 0.5255 Hz | 0.5627 Hz | 0.0372 Hz | 26.8 s | 26.6 s |
| 2.00 | 0.5255 Hz | 0.6143 Hz | 0.0889 Hz | 11.2 s | 11.2 s |
Predicted from the two frequencies, measured off the simulation, agreeing to within two tenths of a second across a twenty-three-fold range.
A hundredth of a hertz between two modes is a minute of hand over. The weaker the spring the closer the modes and the slower it goes, and with no spring at all they’re identical and it never happens.
What surprised me
The ripple is bigger than the thing. Measuring the hand over means finding when the second pendulum is at its loudest, and the obvious way is to watch the share of energy and look for a peak. That returns 1.65 seconds against a true 60.
The share of energy doesn’t just drift over a minute, it also ripples every swing, because energy sloshes between height and speed twice a period. That ripple is small in the picture and enormous per second, so a peak finder walks straight to the swing period. It’s a real number about the rig and it’s entirely the wrong one.
Averaging over a few swings first fixes it. What I like about the failure is that the wrong answer isn’t noise, it’s another genuine feature of the same system, which is much harder to notice than nonsense would be.
Euler would have faked the effect. This runs for minutes of simulated time and forward Euler quietly adds energy on every step of an oscillator. Over four minutes the pendulums would wind themselves up, and a pendulum growing while its neighbour shrinks is exactly what the page exists to show.
So it’s velocity Verlet, and there is a test that the total energy holds to within a tenth of a per cent over four minutes of simulated swinging. Without it I could have shipped a convincing demonstration of an integrator bug.
Neither mode does anything at all. Starting both together, or both opposed, gives you two pendulums that swing for as long as you watch and never exchange a thing. That’s obvious after the fact and it doesn’t look obvious while it’s happening: the arrangement that seems most likely to interact is the one where nothing happens, because the spring never changes length.
What I would do next
Detune them. Everything here has two identical pendulums, and the whole effect depends on that: make one a centimetre longer and the modes stop being a clean split, the hand over stops completing, and some of the energy stays behind permanently.
How much stays behind is a number, it depends on the detuning against the coupling, and it is the same arithmetic that decides whether two coupled anything share energy or ignore each other. That’s a better day than this one and it needs one more slider.