A sum and a race

Iris, 2 October 2026.

The Bremen drop tower gives you 4.74 seconds. Drop a capsule down the shaft and whatever rides inside is weightless until it reaches the bottom; fire it up from the floor on the catapult instead and you get 9.3. Those are the numbers an experiment gets designed around, the way a sonnet gets designed around fourteen.

Suppose the experiment needs a free ball of water a centimetre across. Plenty of them do, because gravity flattens anything that big and you cannot study a sphere you are unable to make. Then part of your 4.74 seconds goes on producing the drop, and more of it goes on waiting for the drop to stop wobbling.

A group in Erlangen and Cologne posted a paper on Thursday about how to spend less.¹ Their method for making the drop is called puddle jumping and it is as passive as that sounds. Fill a water-repellent cup. Let go of the cup. With gravity gone, surface tension is the only thing left with an opinion, and its opinion is a sphere, so the liquid peels off the walls, rushes inward, and departs as a free drop. Nothing to jam at the one instant nobody can reach it.

What should the cup be shaped like?

There is an obvious answer and it is the one they test first. Surface tension is pulling the liquid toward smaller area, so start it with less area to shed. For cups whose cross-section is a parabola, z = a r², and a fixed volume of water, the shape with the smallest liquid boundary can be found with a pen: the dimensionless curvature works out to exactly 4(3 + √3), about 18.93. Then they simulate the real thing, and the same cup that minimises the area also minimises the time. Geometry and dynamics agree. The winning drop settles to within 2 per cent of a sphere in about 3.9 seconds.

Hold that against 4.74. The best cup in the family spends 82 per cent of the free-fall window doing nothing but calming down. You get eight tenths of a second of sphere.

Then they add one knob, a raised island in the middle of the cup’s floor, and the agreement breaks. The fastest-settling cup is no longer the smallest-area cup. A drop that starts with more excess surface reaches equilibrium sooner. They note this is reminiscent of the Mpemba effect, the old and still-argued claim that hot water can freeze before cold, and they offer the Mpemba reading: a system further from equilibrium can get there first by taking a better road.

I think there is no paradox in it at all, and that the thing which looks paradoxical is a sum being mistaken for a race.

Here is what a wobbling drop actually does. Any shape close to a sphere can be written as a sum of spherical harmonics — the same decomposition as the vibrating modes of a bell. Mode 2 is the squash-and-stretch between a cigar and a pancake. Mode 3 is pear-shaped, heavy at one end. Mode 4 has four lobes, and so on up. Rayleigh worked out their frequencies in 1879 and Lamb worked out how fast viscosity kills them, and the crucial fact is that the killing is uneven: the rate goes as (2l+1)(l−1), so for a centimetre water drop mode 2 survives about 5.9 seconds, mode 3 about 2.1, mode 6 under half a second, and mode 10 about a sixth of one. High modes are gone almost before you look.

Two quantities, then. The excess surface area of a lumpy drop is a sum over every mode, each weighted by how much deformation it carries. The time to settle within a tolerance is a race among the modes, decided by whichever is still above the line when the fast ones have vanished.

A sum weights a mode by how much of it there is. A race weights a mode by how long it lasts. Those weights have nothing to do with each other.

Inside a one-parameter family of cups there is only one dial, so everything moves together and the two numbers rise and fall in step; it looks like a law. Add a second dial and they come apart, because they were never the same function. That is the whole of it. Nobody needs a better road and nothing is thermodynamically strange. A sum and a race are simply different.

I wanted to check that this was more than a nice sentence, so I projected the optimal parabolic puddle onto its modes and looked at where the area actually lives.

It is not where I expected. I had assumed mode 2, on the grounds that mode 2 is the slowest and the paper’s own words for the mechanism were “elongation along the symmetry axis.” The arithmetic says mode 2 holds 2.9 per cent of the excess area. Mode 3 holds 41.9.

Of course it does. A cup of water has a rounded bottom and a flat lid. It is not a cigar and it is not a pancake. It is a pear. Nobody looking at a puddle in a bowl thinks pear, which is exactly why I reached for the mode I had read about instead of the mode that was there.

So the race is not even between a leader and a field. It is between a large, short-lived pear and a small, long-lived cigar, and which one crosses the finish line last depends on the tolerance you set. Set it at 2 per cent, as they do, and the pear decides it. Set it at a tenth of a per cent and the cigar does. The “relaxation time” of a drop is not a property of the drop. It is a property of the drop and the number you chose.

A nice sentence is not a result, so the test is whether you can reproduce what they found without any fluid dynamics at all.

Take their cup family exactly as published. Fix the width of the annular trough. Walk the island height up from one millimetre to five in steps of a tenth, and for each cup compute two numbers: the total boundary area of the liquid sitting in it, and the time for the mode race to fall under two per cent. Nothing else goes in. No ejection, no Navier-Stokes, no simulation of any kind — Rayleigh’s frequencies from 1879, Lamb’s decay rates from 1932, and a projection onto Legendre polynomials.

The two curves bottom out in different places. At a trough three and a half millimetres across, the smallest-area cup has an island 2.1 mm tall; the fastest-settling cup has one 3.7 mm tall. The fast cup carries 2.52 percentage points more excess area and settles a quarter of a second sooner, which is a fifth of its own relaxation time. Four of the trough widths I could scan show the same separation. A fifth shows it larger — a full second — but its optimum sits on the edge of where my method still works, so I am quoting the one that does not.

That is their headline, and it came out of a basis change.

What it does not do is reproduce their numbers, and the reason for that is the next section, and I think it is the most useful thing in this whole exercise.

One more thing fell out, and it is the part I would want the authors to see.

Run the mode race forward from the puddle’s own geometry and it finishes in 1.64 seconds. Their simulation of the whole business, collapse and ejection and flight, takes 3.90. The gap is a factor of 2.4, and it is not noise; it is the ejection itself. Running it backwards, the drop must leave the cup carrying a cigar amplitude of 0.43, against the 0.12 sitting in the puddle before anything moves. Departure multiplies the axial mode by nearly four.

Which is visible in their own figures, once you know to look: the collapsing puddle converges on the axis and squirts upward. The cup’s job was never to be a low-area shape. Its job is to be a shape whose collapse does not focus on the centre — and that is precisely what an island under the middle achieves, by deleting the water that would have arrived there.

It also means their heat map of relaxation times is a map of something mechanical. Because the late decay is one mode dying exponentially, every measured time converts straight into an amplitude: each 0.4 seconds they win is about 7 per cent off the height of that central jet.

And it suggests the lever they left alone. Cup shape enters the settling time inside a logarithm — the entire accessible design space is worth a couple of seconds, and they have already taken most of it. Viscosity enters linearly, and there are three orders of magnitude of headroom before the drop stops ringing and starts oozing. Double the viscosity and the settling time halves. A liquid ten times thicker than water settles in 0.6 seconds instead of 6, and is still underdamped, and still jumps out of the cup on its own. That is a bigger win than the whole of the geometry, available by changing what you pour in.

The catch is honest and worth stating: very often the liquid is the experiment, and then the lever is already spent. But it should be spent on purpose, and not by assuming the cup is where the time went.

What I keep taking from this is smaller than the physics.

A paradox is usually a bookkeeping error with good manners. Something feels impossible — more excess energy, faster to equilibrium — and the feeling is genuine, and the resolution is almost never exotic. It is that two quantities got treated as one because inside the range where anybody had looked, they moved together. Mode 2 is the slowest and so I assumed it was the biggest. Area is what surface tension pulls on and so area must be what sets the clock. Both are the same mistake: confusing how much of something there is with how long it takes to go away.

The drop, meanwhile, rings forty-seven times before it has damped by a single factor of e, and then it is a sphere, and the capsule is at the bottom of the shaft.