Whichever ceiling is lower
Almost everything known about bursting bubbles was learned at a flat surface. A bubble rises, its cap thins, the film ruptures, and the open cavity collapses under its own curvature; capillary waves run down the wall of the hole and converge at the bottom, and out of that focus comes a thin jet that breaks into a few droplets. This is where sea salt comes from. It is where a great deal of what the ocean gives the atmosphere comes from.
The flat surface is a fiction. Alfonso Gañán-Calvo put it plainly in a paper posted two days ago: it is “an idealisation that the sea surface satisfies almost nowhere.” A breaking wave does not present a plane. It presents a crest in three states at once — droplets already in the air, a face still coherent and streaked with entrained gas, and a foreground that is not a surface at all but a volume of interpenetrating liquid and gas. Out of that volume, drops get torn. Each one leaves with whatever gas it happened to be holding.
So the honest object is not a bubble at a bath. It is a bubble inside a drop.
Two things change when you shrink the world around it, and both of them help. The drop has an outer surface of its own, which is curved, which means it is already pressing inward; that pressure adds to the bubble’s, so the collapse is driven harder. And there is only so much liquid — beyond the drop there is nothing to be moved — so there is less inertia to overcome. Push harder, move less. Bubbles that a flat surface would have shut out entirely, because the liquid was too viscous or they were too small, throw droplets when you enclose them. Confinement is not a constraint on the event. It is what permits it.
But the question I keep turning over is the second one in the paper, which is how much such a thing throws.
There are two ceilings on it, and they are not the same kind of thing.
The first is geometric, and it is the one that operates at a flat bath, where liquid is unlimited and yet the ejected mass is not. What limits it there is the cavity. The hole is what focuses the waves, so the hole is what sets the scale of the jet, and the gas volume therefore acts as a template rather than as a reservoir — nothing about the gas gets thrown, but its size decides how much liquid the collapse can set moving. Measured, it comes to about one per cent of the bubble’s volume, which is what a century of jet-drop counting already said: a handful of droplets a tenth of the bubble’s radius across, each a thousandth of its volume.
The second ceiling is about stuff. You cannot throw more water than the drop contains. At a flat bath this is inactive and invisible; in a thin shell it is the only thing that matters, and the same one per cent gets taken of the liquid instead.
Now, two ceilings, and the event obeys whichever is lower. Write down the simplest smooth function that becomes each of them in turn and introduces no second constant, and there is essentially one candidate:
M = C · V_gas V_liq / (V_gas + V_liq).
Which is a harmonic combination — the same shape as resistors in parallel, as capacitors in series, as the reduced mass of a two-body problem. It is what you always get when two limits are in series and either one alone can be the bottleneck. The big one drops out and the small one governs, smoothly, with no seam where the changeover happens.
The part that is not obvious is that a single C serves both ends.
There was no reason it had to. One ceiling is a statement about wave focusing and cavity geometry; the other is a statement about how much water is in the bag. Nothing connects them. A constant that converts “the collapse can mobilise this much” into ejected liquid, and a constant that converts “there is only this much” into ejected liquid, are different constants with different jobs, and if they had come out at 0.013 and 0.4 nobody would have blinked.
They come out the same. Averaged column by column, C sits between 0.010 and
0.019 with no trend across more than three decades of liquid-to-gas ratio. That
flatness is the actual evidence, and it is better evidence than a good fit
would be, because a fit can be forced and a constant that refuses to drift
cannot. It says the two ceilings really are the operative pair — that whichever
one is binding, the machinery downstream of it converts the constraint into
thrown water at one and the same efficiency.
One more thing, and it is the part I want to keep.
Everyone wants the droplet count. It is what a spray model asks for, it is what gets measured in a laboratory, it is the number in the abstract of every paper in this field. Under mesh refinement it does not converge. The ejected mass does. So the paper reports the mass, builds the law on the mass, and puts the count aside — while noting that a single threshold hides at least three different ways an ejection can fail, so that the boundary is sharp although the microscopic physics behind it is not unique.
That is a discipline, and it is rarer than it sounds. When two natural observables disagree under refinement, you are allowed a law about the one that converges, and only that one. The other is telling you about your grid.
The fraction of its own liquid that a hollow drop throws runs from about three parts in a hundred thousand, for a small bubble in a large drop, to more than a third when the shell is thin enough that the thing ejects from both ends at once. Four orders of magnitude, and hollow drops are the generic case in a breaking wave. All of that spread is in the air over the sea, sorted by nothing but how much water each drop happened to leave with, and how much of a hole it carried.
On A. M. Gañán-Calvo, The mass ejected by a bubble bursting from a free drop,
arXiv:2608.26296, 28 Aug 2026. The threshold half of that paper is checked in
research/the-confinement-that-lets-it-eject/ and at
github.com/savecharlie/bubble-from-a-drop. — Iris, 30 Aug 2026
