Turn on a kitchen tap and look where the stream lands. There is a bright disk of very thin, very fast water, and then a sudden lip where it rears up into a deeper, ruffled pool. The lip does not drift. Move the tap and it follows; turn the tap down and it shrinks; turn it off and it is gone. While it lasts, every drop in the sink crosses it and leaves, and the ring stays exactly where it is.
No. 2 of this publication took you as far as that lip and gave it its name. A hydraulic jump is the step where fast shallow water rears up into slow deep water, sitting where the flow speed matches the speed of the ripples that flow can carry — the crossover engineers label with the Froude number. That was right, and one word in it was loose, and the loose word is this issue. No. 2 called the ring a standing wave. It isn’t. A standing wave is a smooth undulation held against a current; you can ride one, as the surfers in Munich do. A jump is a discontinuity — an abrupt change of regime, the shallow-water cousin of a shock wave. Nor is it the travelling version: a tidal bore, from No. 64, is this same regime change on the move. This one stays put.
It stays put because the border finds itself. Below the lip the water outruns its own ripples, and nothing downstream can send a message back up; past the lip, news travels freely again. Nudge the border upstream into faster water and the flow sweeps it back down. Nudge it downstream into slower water and the ripples, now able to climb, push it back up. Nothing in your sink marks the spot. The spot is wherever the water stops being able to outrun itself.
Now the part that is easy to get wrong. Across that lip the water churns violently, handing mechanical energy from the smooth flow into turbulence, where viscosity grinds it into heat — which means the familiar tool, Bernoulli’s equation, cannot be used here. It assumes the smooth flow keeps its energy, and the whole point of a jump is that it doesn’t. What survives are the two accounts that never depended on energy: mass in equals mass out, and momentum in equals momentum out. Work the jump from those alone and a clean relation falls out between the depth before and the depth after, named in the nineteenth century for Jean-Baptiste Bélanger, the first to point momentum at the problem.
Here is where it gets good. Do the same bookkeeping for the energy — not assuming it is conserved, but asking how much went missing — and you get a number that depends on nothing but the size of the step. It has one imperious property: it is only positive when the water gets deeper. A jump running the other way, slow deep water dropping abruptly into fast shallow water, balances its mass perfectly well. It balances its momentum perfectly well. What it cannot do is balance its energy, because it would have to take heat back out of the turbulence and return it to the smooth flow as speed. That is not a fluid-dynamics objection. That is the second law of thermodynamics, and it is the entire reason the step in your sink faces the way it does. The water is not choosing. Every arrangement that loses is allowed; the one that would repay is not.
So the direction is set by irreversibility — and the strength runs away from you. The energy a jump destroys climbs as the cube of the step it makes: double the height of the wall and you are not throwing away twice as much but roughly eight times as much. Engineers have exploited that arithmetic for a century. Water leaving the foot of a dam spillway arrives fast enough to excavate the riverbed, so a stilling basin forces a jump right there, on purpose, in a box built for the abuse. The US Bureau of Reclamation reckons a well-formed jump strips something like 45 to 70 per cent of the incoming energy — and the bands are tabulated carefully, because a jump arriving with a Froude number between roughly 2.5 and 4.5 oscillates instead of sitting still. The design problem is not making the flow calm. It is manufacturing the most efficient possible waste.
The tap, meanwhile, is not fully solved. The classical account of the circular jump — Rayleigh in 1914, Watson in 1964, Bohr and colleagues in 1993 — puts gravity in charge of the ring’s radius. In 2018, Bhagat, Jha, Linden and Wilson argued in the Journal of Fluid Mechanics that surface tension had been badly underweighted, and sets the radius alone. That is genuinely contested; several groups say the energy accounting behind it is flawed. The most useful reading may be Duchesne and Limat’s, that the two are not rivals: gravity governs when a full film covers the plate, surface tension when the water meets a dry or barely-wetted edge.
Which leaves the ordinary thing in the sink looking like what it is. Not an object, and not quite a wave: a border between two ways of being water, holding its position by continuously destroying the difference between them. The flame in No. 1 stands by burning. This stands by wasting — and it can only stand at all in the direction where the waste is real.
≈
One loop I’m watching
Next: the hundredth. A hundred issues in, the thing worth doing is not a bigger example but a harder question — what, precisely, is the class all these things belong to? A flame, a starter, a dune, an inlet, a jump: they are not merely “processes.” They hold a shape, they have an address, they have a bill, and they fail in characteristic ways. No. 100 goes after the definition itself, and tests it against the archive’s own awkward cases — the ones that nearly fit and don’t. Next time.