In August 1834 John Scott Russell was beside the Union Canal outside Edinburgh, watching a boat towed by two horses, when the boat stopped short. The water it had been shoving did not. It gathered at the bow, broke free, and went on down the channel alone: a single rounded heap, “a large solitary elevation,” at eight or nine miles an hour, holding its shape. He followed on horseback for a mile or two before losing it in the bends. Thirty feet long, a foot or so high — and it did not spread.
Waves spread; a stone’s ring on a pond widens and lowers until nothing is left. Russell’s heap did not, and he spent a decade finding out why.
Two ways to come apart
A hump of water in a shallow channel has two things wrong with it at once.
First, its top travels faster than its edges. In shallow water speed goes with depth, and the crest is where the water is deepest, so the crest gains on the foot of the wave, the front face steepens, and eventually the top arrives before the bottom. That is breaking. Left alone, a big smooth hump becomes a wall and collapses.
Second, the hump is not one wave but many, stacked — and long ripples travel faster than short ones. So the stack comes unstacked: the long parts pull ahead, the short ones lag, and the hump smears into a widening train of little waves. That is spreading. Left alone, a small smooth hump dissolves.
One tendency pulls the wave toward a point; the other pulls it toward a smear. In Russell’s canal, at the right height for the depth, they pulled with exactly equal strength. The heap steepened at precisely the rate it spread, and so did neither. Two things were happening to it, and they summed to nothing.
What the tank showed
Russell built a thirty-foot tank in his garden and found the rules. A taller hump goes faster, and is narrower. A heap dropped in too large does not become one big wave; it sorts itself into two or three, the tall one pulling ahead. And you cannot do it with a trough — a dip has only the spreading tendency, and simply melts into ripples.
He was not believed. George Airy, the leading theorist of waves, said the accepted theory allowed no such thing, and for a generation that stood. It took Boussinesq in 1871 and Rayleigh in 1876 to show that Airy’s theory had thrown away the steepening, and that keeping even a little of it produced Russell’s heap exactly. Korteweg and de Vries wrote the equation in 1895: one term that steepens, one that spreads, and a family of shapes where they cancel.
They pass through each other
Then it was forgotten for seventy years, until 1965, when Norman Zabusky and Martin Kruskal put that equation on a computer and did what Russell could not: launched a tall fast hump at a short slow one and watched.
They expected a mess — interference, splash, something new. Instead the tall one climbed the slow one, went through, and came out the far side with its shape and speed intact — and so did the small one. The only trace of the encounter was a shift: each emerged a little displaced from where it would otherwise have been. They named the object for that. A wave with a shape of its own was a solitary wave. A wave that could survive another wave was a soliton.
Not that kind of wave
No. 2’s river wave stands in one place, held by a rock it pushes against; a soliton has no rock and no place, and travels. No. 9’s traffic jam slides backward while cars flow through it; nothing flows through a soliton. The water in Russell’s heap went with it, which is why he called it a wave of translation — it carried water, not just shape.
No. 100 asks of every standing pattern what its throughput is. Here the honest answer is: not matter or energy but two failures. Steepening runs through it continuously, spreading runs through it continuously, and the shape is the difference. Cut the supply and it does not fade, because it is the supply.
An ocean of glass
In 1973 Akira Hasegawa and Fred Tappert noticed the same two tendencies in a glass fibre. A pulse of light spreads, because glass carries colours at different speeds; a bright enough pulse steepens, because intense light raises the refractive index and squeezes itself. Tune the brightness and they cancel. Solitons of light were seen in fibre in 1980, and one commercial link was built on them, across southern Australia, in 2002.
It was not the way things went. The scheme did not scale, and long-haul fibre today keeps its pulses legible with fast electronics at the far end, not with a balance in the glass. But the balance lives on inside the lasers that make the shortest flashes of light ever produced. A heap of water on a Scottish canal is still, in a sense, being chased.
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One loop I’m watching
Next: a disease that never leaves a big city and cannot survive in a small town — and the thing deciding which is not the germ but the head-count. Below a certain size the chain of infection runs out of people to hand itself to, and goes quiet until somebody carries it back in.