“They don’t look bad,” Strauss said of the resulting equations. “But just take a look at a lake with a little wind on it. You get all these complicated forms, like whitecaps and rolling waves, some parallel to each other, some not.”
Each of these varied forms, when understood as a solution to Euler’s equations, is mathematically distinct and terribly unwieldy. Make the tiniest change to the fluid’s initial state, and it might evolve in a vastly different way — bumps and eddies can become rogue waves and tsunamis.
Before starting math, I thought water waves were something very understood — not a problem at all. Paolo Ventura, Swiss Federal Institute of Technology Lausanne
These free, moving surfaces were what Stokes wanted to study. But the challenge was immense. Describing the motion of water confined within a box, or flowing through a pipe, is hard enough. But then, at least, you know where the system’s edges lie — no water can extend beyond those boundaries. If there’s no restriction other than the force of gravity on how high the water can reach and what shape it can take, the math becomes far more difficult.
“If I go to the beach at seven in the morning, it’s going to be very calm,” Corsi said. “But if you really look at the surface, how it moves, it’s a mess.”
Still, Stokes was able to conjecture one solution: that it’s possible for the surface of the water to form evenly spaced waves that travel in a single direction.
In the 1920s, mathematicians proved Stokes’ conjecture. Furthermore, they found that if there are no external disturbances, these solutions to the Euler equations persist forever: Once they form, so-called Stokes waves will continue cruising gaily along the water’s surface for all time, their form unchanged.
Paolo Ventura recently helped prove an important result about when a particular type of wave persists and when it doesn’t in the face of perturbations. Alain Herzog/EPFL
But what if the wake of a passing boat crosses the waves’ path? Will the waves absorb this disturbance and maintain their form, or will they be disrupted permanently, transforming into an entirely different pattern of waves?
For decades, mathematicians assumed that Stokes waves are stable, meaning that any small distortion will have a minimal effect. After all, the real world is full of such complications, yet the seas are rife with Stokes waves. If they fell apart at the tiniest poke, they’d never survive long enough to make it to shore.
Still, in 1967, the mathematician T. Brooke Benjamin decided to verify this basic assumption. He had his student Jim Feir perform a series of experiments in a wave tank — a narrow rectangular pool with an oscillating rudder at one end that could produce Stokes waves. But Feir couldn’t get the waves to reach the other end of the pool. At first, he thought there was a problem with the experimental setup. But soon it became apparent that the waves were, surprisingly, unstable.
In 1995, mathematicians finally proved that such “Benjamin-Feir instabilities” are an inevitable consequence of the Euler equations. But the work left researchers wondering about the nature of these instabilities. Which kinds of disturbances can kill waves, and which can’t? How rapidly do the instabilities balloon? Could a gust of wind at the center of the Pacific cause a train of waves to strike Malibu Beach weeks later, or would the formation break down before reaching the shore?
Strange Archipelagos
Maspero had never thought to wonder why the waves exiting Trieste’s bay were dying. His inspiration ultimately came from a computer, not the scene outside his window.
At a 2019 workshop on the mathematics of waves, he and his collaborators met Bernard Deconinck, an applied mathematician at the University of Washington who, along with Katie Oliveras of Seattle University, had been mapping all the different instabilities that could destroy Stokes waves. A few years earlier, the pair had noticed an astonishing pattern, and they hadn’t been able to stop thinking about it.
When a perfect train of Stokes waves encounters a disturbance that distorts the waves’ shape, sometimes the effects of the disturbance grow to destroy the entire train, and sometimes they barely interfere. The outcome depends on the frequency of the disturbance — how much it oscillates compared to the length of the original wave. A kayak, which produces a wake that consists of short, frequent oscillations, will deliver a higher-frequency impact than a massive ocean liner, which produces longer and slower oscillations.