The Hidden Math of Ocean Waves Crashes Into View

Summary

“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. But what if the wake of a passing boat crosses the waves’ path? Will the waves absorb this...