In general, mathematicians expect waves to recover more easily from higher-frequency disruptions like the kayak’s, because their impacts are limited to a smaller region of a passing wave at any given moment. The wake of the ocean liner, on the other hand, can affect the entire wave at once, permanently disrupting it. Benjamin-Feir instabilities are caused by low-frequency disruptions. In 2011, Deconinck and Oliveras simulated different disturbances with higher and higher frequencies and watched what happened to the Stokes waves. As they expected, for disturbances above a certain frequency, the waves persevered. But as the pair continued to dial up the frequency, they suddenly began to see destruction again. At first, Oliveras worried that there was a bug in the computer program. “Part of me was like, this can’t be right,” she said. “But the more I dug, the more it persisted.” In fact, as the frequency of the disturbance increased, an alternating pattern emerged. First there was an interval of frequencies where the waves became unstable. This was followed by an interval of stability, which was followed by yet another interval of instability, and so on. Deconinck and Oliveras published their finding as a counterintuitive conjecture: that this archipelago of instabilities stretches off to infinity. They called all the unstable intervals “isole” — the Italian word for “islands.” It was strange. The pair had no explanation for why instabilities would appear again, let alone infinitely many times. They at least wanted a proof that their startling observation was correct.
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mathematical physics The Hidden Math of Ocean Waves Crashes Into View By Joseph Howlett October 15, 2025 The math of even the simplest ocean waves is notoriously uncooperative. A team of Italian mathematicians has made major advances toward understanding it. Save Article Read Later
mathematical physics The Hidden Math of Ocean Waves Crashes Into View By Joseph Howlett October 15, 2025 The math of even the simplest ocean waves is notoriously uncooperative. A team of Italian mathematicians has made major advances toward understanding it. Save Article Read Later
The math of even the simplest ocean waves is notoriously uncooperative. A team of Italian mathematicians has made major advances toward understanding it. Save Article Read Later
The math of even the simplest ocean waves is notoriously uncooperative. A team of Italian mathematicians has made major advances toward understanding it.
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Alberto Maspero (right) and Massimiliano Berti in Trieste, Italy, where they study the mathematics of ocean waves. Their group has been leading a recent effort to prove major conjectures in the field. Federico Murgante
Ocean waves can form complicated patterns that are almost prohibitively difficult to study mathematically. “Square waves” like the ones shown above form when two different wave systems collide. Michel Griffon
Livia Corsi is fascinated by the messy, unpredictable nature of the ocean and the mathematics that governs it. Michela Procesi
Bernard Deconinck (left) and Katie Oliveras uncovered a strange pattern in computational studies of wave stability. Courtesy of Bernard Deconinck; Courtesy of Katie Oliveras
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Save Article Read Later
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In general, mathematicians expect waves to recover more easily from higher-frequency disruptions like the kayak’s, because their impacts are limited to a smaller region of a passing wave at any given moment. The wake of the ocean liner, on the other hand, can affect the entire wave at once, permanently disrupting it. Benjamin-Feir instabilities are caused by low-frequency disruptions. In 2011, Deconinck and Oliveras simulated different disturbances with higher and higher frequencies and watched what happened to the Stokes waves. As they expected, for disturbances above a certain frequency, the waves persevered. But as the pair continued to dial up the frequency, they suddenly began to see destruction again. At first, Oliveras worried that there was a bug in the computer program. “Part of me was like, this can’t be right,” she said. “But the more I dug, the more it persisted.” In fact, as the frequency of the disturbance increased, an alternating pattern emerged. First there was an interval of frequencies where the waves became unstable. This was followed by an interval of stability, which was followed by yet another interval of instability, and so on. Deconinck and Oliveras published their finding as a counterintuitive conjecture: that this archipelago of instabilities stretches off to infinity. They called all the unstable intervals “isole” — the Italian word for “islands.” It was strange. The pair had no explanation for why instabilities would appear again, let alone infinitely many times. They at least wanted a proof that their startling observation was correct.
In general, mathematicians expect waves to recover more easily from higher-frequency disruptions like the kayak’s, because their impacts are limited to a smaller region of a passing wave at any given moment. The wake of the ocean liner, on the other hand, can affect the entire wave at once, permanently disrupting it. Benjamin-Feir instabilities are caused by low-frequency disruptions. In 2011, Deconinck and Oliveras simulated different disturbances with higher and higher frequencies and watched what happened to the Stokes waves. As they expected, for disturbances above a certain frequency, the waves persevered. But as the pair continued to dial up the frequency, they suddenly began to see destruction again. At first, Oliveras worried that there was a bug in the computer program. “Part of me was like, this can’t be right,” she said. “But the more I dug, the more it persisted.” In fact, as the frequency of the disturbance increased, an alternating pattern emerged. First there was an interval of frequencies where the waves became unstable. This was followed by an interval of stability, which was followed by yet another interval of instability, and so on. Deconinck and Oliveras published their finding as a counterintuitive conjecture: that this archipelago of instabilities stretches off to infinity. They called all the unstable intervals “isole” — the Italian word for “islands.” It was strange. The pair had no explanation for why instabilities would appear again, let alone infinitely many times. They at least wanted a proof that their startling observation was correct.
In general, mathematicians expect waves to recover more easily from higher-frequency disruptions like the kayak’s, because their impacts are limited to a smaller region of a passing wave at any given moment. The wake of the ocean liner, on the other hand, can affect the entire wave at once, permanently disrupting it. Benjamin-Feir instabilities are caused by low-frequency disruptions. In 2011, Deconinck and Oliveras simulated different disturbances with higher and higher frequencies and watched what happened to the Stokes waves. As they expected, for disturbances above a certain frequency, the waves persevered. But as the pair continued to dial up the frequency, they suddenly began to see destruction again. At first, Oliveras worried that there was a bug in the computer program. “Part of me was like, this can’t be right,” she said. “But the more I dug, the more it persisted.” In fact, as the frequency of the disturbance increased, an alternating pattern emerged. First there was an interval of frequencies where the waves became unstable. This was followed by an interval of stability, which was followed by yet another interval of instability, and so on. Deconinck and Oliveras published their finding as a counterintuitive conjecture: that this archipelago of instabilities stretches off to infinity. They called all the unstable intervals “isole” — the Italian word for “islands.” It was strange. The pair had no explanation for why instabilities would appear again, let alone infinitely many times. They at least wanted a proof that their startling observation was correct.
By Joseph Howlett October 15, 2025
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