Tag - Waves

Gigliola Staffilani: The Study of Wave Interactions: Where Beautiful Mathematical Ideas Come Together

25th January, 2023

Phenomena involving interactions of waves happen at different scales and in different media: from gravitational waves to the waves on the surface of the ocean, from our milk and coffee in the morning to infinitesimal particles that behave like wave packets in quantum physics. These phenomena are difficult to study in a rigorous mathematical manner, but maybe because of this challenge mathematicians have developed interdisciplinary approaches that are powerful and beautiful. In the first lecture, which will be colloquium style, I will describe some of these approaches and show for example how the need to understand certain multilinear and periodic wave interactions provided also the tools to prove a famous conjecture in number theory, or how classical tools in probability gave the right framework to still have viable theories behind certain deterministic counterexamples. In the second and third lecture I will open a small window into the concept of weak wave turbulence. I will start with the deterministic approach of Bourgain, involving the study of long time asymptotic of higher Sobolev norms of solutions of dispersive equations, and I will end with the rigorous derivation of a 3-wave kinetic equation.

Qingtang Su: The non-linear modulational instability of the Stokes waves in 2d water waves

1st September, 2022

The Stokes waves are periodic symmetric steady water waves travelling at a constant speed, which plays a fundamental role in the study of water waves. It was observed by Benjamin and Feir in 1967 that the Stokes waves are subject to the modulational instability. However, the rigorous mathematical proof was missing for a long time. In this talk, we will discuss how to prove the non-linear modulational instability of the Stokes waves in the context of 2d full water waves.

Didier Clamond: Water wave determination from seabed measurements

25th January, 2022

Surface waves determination from pressure measurements at the seabed leads to an ill-posed inversion problem. For two-dimensional steady waves in irrotational motion, using elementary complex analysis, the problem can be solved exactly in an analytic implicit form. For practical purposes, the implicit relations must be solved iteratively. We show that simple fixed-point iterations converge, even for extreme waves with angular crests.