# Applied Math Seminar

#### A tale of 2 resonances for waves in fluids

**Speaker:**
Paul Milewski, Uinversity of Bath

**Location:**
Warren Weaver Hall 1302

**Date:**
Friday, March 31, 2023, 2:30 p.m.

**Synopsis:**

We consider 2 problems where resonances (or lack of an expected resonance) yields insight into problems of waves in fluids. The first problem concerns whether mode-2 solitary waves exist in stratified flows. Except in degenerate cases, they are not expected to exist because of a resonance with shorter wavelength mode-1 waves which would lead to energy radiation. We show that, in interfacial models, such waves do exist at particular (discrete) amplitudes. The second problem concerns surface gravity waves in a cylindrical container. While this is a classic problem, it appears that the existence of general triad resonances was unknown (although Miles found certain 1:2 resonances in circular cylinders) perhaps because they do not exist in unbounded problems and in rectangular domains. We give a complete characterisation of such resonances, given the spectrum of the Laplacian on the cross-section of the cylinder. This demonstrates how boundaries can alter fundamental wave resonance properties.