Matthieu Ménard: Spectral gaps for linearized water waves above a small periodic bottom

Abstract:

We consider the movement of a free surface of a two-dimensional fluid over a variable bottom. We assume that the bottom has a periodic profile and we study the water wave system linearized near equilibrium. The latter reduces to a spectral problem for the Dirichlet–Neumann operator in a fluid domain with a periodic bottom and a flat surface elevation. We show that the spectral problem admits a Floquet-Bloch decomposition in terms of spectral band functions and their associated band-parametrized eigenfunctions. We find that, generically, the spectrum consists of a series of bands separated by spectral gaps which are zones of forbidden energies.

This is a joint work with Christophe Lacave and Catherine Sulem.