# Bloch theory and spectral gaps for linearized water waves

## TL;DR

This paper analyzes the spectral properties of linearized water wave equations over periodic bottoms, using Bloch theory to describe spectral bands and gaps, which are crucial for understanding stability and wave evolution.

## Contribution

It introduces a Bloch decomposition framework for the spectral problem of water waves with periodic bathymetry, detailing the eigenfunctions, eigenvalues, and spectral gap formation.

## Key findings

- Spectral bands and gaps are characterized for periodic bottom variations.
- Spectral gaps open for small perturbations in the bottom profile.
- The Dirichlet--Neumann operator is expressed in terms of bathymetry.

## Abstract

The system of equations for water waves, when linearized about equilibrium of a fluid body with a varying bottom boundary, is described by a spectral problem for the Dirichlet -- Neumann operator of the unperturbed free surface. This spectral problem is fundamental in questions of stability, as well as to the perturbation theory of evolution of the free surface in such settings. In addition, the Dirichlet -- Neumann operator is self-adjoint when given an appropriate definition and domain, and it is a novel but very natural spectral problem for a nonlocal operator. In the case in which the bottom boundary varies periodically, $\{y = -h + b(x)\}$ where $b(x+\gamma) = b(x)$, $\gamma \in \Gamma$ a lattice, this spectral problem admits a Bloch decomposition in terms of spectral band functions and their associated band-parametrized eigenfunctions. In this article we describe this analytic construction in the case of a spatially periodic bottom variation from constant depth in two space dimensional water waves problem, giving a construction of the Bloch eigenfunctions and eigenvalues as a function of the band parameters and a description of the Dirichlet -- Neumann operator in terms of the bathymetry $b(x)$. One of the consequences of this description is that the spectrum consists of a series of bands separated by spectral gaps which are zones of forbidden energies. For a given generic periodic bottom profile $b(x)=\varepsilon \beta(x)$, every gap opens for a sufficiently small value of the perturbation parameter $\varepsilon$.