Bloch Theory and Spectral Gaps for Linearized Water Waves

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,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 description of the Dirichlet -Neumann operator in terms of the bathymetry b(x) and a construction of the Bloch eigenfunctions and eigenvalues as a function of the band parameters. 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) = εβ(x), every gap opens for a sufficiently small value of the perturbation parameter ε.

Linear Modes for Channels of Constant Cross-Section and Approximate Dirichlet–Neumann Operators

“…In that case the definition and computations of the Dirichlet-Neumann operator follow the construction of [7], but cannot take into account the sloping beach boundary. Note also that the 2π −periodic modes obtained using the 2π −periodic A G 0 (β 30 • ), A 1 (β 30 • ), are special cases of the Floquet-Bloch modes for the 2π −periodic depth profile, see [9] for a study of the the band structure and Floquet-Bloch modes of A 1 (β) with another periodic profile.…”

“…that we use below. This operator was used recently in Craig et al [9] to calculate bands for periodic depth variation. Higher order expansions in β have been considered in the numerical studies of [15][16][17].…”

“…6 computed using different spatial resolutions. We considered spectral truncations with K = 2 4 to 2 9 Fourier cosine/sine modes for even/odd modes, respectively, and see evidence for convergence as the resolution increases. The value K = 2 6 used in Fig.…”

“…Computations of eigenvalues and eigenvectors used the Matlab implementation of the QR algorithm, see [26] also used in LAPACK. We used discretizations with K = 2 4 to 2 9 , results in figures use K = 2 6 . Computations by the MacBook Pro 3.1 Ghz Intel Core i5 take up to two seconds.…”

The KdV approximation for a system with unstable resonances

“…Remark Although the BKG system looks less complicated than the water wave problem, for the KdV approximation of the BKG system, some features occur, which are not present for the water wave problem over a flat bottom, namely, the occurrence of quadratic resonances, like they occur for the water wave problem over a periodic bottom. The linearized water wave problem over a periodic bottom, which is solved by Bloch modes, has been analyzed in Craig et al…”