Generalized eigenfunction expansions for conservative scattering problems with an application to water waves

Abstract

This paper is devoted to a spectral description of wave propagation phenomena in conservative unbounded media, or, more precisely, the fact that a time-dependent wave can often be represented by a continuous superposition of time-harmonic waves. We are concerned here with the question of the perturbation of such a generalized eigenfunction expansion in the context of scattering problems: if such a property holds for a free situation, i.e. an unperturbed propagative medium, what does it become under perturbation, i.e. in the presence of scatterers? The question has been widely studied in many particular situations. The aim of this paper is to collect some of them in an abstract framework and exhibit sufficient conditions for a perturbation result. We investigate the physical meaning of these conditions which essentially consist in, on the one hand, a stable limiting absorption principle for the free problem, and on the other hand, a compactness (or short-range) property of the perturbed problem. This approach is illustrated by the scattering of linear water waves by a floating body. The above properties are obtained with the help of integral representations, which allow us to deduce the asymptotic behaviour of time-harmonic waves from that of the Green function of the free problem. The results are not new: the main improvement lies in the structure of the proof, which clearly distinguishes the properties related to the free problem from those which involve the perturbation.

On the absence of trapped modes in locally perturbed open waveguides

“…This result is part of the following proposition, which collects the main properties of U . Its proof dates back to the early seventies [18] and can be generalized for numerous scattering problems [13].…”

Generalized eigenfunction method for floating bodies

“…In the case of the scattering problem Hazard & Loret (2007) describe how the eigenfunctions satisfy the same normalization condition with and without the scatterers. It follows that the eigenfunctions satisfy the same normalization condition with and without the body and hence the contribution by the body motion can be ignored.…”

“…This method has been developed for rigid bodies by Hazard & Lenoir (2002); Meylan (2009), for elastic bodies by Meylan (2002); Hazard & Meylan (2007) and previously for floating bodies by Hazard & Loret (2007). Note that this later paper was very theoretically focused and no numerical simulations were presented and the method was not developed with such calculations in mind.…”

The Singularity Expansion Method applied to the transient motions of a floating elastic plate

“…3 The method which consists in diagonalizing the operator A in order to solve the transient problem (2) is the so-called Generalized Eigenfunction Expansions, see e.g. Hazard and Loret [18] as well as references therein.…”