Approximation by finite differences of the propagation of acoustic waves in stratified media

Abstract

Summary.In this paper, we analyze the approximation of acoustic waves in a two layered media by a finite diffrences variational scheme. We examine in particular the approximation of the guided waves. We point out the existence of purely numerical parasitic phenomena and quantify the numerical dispersion relative to guided waves.

Spectral theory for Maxwell’s equations at the interface of a metamaterial. Part I: Generalized Fourier transform

“…Remark 10. Notice that in comparison with previous studies on stratified media which inspire our approach [6,15,34,36], the essential difference lies in the fact that our Sturm-Liouville equation (29) depends nonlinearly on the spectral variable ζ, which is a consequence of the frequency dispersion in a Drude material. This dependence considerably complicates the spectral analysis of A k .…”

“…Theorem 5. For any self-adjoint operator A on a Hilbert space H, there exists a spectral measure E which diagonalizes A in the sense of (14) and (15). Remark 6.…”

“…Our negative material is described by a nondissipative Drude model, which is the simplest model of negative material. The technique we use to construct the generalized Fourier transform is inspired by previous studies in the context of stratified media [6,15,34,36]. Compared to these studies, the difficulty of the present work relies in the fact that in the Drude material, ε and µ depend on the frequency and become negative for low frequencies.…”

“…It will be used in a forthcoming paper [5] to study the validity of a limiting amplitude principle in our medium, but the results we obtain in the present paper are not limited to this purpose. The generalized Fourier transform is also the main tool to study scattering problems as in [7,16,34,35,36], numerical methods in stratified media as in [15] and has many other applications. In Let us mention that both present and forthcoming papers are an advanced version of the preliminary study presented in [4].…”