High frequency limit of Helmholtz equations: refraction by sharp interfaces
type: Item
instance of: scholarly article
title: High frequency limit of Helmholtz equations: refraction by sharp interfaces
zbMATH Open document ID: 1119.35096
author: Elise Fouassier
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 2007-03-15T00:00:00Z
review text: The author composes the high frequency limit of the Helmholtz equation with source term, in the case of a refraction index that is discontinuous along a sharp interface between two unbounded media. The asymptotic propagation of energy is studied using Wigner measures. Her result is twofold. First, in the general case, assuming some geometrical hypotheses on the index and assuming that the interface does not capture energy asymptotically, she proves that the limiting Wigner measure satisfies a stationary transport equation with source term. As a consequence, the Wigner measure is characterized as the integral, along the rays of geometrical optics and up to infinite time, of the energy source. This result encodes the refraction phenomenon. Secondly, the author studies the particular case when the index is constant in each medium, for which the analysis goes further: she proves that the interface does not capture energy asymptotically in this case. The content of the chapters is as follows: In chapter 2 the so-called Wigner transform is introduced. In chapter 3 uniform bounds on a certain sequence \((\mu^{\varepsilon})\) are established. In chapter 4, the transport equations are established. In chapter 5 the refraction results are proved in the case of two homogeneous media. Chapter 6 is devoted to the proof of the main result in the case of a general index of refraction. The proof uses the so-called Snell-Descartes semigroup. Chapter 7 is devoted to the proof of the radiation conditions at infinity in the homogeneous case.
reviewed by: Thomas Ernst
Mathematics Subject Classification ID: 35Q60
Mathematics Subject Classification ID: 35J05
Mathematics Subject Classification ID: 78A45
zbMATH DE Number: 5134238
zbMATH Keywords: Wigner measure
zbMATH Keywords: transmission problem
zbMATH Keywords: Wigner transform
zbMATH Keywords: refraction
zbMATH Keywords: Snell-Descartes semigroup.
MaRDI profile type: Publication
full work available at URL: j.matpur.2006.11.002
OpenAlex ID: W1966123669
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DOI: 10.1016/J.MATPUR.2006.11.002
description: scientific article; zbMATH DE number 5134238