Time harmonic wave propagation in one dimensional weakly randomly perturbed periodic media
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ORIGINAL PAPER
Time harmonic wave propagation in one dimensional weakly randomly perturbed periodic media Sonia Fliss1
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Laure Giovangigli1
Received: 9 March 2020 / Accepted: 25 August 2020 Ó Springer Nature Switzerland AG 2020
Abstract In this work we consider the solution of the time harmonic wave equation in a one dimensional periodic medium with weak random perturbations. More precisely, we study two types of weak perturbations: (1) the case of stationary, ergodic and oscillating coefficients, the typical size of the oscillations being small compared to the wavelength and (2) the case of rare random perturbations of the medium, where each period has a small probability to have its coefficients modified, independently of the other periods. Our goal is to derive an asymptotic approximation of the solution with respect to the small parameter. This can be used in order to construct absorbing boundary conditions for such media. Keywords Wave equation Random media Periodic media Mathematics Subject Classification 35R60 35B27 35L05
1 Introduction and model problem The propagation of waves in periodic media has seen a regain of interest for many important applications, particularly in optics for micro and nano-technology. However, in real applications, the media are often not perfectly periodic and the perturbations can be partially known. The use of randomness to model this partial knowledge is particularly well suited. We want to propose a numerical method for computing the propagation of waves in such media. More precisely, we want to reduce the pure numerical computation to a bounded region, typically a region where the medium is well-known (i.e. not random). It is then necessary to construct transparent or absorbing boundary conditions to impose at the This article is part of the topical collection ‘‘Waves 2019 – invited papers’’ edited by Manfred Kaltenbacher and Markus Melenk. & Sonia Fliss [email protected] Laure Giovangigli [email protected] 1
POEMS, Ensta Paris, 828 boulevard des Mare´chaux, 91128 Palaiseau Cedex, France SN Partial Differential Equations and Applications
40 Page 2 of 36
SN Partial Differ. Equ. Appl. (2020)1:40
boundary of the computational domain. These conditions should reflect the best possible the wave propagation in the exterior medium. They could then be used, for instance, to obtain the statistics or to quantify the uncertainty of the field in the computational domain (via the simulation of numerous realizations of the field). This paper is a first contribution to the construction of such boundary conditions in particular situations. We are interested in this paper in the one-dimensional time harmonic wave equation in infinite media which are periodic with weak random perturbations. More precisely, let ðX; F ; PÞ be a probability space. We consider a random medium occupying R, characterized by the following coefficients for a.e. ðx; xÞ 2 R X;
je ðx; xÞ ¼ jper ðxÞ þ j~e ðx; xÞ; qe ðx; xÞ ¼ qper ðxÞ þ
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