Time-harmonic wave propagation in junctions of two periodic half-spaces

TL;DR

This work tackles time-harmonic wave propagation in a junction of two periodic half-spaces by lifting the 2D problem to a 3D augmented, strictly periodically structured problem along the interface. The key idea is to exploit hidden quasiperiodicity via a cut matrix $\mathbb{\Theta}$, transforming the original problem into a periodic augmented PDE with an elliptically degenerate principal part, which is then handled with a specialized anisotropic Sobolev framework and a partial Floquet-Bloch transform. The solution reduces to a family of waveguide problems connected through a Dirichlet-to-Neumann interface formulation governed by a Riccati propagation operator, with local Dirichlet cell problems providing the DtN data. An efficient quasi-2D discretization leverages the anisotropic structure to solve many 2D cell problems in parallel, yielding accurate numerical solutions in homogeneous, rational, and irrational periodic settings. The methodology opens avenues for extending to source terms, general periodic junctions, and non-absorbing regimes, offering a robust tool for designing devices exploiting band gaps and wave localization in complex quasiperiodic media.

Abstract

We are interested in the Helmholtz equation in a junction of two periodic half-spaces. When the overall medium is periodic in the direction of the interface, Fliss and Joly (2019) proposed a method which consists in applying a partial Floquet-Bloch transform along the interface, to obtain a family of waveguide problems parameterized by the Floquet variable. In this paper, we consider two model configurations where the medium is no longer periodic in the direction of the interface. Inspired by the works of Gérard-Varet and Masmoudi (2011, 2012), and Blanc, Le Bris, and Lions (2015), we use the fact that the overall medium has a so-called quasiperiodic structure, in the sense that it is the restriction of a higher dimensional periodic medium. Accordingly, the Helmholtz equation is lifted onto a higher dimensional problem with coefficients that are periodic along the interface. This periodicity property allows us to adapt the tools previously developed for periodic media. However, the augmented PDE is elliptically degenerate (in the sense of the principal part of its differential operator) and thus more delicate to analyse.

Figures (28)

• Figure 1: Juxtaposition of arbitrary periodic half-spaces
• Figure 2: Reduction of the domains through the steps \ref{['H2Dmod:item:lifting_step_1']}, \ref{['H2Dmod:item:lifting_step_2']}, and \ref{['H2Dmod:item:lifting_step_3']}.
• Figure 5: Left: Schematic representation of the augmented structure for the Configuration \ref{['H2Dmod:item:config_a']}--type medium represented in Figure \ref{['H2Dmod:fig:configuration_a']}. Right: Close-up view of a periodicity cell.
• Figure 6: Left: Schematic representation of the augmented structure for the Configuration \ref{['H2Dmod:item:config_b']}--type medium represented in Figure \ref{['H2Dmod:fig:configuration_b']} (some cylinders have been made more transparent for readability). Right: Close-up view of a periodicity cell.
• Figure 7: Left: the rectangle-based cylindrical domain $\textcolor{surligneur}{\mathcal{O}}_{\scaleobj{0.875}{\#}}$ and the lateral set ${\color{surligneur}\mathbb{\Gamma}}_{\scaleobj{0.875}{\#}}$ given by \ref{['H2Dmod:eq:cylindrical_domain']}. Right: the domains $Q$ and $\widehat{I}$ given by \ref{['H2Dmod:eq:2D_1D_cylindrical_domains']}. $I_{x} = \mathbb{R}_-$ and $I_1 \subset \mathbb{R}_+$.

Theorems & Definitions (52)

• Remark 1.1
• Definition 3.1: Periodic extension
• Proposition 3.2
• Proposition 3.3
• Corollary 3.4
• Proposition 3.5
• Remark 3.6
• Proposition 3.7
• Proposition 3.8
• Proof