Wave propagation in one-dimensional quasiperiodic media

TL;DR

This paper investigates wave propagation in one-dimensional quasiperiodic media by transforming the Helmholtz equation into a higher-dimensional periodic problem, addressing mathematical and numerical challenges unique to quasiperiodic structures.

Contribution

It extends the Dirichlet-to-Neumann (DtN) method to quasiperiodic media by lifting the problem to higher dimensions and analyzing the degenerate PDE involved.

Findings

Extended DtN method to quasiperiodic media.

Analyzed the degenerate PDE's mathematical properties.

Developed numerical techniques involving cell problems and Riccati equations.

Abstract

This work is devoted to the resolution of the Helmholtz equation $-(\mu u')' - \rho \omega^2 u = f$ in a one-dimensional unbounded medium. We assume the coefficients of this equation to be local perturbations of quasiperiodic functions, namely the traces along a particular line of higher-dimensional periodic functions. Using the definition of quasiperiodicity, the problem is lifted onto a higher-dimensional problem with periodic coefficients. The periodicity of the augmented problem allows us to extend the ideas of the DtN-based method developed for the elliptic case. However, the associated mathematical and numerical analysis of the method are more delicate because the augmented PDE is degenerate, in the sense that the principal part of its operator is no longer elliptic. We also study the numerical resolution of this PDE, which relies on the resolution of Dirichlet cell problems as well as a constrained Riccati equation.