Pierre Amenoagbadji, Sonia Fliss, Patrick Joly, Wave propagation in one-dimensional quasiperiodic media, Vol. 2023 (2023), Article ID 17, pp. 1-43

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DOI: 10.23952/cot.2023.17

Received September 2, 2022; Accepted December 14, 2022; Published April 7, 2023


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 \textit{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 in [10,19] 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.


How to Cite this Article:
P. Amenoagbadji, S. Fliss, P. Joly, Wave propagation in one-dimensional quasiperiodic media, Commun. Optim. Theory 2023 (2023) 17.