Floquet$-$Bloch analysis of wave propagation with time-periodic coefficients

Abstract

This paper presents a numerical investigation of acoustic wave propagation in an obstacle with periodically time-modulated material parameters. We focus on the numerical construction of Floquet$-$Bloch solutions, which are quasi-periodic kernel elements of the hyperbolic operator appearing on the left-hand side of the acoustic wave equation. Using the temporal Fourier expansion yields a system of coupled harmonics, which can be truncated. Rewriting this system then provides different (generally nonlinear) eigenvalue formulations for discretized Floquet$-$Bloch solutions. Deriving energy estimates and the necessary conditions for Riesz$-$Schauder theory show basic properties of the occurring Floquet exponents. To derive fully discrete schemes, we employ a general Galerkin space discretization. Under assumptions on the relation of the temporal Fourier truncation and the Galerkin space discretization, we prove that the approximated Floquet exponents exhibit the same limitations as their continuous counterparts. Moreover, the approximated modes are shown to satisfy the defining properties of Floquet$-$Bloch solutions, with a defect that tends to zero as the number of harmonics approaches infinity. Numerical experiments demonstrate the effectiveness of the proposed approach and illustrate the theoretical findings.