Abstract:

Time-modulated metamaterials have emerged as a focal point in the pursuit of next-generation materials. The
talk discusses numerical investigation of acoustic wave propagation in obstacles with periodically time-
modulated material parameters. The first part of the talk discusses 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. Instead, or in addition, we employ a general Galerkin space discretization to discretize
in space. Some basic properties of the fully discretized modes can be shown and the convergence to the space
or time-discretization can be established. Complications in the theory, however, limit the connection to the fully
continuous modulated wave equation when formulated as an initial value problem. A modification of the
Floquet–Bloch approach yields modulated Fourier expansion for fast-time modulated media. For modulations
with small amplitude, the solution of the time-modulated acoustic wave equation can be characterized by
several slowly varying coefficient functions. As a consequence, we can derive fully discrete schemes that
converge, under assumptions on the modulation, independently of the rapid oscillation of the physical
parameters. Numerical experiments illustrate both approaches.