The modulated Fourier expansion for waves propagating through time-modulated media

TL;DR

This paper introduces a modulated Fourier expansion method for analyzing waves in time-modulated media, providing a new mathematical framework that handles non-classical properties and enables stable, accurate numerical simulations.

Contribution

It develops a novel modulated Fourier expansion approach for the time-modulated acoustic wave equation, overcoming limitations of classical methods and enabling efficient numerical integration.

Findings

The coupled PDE system has smoothly varying solutions for small-amplitude fast modulations.

Discretized integrators are stable and accurate with larger time steps.

Numerical experiments confirm theoretical predictions and method effectiveness.

Abstract

Controlling waves by actively changing the material parameters of a medium enables the development of new acoustic and electrical devices. Modulating the material breaks classical properties like reciprocity and the conservation of energy, which complicates the mathematical analysis. Without a limiting amplitude principle, time-harmonic formulations are generally inapplicable. The present manuscript develops an alternative tool for the time-modulated acoustic wave equation, that is based on a modulated Fourier expansion (MFE). The solution is characterized by multiple smoothly varying coefficient functions, which solve a coupled system of evolutionary partial differential equations with temporally constant coefficients. For small-amplitude fast-time modulations, this system of evolutionary partial differential equations is shown to possess a smoothly varying solution, which characterizes the exact solution up to a small defect. Discretization of the derived coupled system yields integrators that are stable and accurate when larger time steps are used, compared to those schemes that are applied to the time-modulated acoustic wave equation directly. Numerical experiments illustrate the theoretical results and the use of the approach.