Propagation and non-reciprocity in time-modulated diffusion through the lens of high-order homogenization

TL;DR

This paper develops a second-order homogenization method for time-modulated diffusion in laminates, revealing non-reciprocal propagation effects that are not captured by first-order models, and verifies these effects through numerical simulations.

Contribution

It introduces a second-order homogenization approach that accurately captures non-reciprocal wave propagation in time-modulated diffusion systems, correcting limitations of first-order models.

Findings

Second-order homogenization reveals non-zero non-reciprocal terms.

Numerical simulations confirm the non-reciprocal propagation predicted.

First-order models fail to capture non-reciprocity in time-modulated diffusion.

Abstract

The homogenization procedure developed here is conducted on a laminate with periodic space-time modulation on the fine scale: at leading order, this modulation creates convection in the low-wavelength regime if both parameters are modulated. However, if only one parameter is modulated, which is more realistic, this convective term disappears and one recovers a standard diffusion equation with effective homogeneous parameters; this does not describe the non-reciprocity and the propagation of the field observed from exact dispersion diagrams. This inconsistency is corrected here by considering second-order homogenization which results in a non-reciprocal propagation term that is proved to be non-zero for any laminate and verified via numerical simulation. The same methodology is also applied to the case when the density is modulated in the heat equation, leading therefore to a corrective advective term which cancels out non-reciprocity at the leading order but not at the second order.