On the dynamic homogenization of periodic media: Willis' approach versus two-scale paradigm

TL;DR

This paper compares Willis' homogenization framework and the two-scale approach for wave propagation in periodic media, revealing a fundamental modulation factor difference and emphasizing the importance of source term homogenization.

Contribution

It establishes a detailed low-frequency, long-wavelength dispersive expansion of Willis' model and clarifies the relationship and differences with the two-scale approach, including source term considerations.

Findings

The two descriptions differ by a polynomial modulation factor.

The two-scale expansion neglects source term homogenization.

The modulation factor is derived for dipole sources.

Abstract

When considering an effective i.e. homogenized description of waves in periodic media that transcends the usual quasi-static approximation, there are generally two schools of thought: (i) the two-scale approach that is prevalent in mathematics, and (ii) the Willis' homogenization framework that has been gaining popularity in engineering and physical sciences. Notwithstanding a mounting body of literature on the two competing paradigms, a clear understanding of their relationship is still lacking. In this study we deploy an effective impedance of the scalar wave equation as a lens for comparison and establish a low-frequency, long-wavelength (LF-LW) dispersive expansion of the Willis effective model, including terms up to the second order. Despite the intuitive expectation that such obtained effective impedance coincides with its two-scale counterpart, we find that the two descriptions differ by a modulation factor which is, up to the second order, expressible as a polynomial in frequency and wavenumber. We track down this inconsistency to the fact that the two-scale expansion is commonly restricted to the free-wave solutions and thus fails to account for the body source term which, as it turns out, must also be homogenized -- by the reciprocal of the featured modulation factor. In the analysis, we also (i) reformulate for generality the Willis' effective description in terms of the eigenfunction approach, and (ii) obtain the corresponding modulation factor for dipole body sources, which may be relevant to some recent efforts to manipulate waves in metamaterials.