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

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 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 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 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 body sources, which may be relevant to some recent efforts to manipulate waves in metamaterials.

Symmetry breaking creates electro-momentum coupling in piezoelectric metamaterials

“…A series of theoretical studies were carried out to characterize Willis coupling and understand its physical origins [18][19][20][21][22][23][24][25][26]. Guided by accompanying predictions that Willis coupling is connected with unusual phenomena such as asymmetric reflections and unidirectional transmission, recent experimental realizations of Willis metamaterials that demonstrate these phenomena were reported [27][28][29][30][31][32][33][34][35].…”

A rational framework for dynamic homogenization at finite wavelengths and frequencies

“…If further a LW-LF asymptotic model of the problem is sought, the effective, i.e. homogenized, field ũ is expanded about (k = 0, ω = 0) [8].…”

Leading and Second Order Homogenization of an Elastic Scattering Problem for Highly Oscillating Anisotropic Medium

“…There are recent interests on higher-order two-scale homogenization of wave propagation in periodic meida [1,7,9,18,23]. In the case that the periodic structure was only supported in a bounded domain, contrary to the case that the periodic structure occupies R d , the boundary correctors played a role both in the leading-order and second-order homogenization as demonstrated in [7] for scalar wave equation.…”

“…A dispersive model for scalar wave equation was derived using Floquet-Bloch theory and higher-order asymptotic of the Bloch variety [21]. Alternatively higher-order two-scale homogenization enables to demonstrate the dispersive effective of wave propagation in periodic meida [1,7,9,18,23]. The higherorder homogenization in particular sheds light on sensing the microstructure through dispersion [15].…”

“…To gain further understanding of the wave dispersion, a higher-order homogenization has been taken into account. A higher-order homogenization was derived for the scalar wave equation in [9,18,23], where a fourth-order partial differential equation was formally derived and the dispersive effect was hence demonstrated. [8] considered a higher-order homogenization of the elastic wave for non-periodic layered media that transcends the usual quasi-staic regime.…”