## 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 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.

Original language | English (US) |
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Article number | 20170638 |

Journal | Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences |

Volume | 474 |

Issue number | 2213 |

DOIs | |

State | Published - May 2018 |

### Bibliographical note

Funding Information:Data accessibility. The data produced in this work are included as the electronic supplementary material. Authors’ contributions. B.G. conceived the problem, guided the developments, and interpreted the results. S.M. performed most derivations and all numerical simulations. Competing interests. We have no competing interests. Funding. This work was supported by the Institute for Mathematics and its Applications (IMA), University of Minnesota, through a postdoctoral fellowship to S.M.

Funding Information:

Acknowledgements. The authors kindly acknowledge the support provided by the University of Minnesota Supercomputing Institute.

## Keywords

- Dynamic homogenization
- Two-scale homogenization
- Waves in periodic media
- Willis' effective model