## Abstract

In this study, we establish an inclusive paradigm for the homogenization of scalar wave motion in periodic media (including the source term) at finite frequencies and wavenumbers spanning the first Brillouin zone. We take the eigenvalue problem for the unit cell of periodicity as a point of departure, and we consider the projection of germane Bloch wave function onto a suitable eigenfunction as descriptor of effective wave motion. For generality the finite wavenumber, finite frequency homogenization is pursued in R^{d} via second-order asymptotic expansion about the apexes of 'wavenumber quadrants' comprising the first Brillouin zone, at frequencies near given (acoustic or optical) dispersion branch. We also consider the junctures of dispersion branches and 'dense' clusters thereof, where the asymptotic analysis reveals several distinct regimes driven by the parity and symmetries of the germane eigenfunction basis. In the case of junctures, one of these asymptotic regimes is shown to describe the so-called Dirac points that are relevant to the phenomenon of topological insulation. On the other hand, the effective model for nearby solution branches is found to invariably entail a Dirac-like system of equations that describes the interacting dispersion surfaces as 'blunted cones'. For all cases considered, the effective description turns out to admit the same general framework, with differences largely being limited to (i) the eigenfunction basis, (ii) the reference cell of medium periodicity, and (iii) the wavenumber-frequency scaling law underpinning the asymptotic expansion. We illustrate the analytical developments by several examples, including Green's function near the edge of a band gap and clusters of nearby dispersion surfaces.

Original language | English (US) |
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Article number | 20180547 |

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

Volume | 475 |

Issue number | 2223 |

DOIs | |

State | Published - Mar 1 2019 |

### Bibliographical note

Funding Information:Data accessibility. The data produced are included as electronic supplementary material. Authors’ contributions. B.G. conceived the problem and led the developments. S.M. provided important inputs in formulating the asymptotic solution. O.O.I. performed numerical simulations and aided the mathematical formulation with keen insights. Competing interests. We declare we have no competing interests. Funding. This study was partially supported through the endowed Shimizu Professorship. S.M. was partially supported by NSF grant no. DMS-1439786 while in residence at ICERM Brown University during autumn 2017 and AFOSR award no. FA9550-18-1-0131.

## Keywords

- Dirac points
- Dynamic homogenization
- Finite frequency
- Finite wavenumber
- Nearby eigenvalues
- Waves in periodic media