## Abstract

In this work, we establish an inclusive paradigm for the homogenization of scalar wave motion in periodic media 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 (FW-FF) homogenization is pursued R^{d} via second-order asymptotic expansion about the apexes of “wavenumber quadrants” comprising the first Brillouin zone, at frequencies near given 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”. We illustrate the analysis by considering wave dispersion through a chessboard-like medium featuring junctures and clusters of nearby dispersion surfaces.

Original language | English (US) |
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Title of host publication | Advances in Engineering Materials, Structures and Systems |

Subtitle of host publication | Innovations, Mechanics and Applications - Proceedings of the 7th International Conference on Structural Engineering, Mechanics and Computation, 2019 |

Editors | Alphose Zingoni |

Publisher | CRC Press/Balkema |

Pages | 362-367 |

Number of pages | 6 |

ISBN (Print) | 9781138386969 |

DOIs | |

State | Published - 2019 |

Event | 7th International Conference on Structural Engineering, Mechanics and Computation, 2019 - Cape Town, South Africa Duration: Sep 2 2019 → Sep 4 2019 |

### Publication series

Name | Advances in Engineering Materials, Structures and Systems: Innovations, Mechanics and Applications - Proceedings of the 7th International Conference on Structural Engineering, Mechanics and Computation, 2019 |
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### Conference

Conference | 7th International Conference on Structural Engineering, Mechanics and Computation, 2019 |
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Country | South Africa |

City | Cape Town |

Period | 9/2/19 → 9/4/19 |

### Bibliographical note

Publisher Copyright:© 2019 Taylor & Francis Group, London, UK.