A rational framework for dynamic homogenization at finite wavelengths and frequencies

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.