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In this study, we establish an inclusive paradigm for the homogenization ofscalar wave motion in periodic media (with or without the source term) atfinite frequencies and wavelengths spanning the first Brioullin zone. We takethe eigenvalue problem for the unit cell of periodicity as a point ofdeparture, and we consider the projection of germane Bloch wave function onto asuitable eigenfunction as descriptor of effective wave motion. For generalitythe finite wavenumber, finite frequency (FW-FF) homogenization is pursuedin~$\mathbb{R}^d$ via second-order asymptotic expansion about the apexes of"wavenumber quadrants" comprising the first Brioullin zone, at frequencies neargiven (optical) dispersion branch. We also consider the degenerate situationsof crossing or merging dispersion branches with arbitrary multiplicity, wherethe effective description of wave motion reveals several distinct asymptoticregimes depending on the symmetries of the eigenfunction basis affiliated witha repeated eigenvalue. One of these regimes -- for whose occurence we expose asufficient condition -- is shown to describe the so-called Dirac points, i.e.conical contacts between dispersion surfaces, that are relevant to thephenomenon of topological insulation. For all cases considered, the effectivedescription turns out to admit the same general framework, with differenceslargely being limited to (i) the basis eigenfunction, (ii)~the reference cellof medium periodicity, and (iii)~the wavenumber-frequency scaling lawunderpinning the asymptotic expansion. We illustrate the utility of ouranalysis by several examples, including an asymptotic description of theGreen's function near the edge of a band gap.

A rational framework for dynamic homogenization at finite wavelengths and frequencies