  In this study, we establish an inclusive paradigm for the homogenization of
scalar wave motion in periodic media (with or without the source term) at
finite frequencies and wavelengths spanning the first Brioullin 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
in~$\mathbb{R}^d$ via second-order asymptotic expansion about the apexes of
"wavenumber quadrants" comprising the first Brioullin zone, at frequencies near
given (optical) dispersion branch. We also consider the degenerate situations
of crossing or merging dispersion branches with arbitrary multiplicity, where
the effective description of wave motion reveals several distinct asymptotic
regimes depending on the symmetries of the eigenfunction basis affiliated with
a repeated eigenvalue. One of these regimes -- for whose occurence we expose a
sufficient condition -- is shown to describe the so-called Dirac points, i.e.
conical contacts between dispersion surfaces, that are relevant to the
phenomenon of topological insulation. For all cases considered, the effective
description turns out to admit the same general framework, with differences
largely being limited to (i) the basis eigenfunction, (ii)~the reference cell
of medium periodicity, and (iii)~the wavenumber-frequency scaling law
underpinning the asymptotic expansion. We illustrate the utility of our
analysis by several examples, including an asymptotic description of the
Green's function near the edge of a band gap.
