  We consider homogenization of the scalar wave equation in periodic media at
finite wavenumbers and frequencies, with the focus on continua characterized
by: (a) arbitrary Bravais lattice in $\mathbb{R}^d$, $d\!\geqslant\!2$, and (b)
exclusions i.e. "voids" that are subject to homogenous (Neumann or Dirichlet)
boundary conditions. Making use of the Bloch wave expansion, we pursue this
goal via asymptotic ansatz featuring the "spectral distance" from a given
wavenumber-eigenfrequency pair (within the first Brillouin zone) as the
perturbation parameter. We then introduce the effective wave motion via
projection(s) of the scalar wavefield onto the Bloch eigenfunction(s) for the
unit cell of periodicity, evaluated at the origin of a spectral neighborhood.
For generality, we account for the presence of the source term in the wave
equation and we consider -- at a given wavenumber -- generic cases of isolated,
repeated, and nearby eigenvalues. In this way we obtain a palette of effective
models, featuring both wave- and Dirac-type behaviors, whose applicability is
controlled by the local band structure and eigenfunction basis. In all spectral
regimes, we pursue the homogenized description up to at least first order of
expansion, featuring asymptotic corrections of the homogenized Bloch-wave
operator and the homogenized source term. Inherently, such framework provides a
convenient platform for the synthesis of a wide range of wave phenomena in
metamaterials and phononic crystals. The proposed homogenization framework is
illustrated by approximating asymptotically the dispersion relationships for
(i) Kagome lattice featuring hexagonal Neumann exclusions, and (ii) "pinned"
square lattice with circular Dirichlet exclusions. We complete the numerical
portrayal of analytical developments by studying the response of a Kagome
lattice due to a dipole-like source term acting near the edge of a band gap.
