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 Rd, d>1, and (b) exclusions i.e. “holes” that are subject to homogeneous Neumann and 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 (situated anywhere 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 – the cases of isolated, repeated, and nearby eigenvalues. In this way we obtain a palette of effective models, featuring both wave- and Dirac-type field equations, whose applicability is controlled by the local band and eigenfunction structure. 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 a framework provides convenient platform for the synthesis of a wide range of intriguing wave phenomena, including negative refraction and topologically protected states 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 Gaussian source term acting near the edge of a band gap.
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