Effective wave motion in periodic discontinua near spectral singularities at finite frequencies and wavenumbers

We consider the effective wave motion, at spectral singularities such as corners of the Brillouin zone and Dirac points, in periodic continua intercepted by compliant interfaces that pertain to e.g. masonry and fractured materials. We assume the Bloch-wave form of the scalar wave equation (describing anti-plane shear waves) as a point of departure, and we seek an asymptotic expansion about a reference point in the wavenumber-frequency space using wavenumber separation as the perturbation parameter. Using the concept of broken Sobolev spaces to cater for the presence of kinematic discontinuities, we next define the “mean” wave motion via inner product between the Bloch wave and an eigenfunction (at specified wavenumber and frequency) for the unit cell of periodicity. With such projection-expansion approach, we obtain an effective field equation, for an arbitrary dispersion branch, near apexes of “wavenumber quadrants” featured by the first Brillouin zone. For completeness, we investigate asymptotic configurations featuring both (a) isolated, (b) repeated, and (c) nearby eigenvalues. In the case of repeated eigenvalues, we find that the “mean” wave motion is governed by a system of wave equations and Dirac equations, whose size is given by the eigenvalue multiplicity, and whose structure is determined by the participating eigenfunctions, the affiliated cell functions, and the direction of wavenumber perturbation. One of these structures is shown to describe the so-called Dirac points – apexes of locally conical dispersion surfaces – that are relevant to the generation of topologically protected waves. In situations featuring clusters of tightly spaced eigenvalues, the effective model is found to entail a Dirac-like system of equations that generates “blunted” conical dispersion surfaces. We illustrate the analysis by numerical simulations for two periodic configurations in R² that showcase the asymptotic developments in terms of (i) wave dispersion, (ii) forced wave motion, and (iii) frequency- and wavenumber-dependent phonon behavior.