Platonic crystals: Bloch bands, neutrality and defects

We consider the properties of flexural waves in a periodically-constrained thin elastic plate. A square array of rigid pins is embedded into the plate. We show that such a system supports a stop band commencing at zero frequency, and we evaluate analytically the frequency limits of its first three bands. We show that each band has neutral directions, in which its dispersion relation and group velocity coincide with those of two corresponding plane waves. We construct the stop band Green’s functions and analyse strongly localized defect modes, and we also find the frequency of the resonant defect mode lying in the first band gap.