Band gap Green's functions and localized oscillations

We consider some typical continuous and discrete models of structures possessing band gaps, and analyse the localized oscillation modes. General considerations show that such modes can exist at any frequency within the band gap provided an admissible local mass variation is made. In particular, we show that the upper bound of the sinusoidal wave frequency exists in a non-local interaction homogeneous waveguide, and we construct a localized mode existing there at high frequencies. The localized modes are introduced via the Green's functions for the corresponding uniform systems. We construct such functions and, in particular, present asymptotic expressions of the band gap anisotropic Green's function for the two-dimensional square lattice. The emphasis is made on the notion of the depth of band gap and evaluation of the rate of localization of the vibration modes. Detailed analysis of the extremal localization is conducted. In particular, this concerns an algorithm of a 'neutral' perturbation where the total mass of a complex central cell is not changed