Some general properties of the band-gap structure for quasi-periodic 1D lattices

Periodicity and quasi-periodicity endow heterogeneous materials and structures of peculiar wave transmission properties. In this work we determine the transmission spectrum for a general class of one dimensional metamaterials by means of an analytical asymptotic approach. We show that the quasi-periodic arrangement introduces (topologically non trivial) frequency band-gaps in addition to those related to the locally resonant (and/or scattering) phenomenon. The analytical prediction of the spectrum, as a function of the parameter governing the quasi-periodicity, is validated by comparison with the one numerically computed. It is also shown that inside the topologically non-trivial band gaps localized modes of a finite system exist.