Super band gaps and periodic approximants of generalised Fibonacci tilings

Abstract

We present mathematical theory for understanding the transmission spectra of heterogeneous materials formed by generalised Fibonacci tilings. Our results, firstly, characterise super band gaps, which are spectral gaps that exist for any periodic approximant of the quasicrystalline material. This theory, secondly, establishes the veracity of these periodic approximants, in the sense that they faithfully reproduce the main spectral gaps. We characterise super band gaps in terms of a growth condition on the traces of the associated transfer matrices. Our theory includes a large family of generalised Fibonacci tilings, including both precious mean and metal mean patterns. We demonstrate our fundamental results through the analysis of three different one-dimensional wave phenomena: compressional waves in a discrete mass-spring system, axial waves in structured rods and flexural waves in multi-supported beams. In all three cases, the theory is shown to give accurate predictions of the super band gaps, with negligible computational cost and with significantly greater precision than previous estimates.