Super band gaps and periodic approximants of generalised Fibonacci tilings

TL;DR

This paper develops a mathematical framework to understand super band gaps in generalized Fibonacci tilings, showing that periodic approximants accurately reproduce spectral gaps across various wave phenomena with high precision.

Contribution

It introduces a new theory characterizing super band gaps and validates the use of periodic approximants for accurate spectral predictions in generalized Fibonacci tilings.

Findings

Super band gaps are characterized by growth conditions on transfer matrix traces.

Periodic approximants faithfully reproduce main spectral gaps.

The theory accurately predicts super band gaps across different wave systems.

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.