Super band gaps and periodic approximants of generalised Fibonacci tilings

Davies, Bryn and Morini, Lorenzo ORCID: https://orcid.org/0000-0001-7155-5036 2024. Super band gaps and periodic approximants of generalised Fibonacci tilings. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 480 (2285) , 20230663. 10.1098/rspa.2023.0663

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Abstract

We present mathematical theory for self-similarity induced spectral gaps in the spectra of systems generated by generalised Fibonacci tilings. Our results characterise super band gaps, which are spectral gaps that exist for all sufficiently large periodic systems in a Fibonacci-generated sequence. 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 apply our analytic results to characterise spectra in three different settings: compressional waves in a discrete mass-spring system, axial waves in structured rods and flexural waves in multi-supported beams. The theory is shown to give accurate predictions of the super band gaps, with minimal computational cost and significantly greater precision than previous estimates. It also provides a mathematical foundation for using periodic approximants (supercells) to predict the transmission gaps of quasicrystalline samples, as we verify numerically.

Item Type:	Article
Date Type:	Publication
Status:	Published
Schools:	Engineering
Additional Information:	License information from Publisher: LICENSE 1: URL: http://creativecommons.org/licenses/by/4.0/, Type: open-access
Publisher:	The Royal Society
ISSN:	1364-5021
Date of First Compliant Deposit:	13 March 2024
Date of Acceptance:	5 February 2024
Last Modified:	13 Mar 2024 10:15
URI:	https://orca.cardiff.ac.uk/id/eprint/167176

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