Asymptotic modelling of phononic box crystals
File(s)
Accepted version
Author(s)
Vanel, Alice L
Craster, Richard V
Schnitzer, Ory
Type
Journal Article
Abstract
We introduce phononic box crystals, namely arrays of adjoined perforated boxes, as a three-dimensional prototype for an unusual class of subwavelength metamaterials based on directly coupling resonating elements. In this case, when the holes coupling the boxes are small, we create networks of Helmholtz resonators with nearest-neighbour interactions. We use matched asymptotic expansions, in the small hole limit, to derive simple, yet asymptotically accurate, discrete wave equations governing the pressure field. These network equations readily furnish analytical dispersion relations for box arrays, slabs and crystals, that agree favourably with finite-element simulations of the physical problem. Our results reveal that the entire acoustic branch is uniformly squeezed into a subwavelength regime; consequently, phononic box crystals exhibit nonlinear-dispersion effects (such as dynamic anisotropy) in a relatively wide band, as well as a high effective refractive index in the long-wavelength limit. We also study the sound field produced by sources placed within one of the boxes by comparing and contrasting monopole- with dipole-type forcing; for the former the pressure field is asymptotically enhanced whilst for the latter there is no asymptotic enhancement and the translation from the microscale to the discrete description entails evaluating singular limits, using a regularized and efficient scheme, of the Neumann's Green's function for a cube. We conclude with an example of using our asymptotic framework to calculate localized modes trapped within a defected box array.
Date Issued
2019-03-07
Date Acceptance
2019-01-23
Citation
SIAM Journal on Applied Mathematics, 2019, 79 (2), pp.506-524
ISSN
0036-1399
Publisher
Society for Industrial and Applied Mathematics
Start Page
506
End Page
524
Journal / Book Title
SIAM Journal on Applied Mathematics
Volume
79
Issue
2
Copyright Statement
© 2019, Society for Industrial and Applied Mathematics
Sponsor
Engineering & Physical Science Research Council (EPSRC)
The Leverhulme Trust
Identifier
http://arxiv.org/abs/1808.08245v1
Grant Number
EP/R041458/1
RPG-2016-365
Subjects
physics.app-ph
physics.app-ph
math.AP
Publication Status
Published
Date Publish Online
2019-03-07