Asymptotic Modeling of Phononic Box Crystals

Abstract

We introduce phononic box crystals, namely arrays of adjoined perforated boxes, as a threedimensional 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 finiteelement 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.

Asymptotic modeling of Helmholtz resonators including thermoviscous effects

“…Using Foldy's method, we extended this model to allow for multiple surfaceembedded resonators, arbitrarily distributed and not necessarily identical, subjected to an arbitrary incident field. Many other straightforward extensions of the model are possible, e.g., non-cylindrical necks, multiple-neck resonators, resonators in free space, resonators connected to channels [8] and resonator networks [38]. In these scenarios, thermoviscous effects could be systematically modeled along the lines of the present paper.…”

“…A3)where î,  and k form an orthonormal basis and {d 1 , d 2 , d 3 } ∈ Z 3 . A formal expression for that Green's function, which is consistent with (4.8)-(4.11), is given by[38,39] is over the set of reciprocal lattice vectorsm = 1 2 m 1 î + m 2  + m 3 k (A5)with {m 1 , m 2 , m 3 } ∈ Z 3 ; the dash indicates that the term corresponding to m = 0 should be omitted from the summation.…”

A mass-spring analogy for modeling the acoustic behaviour of a metamaterial

“…This model is based on a mass-spring analogy. Mass-spring analogy has already been used to model metamaterials [9]- [11]. Viscous and thermal losses are taken into account in the model by considering effective fluid in the main pore and in the pancake cavity with effective fluid method (Johnson-Champoux-Allard model [12]).…”

Asymptotically exact photonic approximations of chiral symmetric topological tight-binding models

“…Although in this article we do not concentrate upon the asymptotic theory of mapping continuum systems to discrete models we note that there is considerable advantage in being able to accurately map between them: the entire machinery and theory for topological tight-binding systems then carries across into continuum systems. In simpler settings of connected acoustic tubes and straight channels [30][31][32] illustrate the power of being able to translate back-and-forth between continuum and discrete networks; we use the asymptotic methodology of [26,27] showing that curved thin channels can be employed for closely spaced cylinders (and other smooth objects) and noting that a three-dimensional network extension [33] is also available.…”