  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.
