  A new electromagnetic cavity structure, a lattice of 3D cavities consisting
of an array of posts and gaps is presented. The individual cavity elements are
based on the cylindrical re-entrant (or Klystron) cavity. We show that these
cavities can also can be thought of as 3D split-ring resonators, which is
confirmed by applying symmetry transformations, each of which is an
electromagnetic resonator with spatially separated magnetic and electric field.
The characteristics of the cavity is used to mimic phonon behaviour of a one
dimensional chain of atoms. It is demonstrated how magnetic field coupling can
lead to phonon-like dispersion curves with acoustical and optical branches. The
system is able to reproduce a number of effects typical to one-dimensional
lattices exhibiting acoustic vibration, such as band gaps, phonon trapping, and
effects of impurities. In addition, quasicrystal emulations predict the results
expected from this class of ordered structures. The system is easily scalable
to simulate 2D and 3D lattices and shows a new way to engineer arrays of
coupled microwave resonators with a variety of possible applications to hybrid
quantum systems proposed.
