implement functionalities such as qubit readout and inter-qubit coupling. Here we consider the case of a rectangular cavity shorted by a periodic array of conducting cylinders. This is a potential enclosure geometry for large-scale quantum chips with many qubits. We develop simple, accurate models for the TM modes of the cavity, over a wide range of cylinder spacing and radii, using a plasma model and a coupled cavity array circuit model. We compare predictions with finite-element simulation and find good agreement. We investigate inter-qubit couplings mediated by such cavities for circuits at the 100-qubit scale, and discuss additional applications to circuit QED.