Resonance gaps and slow sound in three-dimensional phononic crystals: Rod-in-a-box paradigm

We introduce a rod-in-a-box model for acoustic resonators. For resonators small compared to the acoustic wavelength, an elastostatic equilibrium approximation yields closed-form expressions for their frequency-dependent effective masses and moments of inertia. The low-frequency bands and gaps are recaptured by an intuitive $6\ifmmode\times\else\texttimes\fi{}6$ matrix eigenvalue equation, yielding a band structure within $2.3%$ agreement with the finite-element method. Our model is generalized to complex dumbbell-shaped resonators, revealing a dense collection of flat ``slow sound'' bands near the resonance band gap.