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 $6\times 6$ 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.

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• Band gap
• Elasticity
• Flat bands
• Phononic crystals
• Acoustic metamaterials
• Band structure methods
• Effective medium approach