Wave propagation in a periodic structure, formed by membrane elements on nonlinear elastic supports, is studied using a finite element discretization of a single unit cell followed by a perturbation analysis. The study is motivated in part by the need to study the dynamic behavior of micro-machined ultrasonic transducers (CMUTs). The requisite small parameter in the system arises from the ratio of the membrane to flexible support stiffness. The perturbation approach recovers linear Bloch formalism at first order, and amplitude-dependent dispersion corrections at higher orders. The procedure is used to generate weakly nonlinear band diagrams, which can in turn be used to identify amplitude-dependent bandgaps and group velocities. These diagrams also reveal that the strongest amplitude dependency occurs in high-frequency optical modes. Ultimately, the predicted dispersion behavior will be useful in assessing inter-element coupling and identifying effective excitation strategies for actuating CMUTs.

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