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The vibrational properties of out-of-plane elastic waves in hexagonal monolayer granular membranes were studied theoretically. The predicted propagation modes involve an out-of-plane displacement and two rotations with axes in the membrane plane. Shear and bending rigidities at the contact between beads were considered. Both the cases of freely suspended membranes and membranes coupled to a rigid substrate were analyzed. Dispersion relations and the existence of band gaps are presented and discussed for various contact properties. For freely suspended membranes with sufficient contact bending rigidity, it is shown that complete band gaps exist. The results obtained may be of interest for testing with acoustic waves the elasticity of recently developed granular membranes composed of nanoparticles (of interest because of their phoxonic properties) and more generally for the control of designing devices for membrane wave propagation.

Elastic waves in phononic monolayer granular membranes