Attenuation limits in longitudinal phononic crystals

The acoustic attenuation inside the bandgaps is, together with the bandgap width, a fundamental design parameter for phononic-crystal-based systems. We discuss approximate expressions for the maximum attenuation inside the bandgaps of one-dimensional longitudinal phononic crystals and its dependence on the acoustic contrast and the fractional bandwidth. We provide different approximations at small and large fractional bandwidths, computed from the trace of the transmission matrix of the crystal elementary cell. We show that, for relatively small gaps, the attenuation is roughly proportional to the fractional bandwidth, in analogy with the flexural case. For larger gaps, a large attenuation can be obtained only for high (and possibly impractical) acoustic contrasts. Approximate expressions are validated through comparison with FEM results. We also derive asymptotic upper limits for the bandgap borders and show that high contrasts do not necessarily lead to wide bandgaps, a fact connected to geometrical phase inversion for the acoustic wave in the crystal. We finally compare the attenuation of flexural and longitudinal waves at a fixed fractional bandwidth and derive regions of optimum attenuation for the two propagation modes.