Phononic crystal as a homogeneous viscous metamaterial

Abstract

A homogenization theory, representing the low-frequency limit, is developed for a phononic crystal of cylinders embedded in a viscous fluid. The decay coefficient of sound due to viscosity is calculated analytically for any two-dimensional Bravais lattice and cross section of the cylinders. It is shown that due to the formation of a viscous boundary layer around each cylinder, the losses are enhanced by two to three orders of magnitude as compared to the losses in the free fluid. Also, the decay coefficient in a phononic crystal scales with frequency as sqrt[ω], unlike ω^{2} scaling known for free viscous fluid. In the low-frequency limit a phononic crystal with asymmetric unit cell behaves like a dissipative homogeneous metafluid with anisotropic viscosity.