Phononic crystal as a homogeneous viscous metamaterial

Abstract

At low frequencies, a phononic crystal behaves like a homogeneous medium. Different variants of homogenization technique have been recently proposed to calculate the effective elastic parameters of periodic medium. Here, we develop a homogenization theory for a phononic crystal of solid rods in a viscous fluid. Using the plane wave expansion method, we derive analytical formula for the decay coefficient of low frequency sound propagating in a 2D Bravais lattice with arbitrary unit cell. It is shown that due to the formation of a viscous boundary layer around each cylinder, the losses are enhanced by two to four orders of magnitude as compared to the losses in the free fluid. This enhancement depends on the filling fraction of solid rod and it becomes very strong for almost touching scatterers when sound propagates through narrow slits between the neighbouring rods. Also, the decay coefficient in a phononic crystal scales with frequency as ω2, unlike scaling √ω known for free viscous fluid. The enhanced viscous losses are associated with high effective viscosity of phononic crystal. Like other effective parameters the effective viscosity exhibits strong anisotropy, if the unit cell is asymmetric. The proposed theory can be used for evaluation of efficiency of acoustic devices based on phononic crystals. [This study is supported by NSF Grant No. 1741677.]