Effective models of metasurface/metainterface made of three-dimensional Helmholtz resonators

This paper studies the homogenized behavior of a meta-surface/meta-interface made of three-dimensional Helmholtz resonators of any cavity and neck shape, providing an effective model in the time domain obtained by using a three-scale asymptotic technique capable of encapsulating both resonance and boundary layer effects. The effective wave propagation is reduced to a three-dimensional isotropic effective propagation with a cavity pressure that solves a mass-spring equation taking into account an "added neck length". We provide a precise method for determining the so-called "added length" by solving a unique problem of a perfect fluid flowing in a duct opening onto free space with a unit flow rate. The solution to the problem depends only on the geometry of the cross-section of the neck.