Nonlocal boundary conditions for corrugated acoustic metasurface with strong near-field interactions

The propagation of long-wavelength sound in the presence of a metasurface made by arranging acoustic resonators periodically upon or slightly above an impervious substrate is studied. The method of two-scale asymptotic homogenization is used to derive effective boundary conditions, which account for both the surface corrugation and the low-frequency resonance. This method is applied to periodic arrays of resonators of any shape operating in the long-wavelength regime. The approach relies on the existence of a locally periodic boundary layer developed in the vicinity of the metasurface, where strong near-field interactions of the resonators with each other and with the substrate take place. These local effects give rise to an effective surface admittance supplemented by nonlocal contributions from the simple and double gradients of the pressure at the surface. These phenomena are illustrated for the periodic array of cylindrical Helmholtz resonators with an extended inner duct. Effects of the centre-to-centre spacing and orientation of the resonators' opening on the nonlocality and apparent resonance frequency are studied. The model could be used to design metasurfaces with specific effective boundary conditions required for particular applications.