Resonant Metasurfaces with a Tangential Impedance

Metasurfaces formed by monopole and dipole resonators are studied theoretically. The monopole resonators are Helmholtz resonators or membranes vibrating on the first eigenfrequency; the dipole ones are spheres on springs or membranes vibrating on the second eigenfrequency. It is shown that acoustic properties of the metasurface formed by the built-in monopole resonators can be described by an equivalent impedance, which characterizes a normal forcing to the surface, whereas this impedance is not suitable for the metasurface formed by the dipole resonators, because motion of the metasurface is excited by a forcing tangential to the surface. For such boundaries, a new characteristic named “tangential impedance” is proposed. This is a ratio of the second derivative of the sound pressure along a coordinate tangential to the boundary to the normal velocity of the boundary. The dipole metasurface can be described by the equivalent tangential impedance. Reflection and absorption coefficients of the surface with the tangential impedance are found for a harmonic plane wave in dependance of an incidence angle. It is found that the angular dependences of the coefficients are very different for the monopole and dipole metasurfaces.