Resonant Metasurfaces with a Tangential Impedance

Abstract

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.

Acoustic properties of surfaces covered by multipole resonators

“…So, the multiple scatterer is a resonator and below we will focus on this specific case. However, equations (13)(14)(15) remain valid for any scatterer not just those that are resonant, and can be used to describe the multiple scatterers regardless of how they are formed. The main goal is to determine the impedance 𝑍 𝑁 , which is heavily influenced by the structure of the scatterer.…”

“…For 𝑁 = 0 the impedance 𝑍 0 ′ is the ordinary impedance, which is a ratio of the sound pressure to the normal velocity. For 𝑁 ≥ 1 the motion of the surface is excited by a tangential forcing, therefore, the condition (29) can be called the tangential impedance [13,20]. Boundary conditions similar to (29) are widely known in electromagnetism [33,34], they belong to high order impedance boundary conditions or HOIBC.…”

“…The simplest physical model of the dipole resonator is a rigid sphere on a spring [12]. Also, the dipole resonator can be implemented by a membrane in a ring oscillating at the first eigenmode [12] or a membrane embedded in a rigid baffle oscillating at the second eigenmode [13]. Two coupled Helmholtz resonators have a mode, for which they oscillate with opposite phases [14], hence two monopole resonators can form the dipole one.…”

Acoustic Properties of Surfaces Covered by Multipole Resonators

“…So, the multipole scatterer can be considered as a resonator and below we will focus on this specific case. However, Equations (13)(14)(15) remain valid for any scatterer, not just those that are resonant, and can be used to describe multipole scatterers regardless of how they are formed. The main goal is to determine the impedance 𝑍 , which is strongly influenced by the structure of the scatterer.…”

“…The simplest physical model of the dipole resonator is a rigid sphere on a spring [12]. Also, the dipole resonator can be implemented by a membrane in a ring oscillating at the first eigenmode [12] or a membrane embedded in a rigid baffle oscillating at the second eigenmode [13]. Alternatively, the dipole-type metasurface can be formed by a set of short beams clamped on one side [14].…”