Spherical harmonic formulations for passive and active control of acoustic scattering around the sphere

Acoustic cloaking can be implemented using different strategies: passive, active, or a combination of both. The literature on this topic is rich when it comes to the passive methods, specifically on metamaterial designs. However, active and hybrid methods are less well explored. In this work, spherical harmonic decomposition is used to calculate the scattering from a spherical obstacle subjected to an incident, monochromatic plane-wave. Its surface is defined through boundary conditions equivalent to a uniform, locally reacting impedance (Robin condition). Secondary sources represented by point-monopoles on the sphere are used to form an active control system. In the absence of active control, at low-frequencies, the least scattering is achieved if the surface impedance is negative, purely imaginary and has a large absolute value. At high-frequencies, the best performance is realised with a real, positive impedance that is close to that of the exterior propagation medium. When adding active control, only the low-frequency regime shows improvement, given any impedance, and it is most beneficial to enhance for the previous case that exhibited the best passive performance. Thus, a promising hybrid cloaking approach is a combination of: a stiffened-controlled boundary with multi-channel active control at low-frequencies, and a passive, absorptive boundary at high-frequencies