Active cloaking of resonant coated inclusions for waves in membranes and Kirchhoff plates

The dynamic response of a coated inclusion is considered in the context of active cloaking. The active cloak is achieved for a coated inclusion in the presence of flexural and membrane waves. In this article, we investigate the design of an active cloak for a coated inclusion in three frequency regimes: the low-frequency range, the intermediate range and the higher frequency range in which scattering resonances occur. In the first of these ranges, we extend previous work, which resulted in a simple mass-compensation design for the monopole scatterer, while in the second and third ranges, a combination of the use of an appropriate coating and the appropriate choice of the amplitudes of the active cloaking sources is necessary. We show that such cloaking can indeed be effective in the region of strong scattering resonances. We give closed form analytic expressions for the required amplitudes of the active cloaking sources in the three frequency regions and provide asymptotic estimates and numerical illustrations.