Active cloaking of inclusions for flexural waves in thin elastic plates

Abstract

We present a new method to create an active cloak for a rigid inclusion in a thin plate, and analyse flexural waves within such a plate governed by the Kirchhoff plate equation. We consider scattering of both a plane wave and a cylindrical wave by a single clamped inclusion of circular shape. In order to cloak the inclusion, we place control sources at small distances from the scatterer and choose their intensities to eliminate propagating orders of the scattered wave, thus reconstructing the respective incident wave. We then vary the number and position of the control sources to obtain the most effective configuration for cloaking the circular inclusion. Finally, we successfully cloak an arbitrarily shaped scatterer in a thin plate by deriving a semi-analytical, asymptotic algorithm.

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

“…Note that in this particular case A (e) n = i n since 15) and the B (e) n are identically zero. It can be shown that the general expression for the scattered field is (see, for example, [18], [21], [22])…”

“…9 that the multipole field components of higher order become non-negligible, and hence passive reduction of scattering would not be appropriate any longer. Active cloaking by point sources, as described in [18], enables us to make the multipole coefficients vanish to the required order.…”

“…In a recent paper [18], an analytical model of an approximate cloak for flexural waves was developed by introducing a finite number of active sources, whose complex amplitudes were chosen to suppress the scattered field resulting from an interaction of an incident wave with a finite obstacle. The multipole representation was used for the solution, and the required number of multipoles in the scattered field were constrained to be zero.…”

Analytic materials

“…This gave birth to the subject of active exterior cloaking for conductivity [48,52], for acoustics in twodimensions and three-dimensions [48,49,51,53,129] and for elastodynamics [130]. In a beautiful twist of the idea, O'Neill, Selsil, McPhedran, Movchan, Movchan, and Moggach [133,134] found that for the plate equation one could get excellent exterior cloaking without enormous fields if one relaxed the requirement that the cloak create a quiet zone and instead require only that it cloak a given object. The main reason that enormous fields are not needed is that the plate equation has the desirable property that the Green's function is bounded.…”

Composites with invisible inclusions: Eigenvalues of ℝ-linear problem

“…Recently, materials having negative physical constants were discovered. It concerns dielectric-magnetic materials displaying a negative index of refraction [1,2,9,23,24]. Mathematical modelling of metamaterials and neutral (invisible) inclusions were discussed in [12,14,17] and works cited therein.…”