Active manipulation of fields modeled by the Helmholtz equation. (English) Zbl 1305.78011
Summary: We extend the results proposed in [D. Onofrei, Inverse Probl. 28, No. 10, Article ID 105009, 15 p. (2012; Zbl 1259.78017)] and study the problem of active control in the context of a scalar Helmholtz equation. Given a source region \(D_a\) and \(\{v_0,v_1,\dots,v_n\}\), a set of solutions of the homogeneous scalar Helmholtz equation in \(n\) mutually disjoint “control” regions \(\{D_0,D_1,\dots,D_n\}\) of \(\mathbb R^2\) or \(\mathbb R^3\), respectively, the main objective of this paper is to characterize the necessary boundary data on \(\partial D_a\) so that the solution to the corresponding exterior scalar Helmholtz problem will closely approximate \(v_i\) in \(D_i\), respectively, for each \(i\in\{0,\dots,n\}\). Building up on the previous ideas presented in [loc. cit.] we show the existence of a class of solutions to the problem, provide numerical support of the results in 2D and 3D and discuss the existence of a minimal energy solution and its stability.
MSC:
| 78A45 | Diffraction, scattering |
| 35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
Keywords:
active manipulation; Helmholtz equation; layer potentials; integral equation; active exterior cloakingCitations:
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