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On inverse scattering by screens. (English) Zbl 0901.35106
The authors prove uniqueness of a (not closed) surface \(\Gamma\in C^1\) (screen) in \(\mathbb{R}^3\) and of the acoustic impedance coefficient \(Z(x)\) (Re \(Z\geq 0)\) when the scattering amplitude \({\mathcal A}\) is given at one (real) frequency \(k\) for all incident directions \(d\) and all directions \(\sigma\) of the receiver. In more detail, \[ (\Delta+ k^2)u=0\quad \text{in } \mathbb{R}^3 \setminus \Gamma,\quad \partial_\nu u+ iZku= 0\quad\text{on } \Gamma, \]
\[ u(x)= e^{ikd \cdot x} +e^{ik | x|}/ | x| {\mathcal A} (\sigma,d;k) +O(1/ | x|^2). \] In proofs they modify the method of singular solutions due to Kirsch, Kress and the reviewer. The authors present a quite efficient quasi-Newton-type numerical scheme for the reconstruction of the plane \(\Gamma\).
Reviewer: V.Isakov (Wichita)

MSC:
35R30 Inverse problems for PDEs
35P25 Scattering theory for PDEs
76Q05 Hydro- and aero-acoustics
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