The inverse scattering problem for time-harmonic acoustic waves. (English) Zbl 0553.35087
The problem is to find the shape of an obstacle from the scattering data. The obstacle is assumed to be sound-soft, bounded, and connected. The incident wave is a scalar plane wave. Some information on the direct scattering problem is given; existence, uniqueness, the T-matrix method and properties of scattering amplitude are briefly discussed. Schiffer’s uniqueness theorem for the inverse problem is stated and proved.
Remarks: 1) An important omisson is that the high-frequency inversion procedures are not discussed. They are actually used for solving inverse problems. 2) The author writes on p. 330 ”remarkably the null-field equations are uniquely solvable for all \(k>0''\). The solvability of these equations for all \(k>0\) is an immediate consequence of the solvability for all \(k>0\) of the exterior boundary problem for Helmholtz’s equation, while uniqueness follows from the well-known result of I. N. Vekua [Dokl. Akad. Nauk SSSR, Nov. Ser. 90, 715-718 (1953; Zbl 0051.049)] not mentioned by the author.
3) In the proof of theorem 6 one could add that there are no two (or more) disjoint obstacles which generate the same scattering data \(f(n_ 0,\nu,k_ 0),\) \(n_ 0\in S^ 2\) and \(k_ 0>0\) are fixed, \(\nu \in S^ 2\). The scattering data \(f(n_ j,\nu,k_ 0,\) \(k_ 0>0,\nu \in \tilde S^ 2\), where \(\tilde S^ 2\) is an open set in the unit sphere \(S^ 2\), \(\{n_ j\}\) is an infinite linearly independent system of unit vectors, determines the obstacle uniquely. 4) The assumption about connectedness of the obstacle is not necessary.
Remarks: 1) An important omisson is that the high-frequency inversion procedures are not discussed. They are actually used for solving inverse problems. 2) The author writes on p. 330 ”remarkably the null-field equations are uniquely solvable for all \(k>0''\). The solvability of these equations for all \(k>0\) is an immediate consequence of the solvability for all \(k>0\) of the exterior boundary problem for Helmholtz’s equation, while uniqueness follows from the well-known result of I. N. Vekua [Dokl. Akad. Nauk SSSR, Nov. Ser. 90, 715-718 (1953; Zbl 0051.049)] not mentioned by the author.
3) In the proof of theorem 6 one could add that there are no two (or more) disjoint obstacles which generate the same scattering data \(f(n_ 0,\nu,k_ 0),\) \(n_ 0\in S^ 2\) and \(k_ 0>0\) are fixed, \(\nu \in S^ 2\). The scattering data \(f(n_ j,\nu,k_ 0,\) \(k_ 0>0,\nu \in \tilde S^ 2\), where \(\tilde S^ 2\) is an open set in the unit sphere \(S^ 2\), \(\{n_ j\}\) is an infinite linearly independent system of unit vectors, determines the obstacle uniquely. 4) The assumption about connectedness of the obstacle is not necessary.
Reviewer: A.G.Ramm
