On the \(T\)-matrix for scattering by small obstacles
Acoustic wave scattering by small bodies is discussed. A formula for the elements of the \(T\)-matrix from the paper [\textit{D. K. Dacol} and \textit{D. G. Roy}, J. Math. Phys. 44, No. 5, 2133--2148 (2003; Zbl 1062.76048)], is used for small \(ka\), where \(a\) is the diameter of the small body. From this formula and asymptotic formulas for the scattering amplitude for small bodies the author derives asymptotic formulas for \(T\)-matrix for small bodies. The book [\textit{A. G. Ramm}, Wave scattering by small bodies of arbitrary shapes. River Edge, NJ: World Scientific. xvii (2005; Zbl 1081.78001)] contains formulas for the scattering amplitude for acoustic and electromagnetic wave scattering by small bodies of arbitrary shapes. This book and the relevant results from it are not mentioned. In this book one can also find analytic formulas for calculating the electric capacitances of conductors of arbitrary shapes and magnetic polarizability tensors for bodies of arbitrary shapes with any desired accuracy. These formulas are crucial for Sections 4 and 5 of the paper under review, where no formulas for the electric capacitances and magnetic polarizability tensors are given. In the literature analytic formulas for these quantities are found only for ellipsoids and their limiting forms, for example, disks, needles. There are many numerical results for electric capacitances of conductors of various shapes (see references in Zbl 1081.78001). The reader should be aware that in this paper a non-standard definition of the scattering amplitude is used. In particular, the scattering amplitude, used in this paper, is dimensionless, while the standard scattering amplitude, for example, in Section 4, would have dimension of length. In the formula for the scattering amplitude on line 3 of Section 4 it is better to write \(O ((ka)^2)\), rather than \(O(k^2)\), because otherwise the terms in this formula have different dimensions.
scientific article; zbMATH DE number 5161198