Mathematical Problems in Wave Propagation Theory, 1970
Seminars in mathematics Series, Vol. 9

The papers comprising this collection are directly or indirectly related to an important branch of mathematical physics - the mathematical theory of wave propagation and diffraction. The paper by V. M. Babich is concerned with the application of the parabolic-equation method (of Academician V. A. Fok and M. A, Leontovich) to the problem of the asymptotic behavior of eigenfunc­ tions concentrated in a neighborhood of a closed geodesie in a Riemannian space. The techniques used in this paper have been föund useful in solving certain problems in the theory of open resonators. The topic of G. P. Astrakhantsev's paper is similar to that of the paper by V. M. Babich. Here also the parabolic-equation method is used to find the asymptotic solution of the elasticity equations which describes Love waves concentrated in a neighborhood of some surface ray. The paper of T. F. Pankratova is concerned with finding the asymptotic behavior of th~ eigenfunc­ tions of the Laplace operator from the exact solution for the surface of a triaxial ellipsoid and the re­ gion exterior to it. The first three papers of B. G. Nikolaev are somewhat apart from the central theme of the col­ lection; they treat the integral transforms with respect to associated Legendre functions of first kind and their applications. Examples of such applications are the use of this transform for the solution of integral equations with symmetrie kernels and for the solution of certain problems in the theory of electrical prospecting.

Sharply Directed Propagation of Love-Type Surface Waves.- Eigenfunctions Concentrated in a Neighborhood of a Closed Geodesic.- The Calculation of Interference Waves for Diffraction by a Cylinder and a Sphere.- The Expansion of an Arbitrary Function in Terms of an Integral of Associated Legendre Functions of First Kind with Complex Index.- Application of an Integral Transform with Generalized Legendre Kernel to the Solution of Integral Equations with Symmetric Kernels.- Solution of Three-Dimensional Problems for the Hyperboloid of Revolution and the Lens in Electrical Prospecting.- Application of the Laplace Method to the Construction of Solutions of the Helmholtz Equation.- The Problem of Constructing Solutions of the Neumann Problem for the Stationary Diffraction of Waves from a Half Space Separated by an Inclined Boundary into Two Angular Regions with Different Wave Propagation Speeds.- Eigenfunctions of the Laplace Operator on the Surface of a Triaxial Ellipsoid and in the Region Exterior to It.- Calculation of the Wave Fields for Multiple Waves near the Points of Origin.