Применение потенциалов Дебая для расчета первичного поля в изотропной упругой среде, возбуждаемой вертикальным источником

Abstract

An analytical solution of the problem of exciting an isotropic elastic medium by a vertical point source is proposed. This problem is a model one and is also a part of the solution of other model problems such as wave excitation in a plane-layered medium. The solution is based on the method of Debye potentials, which is widely used in solving various problems of electrodynamics. The range of these problems includes radiation from an antenna of a fairly arbitrary kind over a plane-layered dielectric medium, consideration of the effect an intricate fairing of an aircraft has on the operation of the radar station antenna system (the effect on the antenna pattern and the direction-finding characteristics of the system as a whole), designing slit radiators with a feeding resonator, and radiation from horn and waveguide antennas into a magnetodielectric semi-space. In many cases, it is possible to obtain an analytical solution of the problem; if an analytical solution is not available, numerical methods developed using the Debye potentials are effective from the computational point of view. The significance of an analytical solution lies in the simplicity and clarity of the interpretation and understanding of the wave propagation processes in contrast to a numerical solution. However, the fundamental difference between the problems of exciting elastic bodies from electrodynamic ones is in that two types of waves, namely, transverse and longitudinal ones, propagate in elastic media. This feature substantially complicates both the analytical solution and the elastic wave propagation process in comparison with electromagnetic waves. In view of this specific feature, a technique for calculating the excitation of an isotropic elastic medium using the apparatus of Debye potentials is proposed. It is shown that the displacement field of an isotropic elastic medium excited by a vertical source is represented as the sum of longitudinal and transverse wave fields, and symmetrically in the plane perpendicular to the source axis. Simple analytic expressions for calculating the displacement field at an arbitrary point in space are obtained. The results of calculations using analytical formulas are compared with the data of numerical modeling in the COMSOL Multiphysics 3.5a computer program. In view of the fact that the COMSOL Multiphysics 3.5a computer program is intended for calculating structures with finite dimensions, an original model of unlimited elastic medium in the form of a large-radius sphere (around 20 wavelengths) with radially-symmetrical absorption was developed.