Abstract
The use of the variational method for searching the ray path of a high-frequency electromagnetic wave between two fixed points is proposed, the knowledge of which is required, e.g. in ionospheric research using measurements on radio signals from Earth's satellites. The problem has been solved for the case of a lossless ionospheric plasma with electron density gradient in arbitrary direction, and for the dipole geomagnetic field. To solve the problem, a curvilinear coordinate system has been used in which the vector of the geomagnetic field intensity lies in the ( xy)-plane at every point of the path. Analysis of this system has been made showing that in most cases, it can be considered as nearly Cartesian. The ray path can then be expressed with sufficient accuracy by two second-order differential equations that are solved by the Galerkin's variational method by means of a system of suitable functions. The form of the functions has been derived and their properties are shown by which a quickly converging solution is guaranteed.
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