A.S. Kryukovsky

The features of the influence of the Earth’s ionosphere on the propagation of high-frequency radio waves are considered by the method of bicharacteristics, which is relevant in connection with the design of space antennas with a synthesized aperture. The estimates of the Faraday rotation angle are obtained as a function of the frequency of the transmitter. Phase races and refractive errors were investigated.

Necessary and sufficient conditions for the formation of wave catastrophes of the cuspoid series are obtained in the case when the dimension of the internal parameters of the problem is equal to two, which corresponds to one-dimensional focusing of wave fields in three-dimensional space.

Here is developed a method for constructing local asymptotics describing the focusing of cuspoid-type electromagnetic fields, which makes it possible to obtain the coefficients of unfolding and the phase of a traveling wave in the form of segments of power series. The first (linear) and second (quadratic) approximation for a caustic-type wave catastrophe (“cusp”) are constructed.

A method for constructing local asymptotics describing the focusing of electromagnetic fields in the form of a finite sum of special functions of wave catastrophes, the arguments of which, as well as the amplitude and phase factors are represented as segments of power series is proposed. The cases of degenerate saddle (stationary) points of the phase function of the integrand corresponding to the topological singularities of catastrophe theory are considered. In the case of the most common topological singularities of the cuspoid and umbilic types, the first approximation for the asymptotics of rapidly oscillating integrals that uniformly describe the wave fields in the focal regions is obtained. The results of the work allow analyzing short-wave solutions of diffraction and quantum-mechanical problems, constructed, in particular, by the method of the Maslov’s canonical operator.

A numerical simulation of the propagation of frequency-modulated signals in an anisotropic medium, taking into account the infl uence of frequency of electron collisions on deflecting absorption is carried out. The projections of ray trajectories onto different coordinate planes are studied. The influence of the ray trajectory divergences and absorption on the attenuation of the amplitude of radio signals is investigated. The comparison of results of calculations for ordinary and extraordinary waves is fulfilled. The calculations are performed for day and night models of the electron concentration and collision frequency of high-latitude ionosphere.

In this paper, the features of the Earth’s ionosphere influence on the propagation of radio waves of the high-frequency P-band are considered by the method of bicharacteristics. The estimates of the angular values of the Faraday rotation as a function of the mutual orientation of the ray trajectories and the external magnetic field intensity vector are obtained.

A review of the main results devoted to the application of the wave theory of catastrophes to problems of propagation, focusing, and diffraction of radio waves is reviewed. Ray and caustic structures as well as wave fields in the focal regions are considered.

The cuspoid focusing such as A3, formed by parabolic wave front is considered. The methods of definition of parameters of universal deformation of a singularity – method of a global asymptotic (or a saddle point method) – and first approximation of a local asymptotic method are compared. The amplitude and phase structures of a wave field in a vicinity of a singularity are constructed.

The classifi cation and both necessary and sufficient conditions of formation of edge catastrophes with difficult caustic structures (as zero-modal, and uni-modal series) in space-time under condition of propagation of electromagnetic waves in plasma media with a strong frequency dispersion are given. The uniform asymptotic solutions of the wave equations in regions containing special functions of space-time wave catastrophes, corresponding to reference structures of electromagnetic fields are constructed.