**MATHEMATICAL MODELING OF PROPAGATION OF FREQUENCY-MODULATED ELECTROMAGNETIC WAVES IN THE IONOSPHERE FOR THE PROBLEMS OF RADIOCOMMUNICATION**

**Yuliya I. Bova,** *Russian New University (RosNOU), Moscow, Russia, julia_bova@mail.ru*

**Andrey S. Kryukovsky,**

*Russian New University (RosNOU), Moscow, Russia,*

**kryukovsky56@yandex.ru****Dmitry S. Lukin,**

*Russian New University (RosNOU), Moscow, Russia,*

**luknet1@yandex.ru****Abstract**

Decameter radio waves are widely used to provide long-range radio communications, radio navigation, radar, over-the-horizon radio sounding, and to study the structure of the Earth’s ionosphere too. Despite a significant number of publications, the propagation of linearly frequency-modulated signals in a plasma has not been studied enough. In this paper, special attention is paid to the study of the attenuation of a radio signal due to divergence and deflecting absorption using the example of models of electron concentration and frequency of electron collisions for the high-latitude ionosphere. A numerical simulation of the propagation of frequency-modulated signals in an anisotropic medium (Earth’s ionosphere) was performed, taking into account the effect of the frequency of electron collisions on the deflecting absorption. The projections of ray paths onto different coordinate planes are considered. The features of the propagation of linearly frequency-modulated signals in a two-layer ionosphere in the case of the o- wave and e-wave are compared. The influence of the divergence of the ray paths and absorption of radio waves on the attenuation of the amplitude of the radio signal for ordinary and extraordinary waves is investigated. The calculations were performed for the day and night models of the electron concentration and frequency of collisions of the high-latitude ionosphere. In the calculations, a bicharacteristic system of equations was used. To determine the attenuation of the electric field strength along the trajectory, the divergence of the ray flux was calculated on the basis of an extended bicharacteristic system. The effective electric field strength of an isotropic radiator at the receiving point after reflection from the ionosphere was estimated. The positions of caustics along ray paths are investigated. To describe the effective dielectric constant of the medium in the case of an inhomogeneous anisotropic ionosphere, the Appleton formula was used, which as a parameter includes the ratio of the effective frequency of electron collisions to the circular operating frequency.

**Keywords:** numerical modeling, propagation of signals, ionosphere, bicharacteristic system, rays, ordinary and extraordinary waves, absorption, divergence.

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