# Vestnik КRAUNC. Fiz.-Mat. nauki. 2023. vol. 44. no. 3. P. 104-120. ISSN 2079-6641

**https://doi.org/10.26117/2079-6641-2023-44-3-104-120**

**Research Article**

**Full text in Russian**

**MSC 78A40, 93A30, 65M06**

**Computer Simulation of the Propagation of a Plane Electromagnetic Wave in a Waveguide Formed by the Earth’s Surface and the Ionosphere nder the Condition of Inhomogeneous Boundary Conductivity**

**D. A. Tverdyi^\ast, E. I. Malkin**

Institute for Cosmophysical Research and Radio Wave Propagation FEB RAS, 684034, Paratunka, Mirnaya st., 7., Russia

Abstract. The article presents the results of computer simulation of the propagation of a plane electromagnetic wave. Lightning discharges are natural sources of pulsed electromagnetic radiation (atmosphere). Atmospheric propagates in the Earth-Ionosphere waveguide as a plane electromagnetic wave with a wide frequency spectrum with an intensity maximum in the range of 4-10 kHz. After earthquakes with magnitudes of the order of 7 or more, the saturation of groundwater with secondary minerals increases, which leads to a local increase in the conductivity of the earth in these areas. What determines the electrophysical properties of the earth, as the parameters of the lower boundary of the Earth-Ionosphere waveguide. Which affect the characteristics of electromagnetic waves propagating in the waveguide. It is assumed that, by studying the parameters of the atmosferic, it is possible to establish the presence of an inhomogeneity in the conductivity of the waveguide wall. Based on the system of Maxwell equations with boundary conditions in the form of a Perfectly matched layer, a mathematical model of the process is set. The boundary conditions of the model determine the region of propagation of an electromagnetic broadband signal as a waveguide with inhomogeneous boundary conductivity. The system of model equations is solved by the numerical Finite-Difference Time-domain method. In order to solve the problem and conduct computer modeling, a software package was developed in MATLAB environment. In order to verify the assumption, a number of computer simulations were carried out. As a result, it is shown that there is a backscattering of electromagnetic wave on the waveguide trace, arising as a consequence of reflection of the wave in its interaction with the inhomogeneity of the conductivity of the lower boundary of the waveguide. It is shown that with the help of mathematical modeling of the process of atmospheric propagation and its interactions with inhomogeneity in the waveguide it is possible to establish the presence of inhomogeneity and its relationship to the characteristics of radiation.

*Key words: mathematical modeling, dynamic processes, Maxwell’s equations, atmospheric, plane EM **wave, waveguide, conduction inhomogeneity, computer simulation, MATLAB, inverse wave, numerical **solution, FDTD, PML, ABC*

Received: 19.10.2023; Revised: 27.10.2023; Accepted: 29.10.2023; First online: 02.11.2023

**For citation.** Tverdyi D. A., Malkin E. I. Computer simulation of the propagation of a plane electromagnetic wave in a

waveguide formed by the Earth’s surface and the ionosphere under the condition of inhomogeneous boundary conductivity. Vestnik KRAUNC. Fiz.-mat. nauki. 2023, 44: 3, 104-120. EDN: XZYCQD. https://doi.org/10.26117/2079-6641-2023-44-3-104-120.

**Funding.** The work was carried out within the framework of the State task on the topic (2021—2023) “Physical processes in the system of near space and geospheres under solar and lithospheric impact registration number AAAA-A21-121011290003-0

**Competing interests.** There are no conflicts of interest regarding authorship and publication.

**Contribution and Responsibility.** All authors contributed to this article. Authors are solely responsible for providing

the final version of the article in print. The final version of the manuscript was approved by all authors.

**^\astCorrespondence**: E-mail: tverdyi@ikir.ru

The content is published under the terms of the Creative Commons Attribution 4.0 International License

© Tverdyi D. A., Malkin E. I., 2023

© Institute of Cosmophysical Research and Radio Wave Propagation, 2023 (original layout, design, compilation)

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#### Information about authors

**Tverdyi Dmitrii Alexsandrovich** – Ph. D. (Phys. & Math.), Researcher, laboratory of electromagnetic propogation Institute of Cosmophysical Research and Radio Wave Propagation FEB RAS, Paratunka, Russia, ORCID 0000-0001-6983-5258.

**Malkin Evgeniy Ilich** – Researcher, Laboratory of Electromagnetic Radiation Institute of Cosmophysical Research and Radio Wave Propagation FEB RAS, Paratunka, Russia, ORCID 0000-0001-8037-1335.