Irradiation of media by transient field: analytical solving of the problem
Abstract
Background. Application of transient electromagnetic waves for the remote sensing problems is perspective because of large information capacity of these fields and ability to penetrate into the medium with losses deeply. Impulse ultra wideband ground penetrating radar is the practical implementation of these possibilities. Improving of its parameters for the wide application requires not only exact solution of the corresponding electrodynamics’ problems, which is currently provided by the direct numerical methods of computations, but also analytical solutions that allow to get more general information about physical processes of transformations of electromagnetic waves. This information will be more useful in the case of obtaining of solutions in time domain. Therefore, obtaining of the analytical solution at least for the simplest model of radiator, like the plane disk with circular aperture, is quite relevant, for which this article is dedicated.
Objectives. Solve analytically the problem of time domain electrodynamics of impulse electromagnetic wave distribution from the one medium into another lossless medium. To achieve this goal it is necessary to find unknown coefficients from the general solutions of Klein-Gordon equations that are multipliers in evolutionary coefficients by applying of the boundary conditions. It is also need to investigate the cases of full propagation and reflection, with the help of which the obtained expressions will be possible to clarify.
Materials and methods. The problem of distribution of transient impulse wave will be solved by the method of evolutionary equations. The general solution of Klein-Gordon equations is obtained by the separation of variables method. Solving of inhomogeneous Klein-Gordon equation is realized by the Riemann function method. Searching of the connection between unknown coefficients from the solved equations will be realized using of boundary conditions for tangential components of the field according to the laws of classical electrodynamics.
Results. With applying of boundary conditions of classical electrodynamics the connection between unknown coefficients from the Klein-Gordon equations that is described different stages of the wave distribution was founded. The general form of searched solution that is similar to Fresnel’s formulas was suggested. Solution for extreme cases of wave distribution such as full propagation and reflection is investigated. On the base of these two cases the conclusion about the general form of solution has been made.
Conclusion. Evolutionary coefficients that characterized electrical and magnetic components of the field were stitched that allowed to find the connection between unknown coefficients of Klein-Gordon equations. Suggested general solution was verificated by means of substitution to the basic formulas that characterized the boundary conditions. For magnetic component the complete agreement of the boundary conditions is achieved, but electrical constituent requires the additional clarification by introducing an additional surface wave.
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References
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