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Time-Dependent One-Dimensional Electromagnetic Wave Propagation in Inhomogeneous Media: Exact Solution in Terms of Transmutations and Neumann Series of Bessel Functions

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Abstract

The time-dependent Maxwell system describing electromagnetic wave propagation in inhomogeneous isotropic media in the one-dimensional case reduces to a Vekua-type equation for bicomplex-valued functions of a hyperbolic variable, see [7]. In [5] using this reduction a representation of a general solution of the system was obtained in terms of a couple of Darboux-associated transmutation operators [8]. In [6] a Fourier–Legendre expansion of transmutation integral kernels was obtained. This expansion is used in the present work for obtaining an exact solution of the problem of the transmission of a normally incident electromagnetic time-dependent plane wave through an arbitrary inhomogeneous layer. The result can be used for efficient computation of the transmitted modulated signals. In particular, it is shown that in the classical situation of a signal represented in terms of a trigonometric Fourier series the solution of the problem can be written in the form of Neumann series of Bessel functions with exact formulas for the coefficients. The representation lends itself to numerical computation.

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Funding

The second and third named authors acknowledge the support from CONACYT, Mexico via the projects 222478 and 284470. V. Kravchenko acknowledges the support from the Regional mathematical center of the Southern Federal University, Russia, where this work was developed during his sabbatical leave from the Cinvestav.

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Authors and Affiliations

  1. Faculty of Engineering, Autonomous University of Queretaro, Cerro de las Campanas s/n, CP 76010, , Querétaro, Qro. , México

    K. V. Khmelnytskaya

  2. Department of Mathematics, Cinvestav, Campus Querétaro, Libramiento Norponiente 2000, CP 76230, , Querétaro, Qro., México

    V. V. Kravchenko & S. M. Torba

Authors
  1. K. V. Khmelnytskaya
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  2. V. V. Kravchenko
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  3. S. M. Torba
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Correspondence to K. V. Khmelnytskaya, V. V. Kravchenko or S. M. Torba.

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Khmelnytskaya, K.V., Kravchenko, V.V. & Torba, S.M. Time-Dependent One-Dimensional Electromagnetic Wave Propagation in Inhomogeneous Media: Exact Solution in Terms of Transmutations and Neumann Series of Bessel Functions. Lobachevskii J Math 41, 785–796 (2020). https://doi.org/10.1134/S1995080220050054

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  • DOI: https://doi.org/10.1134/S1995080220050054

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