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Keywords = Hertz dipole

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11 pages, 302 KiB  
Article
The Half-Space Sommerfeld Problem of a Horizontal Dipole for Magnetic Media
by Seil Sautbekov and Merey Sautbekova
Mathematics 2025, 13(1), 169; https://doi.org/10.3390/math13010169 - 6 Jan 2025
Viewed by 914
Abstract
A Hertz radiator’s Sommerfeld boundary value problem is considered for the case when its electric moment is directed horizontally relative to the plane interface between two media with different values of magnetic permeability. An integral representation of the exact expression for the Hertz [...] Read more.
A Hertz radiator’s Sommerfeld boundary value problem is considered for the case when its electric moment is directed horizontally relative to the plane interface between two media with different values of magnetic permeability. An integral representation of the exact expression for the Hertz potential, which generalizes the classical solution for non-magnetic media, both in cylindrical and spherical coordinate systems, is obtained. The corresponding expressions for the scattered wave fields are given in the form of Sommerfeld integrals. It is shown that the potential components can be represented as the sum of an infinite series in powers of the Green function. Full article
(This article belongs to the Special Issue Computational Methods in Electromagnetics)
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11 pages, 313 KiB  
Article
Calculation of Sommerfeld Integrals in Dipole Radiation Problems
by Seil Sautbekov, Merey Sautbekova, Kuralay Baisalova and Mustakhim Pshikov
Mathematics 2024, 12(2), 298; https://doi.org/10.3390/math12020298 - 17 Jan 2024
Cited by 4 | Viewed by 1719
Abstract
This article proposes asymptotic methods for calculating Sommerfeld integrals, which enable us to calculate the integral using the expansion of a function into an infinite power series at the saddle point, where the role of a rapidly oscillating function under the integral can [...] Read more.
This article proposes asymptotic methods for calculating Sommerfeld integrals, which enable us to calculate the integral using the expansion of a function into an infinite power series at the saddle point, where the role of a rapidly oscillating function under the integral can be fulfilled either by an exponential or by its product by the Hankel function. The proposed types of Sommerfeld integrals are generalized on the basis of integral representations of the Hertz radiator fields in the form of the inverse Hankel transform with the subsequent replacement of the Bessel function by the Hankel function. It is shown that the numerical values of the saddle point are complex. During integration, reference or so-called standard integrals, which contain the main features of the integrand function, were used. As a demonstration of the accuracy of the technique, a previously known asymptotic formula for the Hankel functions was obtained in the form of an infinite series. The proposed method for calculating Sommerfeld integrals can be useful in solving the half-space Sommerfeld problem. The authors present an example in the form of an infinite series for the magnetic field of reflected waves, obtained directly through the Sommerfeld integral (SI). Full article
(This article belongs to the Special Issue Mathematical Modelling of Wave Phenomena)
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