# Resonant Scattering and Generation of WavesThe Mathematical Model

Resonant Scattering and Generation of Waves: The Mathematical Model [Electromagnetic phenomena in a space–time domain \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^4_>:=\mathbb {R}^3\times (0,\infty )$$\end{document} can be governed by the system of macroscopic Maxwell’s differential equations \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\begin{aligned} \begin{array}{r@{\ }c@{\ }l@{\quad }r@{\ }c@{\ }l} \dfrac{1}{c}\dfrac{\partial {\mathbf B }}{\partial t} + \nabla \times {\mathbf E }&{}=&{} 0, &{}\dfrac{1}{c}\dfrac{\partial {\mathbf D }}{\partial t} - \nabla \times {\mathbf H }&{}=&{}-\dfrac{4\pi }{c}{\mathbf J }\,,\\ \nabla \cdot {\mathbf D }&{}=&{} 4\pi \rho , &{} \nabla \cdot {\mathbf B }&{}=&{} 0, \end{array} \end{aligned}\end{document}where the Gaussian unit system is used (see, for example, Born and Wolf in Principles of Optic, Pergamon Press, Oxford, 1970, [1, Sect. 1.1.1], Landau et al. in Electrodynamics of Continuous Media, Elsevier Butterworth-Heinemann, Oxford, 1984, [2, Chap. IX]). Here, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf E },$$\end{document}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf H },$$\end{document}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf D },$$\end{document}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf B }:\,\mathbb {R}^4_>\rightarrow \mathbb {R}^3$$\end{document} denote the unknown vector fields of electric and magnetic field intensity, electric and magnetic induction, respectively, c is a positive constant—the velocity of light. The function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho :\,\mathbb {R}^4_>\rightarrow \mathbb {R}$$\end{document} and the vector field \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf J }:\,\mathbb {R}^4_>\rightarrow \mathbb {R}^3$$\end{document} are called the electric charge density and the electric current density, respectively. These macroscopic quantities are obtained by averaging rapidly varying microscopic quantities over spatial scales that are much larger than the typical material microstructure scales. Details of the averaging procedure can be found in standard electrodynamic textbooks, for instance, in Jackson, Classical Electrodynamics, Wiley, New York, 1999, [3].] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

# Resonant Scattering and Generation of WavesThe Mathematical Model

42 pages

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# References (12)

Publisher
Springer International Publishing
© Springer International Publishing AG, part of Springer Nature 2019
ISBN
978-3-319-96300-6
Pages
1 –43
DOI
10.1007/978-3-319-96301-3_1
Publisher site
See Chapter on Publisher Site

### Abstract

[Electromagnetic phenomena in a space–time domain \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^4_>:=\mathbb {R}^3\times (0,\infty )$$\end{document} can be governed by the system of macroscopic Maxwell’s differential equations \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\begin{aligned} \begin{array}{r@{\ }c@{\ }l@{\quad }r@{\ }c@{\ }l} \dfrac{1}{c}\dfrac{\partial {\mathbf B }}{\partial t} + \nabla \times {\mathbf E }&{}=&{} 0, &{}\dfrac{1}{c}\dfrac{\partial {\mathbf D }}{\partial t} - \nabla \times {\mathbf H }&{}=&{}-\dfrac{4\pi }{c}{\mathbf J }\,,\\ \nabla \cdot {\mathbf D }&{}=&{} 4\pi \rho , &{} \nabla \cdot {\mathbf B }&{}=&{} 0, \end{array} \end{aligned}\end{document}where the Gaussian unit system is used (see, for example, Born and Wolf in Principles of Optic, Pergamon Press, Oxford, 1970, [1, Sect. 1.1.1], Landau et al. in Electrodynamics of Continuous Media, Elsevier Butterworth-Heinemann, Oxford, 1984, [2, Chap. IX]). Here, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf E },$$\end{document}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf H },$$\end{document}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf D },$$\end{document}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf B }:\,\mathbb {R}^4_>\rightarrow \mathbb {R}^3$$\end{document} denote the unknown vector fields of electric and magnetic field intensity, electric and magnetic induction, respectively, c is a positive constant—the velocity of light. The function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho :\,\mathbb {R}^4_>\rightarrow \mathbb {R}$$\end{document} and the vector field \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf J }:\,\mathbb {R}^4_>\rightarrow \mathbb {R}^3$$\end{document} are called the electric charge density and the electric current density, respectively. These macroscopic quantities are obtained by averaging rapidly varying microscopic quantities over spatial scales that are much larger than the typical material microstructure scales. Details of the averaging procedure can be found in standard electrodynamic textbooks, for instance, in Jackson, Classical Electrodynamics, Wiley, New York, 1999, [3].]

Published: Jul 27, 2018

Keywords: Gaussian Unit System; Unknown Vector Field; Nonlinear Dielectric Layer; Longitudinal Homogeneity; Energy Balance Law