Mathematics of wavefields. (English) Zbl 1428.35555
Dutta, Hemen (ed.) et al., Applied mathematical analysis: theory, methods, and applications. Cham: Springer. Stud. Syst. Decis. Control 177, 187-231 (2020).
Summary: Wave propagation and scattering occupy a large part of physical, mathematical and engineering sciences. The purpose of this chapter is to present the basic mathematical theory of certain aspects of wavefields, that is, waves and fields, as they occur under various physical situations. These are considered in both scalar or acoustical and vector or electromagnetic media, that is, in the context of Helmholtz’s and Maxwell’s equations. The major emphasis is on the mathematical aspects of Green’s functions, tensors and operators. In particular, the singularities involved are discussed at length. The basic mathematical concepts, tools and techniques, necessary for the presentation, are summarized in the beginning. It is shown that mathematical analyses reveal many subtleties hidden in the wavefields that would otherwise have gone unnoticed. Detailed derivations of the equations are provided whenever possible and necessary. Also, if there are alternative ways of solving a problem, these have been
presented. Finally, copious remarks and notes are included for better explaining certain points.
For the entire collection see [Zbl 1417.00004].
For the entire collection see [Zbl 1417.00004].
MSC:
| 35Q60 | PDEs in connection with optics and electromagnetic theory |
| 78A45 | Diffraction, scattering |
| 35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
| 76Q05 | Hydro- and aero-acoustics |
| 42A38 | Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type |
| 35J08 | Green’s functions for elliptic equations |
| 26B20 | Integral formulas of real functions of several variables (Stokes, Gauss, Green, etc.) |
| 78A25 | Electromagnetic theory (general) |
Keywords:
scattering; Green’s functions; wave equations; generalized functions; Fourier transform; improper integrals; domain differentionReferences:
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