Mathematical theory of diffraction. Translated from the 1896 German edition and with an introduction by Raymond J. Nagem, Mario Zampolli and Guido Sandri. Appendix I by Joseph Priestley.
type: Item
instance of: scholarly article
title: Mathematical theory of diffraction. Translated from the 1896 German edition and with an introduction by Raymond J. Nagem, Mario Zampolli and Guido Sandri. Appendix I by Joseph Priestley.
zbMATH Open document ID: 1109.78002
author: A. Zommerfel'd
published in: Progress in Mathematical Physics
publication date: 2004-01-06T00:00:00Z
review text: In 1896 \textit{A. Sommerfeld} published his famous habilitation thesis ``Mathematische Theorie der Diffraktion'' [Math. Ann. 47, 317--374 (1896; JFM 27.0706.03)] which has strongly influenced the development of mathematical diffraction theory. This book offers the first publication in English of Sommerfeld's original work, together with extensive notes by the translators, and it can be recommended to anyone interested in the theory of diffraction. The book starts with a short historical introduction on the theory of diffraction, followed by a careful translation of Sommerfeld's paper. In this paper, he derived the first closed-form solution of a diffraction problem, nowadays called the Sommerfeld half-plane problem: the scattering of a plane electromagnetic wave by a perfectly conducting half-plane. Sommerfeld introduced the concept of a generalized reflection principle and the concept of two-valued solutions of the wave equation that are single-valued, bounded and continuous on a two-sheeted Riemann surface with one branch point. As an application of this, his solution of the half-plane problem, expressed in terms of Fresnel integrals, is remarkably simple and explains physical diffraction phenomena that are not covered by the Fresnel-Kirchhoff theory. The translator's notes (64 pages) are very illuminating for the reader by providing a lot of historical remarks, technical details, and a nice account of previous diffraction analyses of Kirchhoff and Poincaré. The book ends with a text by \textit{Joseph Priestly} [The history and present state of discoveries relating to vision, light and colours (1772)] and another paper by Sommerfeld himself [On the mathematical theory of diffraction phenomena. Gött. Nachr. 1894, 338--342 (1894; JFM 25.1621.01)].
Mathematics Subject Classification ID: 78-02
Mathematics Subject Classification ID: 78A45
Mathematics Subject Classification ID: 01A75
Mathematics Subject Classification ID: 35-02
Mathematics Subject Classification ID: 35J05
Mathematics Subject Classification ID: 35Q60
Mathematics Subject Classification ID: 78-03
zbMATH DE Number: 2022554
zbMATH Keywords: mathematical theory of diffraction
zbMATH Keywords: Helmholtz equation
zbMATH Keywords: Sommerfeld half-plane problem
zbMATH Keywords: wave equation
zbMATH Keywords: Riemann surfaces
reviewed by: Johannes Elschner
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2022554