Abstract
This paper deals with the mathematical expressions called Sommerfeld integrals. Introduced by A. Sommerfeld in 1909, they are mathematically equivalent to inverse Hankel transforms. The original historical goal of these integrals was to provide an accurate mathematical description of the electromagnetic phenomena involved in long-distance wireless radio and telegraphy. However, their scope was quickly enlarged thanks to the so-called spectral-domain stratified theory, and now they are ubiquitous in the mathematical models associated to many electromagnetic technologies, ranging from EMC lightning modelization and ground penetrating radar to optical and plasmonic integrated devices and going through the familiar microwave and millimeter-wave planar structures using printed circuit technology. In all these areas, Sommerfeld integrals can provide direct evaluations of the involved electromagnetic fields or they can be used as Green's functions in the frame of integral equation formulations. Other disciplines involving stratified media, like seismology and geological prospection, also benefit from these integrals. After discussing the most canonical Sommerfeld integral, appearing in the so-called Sommerfeld identity, this paper reviews three classical structures, namely, the original Sommerfeld problem involving two semi-infinite media, and the microstrip and stripline geometries. It is shown that Sommerfeld integrals provide a unifying treatment of these three problems and that their mathematical features have a direct translation in terms of the physical properties exhibited by the electromagnetic fields that can exist in them.
Highlights
In 1909, Arnold Johannes Wilhem Sommerfeld (1868–1951) published in the prestigious German journal “Annalen der Physik” a 72-page paper [1], dealing with the propagation of electromagnetic (EM) waves above a lossy flat Earth
This interest for electromagnetic wave phenomena remained for all of his professional life, as witnessed by subsequent papers [3] and by the relevance given to them in his lectures during his 32 years of tenure as a Full Professor of Theoretical Physics at the University of Munich (19061938), where he supervised PhD theses on this subject
We aim at providing a tutorial explanation of the generation and use of Sommerfeld integrals (SI) in the context of stratified media theory, complemented with some useful information about the algorithms commonly used for their numerical evaluation and the discussion of some typical results that can be obtained with these algorithms
Summary
In 1909, Arnold Johannes Wilhem Sommerfeld (1868–1951) published in the prestigious German journal “Annalen der Physik” a 72-page paper [1], dealing with the propagation of electromagnetic (EM) waves above a lossy flat Earth. A mathematician by education, Sommerfeld became very interested in Applied Physics and Engineering during his tenure (1900-1906) as Associate Professor in Applied Mechanics at the Königliche Technische Hochschule Aachen (later RWTH Aachen University) His interest in electrodynamics can even be traced to earlier years with his research on EM waves along wires, that provided the base upon which his 1909 paper was built, as described in Eckert’s detailed biographical book [2]. Sommerfeld himself already saw them as the ideal playground for applying many mathematical tools, like differential equations, integral transforms, vectorial decompositions, complex integration, asymptotic expansions, and approximation theory, to a basic problem in Electrodynamics He was aware of the need to always conclude the theoretical developments with some results of immediate practical application. We aim at providing a tutorial explanation of the generation and use of SIs in the context of stratified media theory, complemented with some useful information about the algorithms commonly used for their numerical evaluation and the discussion of some typical results that can be obtained with these algorithms
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