AUTHOR=Sheppard Colin J. R. , Kou Shan S. , Lin Jiao TITLE=The Green-function transform and wave propagation JOURNAL=Frontiers in Physics VOLUME=2 YEAR=2014 URL=https://www.frontiersin.org/journals/physics/articles/10.3389/fphy.2014.00067 DOI=10.3389/fphy.2014.00067 ISSN=2296-424X ABSTRACT=

We review Fourier methods used in the disciplines of electromagnetism and signal processing, with a view to reconciling differences in approach. In particular, Fourier methods well known in signal processing are applied to three-dimensional wave propagation problems. The Fourier transform of the Green function, when written explicitly in terms of a real-valued spatial frequency, consists of homogeneous and inhomogeneous components. Both parts are necessary to result in a pure out-going wave that satisfies causality. The homogeneous component consists only of propagating waves, but the inhomogeneous component contains both evanescent and propagating terms. Thus, we make a distinction between inhomogeneous waves and evanescent waves. The evanescent component is completely contained in the region of the inhomogeneous component outside the k-space sphere. Further, propagating waves in the Weyl expansion contain both homogeneous and inhomogeneous components. The connection between the Whittaker and Weyl expansions is discussed. A list of relevant spherically symmetric Fourier transforms is given.