The Green-function transform and wave propagation

TL;DR

This paper applies Fourier methods to three-dimensional wave propagation, analyzing the Green function's Fourier transform to distinguish between propagating, evanescent, and inhomogeneous wave components, enhancing understanding of wave behavior.

Contribution

It introduces a detailed Fourier transform analysis of the Green function, clarifying the roles of homogeneous and inhomogeneous components in wave propagation.

Findings

Homogeneous component contains only propagating waves.

Inhomogeneous component includes evanescent and propagating waves.

Evanescent waves are confined outside the k-space sphere.

Abstract

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 inhomogenous 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.