Consistency Requirements for Integral Representations of Green’s Functions—Part I

Rigorous consistency checks are presented for verifying the correctness of integral representations of Green's functions. We focus here on the Green's functions in open regions of the Helmholtz equation. The two consistency checks are: 1) the completeness relation of the Green's function and 2) the field behavior at infinity. The completeness relation is required so that the solution for arbitrary excitation and at arbitrary location for the problem at hand can be constructed exactly. The behavior at infinity must be such that the fields satisfy Sommerfeld's radiation condition even though the constituent spectra in the integral representation are comprised of incoming and outgoing waves. The application of these two requirements is demonstrated for some existing Green's functions of varying complexity-free-space Green's function, Green's function in the exterior region of a perfectly conducting wedge, and Sommerfeld integral representation of the Green's function for a material half-space.