The two-dimensional free-space Green’s function and its derivatives in a tangent-normal system

Abstract

The two-dimensional free-space Green’s function, G(2), and its derivatives, are used extensively in the formulation of scattering and diffraction problems through its presence in single- and double-layer potentials, and their use in integral equations. The vast majority of the results from elementary classical mathematical physics for G(2) is based on Cartesian coordinate-space, either directly as a Hankel function in coordinate-space or through a transform, such as the Weyl transform, also based on Cartesian coordinate-space. However, if the geometry of the problem is not Cartesian, for example in scattering from a rough surface, there are difficulties in using a transform representation for G(2) which depends on Cartesian geometry, as the standard Weyl transform does. Here we formulate transform-space representations using a tangent-normal coordinate system. The result for G(2) is a new Weyl-type tangent-normal transform representation from which the results for the vector derivatives of the single-layer potential, the double-layer potential, and the vector derivatives of the double-layer potential follow quite simply. The latter three results can be expressed in terms of two new spectral functions in tangent-normal space, S1 and S2. The overall results are new representations for G(2) and its derivatives which may be useful in integral equation formulations of scattering problems for non-Cartesian geometries.