Time-Domain Dyadic Green's Function for an Electric Source in a Conductive Plate

We have determined the quasi-static time-domain dyadic Green's function for an electric source in a conductive plate for use in electric field integral equations. Starting with a frequency-domain representation, we constructed the dyadic kernel from electric and magnetic scalar potentials defined with respect to a preferred direction normal to plate. The final time-domain expression has three parts: a free-space term, multiple image terms, and partial reflection terms. The free-space fundamental solution is expressed in terms of a three-dimensional Gaussian bell curve satisfying the diffusion equation. Similarly, the image terms are expressed in the same form with coordinates shifted so that each Gaussian curve centers on an image point. In order to carry out the inverse Laplace transform from the frequency domain to the time domain, we expand the partial reflection functions as asymptotic series before transforming them analytically. The resulting expression for dyadic Green's kernel can be evaluated efficiently and with well-controlled accuracy.