Improved Babylonian Square Root Algorithm-Based Analytical Expressions for the Surface-to-Surface Solution to the Sommerfeld Half-Space Problem

Abstract

In a recent work, analytical explicit expressions have been derived for the Sommerfeld integrals describing the surface fields of a vertical electric dipole placed near a plane air–earth boundary. Such formulas have the advantage of being exact, but unfortunately consist of slowly convergent infinite sums of special functions. As a consequence, accurate evaluation of the fields is obtained at the cost of a non-negligible computational effort. The purpose of this communication is to derive improved explicit expressions for the Sommerfeld integrals, which assure the same degree of accuracy as the preceding ones, but are constituted by fast-convergent sums of Hankel functions.