Efficient Computation of the Sommerfeld Integral in the Half-Space Problem with Application to the Zenneck Wave

In this presentation, a review and numerical comparison is given for some of the most popular numerical techniques that are reported in the literature for the classic Sommerfeld problem of a vertical electric dipole (VED) over the surface of a semi-infinite half-space corresponding to the earth or ocean, as shown in Fig. 1(a) below (shown for the earth) [F. Mesa and D. R. Jackson, “An Appraisal of Numerical Approaches for a VED over the Earth or Ocean,” IEEE Trans. Antennas and Propagation, vol. 70, no. 8, pp. 6957–6972, Aug. 2022]. The numerical cases compared are: (1) Fourier-Bessel transform (integrating along the real axis after deforming around the poles and branch points); (2) Schelkunoff-like approach (integrating along the branch cuts); (3) Integrating along the extreme steepest-descent paths (ESDP); (4) Integrating along a “numerical” SDP path. Different methods are seen to be numerically best depending on the location of the observation point, namely the distance <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$p$</tex> from the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$z$</tex> axis and the vertical height <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$z$</tex> above the interface.