ELF Near-Field Propagation of a Vertical Electric Dipole Due to Lightning Discharges

The Sommerfeld integrals (SIs) representing the electromagnetic (EM) wave propagation of a vertical electric dipole (VED), which has been established over a century, is essential for the understanding of EM radiation scheme of the natural extremely low-frequency (ELF) noises due to lightning discharges. The classical treatments providing with analytical formulas, such as “quasi-static” method and branch-cut integrations, however, rely on their validity of the applicable conditions, leading to considerable discrepancies in near zone for results between theoretical predictions and experimental observations in estimating the ELF radiation process of lightning discharges. By exploiting the assumptions of
$\gamma _{0}\approx {\mathrm{ i}}\lambda $
and
$|\gamma _{0}\gamma _{1}|/k_{0}^{2}\gg 1$
(wavelength-scale), we re-visit the SIs to derive simplified formulas, where the ground medium is treated as a homogeneous highly conducting medium (i.e., all relevant ELF frequency scales below about 30 Hz). In the meanwhile, the exact formulas are derived by an inverse Fourier transformation (IFT) from SIs for ELF transient pulses. Computation and discussion are carried out by obtained formulas for the modeling of the near-field propagation (distance-scale) caused by lightning, which would provide with higher accuracies than using traditional approaches.