Electromagnetic fields in air of traveling-wave currents above the earth

The problem of the electromagnetic field created by a thin, straight conductor of infinite length carrying a forward traveling-wave current with a complex propagation constant γ above a homogeneous and isotropic planar earth of wave number $k_1$ is formulated in terms of contour integrals. In the limit where γ becomes equal to the free-space wave number $k_0,$ the component of the magnetic field in air normal to both conductor and interface is evaluated in closed form in terms of known special functions while the remaining components of the field in air are expressed as series expansions in ${\mathrm{\delta =k}}_0^2{\mathrm{(k}}_1^4{\mathrm{-k}}_0^4{)}^{\text{−1/2}}$ via the application of a contour integration technique. The new analytical formulas involve familiar transcendental functions and are valid at any distance from the source. The analysis sheds light on the intricate nature of low-frequency electromagnetic fields generated by transmission lines in the presence of a conducting or a dielectric half-space.