Time domain solution for the natural sub-ionospheric ELF pulse

We model numerically the sub-ionospheric propagation of an electromagnetic wave in the extremely low frequency (ELF) band from a few hertz to several kilohertz using the time domain presentation for the field. A point vertical electric dipole is the source of the radio wave that arrives at the ground based observer through the spherical Earth-ionosphere cavity. We suppose the radiation moment of the source to be the delta-pulse, hence, the solution obtained is the Green´s function of the problem. A formal representation for such a function is well-known either for the cases of postulated propagation constant or for a given effective surface impedance of the lower ionosphere. The formal solution is constructed using the zonal harmonic series representation (ZHSR) that applies Legendre polynomials describing the angular field dependence and spherical Hankel functions accounting for the radial dependence