Near fields and antipodal fields in the terrestrial Earth¿ionosphere waveguide

Near fields of a vertical electric dipole are expressed as a series of zonal harmonics, or as a summation of waveguide modes. These representations are compared with dipole fields on a conducting ground plane, and with field calculations, using the method of images or ray theory. The series of zonal harmonics and the mode sum provide nearly identical near fields; their difference characterises the contribution of `branch-cut¿ waves, and is negligible in the present computational model. In the near-field range, the method of images gives the simplest field representation for e.l.f.; for v.l.f., a useful approximation is provided by combining the ground wave and a single sky wave. At the upper waveguide boundary, the radial electric fields are finite, but the other field components vanish when approaching the source axis. The field components also exhibit similar variations when approaching the antipode of the source.