Propagation of terrestrial radio waves of long wavelength - theory of zonal harmonics with improved summation techniques

The rigorous mat he mat ical t reatment for the propagation of a radio wave from a H ertzdipole-sou rce curre nt-moment a round a finitely condu cting sphe ri cal earth surrounded b y a con centric electron -ion plasm a can be expressed as a series of zonal harmonics. S uch a solution to the prob le m was obtained for t he terrestrial sphere without a conce nt ric plasma many years ago (190'1-1915). Howeve r, the summ ation of t he seri es, eve n at long wavelen gths o r low fre quencies, was considered to be impractical and t he well-known a nd, indeed , rigorous \' Vatson transformation was introduced (1918).