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

Abstract

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). The •Watson transfo rm ation led to the development of elegan t mathematical techn iques both rigorous and app roximate for t he evaluation of the fields of r a dio waves in t he vicinity of t he ear th. However , it does not n ecessarily follow t hat the Watson t ra nsform ation is t he only way to achieve numerical master y of t he problem. Indeed, i t also does not fo llow t hat the Watson transformation is t he most effici en t approach to the rigorous form of the theory of propagation , especia lly at long wavelengths. This p a per demonstrates that t he field of t he propagated long wavelength radio wave (fre quenci es less than approximately 50 kc/s) can indeed be evaluated by a summation of a seri es of zonal harmonics. Whereas t he Dumber of terms could become quite large (of t he order of 10 k[a wher e a is the radius of t he sphere a nd k[ is the wave number of the m edium between t he concentric plasma and t he earth), the speed with which these terms can be summed on a large-scale co mputer offsets the complicatio ns introduced by the Watson t ransformation as to the rigoro us m at hem atical solu t ion of the problem. The detailed structure of the field in the a bse nce of a concentric plasma is characterized b y t he quite r eg u lar behavior of the ground wave as a func tion of distan ce. Indeed, the steady decrement of t he ground-wave field is modified on ly near the a ntipode, where an interference pattern or standing wave as a fun ction of distance is no te d because of a nother wave's traveling a round t he sphere in the opposite direction. The introduction of t he concentric electron-ion plasma shell traps t he waves lcaking into space, whem reflection frOlu. the plasm a builds up t raveling waves in the direction of increased distance from t he transmitter. Thus, the series of zo na l hannomi cs co mprises individual waves which a re traveling in the radial direction with r es pect to t he center of the sphere and standing in t he direction of increased angular distance around the spher e. These waves, when summed, build up the wave progressing in the direction of increased angu lar distance. Under special circumstances, standing waves can be noted. This is especially obvious near the antipode of the transmitter. The results of the computations indicate that full rigor can be achieved with comparative ease at fre quencies less t h an approximately 50 kc/s, leaving on ly t he assumed m...

Comments on H. Volland's remarks on Austin's formula

“…In the ELF range, these are equivalent to the results of Schumann [1952], while in the VLF band, the results are a refinement of the ones given by Wait [1957]. The latter results are based essentially on a flat-earth model although, for frequencies below 10 kc/s, the error introduced by neglecting curvature is not great [e.g., see J ohler and Berry, 1962Berry, ,1964. However, in the LF band, the idea that the solution may be obtained by perturbing about Re ~ -1 would seem to be in serious conflict with the full theory for the curved waveguide [i.e., Wait, 1962].…”

VLF waves at satellite altitude to investigate Earth electrical conductivity

“…Such a solution requires a huge number of terms to ensure good convergence of the series [Johler, 1962]. The critical parameters are the wave numbers k 0 and k 1 times the Earth radius and the altitude of the receiver composing the argument of the spherical Hankel functions.…”

“…Among the most effective approaches, Johler [1962] proposes an evaluation by summation on high-speed computer using a Kummer's transformation to improve the series convergence. Fock [1945] and earlier Watson [1918] transform the series in an integral and formulate the solution as a residual series thus providing a quicker convergence than before for a dipole and a receiver located on the Earth surface.…”

“…At the same time, with a rigorous mathematical treatment, Fock [1945] establishes a new solution at the Earth surface and derives a suitable expression for numerical computation. Other important works by Johler [1962], Johler [1970], Stewart et al [1983], Wait [1956Wait [ , 1961aWait [ , 1970, and Wait and Spies [1960] provide formulations of LEYE AND TARITS ©2015. American Geophysical Union.…”

Zonal harmonics in low frequency terrestrial radio wave propagation.

“…1) and the higher order terms can be readily evaluated with zonal harmonics. This can be made evident by introducing the Wronskian (Johler and Berry, 1962 ; page 768). [1962] used a more quickly convergent series based on an approximation of Watson [1919], or using (2.…”