The diffraction of electric waves by the earth

Abstract

During the last 15 years, the problem of determining the effect at a distant point of the earth’s surface due to a Hertzian oscillator emitting waves of a definite frequency has been the subject of numerous theoretical investigations. When certain assumptions of a physical character have been made, the problem is of a definitely mathematical type; it is in fact reduced to the problem of finding an approximate formula for the sum of a certain complicated seines of an oscillatory nature; we shall summarise the principal methods which have been devised for dealing with this series.

VLF waves at satellite altitude to investigate Earth electrical conductivity

“…The case is significantly more complicated than the computation of the series at the Earth surface. We tested most of the mathematical formulations proposed in the literature and found that the approach by Watson [1918] and Fock [1945] was one of the most suitable. Watson [1918] represents the series by a contour integral that enclosed the real axis in the complex plane.…”

“…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. More recently, Houdzoumis [1994] applies Poisson's summation formula to transform the series into an integral and derives a new representation of the solution for the same problem.…”

“…Here we present a direct summation technique and an extension of the existing solutions for near-surface receivers to compute the VLF field at the DEMETER altitude (700 km). We derived the standard solution in series and adapted the theory by Watson [1918] for our purpose. We reproduced the geometrical approach by Norton [1941] and the plane Earth case [King and Sandler, 1994] for comparison and testing.…”

“…Yet no practical solution exists to accurately compute the electromagnetic (EM) field at several hundred kilometers above the Earth. While the direct analytical solution to find the EM field generated by a vertical VLF transmitter is well known [Watson, 1918], its numerical application is difficult and considerable attention is paid on this problem since the early days of electromagnetic theory and radio practice to obtain adequate solutions predicting the EM field for many applications. Zenneck [1907] and Sommerfeld [1909] investigate the propagation of the VLF field just above a flat surface.…”

“…Zenneck [1907] and Sommerfeld [1909] investigate the propagation of the VLF field just above a flat surface. Some years later, Watson [1918] considers a perfectly conducting spherical Earth and obtains a solution for both a transmitter and a receiver on the Earth surface. This solution is a series including spherical Bessel and Hankel functions which converges extremely slowly making the solution difficult to use.…”