A review of the analytical techniques for determining the phase and amplitude of a VLF radio wave propagating in the Earth-ionosphere waveguide

The variation of the phase and amplitude of a VLF wave as a function of distance from the transmitter is calculated by each of three analysis techniques for several Earth ionosphere models and the results obtained compared with each other at various radio frequencies. The zonal harmonic analysis method employs the application of Maxwell's equations and the appropriate boundary conditions, allowing the vertical electric field strength to be calculated at any point remote from the transmitter. The form of the solution is a series summation of zonal harmonics. An approximation permitting the series summation to be performed without a computer consists of forming a complex integral for the Hertz vector which is expressed as a sum of residues which, after suitable transformation, become the waveguide modes. The propagation is then described in terms of the propagating modes in the spherical waveguide formed by the Earth as one wall and the ionosphere as the other. In the wave hop technique, the propagation path is divided into three principal regions and computer computation yields solutions not unlike those of geometrical optics.