A formulation is described for finding the surface vertical E field throughout the lateral extent of the spherical Earth-ionosphere waveguide at extremely low frequencies (ELF) below 100 kHz for a completely general model of values of the sine of the waveguide eigenangle. These values are specified at elements of a mesh point array that extends throughout the lateral extent of the spherical shell guide. The solution is based on the condition that the vertical E field at the surface must satisfy the reduced two-dimensional wave equation at every localized group of mesh points. It has been found practical to use a mesh size of 312.5 km halfway to the antipode, which implies a mesh size of 50 km at a distance 1 Mm from the transmitter. Eigensolutions are found for any far range from the transmitter for which there is no transverse variation of properties, and these are combined with the more general solution which applies to the region of transverse inhomogeneity. The latter may include the entire waveguide. An example is given for illustration. It is expected that use of the formulation will make it possible to describe the combined effects of inhomogeneities such as land-sea boundaries around continents and large seas and islands, the day-night terminator, and particle precipitation on the amplitude and phase of the vertical E field.
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