Spherical wave theory for MF, LF, and VLF propagation

If (n) is the set of positive integers, spherical waves like ζn^{(1, 2)} (kr)Pn(cos θ) in r, θ, ϕ spherical coordinates are used to develop a new approach to terrestrial radio wave field calculations. The spherical wave functions of integer order are used as building blocks for the solution of Maxwell's equations and boundary conditions. Instead of a direct summation of the zonal harmonics or n series, a double summation of a modified zonal harmonics series and the geometric series is used to obtain rapid convergence of the former. The most important disadvantage of an n-series summation is the construction of the solution at the ground interface, which generally causes the n series to converge slowly. This difficulty is overcome by the geometric series approach that permits removal of the ground wave from the n series. The ground wave is then calculated by using classical methods. The remaining ionospheric waves are then calculated with the integer-order spherical wave functions. The formulation presented is applicable to VLF, LF, and MF; anisotropic reflection coefficients based on electron and ion density profiles of the ionosphere are introduced into the analysis. Results obtained with a computer simulation of the formulas using integer-order spherical wave functions have been found to be in close agreement with another computer simulation of formulas using complex-order spherical wave functions at VLF. An interesting standing wave as a function of distance along the ionosphere and the ground has been found by using the integer-order techniques.