Spherical Wave Theory for MF, LF, and VLF Propagation

Abstract

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.

Correction [of numerical results in “ELF terrestrial radio wave field calculations with the zonal‐harmonics series”]

“…We have concluded that our suggested mechanisms [Lewis and Johler, 1972a] for alteration of the mode excitations because of ionospheric stratification are not significant under normal conditions. However, unusual conditions may be present at night with interlayer resonance effects occurring between the E and F regions [discussion, p. 3, King, 1973;Johler, 1970b]. Also, we have found no substantiation for our previous contention that certain theoretical results presented by Galejs [1972] are invalid for diffuse ionospheres [Lewis and Joblet, 1972b], and we hereby retract our criticisms.…”

The Impact of the Choice of Frequency and Modulation on Radio Navigation Systems

“…To illustrate this we show how this was accomplished in the LORAN-C and LORAN-D systems. It has been shown both theoretically (see for example, Johler, 1970) and experimentally that the propagation of waves between the ionosphere and the ground is in reality (and with mathematical rigor) made up of a geometric series of waves traveling in space (distance = d) and local time, t',…”

Reply [to “‘On the field representations in the ELF range’”]

“…Before we conclude, it would seem appropriate to comment on the economics of using the zonal harmonic series, as opposed to the mode formulation. For a rigorous modal formulation of a realistic ionosphere model the numerical solution of the mode equation can be quite time consuming [Johler, 1970]. The number of terms required for the summation of the zonal harmonic series is proportional to the frequency.…”