Wave-hop method of calculating propagation at LF and VLF, and stationary phase

Abstract

The wave-hop method of calculating propagation at LF and VLF does not use the stationary-phase relation to determine reflection height and is therefore liable to error, particularly at short distances. Stationary phase is introduced into the wave-hop calculations and examples are given of the results. The errors are frequently very small but this is not always the case. List of principal symbols= complex reflection coefficient = phase of ,,/?,, (with various suffices) = 2n/X, where X = radio wavelength = distance from transmitter to receiver = reflection height from the original program = triangulation heightTheoretical considerationThe wave-hop technique of calculating propagation at LF and VLF has been employed with success for many years following its introduction by Berry and others [1, 2] in 1964. A computer program by Berry and Herman [3] has been in widespread use for this purpose. This program uses a formula for reflection height which is correct for high angles of incidence, but at lower angles of incidence, corresponding to shorter transmission paths, it would appear to be better to obtain reflection height from the angle of incidence calculated by the method of stationary phase, as described by Bain and May [4]. It is the purpose of this paper to bring out the relation between the two approaches and to point out where the wave-hop technique can be relied on. The case of a plane wave, obliquely incident upon a plane horizontally-stratified ionosphere, and a plane earth will be treated here for simplicity. The assumption of a plane horizontally-stratified ionosphere was made both by Berry and Herman, and by Bain and May. The analysis below applies to the single-hop case; where multihops are involved similar arguments apply to the individual hops.Berry and Herman [3] state that their reflection height (termed h r here) is a quasistationary phase height. It is calculated as follows from the relation between the phases of ,,R,, at two different heights. If the phase at zero height is 9 0 and at height h r it is 9 r , then this relation is [5] 9 r = 9 0 + 2kh r cos (1) It is known that the phase of ,,/?,, is nearly constant and equal to n over a range of angles of incidence near grazing if h r is given an appropriate value. To find this value put 9 r = n in eqn. 1, and h r is then given by h r = (n -9 0 )/2k cos /(2)Paper 3665H (E8, Ell), first 138 for / near n/2. (This explanation of the derivation of eqn. 2 was given by Campbell [6].) This value for h r , derived for / near n/2, is used in the wave-hop program for all angles of incidence. Consider now the true stationary-phase height. As shown by Bain and May [4] the relative phase between the downcoming wave at the point R (see Fig. 1) and the hig. 1The transmission path (r)from T to R along a plane earth, and the sky-wave path (2 p) via a reflection from the triangulation height H upgoing wave at T is 0, where (j) = 6 0 -kr sin / (3)and r is the horizontal distance between T and R. 0 o is the phase difference between the downcoming and upgoing waves at R...