On the connection between the ray theory of electric waves and dynamics

Abstract 1. In the study of the transmission of electric waves round the earth (especially in the case of what are now known as short waves of frequencies between 3·3 X 107 and 3·3 X 106, 10, to 100 M in wave-length) we have to consider the behaviour of such waves in the ionised region of the upper atmo­sphere. For the purposes of the analysis of the wave motions, this region may be considered as one in which there is a variable distribution of electronic density represented by Nϵ, say, which is taken as a function of the co-ordinates x, y, z. The electronic density is of major importance, the ions, in general, being so heavy that their reaction on the waves is small compared with that of the electrons. The phase velocity V in the medium is then, as is well known, c/√1 — v02/v2 where v0 is the critical frequency of the medium at any point x, y, z given by v02 = Ne2c2/πm, and in respect of this (the quantity N) is a function of the co-ordinates x, y, z. The group velocity U is c√1 — v02/v2, so that UV = c2.