Radio transmission problems treated by phase integral methods

Abstract

§ 1. In a recent paper the writer has shown that there is a formal analogy between the analytical description of the transmission of electric waves through a medium of variable electronic density, and the Schroedinger wave theory of quantum dynamics; and that, as in quantum dynamics, each electrical problem is characterised by a set of proper values determined by the boundary con­ditions. These proper values which, in quantum dynamics, give the energy levels, in the wireless theory give a discrete set of direction cosines and attenua­tion factors of the various possible waves. A complete analysis of the wave equation is necessary in every case to give the accurate proper values; but in certain conditions, where the wave-length λ is so short that the phase velocity does not change appreciably (compared with c ) in a wave-length, and where the radius of curvature of the rays is small compared with λ, approximate methods may be used by which the proper values may be calculated. These methods correspond with the old Bohr-Sommerfeld phase integral method of calculating the energy levels. In particular, where transmission between two spherical shells is considered, as in the region between the earth and Heaviside layer, the proper values for the system are found to approximate closely to those found for the case where the layers are plane so long as the height of the layer is small compared with the radius of the earth. In the following illustration the full wave analysis is applied to the transmission between plane stratified layers, the existence of the proper values is demonstrated, and their relation with the approximately calculated proper values given. The results indicate clearly the nature of the transmission to be expected and the nature of the attenuation suffered by the waves for various wave-lengths and density distributions in the layer, and these results may be applied in the first approximation to the spherical case when the above limitation obtains.