Asymptotic Solutions for the Normal Modes in the Theory of Microwave Propagation

In this paper are presented several extensions of the ``W.K.B.'' method in the asymptotic solution of differential equations. The developments were made in the course of a systematic study of the application of asymptotic solutions to the normal‐mode theory of microwave propagation. In Section 1 a brief outline is given of the now standard normal‐mode theory of propagation of microwaves in an atmosphere with a horizontally stratified index of refraction. Particular emphasis is given in this study to the case of a ``surface duct.'' The case of ``leaky modes'' is studied in Section 2. Here the aim has been to obtain explicitly the terms after the leading one in the asymptotic expansion of the solution, in order to have an estimate of the order of magnitude of the error introduced by the use of the leading term only. This is achieved in Eq. (20), which is of general applicability to a wide class of physical problems. The first correction term to the phase‐integral solution for the characteristic values of th...