8 - The Airy integral function and the Stokes phenomenon

Summary

Introduction

In ch. 7 it was shown that at most levels in a slowly varying stratified ionosphere, and for radio waves of frequency greater than about 100 kHz, the propagation can be described by approximate solutions of the differential equations, known as W.K.B. solutions. The approximations fail, however, near levels where two roots q of the Booker quartic equation approach equality. In this chapter we begin the study of how to solve the differential equations when this failure occurs. For an isotropic ionosphere there are only two values of q and they are equal and opposite, and given by (7.1), (7.2). The present chapter examines this case. It leads on to a detailed study of the Airy integral function, which is needed also for the solution of other problems. Its use for studying propagation in an anisotropic ionosphere where two qs approach equality is described in § 16.3.

For an isotropic ionosphere, a level where q = 0 is a level of reflection. In § 7.19 it was implied that the W.K.B. solution for an upgoing wave is somehow converted, at the reflection level, into the W.K.B. solution for a downgoing wave with the same amplitude factor, and this led to the expression (7.151) for the reflection coefficient R. The justification for this assertion is examined in this chapter and it is shown in § 8.20 to require only a small modification, as in (7.152).