7 - Slowly varying medium. The W.K.B. solutions

Summary

Introduction

This chapter continues the discussion of radio waves in a stratified ionosphere and uses a coordinate system x, y, z as defined in §6.1. Thus the electric permittivity ε(z) of the plasma is a function only of the height z. It is assumed that the incident wave below the ionosphere is a plane wave (6.1) with S₂ = 0, S₁ = S so that wave normals, at all heights z, are parallel to the plane y = constant, and for all field components (6.48) is satisfied. It was shown in § 6.10 that if the four components E_x, E_y, ℋ_x, ℋ_y of the total field are known at any height z, they can be expressed as the sum of the fields of the four characteristic waves, with factors f₁, f₂, f₃, f₄. In a homogeneous medium the four waves would be progressive waves, and these factors would be exp (− ikq_iz), i = 1, 2, 3, 4. We now enquire how they depend on z in a variable medium. This question is equivalent to asking whether there is, for a variable medium, any analogue of the progressive characteristic waves in a homogeneous medium. The answer is that there is no exact analogue. There are, however, approximate solutions, the W.K.B. solutions, which have many of the properties of progressive waves.

This problem has proved to be of the greatest importance in all those branches of physics concerned with wave propagation.