Full wave solution for whistler wave propagation through the ionosphere in the case of small angles of incidence

Among many problems in whistler study, wave propagation through the ionosphere is one of the most ar-
duous, and the most important at the same time. Both satellite and ground-based investigations of VLF waves
include considerations of this problem, and it has been in the focus of research since the beginning of whistler
study. The main difficulty in description of the problem arises from the fast variation of the lower ionosphere
parameters as compared to typical VLF wave number. This makes irrelevant the consideration in the frame-
work of geometrical optics, which, along with a smooth variation of parameters, is always based on a particular
dispersion relation. Although the full wave analysis in the framework of cold plasma approximation does not
require slow variations of plasma parameters, and does not assume any particular wave mode, the fact that the
wave of a given frequency belongs to different modes in various regions makes numerical solution of the field
equations not simple. More specifically, as is well known, in a cold magnetized plasma, there are, in general, two
wave modes related to a given frequency. Both modes, however, do not necessarily correspond to propagating
waves. In particular, in the frequency range related to whistler waves, the other mode is evanescent, i.e. it
has a negative value of the refractive index squared. It means that one of solutions of the relevant differential
equations is exponentially growing, which makes a straightforward numerical approach to these equations de-
spairing. This well known difficulty in the problem under discussion is usually identified as numerical swamping.
Resolving the problem of numerical swamping becomes, in fact, a key point in numerical study of wave passage
through the ionosphere. As it is typical for work based on numerical simulations, its essential part remains
virtually hidden. Then, every researcher, in order to get quantitative characteristics of the process, such as
transmission and reflection coefficients, needs to go through the whole problem. That is why the number of
publications dealing with VLF wave transmission through the ionosphere does not run short. The purpose of
this work is to give a new approach to the problem, such that its basic equations are regularized analytically
before numerical calculations, while the latter become accomplishable with the help of a routine program. Such
formulation of the task allows presenting all equations and related formulae in an undisguised form, so that
the problem may be solved in a straightforward way, once the ionospheric plasma parameters are given. In
the framework of the developed method, full wave solution for the electromagnetic field was obtained, and the
reflection coefficient was calculated as a function of frequency for various angles of incidence.