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We have studied the transient behaviour of an electromagnetic
wave incident on a spatially uniform dielectric medium in
great detail. Let us now consider
a quite different, but equally important, problem. What is the time
asymptotic steadystate
behaviour of an electromagnetic wave propagating though a spatially
nonuniform dielectric medium?
As a specific example, let us consider the
propagation of radio waves through the Earth's ionosphere. The refractive
index of the ionosphere can be written [see Eq. (4.27)]

(796) 
where is a real positive constant which parameterizes the damping of electron
motion
(in fact, is the collision frequency of free electrons with other
particles in the ionosphere), and

(797) 
is the plasma frequency. In the above formula, is the density of
free electrons in the ionosphere and is the electron mass. We shall
assume that the ionosphere is horizontally stratified, so that ,
where the coordinate measures height above the Earth's surface (n.b.,
the curvature of the Earth is neglected in the following analysis). The ionosphere
actually consists of two main layers; the Elayer, and the Flayer. We shall
concentrate on the lower Elayer, which lies about 100 km above the surface
of the Earth, and is about 50 km thick. The typical daytime number density
of free electrons in the Elayer is
.
At nighttime, the density of free electrons falls to about half this number.
The typical daytime plasma frequency of the Elayer is, therefore, about 5 MHz.
The typical collision frequency of free electrons in the Elayer
is about 0.05 MHz.
According to simplistic theory, any radio wave whose frequency lies below the
daytime plasma frequency,
5 MHz, (i.e., any wave whose wavelength exceeds about 60 m) is reflected
by the ionosphere during the day. Let us investigate in more detail exactly how this
process takes place. Note, incidentally, that for megaHertz
frequency radio waves , so it follows from Eq. (4.162) that
is predominately real (i.e., under
most circumstances, the electron collisions can be neglected).
The problem of radio wave propagation through the ionosphere was
of great practical importance during the first half of the 20th Century, since
at that time longwave radio waves were the principle means of military communication.
Nowadays, the military have far more reliable ways of communicating. Nevertheless,
this subject area is still worth studying because the principle tool used
to deal with the problem of wave propagation through a nonuniform medium, the
socalled W.K.B. approximation, is of great theoretical importance. In particular,
the W.K.B. approximation is very widely used in quantum mechanics (in fact, there
is a great similarity between the problem of wave propagation through
a nonuniform medium and the problem of solving Schrödinger's equation
in the presence of a nonuniform potential).
Maxwell's equations for a wave propagating through a nonuniform,
unmagnetized, dielectric medium are:
where is the nonuniform refractive index of the medium. It is
assumed that all field quantities vary in time like
, where . Note that, in the following, is the
wavenumber in free space, rather than the wavenumber in the dielectric medium.
Next: The W.K.B. approximation
Up: Electromagnetic wave propagation in
Previous: Signal arrival
Richard Fitzpatrick
20020518