Let us start off by examining a very simple case. Consider a plane
of frequency , propagating along the -axis in an unmagnetized plasma
whose refractive index, , is a function of . We assume that
the wave normal is initially aligned along the -axis, and, furthermore, that
the wave starts off polarized in the -direction. It is
easily demonstrated that the wave normal subsequently remains aligned along
the -axis, and also that the polarization
state of the wave does not change.
Thus, the wave is fully described by
The solution to Eq. (555) for the case of a homogeneous plasma, for which
is constant, is straightforward:
In general, if then the solution of Eq. (555) does not remotely resemble
the wave-like solution (557). However, in the limit in which is
a ``slowly varying'' function of (exactly how slowly varying is something which
will be established later on), we expect to recover wave-like solutions.
Let us suppose that is indeed a ``slowly varying'' function, and let us try
substituting the wave solution (557) into Eq. (555). We obtain
The second approximation to the solution is obtained by substituting Eq. (561) into
the right-hand side of Eq. (559):
Let us test to what extent the expression (565) is a good solution
of Eq. (555) by substituting this expression into the left-hand side
of the equation. The result is
Recall, that when a propagating wave is normally incident on an interface, where the refractive index suddenly changes (for instance, when a light wave propagating through air is normally incident on a glass slab), there is generally significant reflection of the wave. However, according to the WKB solutions, (569)-(570), when a propagating wave is normally incident on a medium in which the refractive index changes slowly along the direction of propagation of the wave then the wave is not reflected at all. This is true even if the refractive index varies very substantially along the path of propagation of the wave, as long as it varies slowly. The WKB solutions imply that as the wave propagates through the medium its wave-length gradually changes. In fact, the wave-length at position is approximately . Equations (569)-(570) also imply that the amplitude of the wave gradually changes as it propagates. In fact, the amplitude of the electric field component is inversely proportional to , whereas the amplitude of the magnetic field component is directly proportional to . Note, however, that the energy flux in the -direction, given by the the Poynting vector , remains constant (assuming that is predominately real).
Of course, the WKB solutions (569)-(570) are only approximations. In reality, a wave propagating into a medium in which the refractive index is a slowly varying function of position is subject to a small amount of reflection. However, it is easily demonstrated that the ratio of the reflected amplitude to the incident amplitude is of order . Thus, as long as the refractive index varies on a much longer length-scale than the wave-length of the radiation, the reflected wave is negligibly small. This conclusion remains valid as long as the inequality (568) is satisfied. This inequality obviously breaks down in the vicinity of a point where . We would, therefore, expect strong reflection of the incident wave from such a point. Furthermore, the WKB solutions also break down at a point where , since the amplitude of becomes infinite.