3. Propagation of a Pulse in a Gyrotropic Plasma
[12] Propagation of a pulse in a gyrotropic plasma is described
by a vector wave equation
 | (33) |
where E is the vector of the electric field strength, and
P is the
vector of polarization of a unit volume of the medium. In the
framework of the model of the medium with free charges
[see, e.g.,
Ginzburg, 1967;
Pamyatnykh and Turov, 2000], P is determined
by a combined action of electric and magnetic fields
 | (34) |
where
H0 is the strength of the homogeneous magnetic field in the
medium, and the remaining designations were introduced above.
The solution of the system of equations (33) and (34) is also found
in the approximation of a long pulse (equation (8)). The particular
form of the solution is determined by the magnitude of the angle
formed by the vectors of the propagation direction and magnetic
field direction. We introduce the coordinate system
(x,y,z) with
the unit vectors
i0; j0; k0 and assume that the pulse propagates
along the
z axis. The magnetic field is also along the
z axis, so that
H0= k0 H0.
The plane wave incident on the boundary of the half-space
z
0 is specified as
 | (35) |
A(0;t) is the vector envelope of the pulse at
z=0.
[13] By analogy with equation (4), the field E in the medium
will be sought in the form
 | (36) |
where
A(z;t)= i0Ax(z;t)+ j0Ay(z;t).
[14] In equations (33) and (34), variables are changed (equation (5)).
By analogy with equations (6) and (7), we obtain
 | (37) |
 | (38) |
Let us compare moduli of the first and third terms in the left-hand
side of equation (37). If inequalities (8) and (9) are fulfilled, we
have
 | (39) |
and the first term in the left-hand side of equation (37) can be
neglected.
[15] If the pulse propagates along the magnetic field,
P= i0Px+ j0Py,
and equation (38) is equivalent to a system of two
scalar equations of the form
 | (40) |
 | (41) |
For a cold gyrotropic plasma, at the moment of arrival of the pulse
at point
z', the conditions
 | (42) |
similar to conditions (11) for an isotropic plasma are fulfilled. Let
us substitute the solution of the system of equations (40) and (41)
obtained for the initial conditions given by equation (42) into
equation (37).
Taking into
account the estimate (40), we obtain the
system of equations for the components of the envelope
Ax(z';t') and
Ay(z';t') ( wH=|e| H0/(mc) is the gyrofrequency for
electrons)
 | (43) |
 | (44) |
The solution of the system of equations (43) and (44) is
found by the operational method. We omit rather cumbersome
intermediate calculations and give the final result
 | (45) |
 | (46) |
In the vector form, the obtained solution is presented as
 | (47) |
where
 | (48) |
and the expression for
Ae(z';t') is obtained from the expression
for
A0(z';t') if
+wH is replaced by
-wH in the right-hand side of
equation (48). The result (equation (47)) implies that the pulse that
propagates along the magnetic field is the sum of two circularly
and counter polarized pulses. According to the terminology used in
the theory of plane waves in plasma
[Ginzburg, 1967;
Pamyatnykh and Turov, 2000],
these pulses
can be defined as an ordinary and
extraordinary pulse. Their projections on the
x axis were denoted
above as
Ao(z';t') and
Ae(z';t'), respectively.
[16] Let us illustrate the obtained solution for distortions of a
bi-exponential pulse (equation (16)) as an example. By substituting
the envelope (16) into equation (46) and (47) and changing in the
obtained expressions the variable
m =q/t', we obtain
 | (49) |
 | (50) |
where the
a -dependent terms are given by
 | (51) |
 | (52) |
|
|
Figure 8
|
and the terms that depend on parameter
b are obtained from
equalities (51) and (52) by substituting
a by
b in them. The
process of decay of the initial bi-exponential pulse for the case when
the signal is received at the electric dipole oriented along the
x axis (see equation (49)) is shown in Figure 8. At first the ordinary
and extraordinary pulses mutually interfere, so that the observed
envelope (curve 2) can appreciably differ from the initial one
(curve 1). As the pulses propagate deeper into the medium, they
become fully separated; the extraordinary pulse follows the
ordinary one, the lag becoming larger and larger and absorption
becoming relatively stronger (curves 3). Manifestation of the
Faraday effect in the case when a linearly or circularly polarized
pulse is incident on a gyrotropic medium is a subject of separate
study because its action will be limited in time and space by a full
separation of the initial pulse into two pulses.
[17] Analytical expressions for the velocities of propagation of
the ordinary and extraordinary pulses can be obtained on the basis
of any terms entering the expressions for
Ax and
Ay (see
equations (49) and (50)). Let us use, for instance, equation (51) for
the term
Ax(z';t'; a).
We compute the integrals in equation (51) by
parts successively an infinite number of times using relation (26).
As a result, we get
 | (53) |
We again assume that inequality
2d tp 
1 is fulfilled (see
caption to Figure 8) and use the asymptotic representation (28).
After substitution of equation (28) into equation (53) and
summation of the series, the expression for
Ax(z';t'; a) acquires
the form
 | (54) |
It follows from equation (54) that on the
z' axis the pulse is
concentrated in the vicinity of two points to which the minima of
the moduli of the radicands in the denominators of the right-hand
side correspond. The coordinates of these points satisfy the
condition
 | (55) |
from which we infer that the ordinary pulse propagates with the
velocity
 | (56) |
and the extraordinary pulse propagates with the velocity
 | (57) |
[18] For the parameters of the medium and pulse given in the caption to
Figure 8, the inequality
describing an
ordinary situation for the ionosphere is also fulfilled
[see, e.g.,
Kirillov and Kopeikin, 2003].
In this case the expressions for the
propagation velocities are greatly simplified
 | (58) |
 | (59) |
At
wH=0, equations (58) and (59) transform into equation (32).
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