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where C
1
and C
2
are undetermined constants. Substituting Eq. (
6.36
)forE
x
into
boundary conditions (
6.35
) we come to an algebraic set of equations for constants
C
1
and C
2
. This set of equations has a nontrivial solution under the requirement that
the system determinant equals to zero, whence it follows
exp
2i!l
1
V
A
D
R
C
R
;
(6.37)
where
1
Ǜ
P
1
C
Ǜ
P
1
Ǜ
P
1
C
Ǜ
P
R
C
D
and R
D
(6.38)
denote the reflection coefficients for the northern and southern ionospheres, respec-
tively. These coefficients vary from 1 to
1 with changing Ǜ
P
from zero to infinity.
Decomposing the frequency in Eq. (
6.37
) into its real and imaginary parts, !
D
!
0
C
i!
00
, one finds the solution of Eq. (
6.37
)intheform
!
n
.x/
D
nV
A
.x/=l
1
;
(6.39)
when R
C
R
>0and
!
n
.x/
Df
V
A
.x/=l
1
g
.n
1=2/;
(6.40)
when the inverse inequality, R
C
R
<0,isvalid.Heren is integer, n
D
1;2;:::In
both of these cases the imaginary part of the frequency is given by
V
A
.x/
2l
1
!
00
.x/
D
ln
j
R
C
R
j
:
(6.41)
If k
y
¤
0, the general solution and eigenfunctions of the problem are found
in Appendix F. In this case the normal modes are coupled through the boundary
conditions at the conjugate ionospheres. The sole exception corresponds to two
opposite extreme cases of zeroth and infinite Pedersen conductivities when the
shear Alfvén and FMS modes become independent. In these extreme cases the wave
vector k
n
D
!
n
=V
A
in Eq. (
6.39
) coincides with that given by Eq. (
6.125
).
The general solution of the problem can be expanded in a series of the
orthonormal eigenfunctions, q
n
.
z
/, given by Eq. (
6.126
). Arbitrary perturbations of
E
x
and ıB
y
appear as a sum of modes, each of which changes harmonically in time.
Considering the amplitudes E
xn
and ıB
yn
of the normal oscillation with frequency
!
D
!
n
.x/ and rearranging Eq. (
6.126
) we get
sin k
n
z
k
n
C
i cos k
n
z
k
n
Ǜ
P
dq
n
d
z
;
E
xn
/
q
n
D
;ıB
yn
/
(6.42)
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