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Matrix Presentation of the Box Equations
First, we shall in this section, present the 2D boundary problem (6.1)-(6.3)
and (6.5) in a matrix form. Then, we obtain an infinite system of matrix
equations which we shall reduce to a finite equation system. Dungey's bound-
ary problem, i.e. the problem of Alfven eigenoscillations between two highly
conductive ionospheres, can be written
d
2
d
z
2
+
Q
n
(
x, z
)=0
,
ω
n
c
2
A
(
x, z
)
Q
n
(
x,
0) =
Q
n
(
x, l
z
)=0
.
Since the eigenfunctions of this problem form an orthonormal basis, i.e.
Q
n
(
x, z
)
Q
m
(
x, z
)
c
2
A
=
l
z
Q
n
(
x, z
)
Q
m
(
x, z
)
c
2
A
d
z
=
δ
nm
,
(6.6)
0
the solutions of (6.1)-(6.3) and (6.5) may be sought in the form of a decom-
position over this basis
ξ
y
=
a
m
(
x
)
Q
m
(
x, z
)
,
x
=
c
m
(
x
)
Q
m
(
x, z
)
,
b
=
b
m
(
x
)
Q
m
(
x, z
)
.
(6.7)
The repeated index convention for summation is systematically used here and
below in this chapter. By substituting series (6.7) into (6.1)-(6.3) with the
allowance for
L
Q
m
=
ω
2
ω
2
m
−
Q
m
,
c
2
A
where the operator
L
is defined in (6.4), we have
ω
2
ω
2
m
−
a
m
Q
m
=
−
ik
y
b
m
Q
m
,
(6.8)
c
2
A
ω
2
c
m
Q
m
+
b
m
Q
m
,
ω
2
m
−
b
m
Q
m
=
−
(6.9)
c
2
A
c
m
Q
m
=
c
m
Q
m
,
−
ik
y
a
m
Q
m
+
b
m
Q
m
−
(6.10)
where the prime indicates differentiation with respect to
x.
In order to derive expressions
a
n
(
x
) in terms of
b
n
(
x
) and to resolve the
system of equations with respect to
b
n
(
x
)and
c
n
(
x
), let us multiply (6.8)
by
Q
n
(
x, z
), while (6.9) and (6.10) multiply by
Q
n
(
x, z
)
/c
a
(
x, z
). Then by
integrating the obtained expressions from 0 to
l
z
with conditions (6.6) we find
the relation of
a
n
and
b
m
ik
y
a
n
=
−
ω
n
(
x
)
Q
n
Q
m
b
m
,
(6.11)
ω
2
−
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