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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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