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Fig. 5.3. The decrement γ 1 A and the Q-factor Q 1 = ω A / 2 γ 1 of the fundamental
harmonic depending on the normalized integral conductivity
5.5 FLR-Equations
Let us discuss now the propagation of a hydromagnetic wave excited by ex-
ternal sources. The displacement ξ and longitudinal magnetic field b are
completely determined by (5.7a), (5.7b), (5.7c) and boundary conditions (5.3),
(5.6). We assume that field sources in (5.7a)-(5.7c) are near the magnetopause,
while the inner regions of the magnetosphere are free of external currents.
The simplest way to deduce the solution is by expanding it into a series of the
eigenfunctions Q n ( x ). Substituting the series
ξ y =
m
x =
m
a m ( x ) Q m ( z ) ,
c m ( x ) Q m ( z ) ,
=
m
b B 0
b m ( x ) Q m ( z ) ,
into (5.7a)-(5.7c) and noting that
L A Q m ( z )= k A ( x )
k 2 m Q m ( z ) ,
we obtain
k 2 m a m ( x ) Q m ( z )= ik y
m
j ( d )
k A ( x )
4 π
c
x
B 0 ,
b m ( x ) Q m ( z )
m
b m ( x ) Q m ( z )=
m
j ( d )
k A ( x )
k 2 m c m ( x ) Q m ( z )
4 π
c
y
B 0 ,
m
m {
c m ( x ) Q m ( z )=
ik y a m ( x )+ b m ( x )
}
Q m ( z ) ,
m
where the prime denotes differentiation with respect to x .
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