Geoscience Reference
In-Depth Information
In the non-dissipative systems, the wavenumbers
k
n
are real and fre-
quency
ω
independent. Then the boundary problem (5.72)-(5.73) is the
usual Sturm-Liouville problem. Its eigenvalues are real, its eigenfunctions
form an orthonormalized system, and FMS-plasma perturbations can be pre-
sented as a superposition of non-damping normal oscillations. With dissi-
pation taken into account,
k
n
depends on
ω
and Im
k
n
=0
.
The problem
(5.70)-(5.72) has non-zero solutions only at some discrete values of frequency
ω
nl
=
ω
nl
−
iγ
nl
,l
=1
,
2
,
3
,...
, with non-zero decrement
γ
nl
.
Let us denote by
V
nl
(
x
) the eigenfunctions corresponding to resonance
frequencies
ω
nl
of (5.71), (5.72) and introduce the designations
l
x
l
x
V
nl
(
x
)
c
2
A
(
x
)
N
(1)
nl
N
(2)
nl
V
nl
(
x
)d
x.
=
d
x,
=
0
0
The Green function
G
ωn
(
x
−
x
) of (5.70)-(5.71) is determined by
+
ω
2
k
n
(
ω
)
G
ωn
=
δ
(
x
d
G
ωn
d
x
2
c
2
A
(
x
)
−
−
x
)
,
G
ωn
|
x
=0;
l
x
=0
,
and an expression analogous to (5.66), where
J
−
1
ωn
is replaced by Wronskian
W
ωn
of
ϕ
(1)
ωn
(
x
),
ϕ
(2)
ωn
(
x
) which are the solutions of (5.72). Let these functions
satisfy the boundary conditions
x
=
l
x
ωn
(
x
)
x
=0
=0
,
d
ϕ
(1)
ωn
(
x
)
d
x
ϕ
(1)
=1
,
(5.74)
x
=
l
x
ωn
(
x
)
x
=
l
x
=0
,
d
ϕ
(2)
ωn
(
x
)
d
x
ϕ
(2)
=1
.
(5.75)
ϕ
(2)
ωn
(
x
) is found from the solution of the Cauchy problem (5.72), (5.75) and
can therefore be found at all
ω
. The problem (5.72), (5.74) is not a Cauchy
problem. Therefore it may have solutions only at some
ω
, more precisely
at such
ω
at which a non-trivial solution of (5.72) satisfies the boundary
conditions
x
=
l
x
ωn
(
x
)
x
=0
=0
,
d
ϕ
(1)
ωn
(
x
)
d
x
ϕ
(1)
=0
.
(5.76)
In a suciently close neighborhood of eigenfrequencies
ω
nl
,wehave
d
ϕ
(1)
= 0. Therefore, the conditions (5.71) at
ω
=
ω
nl
can be sat-
isfied by selecting norm
N
(1)
ωn
(
l
x
)
/
d
x
nl
of eigenfunctions
V
nl
(
x
) in such a way as to
satisfy the boundary condition (5.74) and by setting
ϕ
(1)
ωn
(
x
)=
V
nl
(
x
) t
ω
=
ω
nl
.
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