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Here
j
yω
B
0
,
iωξ
x
0
(
x, z
)
ξ
x
1
(
x, z
)
c
2
A
(
x
)
−
4
π
c
f
xω
=
−
−
(5.55)
j
xω
B
0
iωξ
y
0
(
x, z
)
ξ
y
1
(
x, z
)
c
2
A
(
x
)
−
4
π
c
f
yω
=
−
−
,
(5.56)
d
2
d
z
2
+
ω
2
c
2
A
(
x
)
.
L
ω
=
(5.57)
At the next step, let us proceed to presenting perturbations as superposi-
tions of oscillations of individual field-lines. For that expand
ξ
xω
,ξ
yω
and
b
ω
with respect to eigenfunctions
Q
n
(
z
) of Dungey's problem (5.9)-(5.10). We
have
ξ
xω
(
x, z
)=
∞
ξ
xωn
(
x
)
Q
n
(
z
)
n
=0
and analogous expansions for
ξ
yω
and
b
ω
.
Substitute these series into (5.55)-
(5.57) and multiply each of the equations by
Q
n
(
z
) . Integrate the expressions
obtained over change interval
z
and take into account the biorthogonality con-
ditions (5.11), and after performing all these operations, we obtain uncoupled
equations for normal oscillations. For
n
-th mode from (5.55)-(5.57) follows
b
ωn
B
0
L
ωn
ξ
yωn
=
ik
y
+
f
yωn
,
(5.58)
d
d
x
b
ωn
B
0
=
L
ωn
ξ
xωn
+
f
xωn
,
(5.59)
b
ωn
B
0
−
d
d
x
ξ
xωn
=
−
ik
y
ξ
yωn
,
(5.60)
ξ
xn
|
x
=0;
l
x
=0
,
(5.61)
where
l
z
l
z
f
xn
(
x
)=
f
x
(
x, z
)
Q
n
(
z
)d
z,
f
yn
(
x
)=
f
y
(
x, z
)
Q
n
(
z
)d
z.
0
0
Differential operator
L
ωn
reduces to a multiplication operator
k
n
+
ω
2
c
2
A
(
x
)
.
We find from (5.59) and (5.58) that
ξ
x
and
ξ
y
are
L
ωn
=
−
b
ωn
B
0
−
d
d
x
ξ
xωn
=
L
−
1
L
−
1
ωn
f
xωn
,
(5.62a)
ωn
b
ωn
B
0
ξ
yωn
=
ik
y
L
−
1
+
L
−
1
ωn
f
yωn
.
(5.62b)
ωn
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