Geoscience Reference
In-Depth Information
The relevant singular solution of system (6.1)-(6.3) occurring as a result of
the summation of series (6.7) is
b
(
L
)
=
P
(1)
+
q
s
ln
x
P
(3)
Q
n
(
x, z
)
.
(
x, z
)
ns
(
x
)
P
(2)
ns
(
x
)
P
(4)
s
(6.20)
ξ
(
L
)
xs
(
x, z
)
ns
(
x
)
ns
(
x
)
The set of regular solutions corresponding to columns of the left-hand part
of matrix
U
(
x
) is (see (6.19))
⎛
⎞
q
m
b
(
L
m
(
x, z
)
ξ
(
L
)
=
P
(1)
q
s
P
(1)
nm
(
x
)
−
ns
(
x
)
⎝
⎠
Q
n
(
x, z
)
,
m
=
s.
(6.21)
q
m
P
(2)
q
s
P
(2)
nm
(
x
)
−
ns
(
x
)
xm
(
x, z
)
Regular solutions, obtained from columns of the right-hand part of the matrix
U
,are
b
(
R
m
(
x, z
)
ξ
(
R
)
=
P
(3)
Q
n
(
x, z
)
,
nm
(
x
)
P
(4)
m
=1
,
2
,....
(6.22)
xm
(
x, z
)
nm
(
x
)
From (6.7) and (6.11) we find the
y
-component of plasma displacement
ξ
y
(
x, z
)
in the form
ik
y
Q
n
b
Q
n
(
x, z
)
ξ
y
(
x, z
)=
−
,
ω
2
−
ω
n
(
x
)
where
b
=
b
(
L
)
m
or
b
=
b
(
R
m
.
The behavior of plasma disturbances near a resonance point is determined
by the singular solution (6.20). Using the expressions for Taylor series coef-
ficients of matrix functions
P
(
x
)and
P
(
x
)
T
0
and the decomposition
Q
n
(
x, z
)=
Q
n
(
x
0
,z
)+
Q
n
(
x
0
,z
)(
x
−
x
0
)+
···
,
with
u
=
x
−
x
0
,
we obtain
Q
s
+O
u
2
+
⎧
⎨
⎫
⎬
q
s
ω
2
c
a
(
x
0
)
2
c
a
(
x
0
)
b
(
L
)
s
Q
s
u
2
+O
u
3
(
x, z
)=
A
s
,
(6.23)
⎩
×
ln [
k
y
u
]
⎭
(
Q
s
+
R
(
z
))
u
+O
u
2
+
q
s
Q
s
⎧
⎨
⎫
⎬
1+
k
y
u
2
4
+O
u
3
ξ
(
L
)
xs
(
x, z
)=
A
s
,
(6.24)
⎩
×
ln [
k
y
u
]
⎭
⎧
⎨
⎫
⎬
q
s
Q
s
ik
y
u
−
R
(
z
)
ik
y
−
+O(
u
)+
1
2
ik
y
q
s
Q
s
u
+O
u
2
ξ
(
L
)
ys
(
x, z
)=
A
s
.
(6.25)
⎩
⎭
×
ln [
k
y
u
]
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