Geoscience Reference
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wave equations in the inhomogeneous plasma is required. Nevertheless, the
dispersion equations obtained for the homogeneous or the weakly inhomo-
geneous plasma allow us to estimate spatial and temporal scales of plasma
disturbances.
Let a plane wave be propagated in a homogeneous plasma. It is convenient,
as before, to choose the
z
-axis in the direction of the undisturbed magnetic
field
B
0
. Let all wave components be dependent on
ω
and
x
as
∝
−
iωt
+
exp(
ik
x
x
) and independent on
y
. Hence we may symbolically write
∂
∂t
=
∂
∂x
=
ik
x
,
∂
∂y
=0
.
−
iω,
(1.107)
Equation (1.107) shows that all field quantities contain the factor exp(
iωt
+
ik
x
x
)
.
This factor is assumed to be omitted in the same way as the time
factor exp(
−
iωt
) is omitted. Then the remaining terms are functions of
z
and
Maxwell's equations (1.48), (1.49) and (1.54)-(1.56) reduce to
−
d
E
y
d
z
−
=
ik
0
b
x
,
(1.108)
d
E
x
d
z
=
ik
0
b
y
+
ik
x
E
z
,
(1.109)
ik
x
E
y
=
ik
0
b
z
,
(1.110)
d
b
y
d
z
−
=
−
ik
0
(
ε
⊥
E
x
+
ε
E
y
)
,
(1.111)
d
b
x
d
z
=
−
ik
0
(
−
ε
E
x
+
ε
⊥
E
y
)+
ik
x
b
z
,
(1.112)
ik
x
b
y
=
−
ik
0
ε
E
z
,
(1.113)
where
ε
⊥
=
ε
xx
,
ε
=
ε
xy
. Elimination of
E
z
and
b
z
from (1.108)-(1.113)
gives
d
s
d
z
=
ik
0
Ts
,
(1.114)
where
⎛
⎞
k
2
⊥
k
0
ε
⎛
⎞
0
0
0
1
−
⎝
⎠
E
x
⎝
⎠
0
0
1
0
−
E
y
b
x
b
y
T
=
,
s
=
.
(1.115)
k
2
k
0
ε
ε
⊥
−
0
0
ε
⊥
−
ε
0
0
We put here
k
x
=
k
2
because of the axial symmetry with respect to the
⊥
magnetic field.
In a homogeneous medium the coecients of (1.115) are independent
of the coordinate
z
and (1.114) can be solved by finding four eigenvalues
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