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Coupled Waves Phase Mixing
At
k
y
= 0 interaction takes place between Alfven and FMS-waves. From (4.43)
and (4.45) we obtain
b
(
x
)
B
0
1
k
A
(
x
)
∂
∂x
ξ
x
(
x
)=
,
k
2
−
b
(
x
)
B
0
ik
y
k
A
(
x
)
ξ
y
(
x
)=
.
k
2
−
Then (4.44) becomes
k
A
(
x
)
−
k
y
−
k
2
||
d
dx
db
(
x
)
dx
1
k
A
(
x
)
+
b
(
x
) = 0
(4.52)
k
2
||
k
A
(
x
)
k
2
−
−
or
+
k
A
(
x
)
b
(
x
)=0
.
d
2
b
(
x
)
dx
2
dk
A
/dx
k
A
(
x
)
db
(
x
)
dx
k
y
−
k
2
−
−
(4.53)
−
k
2
In order to study the coupling between these two modes, let us consider
first the problem of wave incidence on a stratified plasma medium. Suppose
Alfven velocity
c
A
(
x
) increases monotonically from
c
1
at
x
→−∞
to
c
2
at
x
, with Alfven wave number
k
A
(
x
) decreasing from
k
1
to
k
2
(Fig. 4.3,
top left panel). Let the incident wave amplitude be
b
0
. Introduce reflection
and transmission coecients
R
and
T
. Then at
x
→
+
∞
, the wave magnetic
field
b
is the sum of the incident (
i
) and reflected (
r
) waves
→−∞
b
=
b
(
i
)
+
b
(
r
)
,
where
=exp
x
+
ik
y
y
+
ik
z
b
0
,
b
(
i
)
iωt
+
ik
(1)
x
−
=
R
exp
ik
(1
x
x
+
ik
y
y
+
ik
z
b
0
b
(
r
)
−
iωt
−
and at
x
→
+
∞
=
T
exp
x
+
ik
y
y
+
ik
z
b
0
b
(
t
)
iωt
+
ik
(2)
x
−
=
k
1
,
2
−
with
k
(1
,
2)
k
2
.
The problems of oblique incidence of FMS-waves on a layer and oblique
incidence of an electromagnetic wave of
H
-polarization on a plane-layered
medium are similar ([11], [12]). The electric field
E
in the
H
-polarized wave
is within the plane of incidence, while the magnetic field is perpendicular to
k
y
−
x
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