Geoscience Reference
In-Depth Information
It now remains to express
k
A
3
and
k
S
3
in terms of
k
1
and
k
2
from the
dispersion equation. It is possible to substitute fields
exp
ik
3
x
3
into the set
(7.7)-(7.10). It is, however, more easy to substitute (7.69) into the dispersion
equations
∝
k
Az
=
ω
2
k
Sx
+
k
Sy
+
k
Sz
=
ω
2
.
and
c
2
A
c
2
A
Then
ω
2
c
2
A
−
ω
c
A
sin
I
,
k
(
i,r
)
A
3
k
(
i,r
)
S
3
k
1
−
k
2
,
=
∓
=
k
1
cot
I
∓
(7.73)
where the upper sign refers to the incident waves and the lower to the reflected
waves. Thus, the electric and magnetic components of the Alfven and FMS-
waves above the ionosphere can be expressed as
E
τ
b
τ
=
b
(
i
A
E
(
i
Aτ
b
(
i
)
Aτ
exp
ik
(
i
A
x
3
+
b
(
i
)
E
(
i
)
Sτ
b
(
i
)
Sτ
exp
ik
(
i
S
x
3
S
E
(
r
)
Aτ
b
(
r
)
Aτ
exp
ik
(
r
)
A
E
(
r
)
Sτ
b
(
r
)
Sτ
exp
ik
(
r
)
S
x
3
+
b
(
r
)
S
x
3
,
(7.74)
+
b
(
r
)
A
where
E
τ
=
E
1
E
2
=
E
x
E
y
,
b
τ
=
b
1
b
2
=
b
x
b
y
.
By virtue of normalization (7.68), the coecients
b
(
i
)
A
,b
(
r
)
,b
(
i
)
S
and
b
(
r
)
S
are
the amplitude of the corresponding wave magnetic components transverse to
B
0
.
A
RMatrix
Introduce the (2
×
2) reflection coecients matrix
R
(
k
)as
b
(
r
)
A
b
(
r
)
S
=
R
(
k
)
b
(
i
A
b
(
i
)
S
,
R
=
R
AA
.
R
AS
(7.75)
R
SA
R
SS
The sense of elements
R
ik
in matrix
R
(
k
) is clear from (7.75). If, for instance,
an Alfven wave with a unit magnetic amplitude is incident on the ionosphere,
then the reflected Alfven wave has the amplitude
R
AA
and the reflected FMS-
wave amplitude is
R
SA
. Rewrite (7.74) in matrix form:
E
τ
b
τ
=
U
(
i
)
τ
exp
i
K
(
i
)
x
3
b
(
i
)
+
U
(
r
)
τ
exp
i
K
(
r
)
x
3
b
(
r
)
,
Search WWH ::
Custom Search