Geoscience Reference
In-Depth Information
Therefore, it is possible to write the equation for transverse electric compo-
nents:
∂E
Sx
∂x
+
∂E
Sy
∂y
=0
.
(7.63)
Henceforth, index
S
marks the FMS-wave components. Let the gauge poten-
tials be so that the scalar potential of the FMS-wave vanishes (
ϕ
S
= 0) [14].
Then, (7.55) gives
A
S
=0
and the
S
-transversal electric field is
E
S⊥
=
ik
0
A
S⊥
.
From (7.63) a condition for the potential can be written as
A
S⊥
∂A
Sx
∂x
+
∂A
Sy
∂y
=0
.
Thus, the transversal components of the vector-potential
A
S⊥
canbeex-
pressed by using a scalar potential
ψ
:
∂ψ
∂y
∂ψ
∂x
A
Sx
=
A
Sy
=
−
and all components in the FMS-wave can be expressed in terms of
ψ
(
x
,y
,z
)
as
∂
2
ψ
∂x
∂z
∂
2
ψ
∂y
∂z
∂
2
ψ
∂x
2
−
∂
2
ψ
∂y
2
,
b
Sx
=
,b
Sy
=
,
b
=
b
Sz
=
−
E
Sx
=
ik
0
∂ψ
∂y
ik
0
∂ψ
∂x
,E
Sy
=
−
,
E
Sz
=0
.
(7.64)
Substituting
b
Sx
and
b
Sy
from (7.64) into (7.5), we obtain
2
ψ
+
ω
2
c
2
A
∇
ψ
=0
.
(7.65)
R and T Matrices
Now we want to find the relation between the amplitude of the initial incident
wave of arbitrary polarization with reflected and transmitted waves. For this
purpose we represent Fourier harmonics exp
ik
1
x
1
+
ik
2
x
2
of the electric
and magnetic fields as the sum of four waves: two Alfven waves (incident,
denoted by index '
i
', and reflected '
r
') and two FMS-waves ('
i
' and '
r
'). Let
E
(
i,r
)
A⊥
,
b
(
i,r
)
A⊥
,
E
(
i,r
)
S⊥
,
b
(
i,r
)
S⊥
be vectors ('polarization vectors') directed along
corresponding wave electric and magnetic components. Polarization vectors
in the coordinate system
x
,y
,z
}
{
, associated with
B
0
, are found directly
from (7.58)-(7.65):
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