Geoscience Reference
In-Depth Information
Electromagnetic Perturbations at the Ground Surface
We start with calculation of the jump of the potential A across the E layer. Taking
into account that the value A.0
C
/ can be found from Eq. (
5.127
) and combining
this equation with Eq. (
5.53
) we obtain that
LJ
1
V
AI
0
m.!/
2k
ŒA.0/
D
dž.0/
A.
d/ cosh .kd/
C
sinh
f
k.d
h/
g
: (5.172)
Substituting Eq. (
5.172
) into Eq. (
5.167
) and rearranging leads to
ix
0
Ǜ
H
L
.LJ
1
C
Ǜ
P
/
V
AI
‰.0/
dž.0/
D
f;
(5.173)
where
kV
AI
Ǜ
H
v
r
C
Ǜ
P
v
'
A.
d/ cosh .kd/
C
0
m.!/
2k
sinh
f
k.d
h/
g
:
(5.174)
Here one can see an analogy between Eqs. (
5.173
) and (
5.128
), which was derived
for the plane problem. These two equations differ only in the source functions which
stay on the right-hand sides of these equations.
As has already been intimated, considering an analogy between the plane and
cylindrical problems, Eq. (
5.168
) can be reduced to the equation analogous to
Eq. (
5.129
), i.e.
B
0
f
D
kV
AI
Ǜ
P
v
r
Ǜ
H
v
'
;
Ǜ
H
V
AI
dž.0/
C
.ix
0
Ǜ
P
s/
L
B
0
‰.0/
D
(5.175)
Finally, one should take into account the continuity of the potential dž at
z
D
0.
As it follows from Eq. (
5.132
)
A.
d/ sinh .kd/
cosh Œk.d
h/
:
kc
2
i!
0
m.!/
2k
dž.0/
D
(5.176)
The set of Eqs. (
5.173
), (
5.175
), and (
5.176
) can be solved for A.
d/, dž.0/ and
‰.0/ to yield
0
m.!/ cosh Œk.d
h/
2k sinh .kd/
A.
d/
;
(5.177)
0
Ǜ
H
m.!/ cosh .kh/
2 sinh .kd/
iL
kq
‰.0/
V
AI
Ǜ
2
H
C
Ǜ
P
C
LJ
1
Ǜ
P
v
r
LJ
1
Ǜ
H
v
'
;
B
0
C
(5.178)
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