Geoscience Reference
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where m.!/ stands for Fourier transform of the lightning current moment and
d<
z
<0.
We shall seek for the solution of Eqs. (
5.132
) and (
5.133
) in the form of Bessel
transform, for example,
Z
A.k;
z
;!/
D
A.r;
z
;!/J
0
.kr/rdr;
(5.134)
0
where J
0
.kr/ is the Bessel function of the first kind of zero order and the parameter
k of the Bessel transform plays a role of the perpendicular wave number. For brevity,
here we made use of the same designation for the potential A.r;
z
;!/ and its Bessel
transform A.k;
z
;!/. The same representation is true for the potentials dž and ‰.
Applying the Bessel transform to Eq. (
5.133
) we obtain
0
m.!/
2
@
z
A
k
a
A
D
ı.
z
C
d
h/;
(5.135)
where k
a
D
k
2
!
2
=c
2
, and A
D
A.k;
z
;!/. We shall restrict our study to the
low-frequency limit when k
a
k. Integrating Eq. (
5.135
)from
z
D
h
d
" to
z
D
h
d
C
" and then formally taking "
!
0, we come to the following condition
at
z
D
h
d
0
m.!/
2
Œ@
z
A
D
;
(5.136)
where the square brackets denote the jump of
z
-derivative of A at
z
D
h
d.The
solution of Eq. (
5.135
) in both regions,
z
<h
d and
z
>h
d, should be matched
via the condition (
5.136
). The solution of Eq. (
5.135
) under the requirement that A
is continuous at that boundary has the form
0
m.!/.
z
0
/
2k
sinh
k
z
0
A
D
C
1
exp .k
z
/
C
C
2
exp .
k
z
/
(5.137)
where C
1
and C
2
are arbitrary constants,
z
0
D
z
C
d
h, and .
z
0
/ is the step-
function; i.e.,
D
1 if
z
0
>0and
D
0 if
z
0
<0.
Now we shall treat electromagnetic fields in the ground, which is considered
as a uniform conducting half-space .
z
<
d/ with constant conductivity
g
.For
the axially symmetrical problem Maxwell's equations for the ground are given by
Eqs. (
5.51
) and (
5.52
). Substituting Eq. (
5.130
) and for the TM mode into those
equations, yields
@
z
A
D
0
g
dž;
(5.138)
1
r
@
r
.r@
r
A/
C
i!
0
g
A
D
0:
@
zz
A
C
(5.139)
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