Geoscience Reference
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where the prime denotes a derivative with respect to x. The solution of the problem
must be limited as x
! ˙1
. Thus, we obtain
1
D
A
1
exp .kx/;
2
D
A
2
exp .
kx/:
(10.32)
The undetermined constants A
1
and A
2
can be found by applying the boundary
conditions at x
D
0 where the potential ' and the normal component of current
density should be continuous. Substituting the result of calculations into Eq. (
10.30
),
we get
Z
C
2
C
1
1
C
2
=
1
‰
2
.x;r/
D
p.0;k/J
0
.kr/ exp .
kx/kdk;
(10.33)
0
where
Z
P
0;r
0
J
0
kr
0
r
0
dr
0
:
p.0;k/
D
(10.34)
0
Permuting the indices 1 and 2 provides us with the solution for ‰
1
.x;r/. Assuming
for the moment the absolute convergence of integrals, we change the order of
integration in Eq. (
10.33
) with respect to variables k and r
0
:
Z
P
0;r
0
D
x;r;r
0
r
0
dr
0
:
C
2
C
1
1
C
2
=
1
‰
2
.x;r/
D
(10.35)
0
Here we have introduced the following function
Z
D
x;r;r
0
D
J
0
.kr/J
0
kr
0
exp .
kx/kdk:
(10.36)
0
The function D can be expressed through associated Legendre functions of the
second order. However, in order to derive an asymptotic form of this function at
great values of x, it will be enough to perform integration several times by parts. As
a result, Eq. (
10.35
) reduces to
Z
P
0;r
0
r
0
dr
0
C
2
P .x;r/:
C
2
C
1
.1
C
2
=
1
/x
2
dž
2
.x;r/
D
(10.37)
0
As is seen from Eq. (
10.37
) the electric potential may have a maximum in the
vicinity of the fault plane. It should be noted that the similar effect can take place
at the contact area of two conductors with different conductivities. Assuming for
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