Geoscience Reference
In-Depth Information
The approximate analytical solution given by Eqs. (
7.43
)-(
7.45
) is more conve-
nient for analysis than the general solution in the form of Eqs. (
7.35
)-(
7.37
). Now
we use the short wave approximation for the diffusion zone .r < r
/, in which
the diffusion front r
d
D
2.
m
t/
1=2
propagates ahead of the seismic wave. The
observation point is assumed to be located within interval
C
l
t
r
r
d
:
(7.48)
This implies that the electromagnetic perturbations have already reached the
observation point whereas the seismic wave is still far from this point. This situation
occurs for the time interval
r
2
=.4
m
/
t
r=C
l
:
(7.49)
It follows from these inequalities that the arguments of the exponential and error
functions included in Eq. (
7.46
) for the function G
1
are small in comparison with
unity. Expanding these functions in terms of power series of the small parameters
r
l
=
m
, r
l
=r
d
, r=
m
and r=r
d
and taking into account that step a function in
Eq. (
7.46
) is equal to zero, we obtain
r
2
C
r
l
2
m
2r
l
m
C
8rr
l
1=2
m
r
d
C
:::
G
1
D
(7.50)
The first term in Eq. (
7.50
) is the largest one while the second and the third terms
are necessary for correct calculations of derivatives of the function G
1
. Substituting
Eq. (
7.50
)forG
1
into Eqs. (
7.43
)-(
7.45
) gives in the first approximation
u
0
SB
0
C
l
t cos
2
m
r
3
ıB
r
D
;
(7.51)
tan
2
ıB
D
B
r
;
(7.52)
u
0
SB
0
C
l
sin
4
m
r
2
E
0
D
:
(7.53)
These equations should be considered as an approximation which is valid only
within distance and time intervals (
7.48
) and (
7.49
).
A simple interpretation of the above obtained results can be achieved, as we
calculate the effective magnetic moment of the extrinsic currents caused by the
motion of conductive medium in the region covered by seismic wave. It follows
from Eq. (
7.7
) that the density,
j
e
, of the extrinsic currents is
j
e
D
.
V
B
0
/:
(7.54)
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