Geoscience Reference
In-Depth Information
where V
0
stands for the amplitude of the component V
z
, i is imaginary unity, k
R
D
!=C
R
is acoustic wave number, ! is the wave frequency and
D
k
R
x
!t.Here
we made use of the following abbreviations:
1
1=2
1
1=2
C
R
C
l
C
R
C
t
q
D
;s
D
:
(7.61)
As is seen from Eqs. (
7.59
) and (
7.60
) the amplitude of the mass velocity
exponentially decreases with depth. Thus, the elastic energy of Rayleigh wave is
mainly concentrated in the near surface layer with thickness of the order of .qk
R
/
1
or .sk
R
/
1
.
The half-space .
z
<0/ is assumed to be a conductive medium with constant con-
ductivity . In such a case the electromagnetic perturbations of the external/Earth's
magnetic field
B
0
are described by Eqs. (
7.11
) and (
7.12
). In the atmosphere .
z
>0/
which is considered as an insulator, Eq. (
7.12
) reduces to the form
r
2
ı
B
D
0, while
the electric field
E
can be found from Eq. (
7.8
).
We seek for the solution of these equations in the form
ı
B
D
b
.
z
/ exp .i /;
E
D
e
.
z
/ exp .i /:
(7.62)
Taking into account that all the perturbations are constant as a function of y,
substituting Eqs. (
7.59
)-(
7.62
)for
V
, ı
B
, and
E
into Eq. (
7.12
), and rearranging, we
come to the set of equations
B
0
z
iqB
0x
exp .qk
R
z
/
k
R
V
0
m
b
0
x
p
2
b
x
D
1
C
s
2
exp .sk
R
z
/
;
2s
C
.iB
0x
sB
0
z
/
(7.63)
qB
0x
C
iB
0
z
exp .qk
R
z
/
k
R
V
0
q
m
b
0
z
p
2
b
z
D
1
C
s
2
exp .sk
R
z
/
;
2q
.is
m
B
0
z
C
B
0x
/
(7.64)
ik
R
V
0
1
q
2
q
m
b
0
y
p
2
b
y
D
B
0y
exp .qk
R
z
/;
(7.65)
where B
0x
, B
0y
, and B
0
z
are projections of
B
0
on the coordinate axes, and p
2
D
k
R
i!=
m
. Here the primes denote derivatives with respect to
z
.
The similar equations can be derived for the atmosphere .
z
>0/
b
0
j
k
R
b
j
D
0; j
D
x;y;
z
:
(7.66)
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