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outside this interval. Note that the function @
r
0
g
3
changes more slowly and thus can
be approximately considered as a constant within this interval. In this approximation
the function @
r
0
g
3
, taken at the point r
0
D
C
l
t
0
, can be moved through the integral
sign in Eq. (
7.110
). Integrating the remaining function @
r
0
f
t
1
with respect to r
0
,
we obtain
Z
t
@
r
0
g
3
r;C
l
t
0
;r
0
dt
0
u
0
SC
l
2
3=2
r
g
1
.r;t/
r
0
;
(7.111)
0
where the function @
r
0
g
3
.r;r
0
;r
0
/ is taken at r
0
D
C
l
t
0
.HereS
D
4R
0
stands for
the area of source surface. Since @
r
0
g
3
D
@
r
g
3
,asitfollowsfromEq.(
7.93
), the
expression (
7.111
) can be transformed to the form
S
u
0
4r
@
r
G
1
.r;t/;
g
1
.r;t/
D
(7.112)
where
Z
t
g
3
r;C
l
t
0
;r
0
dt
0
2C
l
1=2
G
1
.r;t/
D
r
0
:
(7.113)
0
The integrals in Eqs. (
7.35
) and (
7.36
) for the GMP can be expressed via the
function G
1
as follows:
4r
@
r
G
1
:
Z
r
r
02
g
1
r
0
;t
dr
0
D
1
r
3
S
u
0
4r
3
.r@
r
G
1
G
1
/
D
S
u
0
(7.114)
r
0
Substituting
Eqs. (
7.112
)
and
(
7.114
)
into
Eqs. (
7.35
)-(
7.37
)
we
arrive
at
Eqs. (
7.43
)—(
7.45
).
Substituting Eq. (
7.93
)forg
3
and Eq. (
7.94
)forr
0
into Eq. (
7.113
), and perform-
ing integration, one can reduce the function G
1
to the form given by Eq. (
7.46
).
Rayleigh Surface Wave in a Conductive Half-Space
The perturbations of the Earth's magnetic field resulted from Rayleigh surface wave
propagation in a conducting ground .
z
<0/ are described by the set of Eqs. (
7.63
)-
(
7.65
) for the amplitudes b
x
, b
y
, and b
z
of magnetic perturbations. These amplitudes
as functions of vertical coordinates
z
have to tend to zero when
z
! 1
.The
solution of the problem can be written as
k
R
V
0
b
x
D
m
f
a
1
exp .p
z
/
C
1
exp .qk
R
z
/
C
2
exp .sk
R
z
/
g
;
(7.115)
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