Geoscience Reference
In-Depth Information
Turning back to the expressions for the Green's matrices
G
m,g
(8.45), we
obtain
G
m,g
=
i
exp
ik
0
√
ε
m,g
x
2
πx
⎧
⎨
∆
T
m,g
√
s
±
∞
√
s
exp (
C
−
1
√
s
m,g
k
m,g
(
s
)
×
k
(1)
S
−
s
)
ds
⎩
−
0
Q
,
√
s
m,g
C
−
1
1+
i
√
πs
m,g
w
(
√
s
m,g
)
±
i
√
π
x
±
(8.53)
where
Q
=
0
Im
√
s
m,g
>
0
,
at
s
m,g
) tIm
√
s
m,g
<
0
.
The last term in the braces is either zero when the singularity of
T
(
k
)isthe
right of the integration contour or equal to res
T
(
k
(1)
S
)
.
Substituting this expression into (8.43), we write the Green matrix
2
πx
C
−
1
exp (
−
G
(
g
)
=
n
=
m,g
G
n
+
∞
G
n
exp
ik
(
n
)
x
+
∞
G
nS
exp
ik
(
n
)
S
x
.
(8.54)
n
=0
n
=1
In the last sum, term
n
= 1 is off because it was already taken into account in
(8.53)
.
The integrals which define
G
m,g
in (8.54), now have no singularities.
The influence of the singularities is accounted by the Kramp function
w
(
V
m,g
).
The impact of waveguide modes with numbers
n
2 on field behavior is
insignificant because of the strong damping of these modes.
≥
Electric Mode
The contribution of electric modes into the ground magnetic field is given in
(8.54) by
∞
G
n
e
ik
(
n
)
x
,
n
=0
where only the term with
n
= 0 is significant. The higher modes, like in the
case of magnetic modes, damp rapidly because
nπ
h
Im
k
(
n
)
≈
.
Besides, the amplitude of the basic mode (determined by the residue) is much
larger than that of higher modes. Write
ζ
i
,κ
S
,∆
S
and
∆
A
near
k
=
k
(0)
=
e
iπ/
4
k
0
ε
a
h
1
/
2
α
,
1
−
γ
(0)
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