Geoscience Reference
In-Depth Information
As
A
extends from
e
to 0, the value of
y
0
varies from 0 to
π
. Correspondingly,
x
0
variesfrom1to
. The solution of (8.B.3) with an accuracy of up to
terms of first-order smallness with respect to
τ
K
, will be of the form:
−∞
Aξ
0
−
1
ξ
=
ξ
0
+
ξ
1
=
ξ
0
+
iτ
K
1)
.
(8.B.7)
ξ
(
ξ
0
−
The expression (8.B.7) has been obtained under not very high ground conduc-
tivities
k d
g
|
1 An analogous investigation of the behavior of roots in all the
regions of the four-sheeted Riemannian surface shows that only the root
k
(1)
S
is of interest, since the other roots either lie in the region of large Im
k
and
therefore correspond to waves damping, or are remote from the integration
contour and are not covered by it.
|
Appendix
Electric mode
Electric mode wavenumbers are found from (8.37). Rewrite (8.37) in the form
κ
g
k
0
ε
g
1
−
γ
1+
α
1
κ
a
h
,
tan(
κ
A
h
)=
−
iβ
(8.C.1)
γ
+
α
k
0
ε
a
κ
g
ε
g
κ
a
1
−
where
α
=
sin
2
I
X
Y
2
X∆
S
,
γ
=
,
=
k
0
hε
a
α.
Expressions for
∆
S
is given in (7.102). Note that
β
1. Let us solve (8.C.1)
by successive approximations with respect to parameter
β.
For the first order
in
β
, we get tan
κ
(0
a
h
≈
0, therefore
nπ
h
κ
(0)
a
(
n
)
≈
(
n
=0
,
±
1
,
±
2
,...
)
.
Substituting
κ
(0)
a
(
n
)
into (8.C.1), we find the first approximation
κ
(0)
g
(
n
)
k
0
ε
g
γ
(0)
(
n
)
1
−
1+
nπ
h
−
α
1
nπ
,
κ
a
(
n
)
≈
iβ
n
=0
,
(8.C.2)
γ
(0)
(
n
)
1
−
where
κ
(0)
g
(
n
)
and
γ
(0)
(
n
)
are obtained by substituting
k
(0)
a
(
n
)
=
inπ/h
into
κ
g
and
γ
.Theterm
α
k
0
ε
a
κ
g
ε
g
κ
a
1
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