Geoscience Reference
In-Depth Information
Roots of a Dispersion Equation
Magnetic Mode
Let us study (8.35) far from
Γ
m
and
Γ
g
which is reduced to (8.36). The
wavenumbers denoted as
k
(
n
)
S
are roots of (8.36) and determine normal
magnetic-mode waves. As shown below, waveguide modes of this type do not
always exist. In the general case, for arbitrary conductivities
σ
g
,
the analysis
of (8.36) is rather complicated. Therefore, consider first the case of the high
conductive ground.
Ground Conductivity σ
g
→∞
Y
(
m
)
In this case, the surface admittance
|
|→∞
and (8.36) reduce to
g
ik
coth
k
+
k
A
−
k
2
+
τ
K
=0
.
(8.38)
It is proved in Appendix 8.A that (8.38) has no roots on the physical
sheet. A root can be found just in the 2-nd and 3-rd quadrants of the non-
physical sheet. The estimate for the damping of these virtual modes shows
that the damping is very strong with Im
k>
3
π/
4
h
, where
h
is an atmospheric
thickness.
Finite Conductive Ground
The last statement is valid for a perfect conductive ground. The following
question is of interest: are there roots of (8.36) in the band 0
<
Im
k<
3
π/
4if
σ
g
? If such roots exist then it can strongly affect the results of numerical
integration. If it appears on a non-physical sheet close to the integration path,
then the root becomes apparent as a poor integrable singularity. If the root
is found on the physical sheet, it manifests itself as a pole corresponding to
the horizontally propagating mode. Analysis of (8.36) for a finite conductive
ground presented in Appendix 8.B proves that the root in fact exists for
moderate conductivities.
Figure 8.6 charts the dependence of the root
k
(1)
S
on the ground conduc-
tivity obtained by a numerical solution of (8.36) for the period
T
= 100 s,
Σ
P
=0
.
116
=
∞
10
9
km/s. The ground was replaced in these computations
by a homogeneous half-space. Comparison of roots found by (8.B.7) and by
the numerical solution (the curve in Fig. 8.6) shows that (8.B.7) is valid at
σ
g
≤
×
10
6
s
−
1
.
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