Geoscience Reference
In-Depth Information
Here we made use of the following abbreviations:
n
rh
.1;2
n
.kr/
o
:
d
dr
u
.1;2/
n
D
(4.18)
The set of Eqs. (
4.16
) and (
4.17
) can be solved for A
n
and B
n
to yield the potential U
C
n
n
C
P
n
.cos /;
X
m.!/
2i"
0
!R
e
1
2
U
D
(4.19)
nD0
where
u
.2
n
.kR
i
/h
.1
n
.kr/
u
.1
n
.kR
i
/h
.2
n
.kr/
u
.1
n
.kR
e
/
u
.2
n
.kR
i
/
u
.1
n
.kR
i
/
u
.2
n
.kR
e
/
C
n
D
:
(4.20)
Substituting Eq. (
4.19
)forU into Eqs. (
4.6
) and (
4.7
) gives the solution of problem.
The structure of the solution given by these equations is very complicated despite
the fact that the simplest model of Earth-Ionosphere resonance cavity has been
used. In what follows we simplify this solution to extract the eigenfrequencies of
the resonator.
4.2
Schumann Resonances
4.2.1
Eigenfrequencies of the Schumann Resonances
It should be emphasized that the set of eigenfrequencies of the Earth-Ionosphere
cavity is its inner property, which depends on the sizes and inner structure of
the resonant cavity. The eigenfrequencies are affected by neither a type of source
nor the way of the cavity excitation. Eigenvalues of the frequency are determined
by singular points of the solution given by Eqs. (
4.19
) and (
4.20
). To find the
eigenfrequencies, it is necessary at this point to examine the poles of the functions
C
n
, that is, the points in complex !-plane where the denominator of Eq. (
4.20
)
vanishes. First, we note that the set of corresponding wave numbers k
D
!=c is
derivable from the following equation set
u
.1
n
.
kR
e
/
u
.2
n
.
kR
i
/
u
.1
n
.
kR
i
/
u
.2
n
.
kR
e
/
D
0;
(4.21)
where n
D
1;2;3:::
When performing the integration of U in complex !-plane, the area surrounding
the poles !
D
!
n
, which are commonly complex, makes a contribution to the
integral via residues of the functions C
n
in the poles. From physical viewpoint,
the real part of the poles !
n
defines eigenfrequencies of the resonator while the
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