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imaginary part of !
n
defines the damping factors of the eigenmodes. As we shall
see, these eigenfrequencies typically lie in the range of f
D
7:5-50 Hz. In such
a case Eq. (
4.21
) contains the small parameter kd, where d
D
R
i
R
e
is the
altitude/thickness of the nonconducting atmospheric layer. Indeed, taking !
D
2f
D
50 Hz and d
D
40-90 km, one obtains kd
D
.0:67-1:5/
10
2
. Taking
into account that kd
1, we use the approximation
d
u
.1;2
n
.kR
e
/
dr
u
.
1;2
n
.kR
i
/
u
.
1;2
n
.kR
e
/
C
kd:
(4.22)
Substituting Eq. (
4.22
) into Eq. (
4.21
) yields
d
u
.2/
n
d
u
.1/
n
dr
D
0;
u
.1/
n
dr
u
.2/
(4.23)
n
where all the functions are taken at r
D
kR
e
. Substituting Eq. (
4.18
)for
u
.
1
n
and
u
.
2
n
into Eq. (
4.23
), we come to the equation, which contains both the functions
h
.
1
n
and h
.
2
n
and their derivatives. Eliminating from this equation the second order
derivatives with the help of Eq. (
4.53
) and rearranging, we obtain
h
.1/
n
!
ǚ
k
2
R
e
n.n
C
1/
dh
.2/
n
dh
.1/
n
dr
dr
h
.2/
D
0:
(4.24)
n
The factor in the round brackets is Wronskian of the functions h
.
1
/
n
and h
.
2
n
, which is
equal to
2i.kR
e
/
2
(Abramowitz and Stegun
1964
). Since this factor is nonzero,
the first factor in Eq. (
4.24
) must be equal to zero. Hence, we arrive at the following
result
c
2
n.n
C
1/
R
e
D
0;
!
2
(4.25)
whence it follows that (Schumann
1952a
,
b
,
1957
; Nickolaenko and Hayakawa
2002
)
!
n
2
D
c
2R
e
Œn.n
C
1/
1=2
;
f
n
D
(4.26)
where n
D
1;2;3::: enumerates the mode number. Substituting R
e
D
6;370 km
into Eq. (
4.26
) gives the following frequencies of the first Schumann's resonances
f
1
D
10:6, f
2
D
18:3, f
3
D
25:9, f
4
D
33:5, f
5
D
41:1;:::, f
n
D
f
1
Œn.n
C
1/=2
1=2
;:::(in Hz).
Schumann was the first who predicted the spectrum of the eigenfrequencies
defined by Eq. (
4.26
), and this resonance spectrum has been termed Schumann
spectrum. In the first approximation this spectrum is independent of the atmosphere
altitude d.
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