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m.!/ı./
2i"
0
!R
e
sin
:
@
r
.rU/
D
(4.12)
Similarly, if the ionosphere is supposed to be a perfect conductor, the tangential
component of the electric field E
becomes zero at the boundary with the iono-
sphere. Whence it follows from Eq. (
4.7
) the boundary condition at r
D
R
i
:
@
r
.rU/
D
0:
(4.13)
4.1.3
Solution of the Problem
Finally, we have arrived at the second order differential Eq. (
4.8
), where the right-
hand side is equal to zero. This equation for the unknown potential function U
should be supplemented by two boundary conditions of Eqs. (
4.12
) and (
4.13
).
In standard mathematical technique, it is recommended to seek for the solution of
Eq. (
4.8
)intheform
n
A
n
h
.1
n
.kr/
C
B
n
h
.2
n
.kr/
o
P
n
.cos /;
X
U
D
(4.14)
n
D
0
where A
n
and B
n
are the undetermined coefficients, h
.1
n
.x/ and h
.2
n
.x/ are the
spherical Bessel functions of the third kind, and P
n
.x/ stands for Legendre poly-
nomials. The definitions of these functions are found in Appendix A. To construct
the solution of the problem, the boundary condition (
4.12
) can be expanded in a
series of the Legendre polynomials. The representation of the delta-function ı./
through Legendre polynomials is given by Eqs. (
4.59
) and (
4.60
). Combining these
equations with Eq. (
4.12
), we come to the following boundary condition at r
D
R
e
n
C
P
n
.cos /:
X
m.!/
2i"
0
!R
e
1
2
@
r
.rU/
D
(4.15)
nD0
Substituting Eq. (
4.14
)forU into Eqs. (
4.13
) and (
4.15
) we get the boundary
condition expanded in a series of the Legendre polynomials. This representation
of the boundary conditions gives a set of equations for undetermined coefficients
A
n
and B
n
A
n
u
.
1
n
.kR
i
/
C
B
n
u
.
2
n
.kR
i
/
D
0;
(4.16)
n
C
:
m.!/
2i"
0
!R
e
1
2
A
n
u
.1
n
.kR
e
/
C
B
n
u
.2
n
.kR
e
/
D
(4.17)
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