Geoscience Reference
In-Depth Information
r
∂
+
2
F
∂ϕ
2
F
1
r
∂
∂
F
r
2
∂
1
2
+
∂
= 0
,
(2.43)
∂
∂
z
2
r
r
while the boundary conditions (2.30) and (2.31) remain intact.
For resolving equation (2.43) we apply the traditional method of variable sepa-
ration, i.e. we shall assume that
F
(
r
,
ϕ
,
z
)=
R
(
r
)
Φ
(
ϕ
)
Z
(
z
)
.
(2.44)
Substitution of expression (2.44) into equation (2.43) results in the following set of
ordinary differential equations:
2
R
r
+
r
2
n
2
R
= 0
,
r
2
∂
+
r
∂
R
−
(2.45)
r
2
∂
∂
2
∂
Φ
∂ϕ
2
+
n
2
Φ
= 0
,
(2.46)
2
Z
∂
k
2
Z
= 0
.
z
2
−
(2.47)
∂
The Bessel equation (2.45) is written with account of the substitution of variable
r
=
rk
(the asterisk '*' is dropped). The solutions of equations (2.45)-(2.47) are
well known and can be written as follows:
∗
R
(
rk
)=
C
1
J
n
(
kr
)+
C
2
Y
n
(
kr
)
,
Φ
(
ϕ
)=
C
3
cos(
n
ϕ
)+
C
4
sin(
n
ϕ
)
,
Z
(
z
)=
C
5
cosh(
kz
)+
C
6
sinh(
kz
)
,
where
J
n
and
Y
n
are Bessel functions of the first and second kinds and of the
n
th
order,
C
i
are arbitrary constants.
Functions
Φ
(
ϕ
) must satisfy the periodicity condition:
Φ
ϕ
Φ
ϕ
π
)
,
(
)=
(
+ 2
from which it follows that parameter n is an integer,
n
= 0
,
2
,...
The condi-
tion, that function
R
(
rk
) be limited at
r
= 0 requires the coefficient of the Bessel
function of the second kind to be equal to zero:
C
2
= 0.
Thus, it is expedient to seek for the general solution of the problem in the form of
a Fourier expansion and of Laplace and Fourier-Bessel transformations [Nikiforov,
Uvarov (1984)]:
±
1
,
±
,
z
,
t
)=
∞
s
+
i
∞
F
(
r
,
ϕ
d
k
d
p
s
−
i
∞
0
J
0
(
kr
)
C
3
2
C
5
(
p
,
k
) cosh(
kz
)+
C
6
(
p
,
k
) sinh(
kz
)
×
exp
{
pt
}
