Environmental Engineering Reference
In-Depth Information
When
m
=
1
,
2
, ···
,
Eigenvalues
l
(
l
+
1
)=
n
(
n
+
1
)
,
n
=
1
,
2
, ··· ,
m
≤
n
;
m
n
P
n
P
n
Eigenfunctions
Θ
is the
Associated
Legendre polynomial of degree n and order m
(
θ
)=
(
cos
θ
)
,
(
x
)
.
3. With
l
=
n
=
0
,
1
,
2
, ···
, the equation of
R
(
r
)
in (2.56), together with
|
R
(
0
)
| <
∞
,
R
(
R
(
|
0 forms another eigenvalue problem. To
solve this problem, introduce a new variable
x
and a new function
y
0
)
| <
∞
and
L
(
R
(
r
)
,
r
))
|
r
=
a
0
=
(
x
)
defined
x
2
R
. The equation of
R
by
x
=
kr
,
y
(
x
)=
(
r
)
in (2.56) is thus transformed into
x
2
2
y
n
1
2
x
2
y
+
xy
+
−
+
=
0
,
which is a Bessel equation. Its solution reads
c
(
1
n
J
n
+
1
√
kr
c
(
2
n
J
R
n
(
r
)=
2
(
kr
)+
)
(
kr
)
,
1
−
(
1
2
n
+
where
c
(
1
)
and
c
(
2
)
are constants that are not all zero. To satisfy
|
R
(
0
)
| <
∞
,we
n
n
have
c
(
2
)
0and
c
(
1
)
=
=
0. Therefore, without taking account of a constant factor,
n
n
2
kr
J
n
+
R
n
(
r
)=
2
(
kr
)=
j
n
(
kr
)
.
1
2
x
J
l
+
Here
j
l
(
is the
spherical Bessel function of the first kind
.The
eigenvalues can be readily obtained by substituting the boundary conditions
R
n
(
x
)=
2
(
x
)
1
0 (first kind),
R
n
(
0 (second kind) or
R
n
(
a
0
)=
a
0
)=
a
0
)+
hR
n
(
a
0
)=
0 (third
kind) into it. The final results are:
a
0
2
μ
(
1
2
)
n
+
k
nl
=
Eigenvalues
λ
=
,
n
=
0
,
1
,
2
, ··· ,
l
=
1
,
2
, ···
l
j
n
r
a
0
μ
(
1
2
)
n
+
Eigenfunctions
j
n
(
k
nl
r
)=
.
l
μ
(
1
2
)
n
+
1
2
J
n
+
,
xJ
n
+
Here
are the positive zero-points of
J
n
+
2
(
x
)
2
(
x
)
−
2
(
x
)
and
1
1
1
l
)+
ha
0
−
2
J
n
+
xJ
n
+
1
for boundary conditions of the first kind, the sec-
ond kind and the third kind, respectively.
2
(
x
2
(
x
)
1
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