Environmental Engineering Reference
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and the boundary-value problem of Helmholtz equations
Δ
+
λ
=
,
(
,
)
|
r
=
a
0
=
,
U
U
0
L
U
U
r
0
(4.61)
U
(
r
,
θ
,
ϕ
+
2
π
)=
U
(
r
,
θ
,
ϕ
)
, |
U
(
0
,
θ
,
ϕ
)
| <
∞
.
k
2
where
λ
=
>
0. Applying separation of variables twice to Eq. 4.61 leads to three
S-L problems
1. The S-L problem regarding
Φ
(
ϕ
)
,
Φ
+
ηΦ
=
0
,
Φ
(
ϕ
+
2
π
)=
Φ
(
ϕ
)
,
(4.62)
is the parameter to be determined.
2. The S-L problem of
where
η
Θ
(
θ
)
,
l
η
sin
2
Θ
(
θ
)+(
θ
)
Θ
(
θ
)+
cot
(
l
+
1
)
−
Θ
(
θ
)=
0
,
(4.63)
θ
0
<
θ
<
π
,
|
Θ
(
θ
)
| <
∞
.
are the undetermined parameters.
3. The S-L problem of
R
Here
l
(
l
+
1
)
(
r
)
r
2
R
+
2
rR
+
λ
)
R
r
2
−
l
(
l
+
1
=
0
,
(4.64)
R
(
L
(
R
,
R
r
)
|
r
=
a
0
=
0
,
|
R
(
0
)
| <
∞
,
|
0
)
| <
∞
.
Here
-equation.
The eigenfunctions of Eq. (4.64) have the same form for all three kinds of bound-
ary conditions, but have zero-points of different functions and different normal
squares (see Section 2.6.2).
For convenience in applications, we list main results regarding these three
S-L problems in Table 4.2.
λ
is the undetermined parameter, which is required by the
T
(
t
)
n
2
1
+
Remark 1.
The readers are referred to Section 2.6.2 for
, ···
)
and
M
nl
. It has also been discussed how to solve Eqs. (4.63) and (4.64) in Sec-
tion 2.6.2.
Remark 2.
Equation (4.61) has nontrivial bounded solutions only for certain values
μ
(
l
=
1
,
2
l
2
n
2
1
a
0
+
of separation constant
λ
.For
λ
=
μ
, we have the corresponding
l
nontrivial bounded solutions
⎛
⎞
n
+
2
1
μ
⎝
⎠
,
P
n
(
l
(
ϕ
+
ϕ
)
θ
)
a
mn
cos
m
b
mn
sin
m
cos
j
n
r
a
0
in which
a
mn
+
b
mn
=
0.
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