Environmental Engineering Reference
In-Depth Information
Remark 2
. To apply Eqs. (2.44) and (2.45), we must determine
M
mn
, the normal
square of eigenfunction sets. Here we develop expressions of
M
mn
for three cases of
boundary conditions.
Rewrite Eq. (2.41) into
r
d
R
d
r
n
2
r
R
d
d
r
k
2
rR
−
+
=
.
(2.46)
Multiplying Eq. (2.46) by
rR
and integrating over
[
0
,
a
0
]
yields
a
0
a
0
k
2
a
0
0
rR
rR
d
r
n
2
RR
d
r
r
2
RR
d
r
−
+
=
,
0
0
or
2
2
R
2
k
2
a
0
r
2
d
R
2
2
a
0
0
=
a
0
0
+
n
2
1
2
(
rR
)
−
k
2
r
2
R
2
2
M
mn
a
0
0
−
=
.
Thus
(
a
0
0
.
2
rR
)
2
2
rR
)
+
(
−
(
nR
)
M
mn
=
(2.47)
2
k
2
2
k
2
2
1. Boundary Condition of the First Kind
Since
R
(
a
0
)=
0and
R
(
0
)=
J
n
(
0
)=
0
(
n
=
1
,
2
, ···
)
, Eq. (2.47) becomes
J
n
2
2
k
mn
k
mn
J
n
(
a
0
k
mn
a
0
)
2
a
0
2
μ
(
n
)
M
mn
=
=
m
2
J
n
−
1
2
J
n
+
1
a
0
a
0
μ
(
n
)
μ
(
n
)
=
=
,
m
m
2
J
n
(
2
n
x
J
n
(
where
J
n
−
1
(
x
)
−
J
n
+
1
(
x
)=
x
)
and
J
n
−
1
(
x
)+
J
n
+
1
(
x
)=
x
)
have been used.
2. Boundary Condition of the Second Kind
For this case,
R
(
a
0
)=
0and
R
(
0
)=
J
n
(
0
)=
0
(
n
=
1
,
2
, ···
)
. Eq. (2.47) reduces to
⎡
2
⎤
n
μ
(
n
)
⎦
J
n
a
0
2
μ
(
n
)
⎣
1
M
mn
=
−
.
m
m
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