Environmental Engineering Reference
In-Depth Information
Thus, we obtain
⎧
⎨
+
∞
∑
u
=
W
Ψ
(
r
,
θ
,
ϕ
,
t
)=
1
(
b
mnl
cos
m
ϕ
+
d
mnl
sin
m
ϕ
)
m
,
n
=
0
,
l
=
j
n
μ
(
r
sin
)
2
n
+
P
n
(
×
cos
θ
)
l
ω
nl
t
,
a
0
⎛
⎝
μ
(
⎞
1
2
)
n
+
1
⎠
l
b
0
nl
=
Ψ
(
r
,
θ
,
ϕ
)
P
n
(
cos
θ
)
j
n
r
2
πω
nl
M
0
nl
a
0
r
≤
a
0
r
2
sin
×
ϕ
,
θ
d
θ
d
r
d
(2.59)
⎛
⎝
μ
(
⎞
⎩
1
2
)
n
+
1
πω
nl
M
mnl
P
n
(
l
⎠
b
mnl
=
Ψ
(
r
,
θ
,
ϕ
)
cos
θ
)
j
n
r
a
0
r
≤
a
0
r
2
cos
m
×
ϕ
sin
θ
d
θ
d
r
d
ϕ
,
⎛
⎝
μ
(
⎞
1
2
)
n
+
1
πω
nl
M
mnl
P
n
(
⎠
l
d
mnl
=
Ψ
(
r
,
θ
,
ϕ
)
cos
θ
)
j
n
r
a
0
r
≤
a
0
r
2
sin
m
×
ϕ
sin
θ
d
θ
d
r
d
ϕ
,
π
0
[
θ
=
(
n
+
m
)
!
2
2
sin
P
n
(
where
M
mnl
=
M
mn
M
nl
,
M
mn
=
cos
θ
)]
θ
d
1
is the
(
n
−
m
)
!
2
n
+
,
M
nl
is the normal square of
j
n
a
0
normal square of
P
n
(
μ
(
1
2
)
n
+
cos
θ
)
r
/
l
(
n
fixed
,
l
=
1
,
2
, ···
)
and can be determined by Eq. (2.47) in Section 2.5 as
⎧
⎨
Boundary
condition
of the first
kind
J
n
+
2
μ
(
1
2
)
n
+
,
3
2
l
,
⎡
⎣
⎤
⎦
Boundary
condition
of the
second kind
n
(
n
+
1
)
μ
(
1
2
)
n
+
J
n
+
a
0
1
−
,
π
2
1
2
l
M
nl
=
μ
(
1
2
)
n
+
μ
(
2
)
⎩
n
+
,
4
l
l
⎡
⎤
Boundary
condition
of the
third kind
⎣
⎦
+
(
ha
0
+
n
)(
ha
0
−
n
−
1
)
μ
(
1
2
)
n
+
J
n
+
1
,
2
2
l
μ
(
)
2
n
+
.
l
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