Environmental Engineering Reference
In-Depth Information
Note that
B
mnl
β
nl
=
D
mnl
β
nl
=
A
mnl
,
C
mnl
,
α
nl
B
mnl
=
−
B
mnl
,
α
nl
D
mnl
=
−
D
mnl
.
Thus
∂
∂
)=
m
,
n
,
l
{
[(
−
B
mnl
sin
β
nl
t
)
cos
m
ϕ
t
W
Φ
(
r
,
θ
,
ϕ
,
t
+(
−
D
mnl
sin
β
nl
t
)
sin
m
ϕ
]+[(
A
mnl
cos
β
nl
t
)
cos
m
ϕ
e
α
nl
t
P
n
(
+(
C
mnl
cos
β
nl
t
)
sin
m
ϕ
]
}
cos
θ
)
j
n
(
k
nl
r
)
.
(6.151)
Substituting Eqs. (6.149) and (6.151) into Eq. (6.148) and using the structure of
W
ψ
(
r
,
θ
,
ϕ
,
t
)
in Eq. (6.146) thus yields the solution of PDS (6.147)
1
W
Φ
(
τ
0
+
∂
B
2
W
k
nl
Φ
(
(
,
θ
,
ϕ
,
)=
,
θ
,
ϕ
,
)+
,
θ
,
ϕ
,
)
.
u
r
t
r
t
r
t
∂
t
6.7.3 Solution from
f
(
r
,
θ
,
ϕ
,
t
)
Theorem 2.
Let
u
=
W
ψ
(
r
,
θ
,
ϕ
,
t
)
be the solution of PDS (6.141). The solution of
⎧
⎨
u
t
τ
0
+
B
2
∂
∂
A
2
u
tt
=
Δ
u
(
r
,
θ
,
ϕ
,
t
)+
t
Δ
u
(
r
,
θ
,
ϕ
,
t
)
+
f
(
r
,
θ
,
ϕ
,
t
)
,
0
<
r
<
a
,
0
<
θ
<
π
,
0
<
ϕ
<
2
π
,
0
<
t
,
(6.152)
⎩
L
(
u
,
u
r
)
|
r
=
a
=
0
,
u
(
r
,
θ
,
ϕ
,
0
)=
0
,
u
t
(
r
,
θ
,
ϕ
,
0
)=
0
is
t
u
(
r
,
θ
,
ϕ
,
t
)=
W
f
τ
(
r
,
θ
,
ϕ
,
t
−
τ
)
d
τ
,
(6.153)
0
where
f
τ
=
f
(
r
,
θ
,
ϕ
,
τ
)
. Thus Theorem 2 in Section 6.1 is also valid for spherical
coordinate systems.
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