Environmental Engineering Reference
In-Depth Information
Substituting Eqs. (6.127), (6.128) and (6.132) into Eq. (6.126) thus yields the solu-
tion of PDS (6.125)
1
W
ϕ
(
τ
0
+
∂
B
2
W
u
(
r
,
θ
,
z
,
t
)=
r
,
θ
,
z
,
t
)+
)
ϕ
(
r
,
θ
,
z
,
t
)
.
(6.133)
(
k
mn
+
λ
k
∂
t
6.6.3 Solution from
f
(
r
,
θ
,
z
,
t
)
Theorem 2.
Let
u
(
r
,
θ
,
z
,
t
)=
W
ψ
(
r
,
θ
,
z
,
t
)
be the solution of
⎧
⎨
u
t
τ
B
2
∂
∂
A
2
0
+
u
tt
=
Δ
u
(
r
,
θ
,
z
,
t
)+
t
Δ
u
(
r
,
θ
,
z
,
t
)
Ω
×
(
,
+
∞
)
,
0
⎩
(
,
,
)
|
∂Ω
=
,
L
u
u
r
u
z
0
u
(
r
,
θ
,
z
,
0
)=
0
,
u
t
(
r
,
θ
,
z
,
0
)=
ψ
(
r
,
θ
,
z
)
.
The solution of
⎧
⎨
u
t
τ
0
+
B
2
∂
∂
A
2
u
tt
=
Δ
u
(
r
,
θ
,
z
,
t
)+
t
Δ
u
(
r
,
θ
,
z
,
t
)
+
f
(
r
,
θ
,
z
,
t
)
,
Ω
×
(
0
,
+
∞
)
,
(6.134)
⎩
L
(
u
,
u
r
,
u
z
)
|
∂Ω
=
0
,
u
(
r
,
θ
,
z
,
0
)=
u
t
(
r
,
θ
,
z
,
0
)=
0
is
t
u
=
W
f
τ
(
r
,
θ
,
z
,
t
−
τ
)
d
τ
,
(6.135)
0
where
f
τ
=
. Therefore, Theorem 2 in Section 6.1 is also valid in cylin-
drical coordinate systems.
f
(
r
,
θ
,
z
,
τ
)
Proof.
By the definition of
W
ψ
(
r
,
θ
,
z
,
t
)
,the
W
f
τ
(
r
,
θ
,
z
,
t
−
τ
)
satisfies
⎧
⎨
2
W
f
τ
∂
1
τ
0
∂
W
f
τ
∂
+
∂
B
2
∂
∂
A
2
=
Δ
W
f
τ
+
t
Δ
W
f
τ
,
(6.136a)
t
t
2
∂Ω
=
L
W
f
τ
,
∂
W
f
τ
∂
r
,
∂
W
f
τ
∂
0
,
(6.136b)
⎩
z
t
=
τ
=
W
f
τ
t
=
τ
=
∂
W
f
τ
∂
0
,
f
(
r
,
θ
,
z
,
τ
)
.
(6.136c)
t
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