Environmental Engineering Reference
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where
ψ
=
ψ
¯
η
,
−
x
ξ
2
,
¯
η
∈
ch
c
A
¯
2
2
D
At
.
ξ
,
¯
ch
=
(
At
)
−
−
(
y
−
η
)
¯
ξ
,
¯
Note that, by a polar coordinate transformation,
1
2
d
ξ
d
η
=
2
π
At
.
2
2
(
At
)
−
(
x
−
ξ
)
−
(
y
−
η
)
D
At
Therefore
1
t
t
e
−
e
−
u
(
x
,
y
,
t
)=
2
τ
0
A
ψ
·
ch2
π
At
=
2
τ
0
t
ψ
·
ch
.
2
π
Finally,
u
t
(
x
,
y
,
0
)=
⎧
⎨
⎫
⎬
t
=
0
1
1
2
∂
∂
t
2τ
0
t
2τ
0
τ
0
e
−
e
−
−
·
K
(
ψ
)
d
ξ
d
η
+
K
(
ψ
)
d
ξ
d
η
2
π
A
⎩
t
⎭
D
At
D
At
⎧
⎨
τ
0
∂
∂
τ
0
t
ψ
·
ch
·
1
1
2
t
t
τ
0
e
−
e
−
=
−
2
K
(
ψ
)
d
ξ
d
η
+
2
2
π
At
2
π
A
⎩
D
At
⎫
⎬
t
=
0
A
+
ψ
·
ch
·
2
π
=
ψ
(
x
,
y
)
,
⎭
where we have used lim
t
0
ψ
=
ψ
(
x
,
y
)
and lim
t
0
ch
=
1.
→
→
2.
The
u
in Eq. (5.69)
Rewrite Eq. (5.69) into
2τ
0
1
1
A
∂
t
t
2τ
0
τ
0
e
−
e
−
u
=
K
(
ϕ
)
d
ξ
d
η
+
K
(
ϕ
)
d
ξ
d
η
,
(5.71)
4
π
A
2
π
∂
t
D
At
D
At
where
ch
A
2
2
2
(
At
)
−
(
x
−
ξ
)
−
(
y
−
η
)
K
(
ϕ
)=
(
ϕ
(
ξ
,
η
)
.
At
)
2
−
(
x
−
ξ
)
2
−
(
y
−
η
)
2
Note that
∂
∂
η
=
∂
∂
K
(
ϕ
)
d
ξ
d
t
(
ψ
·
ch
)
·
2
π
At
+
ψ
·
ch
·
2
π
A
.
(5.72)
t
C
At
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