Environmental Engineering Reference
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which is the sum of three terms
⎧
⎨
+
∞
n
=
1
B
(
1
n
e
α
n
t
sin
β
n
t
sin
n
π
x
u
1
(
x
,
t
)=
,
l
(6.30a)
l
0
ϕ
(
ξ
)
⎩
2
τ
0
l
sin
n
πξ
l
B
(
1
)
=
d
ξ
.
n
β
n
⎧
⎨
β
n
t
e
α
n
t
sin
n
+
∞
n
=
1
B
(
2
)
π
x
u
2
(
x
,
t
)=
α
n
sin
β
n
t
+
β
n
cos
n
l
+
∞
n
=
1
B
(
2
)
n
t
sin
n
π
x
n
e
α
n
t
sin
=
α
β
n
l
(6.30b)
⎩
+
∞
n
=
1
B
(
2
)
e
α
n
t
sin
n
π
x
+
β
n
cos
β
n
t
·
,
n
l
l
0
ϕ
(
ξ
)
2
sin
n
πξ
l
B
(
2
)
=
d
ξ
.
n
l
β
n
⎧
⎨
+
∞
n
=
1
B
(
3
n
e
α
n
t
sin
β
n
t
sin
n
π
x
u
3
(
x
,
t
)=
,
l
(6.30c)
l
0
ϕ
(
ξ
)
⎩
2
B
2
l
sin
n
πξ
l
B
(
3
)
=
−
d
ξ
.
n
β
n
i.e.
u
=
u
1
(
x
,
t
)+
u
2
(
x
,
t
)+
u
3
(
x
,
t
)
.
(6.31)
1
τ
0
+
2
2 ; thus the coefficient
B
n
in Eq. (6.21) is
n
π
B
Note that
α
n
=
−
l
1
τ
0
+
2
l
0
ϕ
(
ξ
)
n
1
π
B
sin
n
πξ
l
B
n
=
d
ξ
.
l
β
n
l
Also,
n
2
l
0
ϕ
(
ξ
)
d
cos
n
l
0
ϕ
(
ξ
)
B
2
π
B
sin
n
πξ
l
n
l
πξ
l
d
ξ
=
−
l
B
2
n
l
l
cos
n
πξ
l
=
−
ϕ
(
ξ
)
0
l
0
ϕ
(
ξ
)
cos
n
πξ
l
−
d
ξ
d
sin
n
B
2
l
πξ
l
0
ϕ
(
ξ
)
=
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