Environmental Engineering Reference
In-Depth Information
Therefore
+
∞
m
,
n
=
0
e
−
t
−
τ
sin
(
2
m
+
1
)
π
x
cos
n
y
l
2
.
π
G
=
2
τ
0
[
a
mn
cos
γ
mn
(
t
−
τ
)+
b
mn
sin
γ
mn
(
t
−
τ
)]
2
l
1
Applying initial conditions leads to
a
mn
=
0
,
1
γ
mn
M
mn
1
τ
sin
(
2
m
+
1
)
π
x
cos
n
π
y
b
mn
=
0
δ
(
x
−
ξ
,
y
−
η
)
d
σ
2
l
1
l
2
(4.31)
D
τ
0
γ
mn
M
mn
sin
(
1
2
m
+
1
)
πξ
cos
n
πη
l
2
=
,
2
l
1
where
M
mn
=
M
m
M
n
.
M
m
and
M
n
are the normal squares of the two eigenfunction
sets
sin
(
and
cos
n
, respectively.
M
mn
=
+
)
π
2
m
1
x
π
y
l
1
2
l
2
2
=
l
1
l
2
4
M
m
M
n
=
.
2
l
1
l
2
Therefore,
+
∞
m
,
n
=
0
τ
0
γ
mn
M
mn
sin
(
1
2
m
+
1
)
πξ
sin
(
2
m
+
1
)
π
x
t
−
τ
2τ
0
e
−
G
=
2
l
1
2
l
1
cos
n
πη
l
2
cos
n
π
y
·
sin
γ
mn
(
t
−
τ
)
.
(4.32)
l
2
t
t
u
3
=
W
f
τ
(
x
,
y
,
t
−
τ
)
d
τ
=
d
τ
G
(
x
,
ξ
;
y
,
η
;
t
−
τ
)
f
(
ξ
,
η
,
τ
)
d
σ
.
(4.33)
0
0
D
−
η
)
τ
0
in Eq. (4.31) by
By replacing the integrand
δ
(
x
−
ξ
,
y
ψ
(
x
,
y
)
and letting
τ
=
0 in Eq. (4.31), we obtain
⎧
⎨
+
∞
m
,
n
=
0
b
mn
sin
(
2
m
+
1
)
π
x
cos
n
π
y
t
e
−
u
2
=
W
ψ
(
x
,
y
,
t
)=
2
τ
0
sin
γ
mn
t
,
2
l
1
l
2
(4.34)
1
γ
mn
M
mn
sin
(
2
m
+
1
)
π
x
cos
n
π
y
⎩
b
mn
=
ψ
(
x
,
y
)
d
σ
.
2
l
1
l
2
D
W
ϕ
(
x
,
y
,
t
)
can be obtained through replacing
ψ
(
x
,
y
)
in Eq. (4.34) by
ϕ
(
x
,
y
)
.Also,
1
τ
W
ϕ
(
0
+
∂
u
1
=
x
,
y
,
t
)
.
(4.35)
∂
t
The solution of PDS (4.29) is, by the principle of superposition,
u
(
x
,
y
,
t
)=
u
1
(
x
,
y
,
t
)+
u
2
(
x
,
y
,
t
)+
u
3
(
x
,
y
,
t
)
.
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