Environmental Engineering Reference
In-Depth Information
tion in PDS (2.34) yields
T
mn
+
a
2
μ
m
l
1
2
T
mn
sin
μ
m
x
l
1
2
μ
n
l
2
sin
μ
n
y
n
∞
m
,
n
=
1
+
l
2
+
ϕ
=
,
0
which leads to the equation of
T
mn
(
t
)
and its general solution
T
mn
+
ω
2
mn
T
mn
=
0
,
T
mn
(
t
)=
a
mn
cos
ω
mn
t
+
b
mn
sin
ω
mn
t
,
where
a
2
μ
m
l
1
2
2
μ
n
l
2
2
mn
ω
=
+
.
Substituting
T
mn
(
t
)
and applying
u
(
x
,
y
,
0
)=
0 leads to
a
mn
=
0. To satisfy the
initial condition
u
t
(
x
,
y
,
0
)=
ψ
(
x
,
y
)
,
b
mn
must be determined such that
sin
μ
n
y
l
2
+
ϕ
n
∞
m
,
n
=
1
b
mn
ω
mn
sin
μ
m
x
=
ψ
(
x
,
y
)
.
l
1
It is straightforward to obtain
b
mn
by orthogonality and the normal square of eigen-
function sets. Thus we have
⎧
⎨
sin
μ
n
y
l
2
+
ϕ
n
sin
∞
m
,
n
=
1
b
mn
sin
μ
m
x
u
=
W
ψ
(
x
,
y
,
t
)=
ω
mn
t
,
l
1
sin
μ
n
y
l
2
+
ϕ
n
d
x
d
y
⎩
1
M
mn
sin
μ
m
x
l
1
b
mn
=
ψ
(
x
,
y
)
.
ω
mn
D
where
M
mn
=
M
m
M
n
,
M
m
and
M
n
are the normal squares of the two eigenfunction
sets, respectively.
Finally, the solution of PDS (2.34) under the boundary conditions (2.35) is, by
the solution structure theorem,
t
=
∂
∂
u
t
W
ϕ
+
W
ψ
(
x
,
y
,
t
)+
W
f
τ
(
x
,
y
,
t
−
τ
)
d
τ
,
(2.36)
0
where
f
τ
=
.
We can also obtain the Green function by considering the case
f
f
(
x
,
y
,
τ
)
(
x
,
y
,
t
)=
δ
(
0.
The solution of PDS (2.34) for the other 80 combinations of boundary conditions
can also be written in the form of Eq. (2.36). However,
W
ψ
(
x
−
ξ
,
y
−
η
,
t
−
τ
)
,
ψ
(
x
,
y
)=
0and
ϕ
(
x
,
y
)=
x
,
y
,
t
)
differs from one
to another.
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