Environmental Engineering Reference
In-Depth Information
1.
The Green function in Eq. (4.32).
The general term of the Green function can
be obtained based on the eigenfunction sets
sin
(
and
cos
n
2
m
+
1
)
π
x
π
y
2
l
1
l
2
(Table 2.1) by the following procedure:
multiplying factor
sin
(
2
m
+
1
)
πξ
2
l
1
cos
n
πη
l
2
sin
(
2
m
+
1
)
π
x
cos
n
π
y
sin
(
2
m
+
1
)
π
x
cos
n
π
y
⇒
2
l
1
l
2
2
l
1
l
2
mu
ltip
lying another factor
e
−
t
−
τ
2τ
0
1
τ
0
γ
mn
M
mn
sin
sin
(
2
m
+
1
)
πξ
cos
n
πη
l
2
γ
mn
(
t
−
τ
)
τ
0
γ
mn
M
mn
sin
(
1
2
m
+
1
)
π
x
t
−
τ
2τ
0
e
−
·
⇒
2
l
1
2
l
1
cos
n
π
y
sin
(
2
m
+
1
)
πξ
cos
n
πη
l
2
·
sin
γ
mn
(
t
−
τ
)
.
l
2
2
l
1
The two operations of multiplication can also be combined together.
2.
The green function in Eq. (4.36)
. The general term of the Green function can be
written out based on the eigenfunction sets
{
sin
α
m
x
}
and
{
cos
β
n
y
}
(Table 2.1)
by the following procedure:
multiplying factor
t
−
τ
2
e
−
τ
0
1
τ
0
γ
mn
M
mn
sin α
m
ξ cos β
n
η
sin
γ
mn
(
t
−
τ
)
⇒
1
τ
0
γ
mn
M
mn
t
−
τ
2
e
−
sin
α
m
x
cos
β
n
y
τ
0
·
sin
α
m
x
cos
β
n
y
sin
α
m
ξ
cos
β
n
η
sin
γ
mn
(
t
−
τ
)
.
We thus list the steps of writing Green functions from eigenfunctions in Table 2.1.
Step 1.
Based on the given boundary conditions, find the eigenfunction sets from
Tab l e 2 . 1
X
i
(
x
)
,
Y
j
(
y
)
,
i
,
j
=
1
,
2
, ··· ,
9
.
When writing
Y
j
(
, replace
x
, the domain of
x
and subscript index
m
in Table 2.1
by
y
, the domain of
y
and subscript index
n
, respectively. If applied, use
x
)
μ
m
and
μ
n
(not
μ
n
) for the zero-points of the corresponding function occurring in
X
i
(
x
)
and
Y
j
(
y
)
, respectively.
Step 2.
Construct the complete and orthogonal group for expanding
G
,
(
,
)=
(
)
(
)
,
,
=
,
, ··· ,
.
u
ij
x
y
X
i
x
Y
j
y
i
j
1
2
9
A
)
9
×
9
is a matrix of order 9. Each of its components corresponds
to a complete and orthogonal group in
D
for one combination of boundary
conditions.
(
x
,
y
)=(
u
ij
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