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
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
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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