Environmental Engineering Reference
In-Depth Information
determined by the initial condition
u
t
(
x
,
0
)=
ψ
(
x
)
. Thus we have
⎧
⎨
)=
m
B
m
e
α
m
t
sin
u
(
x
,
t
)=
W
ψ
(
x
,
t
β
m
t
·
X
m
(
x
)
,
l
0
ψ
(
(6.59)
1
M
m
β
m
⎩
B
m
=
x
)
X
m
(
x
)
d
x
,
where
M
m
is the normal square of
{
X
m
(
x
)
}
available in Table 2.1, i.e.
M
m
=
(
X
m
(
.
Similarly, consider the solution of PDS (6.56)
x
)
,
X
m
(
x
))
)=
m
e
α
m
t
u
(
x
,
t
(
C
m
cos
β
m
t
+
D
m
sin
β
m
t
)
X
m
(
x
)
.
(6.60)
Thus
)=
m
e
α
m
t
u
t
(
x
,
t
[
α
m
(
C
m
cos
β
m
t
+
D
m
sin
β
m
t
)
+(
−
C
m
β
m
sin
β
m
t
+
D
m
β
m
cos
β
m
t
)]
X
m
(
x
)
.
To satisfy the two initial conditions the
C
m
and the
D
m
must be determined such that
l
0
ϕ
(
1
M
m
C
m
=
x
)
X
m
(
x
)
d
x
,
α
m
C
m
+
D
m
β
m
=
0
.
Thus
l
0
ϕ
(
1
τ
0
+
λ
m
B
2
2
D
m
=
−
α
m
β
1
M
m
C
m
=
x
)
X
m
(
x
)
d
x
.
(6.61)
β
m
m
By Eq. (6.59), we have
⎧
⎨
)=
m
D
m
α
m
e
α
m
t
sin
β
m
t
+
e
α
m
t
β
m
t
X
m
(
∂
∂
t
W
ϕ
(
x
,
t
β
m
cos
x
)
l
0
ϕ
(
1
M
m
⎩
D
m
=
x
)
X
m
(
x
)
d
x
.
β
m
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