Environmental Engineering Reference
In-Depth Information
The solution of
⎧
⎨
u
t
τ
0
+
A
2
u
xx
, −
∞
<
u
tt
=
x
<
+
∞
,
0
<
t
,
⎩
u
(
x
,
0
)=
ϕ
(
x
)
,
u
t
(
x
,
0
)=
ψ
(
x
)
may thus be written as
τ
0
ϕ
(
x
−
At
)+
ϕ
(
x
+
At
)
t
e
−
u
=
2
2
1
2
τ
0
I
0
b
2
x
+
At
1
2
A
2
+
(
At
)
−
(
x
−
ξ
)
x
−
At
I
1
b
2
(5.30)
t
2
+
0
b
(
At
)
−
(
x
−
ξ
)
ϕ
(
ξ
)
2
2
2
4
τ
(
At
)
−
(
x
−
ξ
)
I
0
b
2
d
2
+
(
At
)
−
(
x
−
ξ
)
ψ
(
ξ
)
ξ
,
which is similar to the D'Alembert formula of wave equations. The unit of the sec-
ond term in the integrand reads
⎡
⎤
1
2
A
t
T
L
·
1
T
·
Θ
·
⎣
⎦
=
0
b
2
I
1
ϕ
(
ξ
)
d
ξ
L
=
Θ
.
2
4
τ
(
At
)
−
(
x
−
ξ
)
The unit of all the other terms is also
Θ
.
The solution of
⎧
⎨
u
t
τ
0
+
f
(
x
,
t
)
A
2
u
xx
+
u
tt
=
, −
∞
<
x
<
+
∞
,
0
<
t
,
τ
0
(5.31)
⎩
u
(
x
,
0
)=
0
,
u
t
(
x
,
0
)=
0
is
I
0
b
A
2
2
2
(
−
τ
)
−
(
−
ξ
)
t
x
t
x
+
A
(
t
−
τ
)
1
2
A
t
−
τ
2
e
−
u
=
τ
0
d
τ
f
(
ξ
,
τ
)
d
ξ
,
(5.32)
τ
0
0
x
−
A
(
t
−
τ
)
which is similar to the Kirchhoff formula of wave equations.
For convenience, rewrite the equation of PDS (5.31) as
u
t
τ
0
+
A
2
u
xx
T
−
2
u
tt
=
+
f
(
x
,
t
)
,
[
f
]=
Θ
.
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