Environmental Engineering Reference
In-Depth Information
Here
H
(
x
)
is the Heaviside function defined by
0
,
x
<
0
,
H
(
x
)=
1
,
x
>
0
.
Finally, the solution of (5.18) is
I
0
c
t
2
2
a
d
ξ
+
t
L
−
1
U
ξ
√
τ
0
1
2
ξ
,
U
(
ξ
,
t
)=
(
ξ
,
s
)
=
−
(
ξ
−
ξ
)
ψ
ξ
−
t
which is the same as that obtained by the Riemann method.
5.2.3 Some Properties of Solutions
We have obtained solutions of wave, heat-conduction and hyperbolic heat-conduction
equations due to initial disturbances
ϕ
(
x
)
and
ψ
(
x
)
. They are, respectively,
x
+
at
)=
ϕ
(
x
−
at
)+
ϕ
(
x
+
at
)
1
2
a
u
(
x
,
t
+
ψ
(
ξ
)
d
ξ
,
(5.37)
2
x
−
at
+
∞
t
e
−
(
x
−
ξ
)
2
1
2
a
√
π
u
(
x
,
t
)=
V
(
x
,
ξ
,
t
)
ϕ
(
ξ
)
d
ξ
,
V
=
,
(5.38)
4
a
2
t
−
∞
τ
0
ϕ
(
x
−
At
)+
ϕ
(
x
+
At
)
t
e
−
u
(
x
,
t
)=
2
2
1
2
I
0
b
2
x
+
At
1
2
A
2
+
(
At
)
−
(
x
−
ξ
)
τ
x
−
At
0
2
⎤
I
1
b
t
2
⎦
ϕ
(
ξ
)
+
0
b
(
At
)
−
(
x
−
ξ
)
2
2
4
τ
(
At
)
−
(
x
−
ξ
)
I
0
b
2
d
2
+
(
At
)
−
(
x
−
ξ
)
ψ
(
ξ
)
ξ
.
(5.39)
Equation (5.37) shows that vibration propagates in the form of superimposed for-
ward and backward traveling waves that are created by initial disturbances. Waves
travel at a speed
a
. The problems of semi-infinite domains can be solved by using
the method of continuation. When the end is fixed, there is semi-wave loss. When
the end is free, however, the semi-wave loss disappears.
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