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.
 
Search WWH ::




Custom Search