Environmental Engineering Reference
In-Depth Information
Φ
(
x
+
at
)
: backward wave generated by initial displacement
ϕ
(
x
)
,
Φ
(
x
−
at
)
: forward wave generated by initial displacement
ϕ
(
x
)
,
Ψ
(
x
+
at
)
: backward wave generated by initial velocity
ψ
(
x
)
,
−
Ψ
(
x
−
at
)
: forward wave generated by initial velocity
ψ
(
x
)
,
u
1
(
: superposition of forward and backward waves generated by the initial
displacement
x
,
t
)
ϕ
(
x
)
,
u
2
(
x
,
t
)
: superposition of forward and backward waves generated by initial ve-
locity
ψ
(
x
)
,
: superposition of constant waves generated by initial displacement and
velocity.
u
(
x
,
t
)
The general solution (2.64) from the method of characteristics is, in fact, a su-
perposition of forward and backward waves that are fixed by initial conditions. This
is the basis for also calling the method of characteristics the
method of traveling
waves
.
Example.
Consider
u
tt
=
4
u
xx
, −
∞
<
x
<
+
∞
,
0
<
t
,
(2.66)
u
(
x
,
0
)=
ϕ
(
x
)
,
u
t
(
x
,
0
)=
ψ
(
x
)
.
1
2
1. Draw the wave shape at
t
=
⎧
⎨
0
,−
∞
<
x
<
0
,
x
,
0
≤
x
<
2
,
when
ϕ
(
x
)=
2
,
2
≤
x
<
3
,
and
ψ
(
x
)=
0
.
⎩
−
+
,
≤
<
,
x
5
3
x
5
,
≤
<
+
∞
0
5
x
ψ
,
≤
≤
,
x
1
x
x
2
0
2. For
ϕ
(
x
)=
0and
ψ
(
x
)=
show
Ψ
(
x
)
graphically and state
0
,
x
<
x
1
,
x
2
<
x
=
,
,
(
<
<
<
)
the procedure of drawing wave shapes at
t
t
1
t
2
t
3
and
t
4
t
1
t
2
t
3
t
4
.
Solution.
1. Since the wave speed is 2, the sequence of drawing the wave shape is
)
⇒
ϕ
(
x
)
⇒
ϕ
(
x
−
1
)
,
ϕ
(
x
+
1
)
=
ϕ
(
x
+
1
)+
ϕ
(
x
−
1
)
ϕ
(
x
⇒
u
.
2
2
2
2
1
2
is shown in the last figure.
This is shown in Fig. 2.1. The wave shape at
t
=
Search WWH ::
Custom Search