Environmental Engineering Reference
In-Depth Information
so that the influence of the initial disturbances have already propagated
to this point. The region
D
is called
the domain of influence
of interval
with
[
x
1
,
x
2
]
[
x
1
,
x
2
]
.
When the length of the interval
tends to zero so that the interval tends to one
point
x
∗
, in particular, the domain of influence becomes (Fig. 2.5)
[
x
1
,
x
2
]
D
∗
:
x
∗
−
x
∗
+
at
≤
x
≤
at
,
t
>
0
.
Remark.
Characteristic curves
x
play a very important role in
studying wave equations. The CDS can also be specified on those curves. The cor-
responding PDS is called the
Goursat Problem
. Such problems can also be solved
by the method of traveling waves.
±
at
=
c
(
constant
)
Example
. Find the solution of the Goursat Problem
u
tt
a
2
u
xx
=
, −
∞
<
x
<
+
∞
,
0
<
t
,
u
|
x
−
at
=
0
=
ϕ
(
x
)
,
u
|
x
+
at
=
0
=
ψ
(
x
)
,
ϕ
(
0
)=
ψ
(
0
)
.
Solution.
The general solution is, by Eq. (2.64),
u
(
x
,
t
)=
f
(
x
+
at
)+
g
(
x
−
at
)
.
To satisfy the initial conditions,
f
and
g
must be determined such that
f
(
2
x
)+
g
(
0
)=
ϕ
(
x
)
,
f
(
0
)+
g
(
2
x
)=
ψ
(
x
)
or
x
2
x
2
f
(
x
)=
ϕ
−
g
(
0
)
,
g
(
x
)=
ψ
−
f
(
0
)
.
Fig. 2.5
Domain of influence
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