Environmental Engineering Reference
In-Depth Information
Fig. 2.2 Function
Ψ (
x
)
The drawing procedure is
Ψ (
x
) ,− Ψ (
x
)
u
(
x
,
t 1 )= Ψ (
x
+
2 t 1 ) Ψ (
x
2 t 1 )
u
(
x
,
t 2 )= Ψ (
x
+
2 t 2 ) Ψ (
x
2 t 2 )
u
(
x
,
t 3 )= Ψ (
x
+
2 t 3 ) Ψ (
x
2 t 3 )
u
(
x
,
t 4 )= Ψ (
x
+
2 t 4 ) Ψ (
x
2 t 4 ) .
1
4 (
Ψ (
x
)
is graphically shown in Fig. 2.2, where c 0 =
x 2
x 1 ) ψ 0 .
2.7.4 Domains of Dependence, Determinacy and Influence
Domain of Dependence
Let D be a semi-infinite plane:
<
x
< + ,
0
<
t . We note, by the D'Alembert
formula, that at any point
(
x 0 ,
t 0 )
D ,thevalue u
(
x 0 ,
t 0 )
depends only on the initial
values
ϕ (
x
)
and
ψ (
x
)
at points between x 0
at 0 and x 0 +
at 0 and not on initial
values outside of this range. This interval
[
x 0
at 0 ,
x 0 +
at 0 ]
is called the domain of
dependence of point
at 0 are the abscissas of two
intersecting points of Ox -axis and two characteristic curves x
(
x 0 ,
t 0 )
. x 1 =
x 0
at 0 and x 2 =
x 0 +
x 0 = ±
a
(
t
t 0 )
that
are passing though point
(
x 0 ,
t 0 )
.
Domain of Determinacy
For any interval
[
x 1 ,
x 2 ]
on the Ox -axis, consider a triangle in D formed by
[
x 1 ,
x 2 ]
,
the characteristic curve x
=
x 1 +
at passing though point
(
x 1 ,
0
)
and the characteristic
curve x
=
x 2
at passing though point
(
x 2 ,
0
)
, Fig. 2.3. For any point
(
x 0 ,
t 0 )
inside
the triangle, its domain of dependence is always in
[
x 1 ,
x 2 ]
, so that the value of u
(
x
,
t
)
 
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