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
)
Search WWH ::
Custom Search