Environmental Engineering Reference
In-Depth Information
Fig. 2.3
Domain of determinacy
at any point inside the triangle depends only on the initial values in
[
x
1
,
x
2
]
and not
on initial values outside
[
x
1
,
x
2
]
. Therefore the initial values in
[
x
1
,
x
2
]
completely
determine values of
u
inside this triangle. Such a triangle region in
D
is called
the
domain of determinacy
of interval
(
x
,
t
)
[
x
1
,
x
2
]
.
Domain of Influence
ϕ
(
)
ψ
(
)
[
,
]
To illustrate the propagation in
D
of initial disturbances
x
and
x
in
x
1
x
2
,
ϕ
(
)
ψ
(
)
consider initial disturbances such that both
x
and
x
are vanished outside of
[
x
1
,
x
2
]
and non-zero inside of
[
x
1
,
x
2
]
.
Let
D
be the region:
x
1
−
at
≤
x
≤
x
2
+
at
,
0
<
t
(Fig. 2.4). For any point
(
x
0
,
t
0
)
outside of
D
, its domain of dependence is also outside of
so that the effect
of the initial disturbances has not propagated to this point yet. For any point
[
x
1
,
x
2
]
t
0
)
inside of
D
, however, its domain of dependence has always some parts in common
(
x
0
,
Fig. 2.4
Region
D
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