Environmental Engineering Reference
In-Depth Information
d
u
d
t
=
∂u
∂x
d
x
d
t
+
∂u
∂t
d
u
d
t
=
Along these two lines
and
0
d
t
d
x
=
1
+
defines a positive characteristic along which
v
2
c
is
v
+
c
constant
d
t
d
x
=
1
defines a negative characteristic along which
v
−
2
c
is
v
−
c
constant
In principle the terms
v
±
2
c
and
g
(
S
f
−
S
o
) are constant
d
t
d
x
=
1
(2.57)
v
±
c
d(
v
±
2
c
+
g
(
S
f
−
S
o
)
t
)
=
0
(2.58)
d
t
The solution of the method of characteristics is based on the construction
of a network of points in the
x
t
plane where the dependent variables
are computed. From a given set of two points on the network, a new
point may be constructed by drawing a characteristic of one family from
one point that intersects with the characteristics of the other family drawn
from the other point. In general, these directions are not known initially as
−
Domain of influence
t
dx
dt
dx
dt
= v + gh
= v - gh
t
0
x
0
x
Domain of dependence
t
Forward
Characteristic
Backward
Characteristic
t
0
dx
dt
dx
dt
= v - gh
= v + gh
Figure 2.17. Characteristics in
the
x
and
t
plane with an
example of the domain of
influence and of the domain of
dependence.
x
0
x
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