Geoscience Reference
In-Depth Information
Put differently, comparison of Equations (5.103) and (5.104) with (5.105) and (5.106),
respectively, shows that to an imaginary observer, moving along
x
with a velocity defined
by (5.108) and (5.107), it would appear that the cross-sectional area and the flow rate
change as indicated by (5.109). In the absence of lateral inflow, when
i
0, the velocity
defined by (5.108) and (5.107) is the speed of propagation of points where
dh
=
/
dt
=
0 and
where
dq
0, that is the speed of propagation of points of a given value of
h
and
q
.
These observations are somewhat similar to the comments made in connection with
Equations (5.51), (5.55) and (5.64). Thus (5.107), describing the path of the observer,
defines the characteristics of the problem. Because (5.103) and (5.104) are of the first
order, there is only one set of characteristics, namely in the forward direction.
One practical result obtainable with this approach is the celerity of a small monoclinal
rising wave in an open channel. In a channel with an arbitrary cross section, in light of
(5.24), the differential equations (5.103) and (5.104) assume the following form
∂
/
dt
=
A
c
∂
c
k
∂
A
c
∂
t
+
x
=
Q
l
(5.110)
and
∂
Q
∂
c
k
∂
Q
∂
t
+
x
=
c
k
Q
l
(5.111)
where the wave celerity (5.108) is now given by
dQ
dA
c
c
k
=
(5.112)
For a wide channel
R
h
=
(
B
c
h
) to a good approximation; thus, with
V
in
q
or
Q
given by (5.39), both (5.108) and (5.112) yield immediately
h
and
A
c
=
c
k
=
(
a
+
1)
V
(5.113)
+
in which (
a
1) is of the order of 1.5 to 1.7, depending on whether Chezy or GM is
adopted to describe the flow. However, when the cross section does not have a wide
rectangular shape, the wave celerity (5.112) yields, with (5.39),
aQ
B
s
P
w
dP
w
dh
c
k
=
(
a
+
1)
V
−
(5.114)
where
B
s
is the width of the channel at the water surface;
dP
w
/
dh
is the rate of increase
in the wetted perimeter
P
w
with depth, which is zero for a wide channel.
Apparently, the quasi-steady-uniform flow approximation has been used as early as
1857 by Kleitz (1877, p. 172) and his fellow engineers on the Rhone River, and by Breton
in 1867 (Forchheimer, 1930). Equation (5.108) was also applied successfully with gage
heights on the Mississippi and Missouri Rivers by Seddon (1900); it is now sometimes
referred to as the Kleitz-Seddon law. The full implications of the approximation were
investigated by Lighthill and Whitham (1955). They called the wave motion
kinematic
,
because it arises from the elimination of the dynamic aspects of the momentum equation,
namely the first three terms of Equation (5.22), leading to the assumption of (5.101) and
(5.102).
