Geoscience Reference
In-Depth Information
Fig. 7.4 Monoclinal wave, as a
gentle transition
between two uniform
quasi-steady regimes.
Its rate of
propagation is given
by Equation (7.9).
c
s
h
1
h
2
are frictional resistance and gravity. It is often considered the kinematic equivalent of
the dynamic shock, because the wave front constitutes a moving transition between two
regions of essentially uniform flow.
Again, the problem can be analyzed within a reference frame moving at the same
velocity as the wave (Figure 7.4). This yields, as before, Equation (7.5) for the continuity
equation, or
q
2
−
q
1
c
s
=
(7.9)
h
2
−
h
1
In a perfect kinematic system this result suggests a discontinuity in
h
or in its deriva-
tive, because in order to keep
c
s
a constant,
dq
/
dh
must also be constant, even though
q
q
(
h
) is nonlinear. This is the reason why the monoclinal rising wave has also been
referred to as a kinematic shock. In actual rivers, however, flood rises, which are subject
mainly (but never only) to the effects of gravity and friction, usually do not display any
such discontinuities and they are quite continuous and smooth; they usually extend over
large distances and the transition or shock thickness may be considerable. In view of this
ambiguity, the question arises whether the type of motion predicted by Equation (7.9)
can even exist or maintain itself in the real world. The matter can be resolved by consid-
ering the full equation of motion (5.22) to determine under what conditions diffusion and
wave steepening might be in balance, to allow the stability of the uniformly progressive
flow assumed in the derivation of (7.9). By means of a moving coordinate system it
can be shown that the monoclinal wave profile is actually stable, as assumed, provided
h
1
>
=
h
cr
, in which
h
cr
is the critical depth, and provided the wave front extends an
infinite distance downstream. It can also be shown that in most practical cases on large
rivers the wave will rise to 0.90(
h
1
−
h
2
>
S
0
, which
is usually of the order of some tens of kilometers; this means that the monoclinal rising
wave is often well approximated on long rivers. The details of the analysis, although
straightforward, are beyond the present scope and various aspects can be found elsewhere
(Lighthill and Whitham, 1955; Henderson, 1966).
h
2
) within a distance of the order of
h
2
/
