When modeling flood
waves, often the first question that comes to mind is: What type of wave should
Over the past 50 years, the approach that seems to have prevailed in some quarters is the following: "Forget
about the various types of waves; let's use
the complete solution of the St. Venant equations
in We aim to show that the sole use of the mixed wave approach is at best futile, and at worst, wrong; and very likely to lead to wasted time and resources. For added clarity, in the following section we list the various types of waves in current use, while elaborating on their nature and properties.
In one-dimensional
unsteady free-surface flow,
the following four wave types are in general use:
The properties of these wave types have been examined in detail by
π /L) times
the reference channel length L, i.e., the
length of channel that it would take the
equilibrium flow to drop a head equal to its depth.
_{o}
Kinematic waves are those of
Diffusion waves
lie to the right of kinematic waves and to the left
of mixed kinematic-dynamic waves in the wavenumber spectrum (Fig. 2). Unlike kinematic waves, which feature zero diffusion,
diffusion waves have a small but perceptible amount of diffusion. However,
this diffusion is small compared to that of the mixed waves (Fig. 3).
We note that the inclusion of
Figure 2 shows that Seddon's kinematic waves,
lying toward the left of the dimensionless wavenumber spectrum,
feature a constant wave celerity and are, therefore, nondiffusive.
Following the same rationale,
Lagrange's dynamic waves, lying toward the right,
are also nondiffusive. However,
the mixed waves, lying toward the middle-to-right
and featuring sharply varying celerity,
are shown to be strongly diffusive. The amount of diffusion,
characterized by the logarithmic decrement
A kinematic
wave may be indeed regarded as the quintessential flood wave.
Theory tells us that a kinematic wave does not attenuate.
Practical experience would indicate that if a wave attenuates very
quickly, it is most likely Ponce and Simons, 1977Ponce and Windingland, 1985).
At this juncture, we endeavor to quote
Unlike kinematic waves,
diffusion waves are subject to a small amount of diffusion. They lie Ponce, 2024).
The value = 0.01
depicts σ_{*} = 0.17
depicts σ_{*}
Typical flood waves diffuse somewhat; therefore, diffusion waves are indeed a practical
model of flood wave propagation. They complement kinematic waves rather nicely, while finding their best application
in cases where
wave diffusion is appreciable and its calculation is deemed necessary.
The matter of how to best handle the numerical diffusion
has been resolved by
Classical dynamic waves are those of Lagrange (1788). More recently, Fread (1985) and others have referred to the mixed kinematic-dynamic waves as "dynamic" waves, while herein we refer to them simply as "mixed" waves. The semantic confusion is judged to be unfortunate. In an attempt to fix the problem, in this article we use the adjective "dynamic" to refer solely to the Lagrange waves.
Dynamic waves feature
a constant wave celerity for dimensionless wavenumber
≥ 1000 for all Froude numbers (Fig. 2).
This means conclusively, as with kinematic waves, that the dynamic waves of Lagrange are σ_{*}not subject to diffusion.
The dynamic waves of Lagrange are not the typical flood waves.
Their size is too small to constitute a veritable flood risk. Their appplication is restricted
to short wave propagation in irrigation and power canals, where the scale of the
disturbance is such that it may be actually seen,
or perceived, with the naked eye. Unlike flood waves, which are mass waves that feature only
For enhanced clarity, a parenthetical comment regarding the cause of wave diffusion is advisable here.
Diffusion is produced by the interaction of the pressure gradient with
the friction and gravity terms (Table 1, Row 2).
More precisely defined, diffusion is produced by the interaction of the
At this juncture, it remains for us to
discuss the only other wave type left: The mixed kinematic-dynamic wave, for short, the "mixed" wave
of unsteady open-channel flow. Since, by definition, this wave features The answer to
this question is Yes!
The mixed kinematic-dynamic wave is indeed
Within the F < 1),
where the attenuation is shown to be stronger (greater values of δ), the logarithmic
decrement is seen to vary from a low of δ = 0.0021
= 0.001 σ_{*}δ = 180
= 90σ_{*}
Table 2, Line 0 (emphasized with yellow background) shows a very small amount of wave
attenuation, 0.02%, or
Table 2, Line 3a (with yellow background) purposely depicts a wave attenuation of 0.17.
Table 2, Line 4a (with yellow background) purposely depicts a wave attenuation of 2.
Table 2, Line 5a (with yellow background) purposely depicts a wave attenuation of 90. Table 2, Col. 5 shows the indicated wave types, from kinematic, with very small attenuation (0.0002), to diffusion, with small to medium attenuation (0.0021 to 0.1894), to mixed wave, with large to very large attenuation (0.3 to 0.9999). Table 2, Lines 5 and 6 depict a nonexisting wave; the wave having disappeared completely, with its mass going on to form part of the underlying, or equilibrium, flow.
Given that wave attenuation is
We conclude that most, if not all, mixed waves would have effectively lost all their strength
in most cases of practical interest.
They lose their strength rapidly due to their highly diffusive nature,
the latter due to the competition between kinematic and dynamic terms (read forces) that
are comparable in size.
It follows that mixed waves lack a basic property of a flood wave, namely, its
Every rule is likely to call for an exception. In the previous section (Section 7), we presented an elaborate mathematical rationale for why the mixed wave is not likely to apply for the case of a general flood, i.e., one that is subject to very little or no attenuation. Yet we reckon that there is one particular flood wave that actually may diffuse appreciably. This is the case of a dam-breach flood wave.
Typically, the flood wave produced by the breaching of an earthen embankment
is sudden, lasting about 3 hr, a sure candidate for strong wave diffusion.
A case in point:
Of 24 dam failures in the United States documented by
Such flood waves (Fig. 4) are apt to fall under the category of
mixed wave or, at the very least, be a
This section elaborates on ways to model flood waves. It seeks to answer the question: Now that I chose a type of wave, how should I proceed? What actual tool should I use in a real practical situation? This section is divided into three parts: (1) kinematic waves, (2) diffusion waves, and (3) mixed kinematic-dynamic waves. The classical dynamic waves of Lagrange described in Section 6 lie outside of the scope of this section.
Kinematic wave modeling may be performed in two ways.
The first way is to realize that a true kinematic wave does not attenuate; therefore, subsidence, or
diffusion, is out of the question. Still, the wave is actually moving downstream with a certain celerity,
and that speed is subject to calculation.
Indeed, that speed is Seddon's celerity, which states
that the velocity of a flood wave at a given cross-section is equal to the slope
of the rating curve (d
The simplicity of Seddon's celerity is remarkable,
providing a ready tool to assess flood movement with a minimum of computational effort.
The second way to perform kinematic wave modeling is to use a numerical model,
many of which exist in various forms, in the literature and elsewhere. These models, however,
suffer from a decided conundrum: How to properly model a kinematic
wave without introducing a certain amount of numerical diffusion
associated itself with the finite grid size . [Note that a kinematic wave proper is not supposed to have There does not appear to be a way out of this difficulty. At this juncture, the best that can be stated is that, for a sufficiently fine grid resolution, the numerical diffusion should reduce itself to where it may not be of much concern in a given practical application.
Unlike kinematic waves, which are governed by a first-order differential
equation, describing only convection, diffusion waves are governed by a second-order equation, describing convection
The avowed feature of
Mixed waves comprise all terms in the
governing equations of water continuity and motion, that is, the St. Venant equations (Table, 1, Line 3).
The inclusion of the inertia terms is indeed forceful, but is not without its pitfalls.
The resulting wave
is
Another significant pitfall is that the numerical solution of the complete St. Venant equations
represents an order-of-magnitude increase in complexity in the formulation and actual performance of the
numerical analog chosen to model the full equations. A scheme that appears to be
widely favored by practitioners is
the Preissmann box scheme ( θ = 0.5.
In practice, however, center-weighing the Preissmann scheme does not work,
because it leads to strong
numerical instabilities, which eventually render it inoperable. θ > 0.5, typically in the range 0.55-0.60,
to stabilize the scheme by
providing a certain amount of numerical diffusion to control the computation.
Greater values of θ, in the range 0.6-1.0,
provide increasing amounts of numerical diffusion,
but this is always at the expense of increased nonconvergence (Fig. 6). Thus,
the methodology
is seen to degrade to first-order, compromising the original advantage predicated on
the use of a complete "dynamic wave" model (i.e., our mixed wave model).
Yet another significant pitfall of the numerical solution of the St. Venant equations is that the model
We point out that the comments of this subsection
purposely exclude U.S. government
software such as the To sum up, by now it must be widely apparent that the mixed kinematic-dynamic wave is not what its users had originally in mind. The mixed wave is shown to be fraught with difficulties, the least of them being the realization of whether the said wave is there or not for us to calculate it! More commonly, the modeler will face other problems, of both a numerical and physical nature, which will have the net effect of casting doubts on the accuracy and practicality of the overall procedure.
We have analyzed the celerity and attenuation
properties of four types of shallow-water waves currently in use in hydraulic engineering:
(1) kinematic, (2) diffusion, (3) mixed kinematic-dynamic, and
We have sought to answer the question of whether the mixed wave is generally too strongly diffusive
to be considered a practical flood wave. The answer is Note that only in the extremely unusual case of a dam-breach flood wave could we be actually confronted with the case of a mixed flood wave. A dam-breach flood wave is characteristically sudden, poised by Nature to be a mixed wave, an unusual type of flood wave [The experience of the Teton dam failure (Fig. 4) is a case in point]. Professionals in charge of forecasting or hindcasting a dam-breach flood wave would be keen to keep this in mind. For all other flood wave routing applications, the kinematic and diffusion waves should do the job in an accurate and forthright manner.
Notably, since a diffusion wave will actually calculate diffusion, including the case of zero diffusion,
it follows that the solution of a diffusion wave encompasses the solution of a kinematic wave.
Therefore, the diffusion wave is postulated as the flood wave
In general, mixed kinematic-dynamic waves,
herein simply referred to as
Abbott, M. B. 1976.
Chow, V. T. 1959.
Cunge, J. A. 1969.
Fread, D. L. 1985. "Channel Routing," in
Hayami, I. 1951.
HEC-RAS.
Lagrange, J. L. de. 1788.
Lighthill, M. J. and G. B. Whitham. 1955.
McCarthy, G.T. 1938. "The Unit Hydrograph and Flood Routing," unpublished manuscript, presented at a Conference of the North Atlantic Division, U.S. Army Corps of Engineers, June 24. (Cited by V. T, Chow's text "Open-channel Hydraulics," page 607).
Natural Environment Research Council. 1975.
Ponce, V. M. and D. B. Simons. 1977.
Ponce, V. M. and V. Yevjevich. 1978.
Ponce, V. M., H. Indlekofer, and D. B. Simons. 1978.
Ponce, V. M. 1982.
Ponce, V. M. and D. Windingland. 1985.
Ponce, V. M. 1986.
Ponce, V. M. 1995.
Ponce, V. M. 2014a.
Ponce, V. M. 2014b.
Ponce, V. M. 2024.
Seddon, J. A. 1900.
Taher-Shamsi, A., A. V. Shetty, and V. M. Ponce. 2003.
Wylie, C. R. 1966. |

240516 1100 |