1. INTRODUCTION
There is a continuing controversy regarding the nature and applicability
of the kinematic wave model. Researchers and practitioners alike have reported
successes and failures of the model, with papers continuing to appear
in the literature describing what the model can and cannot do (Hromadka
and DeVries 1988). Current areas of concern focus on the following issues:
(1) Whether the kinematic wave can describe physical diffusion, and if so,
under what circumstances; While answers to these questions can be found in the literature, they are dispersed among various sources and not readily accessible. This difficulty appears to be fueling the current controversy (Dawdy 1990; Goldman 1990; Hromadka and DeVries 1990; Merkel 1990; Unkrich and Woolhiser 1990; Woolhiser and Goodrich 1990). Therefore, the aim of the present paper is to review the concept of kinematic wave, delineate its range of applicability, and critically examine its overall modeling philosophy. It is hoped that this review will help focus the attention of researchers and practitioners so that the controversy may be brought to a timely end.
2. BACKGROUND The concept of kinematic wave is well established among the existing methods to solve unsteady, onedimensional, gradually varied openchannel flow problems. In contrast to the dynamic wave, which features a significant inertial component, a kinematic wave is one in which the inertial component is too small to be of any practical importance. In unsteady openchannel flow modeling, a first type of kinematic wave arises when the governing equations are simplified by neglecting the local inertia, convective inertia, pressuregradient, and momentumsource terms (Lighthill and Whitham 1955). A second, less restrictive type can be formulated by neglecting the local inertia, convective inertia, and momentumsource terms, but keeping the pressuregradient term (Hayami 1951). To avoid confusion between these two types of kinematic waves, it is common practice to refer to the first type as kinematic wave proper, and to the second as diffusion wave (Ponce and Simons 1977).
From the physical standpoint, the kinematic wave assumption amounts to
substituting a uniform flow formula (such as Manning's or Chezy's) for the
equation of motion. In essence, it says that as far as momentum is concerned,
the flow can be considered steady. The unsteadiness of the phenomenon,
however, is preserved through the rateofrise term in the continuity
equation (Liggett 1975). From the mathematical standpoint, the kinematic wave assumption results in a considerable simplification of the equation of motion, reducing it to a statement of uniform flow (such as, for instance, the Manning equation). Combining this latter equation with the equation of continuity gives also to a firstorder partial differential equation, referred to as the kinematic wave equation:
in which Q = discharge; c = kinematic wave celerity; q_{L} = lateral inflow;
x = spatial variable; and
in which q = unitwidth discharge; i = effective rainfall intensity; and the other terms are as defined previously.
The kinematic wave celerity is defined as the slope of the rating curve,
either dischargeflow area Equation 1 (and by extension, Eq. 2) is a differential equation of first order; therefore, it can describe convection but not diffusion, which is a secondorder process. In practice, this means that the kinematic wave equation can describe the travel of a flood wave, but not its attenuation as it propagates downstream. Although Eq. 1 cannot describe diffusion, it is a quasilinear equation because the kinematic wave celerity is a function of discharge. This gives kinematic waves the tendency to change in shape as they propagate. If the celerity increases with discharge, the leading face of the wave will steepen; conversely, if the celerity decreases with discharge, the leading face of the wave will flatten out. In overland flow and inbank streamflow, the tendency is for the wave to steepen; in shallowoverbank streamflow, the tendency is for the wave to flatten out.
3. KINEMATIC WAVE SOLUTIONS Solutions to Eq. 1 (or Eq. 2) can be attempted in a variety of ways. Analytical solutions are possible for linearized analogs of the governing equations [see for instance Lighthill and Whitham (1955) and Ponce and Simons (1977)]. These solutions describe the convection of a flow quantity (either Q or q) with the celerity c in the absence of diffusion. Numerical solutions are possible using the method of characteristics or the finite difference method. Early work on the kinematic wave used the method of characteristics. In overland flow applications, the nonlinearity (or rather the quasilinearity) of the phenomena usually led to wave steepening and the eventual development of a kinematic shock. A kinematic shock is a kinematic wave that has steepened to the point where its rising limb has an almost vertical face, wherein the flow develops a singularity and loses its gradually varied property.
4. NUMERICAL DIFFUSION AND DISPERSION
Although kinematic wave solutions using the method of characteristics are
prone to shock development, ostensibly because of their lack of diffusion,
solutions using the finite difference method exhibit a somewhat different
behavior. Finite difference solutions, by virtue of their discrete nature, introduce
appreciable amounts of numerical diffusion and numerical dispersion.
The presence of numerical diffusion and dispersion in a numerical solution
using the finite difference method is at the crux of the controversy surrounding
the kinematic wave model. A numerical scheme is characterized by its
amplitude and phase error portraits (Leendertse 1967).
Finite difference solutions of the kinematic wave equation exhibit varying
amounts of numerical diffusion and dispersion depending on the type of scheme
used to discretize Eq. 1 (or Eq. 2). Numerical diffusion arises due to the neglect of the secondorder term of the corresponding Taylor series expansion of the related discrete analog. Numerical dispersion arises due to the neglect of the thirdorder term of the Taylor series expansion (Cunge 1969; Ponce et al. 1979). Therefore, in a typical application, numerical diffusion is usually about an order of magnitude greater than numerical dispersion. In practice, this means that as a rule, most of the error of numerical solutions can be attributed to numerical diffusion rather than to numerical dispersion. Exceptions are the cases in which the Courant number is significantly less than 1, in which case the amount of numerical dispersion may grow to the point where it compares in size with the amount of numerical diffusion. Numerical diffusion manifest itself as the diffusion or attenuation of the calculated runoff hydryograph. Since the kinematic wave equation has no builtin physical diffusion, it follows that a finite difference solution is actually simulating physical diffusion through numerical diffusion. The fact that the latter is artificial, and intrinsically related to the grid size, can be readily demonstrated by solving the same problem several times, each time halving the spatial and temporal increments [see for instance Ponce (1986)]. Carried to a practical limit, this test leads to the eventual disappearance of the numerical diffusion in question, with the result approaching the analytical solution of the kinematic wave. The recognition of this fact prompted the USDA Soil Conservation Service to retire its convex method of flood routing in the early 1980s. The convex method showed considerable sensitivity to the grid size, with the numerical diffusion vanishing as the grid size was gradually reduced. Numerical dispersion manifests itself as dispersion; that is, as the steepening or flattening of the rising limb of the calculated runoff hydrograph. In certain extreme cases, the numerical dispersion is responsible for the wiggles, or the (usually small) negative outflows observed at the beginning or end of the calculated runoff hydrograph (Hjelmfelt 1985). In practice, these negative outflows are small, and disappear (together with all traces of numerical dispersion) as the space and time steps are chosen such that the Courant number approaches 1.
Since numerical diffusion and dispersion are inherent in the choice of space
and time steps (and their ratio relative to the wave celerity), the result of a
finite difference solution of Eqs. 1 or 2 is per force grid dependent; i.e., the
calculated runoff hydrograph varies with the choice of grid size. Therefore,
it seems pointless to try to "calibrate" a kinematic wave model by varying
a physical parameter such as Manning's n in order to match
calculated results
with observed data.
5. DIFFUSION WAVE MODELING In practice, actual runoff hydrographs do exhibit a certain amount of diffusion. To properly simulate this diffusion it is necessary to extend kinematic wave theory to encompass the related diffusion wave theory. Following Hayami (1951) and Lighthill and Whitham (1955), the diffusion wave equation is derived by neglecting the local inertia, convective inertia, and momentumsource terms in the equation of motion, leading to the following equation for streamflow and channel flow:
and a similar equation for overland flow:
with v = hydraulic diffusivity, defined as follows:
and S_{o} = bottom slope; and other terms are as previously defined. Eqs. 3 and 4 describe the movement of kinematic waves with a diffusion component, for short diffusion waves. Unlike their counterparts Eqs. 1 and 2, Eqs. 3 and 4 are secondorder parabolic differential equations and are, therefore, able to describe physical diffusion, with the diffusion coefficient defined by Eq. 5. The diffusion wave equation can be solved either analytically, leading to Hayami's diffusionanalogy solution for flood waves, or numerically, with the aid of a numerical scheme for parabolic equations, such as the CrankNicolson scheme (Crandall 1956). An alternative is to extend the finite difference solution of the kinematic wave to the realm of diffusion waves by matching physical and numerical diffusivities (Cunge 1969; Dooge 1973). The physical diffusivity is the hydraulic diffusivity given by Eq. 5. The numerical diffusivity is the numerical diffusion coefficient of the discretized kinematic wave model, i.e., the coefficient of the leading (secondorder) error term. When the Muskingum scheme is used to model the kinematic wave, this method is referred to as MuskingumCunge model (Flood 1975; Ponce and Yevjevich 1979). The MuskingumCunge method has the significant advantage over conventional kinematic wave models of being essentially grid independent (Ponce and Theurer 1982; Ponce 1986). Therefore, calibration requires only a fine tuning of the frictional and crosssectional parameters such as Manning's n and flow rating exponent (either β for channel flow or m for overland flow). Then, the choice of spatial and temporal increments is based solely on considerations of problem scale.
6. APPLICABILITY OF KINEMATIC WAVES The question of the applicability of kinematic waves has interested researchers and practitioners alike. Although kinematic waves were originally used for describing river flows (Seddon 1900), it is in the field of overland flow that questions first arose with respect to its applicability and accuracy. Notable among these contributions is that of Woolhiser and Liggett (1967), who identified a parameter k defined as follows:
to characterize the applicability of the kinematic wave to overland flow situations.
In Eq. 6,
Ponce et al. (1978) used an analytical solution of the linearized equation
set (Lighthill and Whitham 1955) to develop criteria for the applicability of
kinematic and diffusion waves to open channel flow. They used sinusoidal
perturbations to the mean flow, with L = wavelength; and T = wave period.
in which u_{o} = reference flow mean velocity; d_{o} = reference flow depth; and
S_{o} = bottom slope. For diffusion waves, a parameter τ/F_{o}, in which F_{o} = reference flow Froude number, is shown to be a better descriptor of the overall accuracy, accounting for both amplitude and phase errors. A practical applicability criterion for diffusion waves is the following (Ponce et al. 1978):
in which g = gravitational acceleration; and the other terms are as previously defined. Ponce et al. (1978) confirmed the conclusions of Lighthill and Whitham (1955), as well as those of many others, that most overland flow situations would satisfy the kinematic wave criterion, and that most flood wave propagation cases in stream channels (excluding those with significant downstream control) would satisfy the diffusion wave criterion. Only in situations with markedly strong dissipative tendencies (for instance, a dambreak flood wave), flow into large reservoirs (with substantial backwater effects), or flow reversals would it be necessary to resort to the dynamic wave to properly describe the propagation of shallowwater waves. 7. OVERLAND FLOW VERSUS UNIT HYDROGRAPH The issue of whether an overland flow kinematic wave solution can replace (and perhaps eventually retire) the unit hydrograph as a practical method of runoff generation remains clouded in controversy. Due to the fundamental differences between these two methods, a resolution of this conflict does not appear to be forthcoming in the near future. The overland flow kinematic wave solution is a deterministic, distributedparameter, hydraulicdataintensive method (requiring geometric and frictional parameters), which is primarily applicable to small catchments [i.e., those less than 1 sq mi (2.5 km^{2 })], for which the idealizations inherent in mathematical modeling can be justified on practical grounds. In other words, for the kinematic wave solution to be useful, the discretization must reflect what is actually occurring in the field. When used indiscriminately, without due regard for problem scale, there is a risk that the amount of lumping introduced may interfere with the deterministic character of the method and its ability to simulate overland flows in a distributed context. In contrast to the overland flow kinematic wave solution, the unit hydrograph is a conceptual model of runoff generation, spatially lumped, and based exclusively on hydrologic data (streamflow measurements). Although originally derived for large catchments (Sherman 1932), the method has been found to have primary applicability to midsize catchments, i.e., those in excess of 1 sq mi (2.5 km^{2 }) and less than 400 sq mi (1,000 km^{2}). While these limits are somewhat arbitrary, they tend to reflect current hydrologic engineering practice. Furthermore, in the proper modeling context (i.e., with catchment subdivision), the applicability of the unit hydrograph can be extended to large catchments. Since the overland flow kinematic wave solution is primarily applicable to small catchments, and the unit hydrograph is primarily applicable to midsize (and, by extension, to large) catchments, it seems that there should be little overlap between these two methods. In practice, however, existing computer models [for instance, the U.S. Army Corps of Engineers HEC1 (HEC1 1985)] provide users with a choice between these two methods to solve any given runoff problem, regardless of scale. This raises the question of which method is better, or more accurate, for a given problem, a question that has no clear answer. The methods are of such different nature and have such different data needs that they are not readily comparable. Perhaps the only defensible argument in this regard is that the kinematic wave solution should increase in accuracy as the catchment size decreases; and the unit hydrograph should increase in practicality as the catchment scale increases. Specific comparisons between the two methods are likely to lead to heated arguments, but the central issue of accuracy is not likely to be settled soon. For one thing, the overland flow kinematic wave solution is based on our currently imperfect knowledge of friction mechanisms, including the estimation of Manning's n and of the rating exponent m describing the mixed laminarturbulent regime characterizing most overland flow situations. Likewise, the unit hydrograph would have to be verified with concurrent rainfallrunoff data, which are not readily available for the typical midsize catchment application. Keeping in mind the question of scale, the kinematic wave solution does have the significant advantage that it can describe spatial and/or temporal rainfall and roughness variations, which the unit hydrograph method, by virtue of its being lumped, cannot do. Therefore, in situations where the scale question can be reasonably compromised, the overland flow solution should provide better detail in the simulation of flood flows, including the description of runoff concentration and diffusion. Therein lies the promise of kinematic waves and the expectation of significant improvements in the accuracy of runoff prediction. As kinematic wave solutions continue to mature, particularly with the advent of a physically meaningful description of runoff diffusion, the way will be paved for the two methods of runoff generation to complement each other. There is an urgent need to develop synthetic unit hydrographs that reach beyond established practice [the Snyder unit hydrograph, to follow Corps of Engineers' practice; or the SCS dimensionless, to follow the Soil Conservation Service, (USDA: SCS 1985)]. Acting on this perceived need, the U.S. Bureau of Reclamation has developed a set of regional synthetic unit hydrographs for use within its jurisdiction (11 western United States) (USBR: Design 1987). In an attempt to overcome the shortcomings of conventional synthetic unit hydrographs, local agencies are engaged in developing synthetic unit hydrographs of the Sgraph type (Sabol 1987, 1990). It is envisioned that under the proper modeling context, the overland flow kinematic wave model may be used as a tool to develop synthetic unit hydrographs without the burden of an extensive (and expensive) network of streamflow data collection [see Overton (1970)]. A precedent for the use of models to synthesize peak flows already exists in U.S. hydrologic practice: the TR55 method (USDA: "Urban" 1986). This SCS method of peak flow generation was developed using the hydrologic catchment model TR20 (USDA: "Computer" 1983) to generate synthetic peak flows that take into account the catchment's concentration properties, regional temporal rainfall distribution, and eventabstraction mechanisms of infiltration and depression storage. The TR55 method improves on the rational method, substituting modeling for empiricism and leading to better and more consistent runoff predictions.
8. KINEMATIC SHOCK The kinematic shock was discussed in detail by Lighthitll and Whitham (1955); and since then, numerous studies have endeavored to analyze its causes and effects. But the subject continues to mystify researchers and practitioners alike (Cunge 1969; Kibler and Woolhiser 1970). The shock arises due to the nonlinear feature of kinematic waves, which under the right set of circumstances can result in the kinematic wave steepening to the point where it becomes for all practical purposes a wall of water. (In overland flow situations, the "wall of water" would be a small discontinuity in the water surface profile.) The shock is a direct consequence of the nonlinear steepening tendency, which is abetted when the following conditions occur (Ponce and Windingland 1985).
The preceding four physical conditions contribute to shock development. When all of them occur at the same time, the shock is very likely to develop. If only one or two of them are present, the shock may not develop. While the kinematic shock has been interpreted differently by many authors [see for instance Cunge (1969) and Kibler and Woolhiser (1970)], there is no doubt that the shock is physical and that it occurs under the proper set of highly selective circumstances. Unfortunately, adequate documentation of the occurrence of kinematic shocks in stream channels is lacking in the literature. Measurements are next to impossible, with sightings being all that diligent observers can settle for. The shock appears to be present in flash floods, with the related killer flood being a nefarious manifestation of the kinematic shock [see for instance Hjalmarson's account of the flood of July 26, 1982, in Tanque Verde Creek, east of Tucson, Arizona, in which the lives of eight unsuspecting bathers were claimed by what was in all likelihood a kinematic shock (Hjalmarson 1985)]. The conditions for kinematic shock development having been identified, the question remains as to whether the kinematic shock is as common in practice as calculations using an overland flow kinematic wave model would seem to indicate. For instance, the shock is a common occurrence in kinematic wave solutions using the method of characteristics. This is understandable, since this method solves the kinematic wave equation without introducing any numerical diffusion. A case in point: Kibler and Woolhiser (1970) used the method of characteristics to study the cascade of planes as a possible hydrologic model, and were able to derive a kinematic shock parameter as a function of geometric and frictional characteristics of two adjacent planes. However, in summarising their findings, Kibler and Woolhiser stated:
The shock is a much less common occurrence in finite difference solutions, particularly when these feature a large numerical diffusion component. For instance, the shock is conspicuously absent from the convex method, which by fully offsetting its derivatives features a substantial amount of numerical diffusion (Ponce et al. 1979). Characteristic solutions intrinsically satisfy the aforementioned first condition; finite difference solutions usually do not. However, as shown by Ponce and Windingland (1985), the shock can indeed develop in finite difference solutions, particularly when the four conditions are met concurrently. In practice, the shock is an uncommon occurrence in natural channels. In overland flow situations, the presence of shocks has been documented under highly selective circumstances, usually in connection with overland flow in long rectangular planes of constant slope (such as intense runoff on steep city streets and parking lots). Spatial rainfall nonuniformities and small topographic irregularities usually generate enough diffusionlike effects to counteract shock development. Therefore, the presence of the shock in a kinematic wave solution, more often than warranted (as in a characteristic solution), must be interpreted as the method's inability to properly account for catchment irregularities. Moreover, the practitioners' preference for kinematic wave finite difference solutions, where the shock is an uncommon occurrence, does not come as a great surprise (Alley and Smith 1982). The resolution of this conflict appears to be in the proper description of runoff diffusion within the context of a kinematic wave solution sensu lato. Diffusion will effectively counteract the nonlinear steepening tendency, which is at the root of shock development. Such an improved kinematic wave formulation should lead only to isolated instances of the shock's presence, and therefore be much more in agreement with physical reality.
9. DYNAMIC EXTENSION TO KINEMATIC WAVES Under the proper set of linearizing assumptions, kinematic wave theory can be extended to the realm of dynamic waves (Ponce 1990). Early work on this subject was done by Dooge (1973), who derived the expression for a dynamic hydraulic diffusivity ν_{d} , for the case of a wide channel with Chezy friction:
For overland flow, a general expression for the dynamic hydraulic diffusivity is:
which reduces to Eq. 9 for m = 3/2. For streamflow and channel flow, an expression for the dynamic hydraulic diffusivity is (Ponce 1986):
Since the hydraulic diffusivity vanishes at the condition of neutral stability, Eqs. 911 do account for dynamic wave behavior (Ponce and Simons 1977). This condition is characterized by the Vedernikov number V = 1. The Vedernikov number (Vedernikov 1945; Powell 1948; Craya 1952; Chow 1959) is:
Given Eqs. 1012, the dynamic hydraulic diffusivity can be expressed in terms of the Vedernikov number as follows:
For β = 1 [i.e., a channel of rapidly expanding top width such that the
wetted perimeter increases in the same proportion as the flow area (a channel
of constant hydraulic radius)], Eq. 12 predicts that
Given Eq. 13, it is possible to extend kinematic wave theory to the realm
of dynamic waves.
10. SUMMARY AND CONCLUSIONS
Kinematic and diffusion wave theories are reviewed prompted by the continuing
controversy regarding their nature and applicability. Kinematic waves
are shown to be nondiffusive but undergo change in shape during propagation
due to nonlinearity. In overland flow and inbank streamflow this feature
gives kinematic waves the capability of steepening, eventually leading
to the formation of the kinematic shock. The kinematic shock is shown to
be rare, and to occur only under a set of highly selective circumstances,
including: (1) A kinematic wave proper; (2) a low basetopeak flow ratio;
(3) a hydraulically wide and sufficiently long channel; and (4) a highFroudenumber
flow. Kinematic wave solutions using finite differences are shown to possess intrinsic amounts of numerical diffusion and dispersion, as a direct consequence of the finite grid size. These numerical effects are artificial, tending to disappear as the grid size is refined. In the limit, as the grid size approaches zero, the numerical effects vanish altogether. In practice, this means that overland flow kinematic wave solutions are grid dependent; that is, the results are a function of grid size, with a typical solution featuring appreciable amounts of numerical diffusion and dispersion. Kinematic wave modeling can be improved by extending kinematic wave theory to the realm of diffusion waves. In this way, the diffusion inherent in many practical runoff computations can be amounted for directly in the modeling, rather than as an afterthought. In this regard, the MuskingumCunge method is particularly attractive, because while remaining within the computational framework of kinematic wave models, it has enough physical information to compare favorably with implicit numerical solutions of the diffusion wave equation. Unlike conventional finite difference kinematic wave models, the MuskingumCunge method is shown to be grid independent, further underscoring its usefulness as a practical model of diffusion waves. The applicability of kinematic and diffusion waves is reviewed. It is concluded, echoing many past authors, that most overland flow situations would satisfy the kinematic wave criterion, and that most flood wave propagation cases in stream channels would satisfy the diffusion wave criterion. Only in situations with markedly strong dissipative tendencies or substantial downstream control would it be necessary to resort to the dynamic wave to properly describe the propagation of shallow water waves. The issue of the choice between kinematic wave and unit hydrograph methods for practical runoff computations is examined with the aid of the concept of catchment scale. The use of the kinematic wave method is indicated primarily for small catchments [those less than 1 sq mi (2.5 km^{2})], particularly in the cases in which it is possible to resolve the physical detail without compromising the deterministic nature of the model. The use of the unit hydlograph method is advocated for midsize catchments; i.e. those greater than 1 sq mi (2.5 km^{2}) but less than 400 sq mi (1,000 km^{2}), in which the kinematic wave method may prove difficult to implement. A case is made for the use of the kinematic wave as a tool for the development of synthetic unit hydrographs. The dynamic extension to kinematic and diffusion wave theory is reviewed with a view to the future. The dynamic extension is shown to properly account for the dependence of the hydraulic diffusivity on the Vedernikov number, allowing the simulation to be responsive to the dynamic effect. This type of modeling would be particularly applicable to channel and flow conditions such that the Vedernikov number is substantially different from zero, for instance, for nearcritical and supercritical inbank flows.
APPENDIX I. REFERENCES Abbott, M. B. 1976. "Computational hydraulics: A short pathology." J. Hydr. Res., 14(4), 271285. Alley, W. M., and P. E. Smith. 1982. "Distributed routing rainfallrunoff model ^{___} version II: User's manual." OpenFile Report 82344, U.S. Geological Survey Water Resources Division, Gulf Coast Hydroscience Center, NSTL Station, Miss. Chow, V. T. 1959. Openchannel hydraulics. McGraw Hill Book Co., Inc., New York, N.Y. "Computer program for project formulationhydrology." (1983). USDA Soil Conservation Service, Tech. Release No. 20 (TR20), Washington, D.C. Cunge, J. A. 1969. "On the subject of a flood propagation computation method (Muskingum method)." J. Hydr. Res., 7(2), 205230. Crandall, S. H. 1956. Engineering analysis: A survey of numerical procedures. McGrawHill Book Co., New York, N.Y. Craya, A. 1952. "The criterion for the possibility of roll wave formation." Proc., Gravity Waves Symp., Circular 521, U.S. National Bureau of Standards, Washington, D.C., 294332. Dawdy, D. R. 1990. Discussion of "Kitthmatic wave routing and computational errror." J. Hydr. Engrg., ASCE, 116(2), 278280. Design of Small Dams, 3rd ed. 1987. U.S. Bureau of Reclamation, Denver, Colo. Dooge, J. C. I. 1973. "Linear theory of hydrologic systems." Tech. Bulletin No. 1468, USDA Agricultural Research Service, Washington, D.C. Dressler, R. F. 1949. "Mathematical solution of the problem of roll waves in inclined open channels." Communications in Pure and Appl. Mathematics, 2, 149194. Flood Studies Report. 1975. Natural Environment Research Council, London, England, Vol. III. Goldman, D. 1990. Discussion of "Kinematic wave routing and computational error." J. Hydr. Engrg., ASCE, 116(2), 280282. Hayami, S. 1951. "On the propagation of flood waves." Bulletin of the Disaster Prevention Research Institute, Disaster Prevention Research Institute, Kyoto, Japan, 1(1), 116. HEC1, Flood Hydrograph Package: Users' Manual. 1985. U.S. Army Corps of Engineers, Hydrologic Engineering Center, Davis, Calif. Hjalmarson, H. W. 1984. "Flash flood is Tanque Verde Creek, Tucson, Arizona," J. Hydr. Engrg., 110(12), 18411852. Hjelmfelt, A. T. 1984. "Negative outflows from Muskingum flood routing." J. Hydr. Engrg., ASCE, 111(6), 10101014. Hromadka, T. V., and J. J. DeVries. 1988. "Kinematic wave and computational error." J. Hydr. Engrg., ASCE, 114(2), 207217. Hromadka, T. V., and J. J. DeVries. 1990. Closure of "Kinematic wave and computational error." J. Hydr. Engrg., ASCE, 116(2), 288289. Jolly, P. J., and V. Yevjevich. 1971. "Amplification criterion of gradually varied, single peaked waves." Hydrol. Paper No. 51, Colorado State University, Fort Collins, Colo. Kibler, D. F., and D. A. Woolhiser. 1970. "The kinematic cascade as a hydrologic model." Hydrol. Paper No. 39, Colorado State University, Fort Collins, Colo. Leendertse, J. J. 1967. "Aspests of a computational model for longperiod water wave propagation." RM.5294.PR, The Rand Corporation, Santa Monica, Calif. Liggett, J. A. 1975. "Basic equations of unsteady flow." Unsteady flow in open channels, K. Mahmood and V. Yevjevich, eds., Water Resources Publications. Fort Collins, Colo., Vol. 1. Lighthill, M. J., and G. B. Whitham. 1955. "On kinematic waves. I: Flood movement in long rivers." Proc., Royal Society, London, England, A229, 281316. Merkel, W. H. 1990. Discussion of "Kinematic wave routing and computational error." J. Hydr. Engrg., ASCE, 116(2), 282284. Morris, E. M., and D. A. Woolhiser. 1980. "Unsteady onedimensional flow over a plane: Partial equilibrium and recession hydrographs." Water Resour. Res., 16(2), 355360. Overton, D. E. 1970. "Route or Convolute?" Water Resour. Res., 6(l), 4352. Ponce, V. M., and Simons, D. B. 1977. "Shallow wave propagation in open channel flow." J. Hydr. Div., ASCE, 103(12), 14611476. Ponce. V. M., R. M. Li, and D. B. Simons. 1978. "Applicability of kinematic and diffusion models." J. Hydr. Div., ASCE, 104(3), 353360. Ponce, V. M., and Yevjevich, V. 1978. "MuskingumCunge method with variable parameters." J. Hydr. Div., ASCE, 104(12), 16631667. Ponce, V. M., Y. H. Chen, and D. B. Simons. 1979. "Unconditional stability in convection computations." J. Hydr. Div., ASCE, 105(9), 10791086. Ponce, V. M., and F. D. Theurer. 1982. "Accuracy criteria in diffusion routing." J. Hydr. Div., ASCE, 108(6), 747757. Ponce, V. M., and D. Windingland. 1985. "Kinematic shock: Sensitivity analysis." J. Hydr. Engrg., ASCE, 111(4), 600611. Ponce, V. M. 1986. "Diffusion wave modeling of catchment dynamics." J. Hydr. Engrg., ASCE, 112(8), 716727. Ponce, V. M. 1990. "Genenilized diffusion wave model with inertial effects." Water Resour. Res., 26(5). 10991101. Powell, R. W. 1948. "Vedernikov criterion for ultrarapid flow." Transactions, American Geophysical Union, 29(6), 882886. Sabol, G. V. 1987. "Development, use, and synthesis of S.graphs." Proc., Engrg. Hydrol. Symp., ASCE. New York, N.Y., 627632. Sabol". G. V. 1990. "Maricopa County hydrology procedure." Proc., Watershed Management Symp., ASCE, New York, N.Y., 423431. SCS National Engineering Handbook, Section 4: Hydrology. 1985. USDA Soil Conservation Service, Washington, D.C. Seddon, J. A. 1900. "River hydraulics." Trans., ASCE, 43, 179229. Sherman, L. K. 1932. "Streamflow from rainfall by unitgraph method." Engrg. News Record, 108, April 7, 501505. Unkrich, C. L., and D. A. Woolhiser. 1990. Discussion of "Kinematic wave routing and computational error." J. Hydr. Engrg., ASCE, 116(2), 284286. "Urban Hydrology for Small Watersheds." 1986. Tech. Release No. 55 (TR55), USDA Soil Conservation Service, Washington, D.C. Vedernikov, V. V. 1945. "Conditions at the front of a translation wave distributing a steady motion of a real fluid." Comptes Rendus (Doklady) de l'Academie des Sciences de l' U.R.S.S., (in French), 48(4). Woolhiser, D. A., and J. A. Liggett. 1967. "Unsteady onedimensional flow over a planethe rising hydrograph." Water Resour. Res., 3(3), 753771. Woolhiser, D. A., and D. C. Goodrich. 1990. Discussion of "Kinematic wave routing and computational error." J. Hydr. Engrg., ASCE, 116(2), 278288. APPENDIX II. NOTATION

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