1. INTRODUCTION When dealing with freesurface flows, the modeler often finds himself resorting to various simplifications of the full equations, in the interest of practicality and/or mathematical tractability. It is therefore not surprising that despite the perceived trend toward computational sophistication, simplified freesurface flow models continue to hold the attention of researchers and practitioners alike. In practice, modeling proves to be both an art and a science, a duality which requires a judicious compromise between achievable accuracy and resource utilization. It is this duality which is at the root of the diversity in freesurface flow modeling approaches. The diffusion wave model is well established in the literature [Hayami, 1951; Lighthill and Whitham, 1955; Cunge, 1969; Dooge, 1973]. However, its capability to simulate inertial behavior has not been fully elucidated to date. As shown herein, under certain simplifying assumptions a generalized diffusion wave equation can be derived from the complete equations of continuity and motion (i.e., the shallowwater equations [Liggett, 1975]). Such a generalized diffusion wave model differs from Hayami's diffusion analogy [Hayami, 1951] in that the former includes inertia while the latter does not. This qualifies the generalized model as a diffusion wave model with inertial effects [Ponce, 1986]. Applications of the generalized diffusion wave theory are envisioned in both analytical and numerical modeling, ranging from the linear theory of hydrologic systems [Dooge, 1973] to strategies for twodimensional depthaveraged freesurface flow computational modeling [Ponce and Yabusaki, 1981]. 2. THEORETICAL DEVELOPMENT The theoretical background of the diffusion wave approach to unsteady freesurface flow can be traced back to Deymie [1938], Hayami [1951], Lighthill and Whitham [1955], and Dooge [1973]. Following Lighthill and Whitham [1955] the linearized analog of the equation of continuity a unitwidth analysis is:
in which the subscripts x and t denote differentiation with respect to space and time, respectively, and the subscript denotes a reference flow value, either flow depth h or mean velocity u. The linearized analog of the equation of motion [Lighthill and Whitham, 1955], assuming Chézy friction, is:
in which g is the gravitational acceleration and S_{0} is the bottom slope. The role of the inertial terms in describing runoff diffusion can be analyzed by a procedure similar to that used by Ponce and Simons [1977]. Accordingly, (2) is recast as:
in which the tracking variables a and b can take values of either 0 or 1, depending on whether the inertia terms are excluded from or included in the formulation. 3. DIFFUSION WAVE EQUATION WITH INERTIA Hyperbolic System The diffusion wave equation with inertia is derived by differentiating (1) with respect to x, (1) with respect to t, and (3) with respect to x. Combining the resulting equations into a secondother equation in terms of flow depth leads to:
In which the double subscripts represent double differentiation. In (4), ν is the noninertial hydraulic diffusivity based on the reference flow [Hayami, 1951]:
and F is the reference flow Froude number:
It can be readily shown that (4) is parabolic for the special case of a = b = 0 and hyperbolic otherwise, confirming that inertial systems are hyperbolic in nature. However. Ponce and Simons [19771 have shown that shallowwater flows are weakly hyperbolic, with the secondary wave (i.e., that associated with the C characteristic [Abbott, 19791) featuring a high degree of dissipation through a wide range of dimensionless wave numbers. Such a mechanical system is readily amenable to conversion to a parabolic analog, which ostensibly can propagate in only one characteristic direction (C+). Parabolic Analog to Hyperbolic System
To derive the generalized diffusion wave equation. a parabolic analog to the hyperbolic system of (4) is derived. To this end, (4) is differentiated with respect to space and, likewise, with respect to time, with the neglect of thirdorder terms justified by the usual license of differential calculus. The resulting secondorder equations are substituted into the right side of (4) to yield:
Equation (7) describes convection and diffusion and is of parabolic type. The left side of (7) describes the convection of the flow depth with a wave celerity c = 1.5u_{0} , applicable to a unitwidth channel governed by Chezy friction. The right side describes the diffusion of the flow depth with a Froudenumberdependent hydraulic diffusivity. This qualifies (7) as a generalized diffusion wave equation which includes inertial effects (in its description of runoff diffusion). Accordingly, the diffusivity of (7) can properly be referred to as inertial hydraulic diffusivity. By excluding or including the inertia terms, (7) specializes into the following models: (1) noninertial, (2) convective inertial, (3) local inertial, and (4) full inertial. The noninertial model [Hayami, 1951; Cunge, 1969] is obtained by setting a = b = 0 in (7):
in which ν is the noninertial hydraulic diffusivity, i.e., (5). The convective inertial model [Ponce and Simons, 1977] is obtained by setting a = 0 and b = 1 in (7):
in which [1 + 0.5 F ^{2} ] ν is the hydraulic diffusivity including convective inertia only. The local inertial model [Ponce and Yabusaki, 1981] is obtained by setting a = 1 and b = 0 in (7):
in which [1  0.75 F ^{2} ]ν is the hydraulic diffusivity including local inertia only. The full inertial model [Dooge, 1973] is obtained by setting a = 1 and b = 1 in (7):
in which [1  0.25 F ^{2} ]ν is the full inertial hydraulic diffusivity. A generalized diffusion wave equation encompassing all four models can he expressed as follows:
in which α is equal to 0 for the noninertial model, 0.5 for the convective inertial model, 0.75 for the local inertial model, and 0.25 for the full inertial model. The hydraulic diffusity of the full inertial model is Froude number dependent, decreasing with an increase in Froude number in the range 02. Assuming a wide channel and Chezy friction, its neutral Froude number (i.e., the Froude number for which the hydraulic diffusivity vanishes) is F = 2, which simulates that of the complete equations [Lighthill and Whitham, 1955; Ponce and Simons, 1977]. The hydraulic diffusivity of the noninertial model is independent of the Froude number, which limits the noninertial model to low Froude number flows. Therefore for high Froude number flows the noninertial model tends to overpredict the hydraulic diffusivity. The hydraulic diffusivity of the convective inertial model is Froude number dependent, increasing with an increase in Froude number. The hydraulic diffusivity of the local inertial model is also Froude number dependent, decreasing with an increase in Froude number in the range 0  (4/3)^{0.5}. Its neutral Froude number is F = (4/3)^{0.5} = 1.15.
It is seen that the convective inertial (α =  0.5) and local inertial (α = 0.75) models have an opposite behavior with regard to hydraulic diffusivity. When combined, the resulting full inertial model has α = 0.25, which is closer to the noninertial model (α = 0) than either convective or local models. Therefore, it is concluded that the noninertial model better approximates the hydraulic diffusivity of the full inertial model. For example, for a channel with u_{0} = 1 m s^{1}, h_{0} = 1 m, and bottom slope
4. SUMMARY AND CONCLUSIONS A generalized diffusion wave equation is derived on the basis of the linear analogs of the complete equations of continuity and motion of freesurface flow. Specializations of this equation lead to the following four types of diffusion wave models, depending on whether the inertia terms (local and convective) are excluded from or included in the formulation: (1) full inertial, (2) local inertial, (3) convective inertial, and (4) noninertial. Analysis of these diffusion wave models reveals substantial differences in their behavior, particularly with regard to the Froude number dependence of the hydraulic diffusivity. The full inertial and local inertial models have neutral Froude numbers, while the convective and noninertial models do not. In addition, the neutral Froude number of the full inertial model under Chezy friction simulates that of the complete equations (F = 2). For low Froude number flows, the noninertial model is shown to be a good approximation to the full inertial model. The noninertial model is a better approximation to the full inertial model than either local or convective models. APPENDIX I. REFERENCES
Abbott, M. B., Computational Hydraulics, 324 pp., Pitman. London, 1979. Cunge. J. A., On the subject of a flood propagation computation method (Muskingum method), J. Hydraul. Res., 7(2), 205230. 1969. Deynne, P., Propagation d'une intumescence allongée (probleme aval), Appl. Mech. Proc. Int. Congr., 5th, 537544, 1938. Dooge, J. C. I., Linear theory of hydrologic systems, U.S. Dep. Agric. Tech. Bull., 1469, 327 pp., 1973. Hayami, S., On the propagation of flood waves, Bull Disaster Prev. Res. Inst., Kyoto Univ., 116 1931. Liggett, J. A., Basic equations of unsteady flow, in Unsteady Flow in Open Channels, vol. I, edited by K. Mahmood and Yevjevich. pp. 29.02. Water Resources Publications, Fort Collins, Colo., 1975. Lighthin, M. J., and G. B. Whitham, On kinematic waves, I, Flood movement in long rivers, Proc. Soc. London, Ser. A, 229, 281316,1935. Ponce. V. M., Diffusion wave modeling of catchment dynamics. J. Hydraul. Eng., 112(8),716727, 1986. Ponce, V. M., and D. B. Simons, Shallow wave propagation in open channel flow, J. Hydraul. Div., Am. Soc. Civ. Eng., 103(HY11). 14611476, 1977. Ponce, V. M., and S. Yabusaki, Modeling circulation in depthaveraged flow, J. Hydraul. Div., Am. Soc. Civ. Eng., 107(HY11). 15011518, 1981.

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