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Sf = So - ~ - v av - 1 av (7) ax gax gat Steady Uniform Flow Kinematic Wave Approx. Steady Nonuniform Flow Diffusion Wave Approximation Steady Nonuniform Flow Quasi-Steady Dynamic Wave Approximation Unsteady Nonuniform Flow Full Dynamic Wave Equation The use of approximations to the full equations for unsteady flow can be justified when specific terms in the momentum equation are small in comparison to the bed slope. This is best illustrated by an example taken from Henderson's book aDen Channel Flow, 1966. Henderson computed values for each of the terms on the right hand side of the momentum equation for a steep alluvial stream: Maanitude (ft/Mil: 26 .5 .12-.25 lav g at .05 Term: So ~ ax vav g ax These figures relate to a very fast rising hydrograph in which the flow increased form 10,000 cfs to 150,000 cfs and decreased again to 10,000 cfs within 24 hours. Even in this case, where changes in depth and velocity with respect to distance and time are relatively large, the last three terms are still small in comparison to the bed slope. For this type of flow situation, an approximation of the full equations would be appropriate. 6.3. Kinematic Wave Approximation. Kinematic flow occurs when gravitational and frictional forces achieve a balance. In reality, a true balance between gravitational and frictional forces never occurs. However, there are flow situations in which gravitational and frictional forces approach an equilibrium. For such conditions, changes in depth and velocity with respect to time and distance are small in magnitude when compared to the bed slope of the channel. Therefore, the terms to the right of the bed slop in equation (7) are assumed to be negligible. This assumption reduces the momentum equation to the 7-67