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<br />following: <br /> <br />Sf = So <br /> <br />(8) <br /> <br />Equation (8) essentially states that the momentum of the flow can be <br />approximated with a uniform flow assumption as described by Manning's or Chezy's <br />equation. Manning's equation can be written in the following form: <br /> <br />Q = aAm <br /> <br />(9) <br /> <br />where a and m are related to flow geometry and surface roughness. <br />Since the momentum equation has been reduced to a simple functional relationship <br />between area and discharge, the movement of a flood wave is described solely by the <br />continuity equation. Writing the continuity equation in the following form: <br /> <br />aA+aQ=q <br />at ax <br /> <br />(10) <br /> <br />Then by combining equations (9) and (10), the governing kinematic wave' <br />equation is obtained as: <br /> <br />aA + amAlm-1J aA = q <br />at ax <br /> <br />(11 ) <br /> <br />Because of the steady uniform flow assumptions, the kinematic wave <br />equations do not allow for hydrograph diffusion, but just simple translation of the <br />hydrograph in time. The kinematic wave equations are usually solved by explicit or <br />implicit finite difference techniques. Any attenuation of the peak flow that is <br />computed using the kinematic wave equations is due to errors inherent in the finite <br />difference solution scheme. <br /> <br />The application of the kinematic wave equation is linked to flow <br />conditions that do not demonstrate appreciable hydrograph attenuation. In general, <br />the kinematic wave approximation works best when applied to steep (10 ftlmi or <br />greater). well defined channels, where the flood wave is gradually varied. The <br />kinematic wave approach is often applied in urban areas because the routing reaches <br />are generally short and well defined (i.e. circular pipes, concrete lined channels, etc...). <br />The kinematic wave equations cannot handle backwater effects since with a kinematic <br />model flow disturbances can only propagate in,the downstream direction. All of the <br />terms in the momentum equation that are used to describe the propagation of the <br />flood wave upstream (backwater effects) have been excluded. <br /> <br />7-68 <br />