## Kinematic Wave Model

The most simple type of distributed hydraulic routing model is the kinematic wave model. It is based on the following simplified form of the momentum equation (2):

in which S0 is the bottom slope of the channel (waterway) and a component of the term, dh/dx = dy/dx — S0, in which dy/dx is assumed to be zero. This assumes that the momentum of the unsteady flow is the same as that of steady, uniform flow described by the Manning equation or a similar expression in which discharge is a single-valued function of depth, i.e., 9Q/9A = dQ/dA = c. Also, since dA/dt = (idA/8Q)(dQ/dt) and Q = AV, Eq. (1) can be expanded into the classical kinematic wave equation, i.e.,

in which the kinematic wave velocity or celerity (c) is defined by Eq. (20).

Solutions for the kinematic wave equation (25) can be achieved using the method of characteristics or directly by finite-difference approximation techniques of either explicit or implicit types (Chow et al., 1988; Hydrologic Engr. Ctr., 1981; Linsley et al., 1986). The kinematic wave equation does not theoretically account for hydro-

graph (wave) attenuation. It is only through the numerical error associated with the finite-difference solution that attenuation of the hydrograph peak is achieved. Kinematic wave models are limited to applications where single-value, stage-discharge ratings exist—where there are no loop ratings—and where backwater effects are insignificant. Since, in kinematic wave models, flow disturbances can propagate only in the downstream direction, reverse (negative) flows cannot be predicted. Kinematic wave models are appropriately used as components of hydrologic watershed models for overland flow routing of runoff; they are not recommended for channel routing unless the hydograph is very slow rising, the channel slope is moderate to steep, and hydrograph attenuation is quite small. The range of application (with expected modeling errors less than 5%) for kinematic models, including the Muskingum method, is given by the following:

in which Tr is the time (in hours) of rise of the wave (hydrograph), i.e., the interval of time from beginning of significant rise to when the peak occurs; S0 is the bottom slope (in ft/ft), q0 is the unit-width discharge (Q/B) (in ft2/s), and n is the Manning roughness coefficient (Fread, 1985, 1992).