In the analysis of most free-surface flows in hydrology it can be assumed that the pressure distribution is hydrostatic normally to the bottom; this in turn allows the adoption of a uniform velocity profile. These two simplifications form the basis of shallow-water theory. In the resulting continuity and momentum equations, also referred to as the Saint Venant equations, the effects of viscosity and turbulence are parameterized in terms of a friction slope. These equations are not easy to solve in general, but important features of free-surface flow can be brought out by solutions of their linearized, diffusion, quasi-steady-uniform flow (or kinematic wave), and lumped kinematic approximations.

This type of flow is usually observed on surfaces with low permeability and has therefore been one of the central problems in urban hydrology; its relevance lies in the design of small engineering structures for roads, highways, airports, and also for some surface irrigation systems. Detailed numerical solutions of the shallow-water equations have shown that the analysis of most overland flow situations can be reliably simplified using the kinematic wave approximation. The basic assumption of this approach is that the friction slope can be taken to be equal to the slope of the underlying surface. Analytical solutions of the resulting equation, albeit for special runoff situations, still provide useful insight regarding the rising and recession hydrographs for more general applications.