The depth–slope product is used to calculate the shear stress at the bed of an open channel containing fluid that is undergoing steady, uniform flow. It is widely used in river engineering, stream restoration, sedimentology, and fluvial geomorphology. It is the product of the water depth and the mean bed slope, along with the acceleration due to gravity and density of the fluid. Formulation The use of the depth–slope product — in computing the bed shear-stress — specifically refers to two assumptions that are widely applicable to natural river channels: that the angle of the channel from horizontal is small enough that it can be approximated as the slope by the small-angle formula, and that the channel is much wider than it is deep, and sidewall effects can be ignored. Although it is a simplistic approach to find the shear stress in what can often be a locally unsteady fluvial system, when averaged over distances of kilometers, these local variations average and the depth–slope product becomes a useful tool to understand shear stress in open channels such as rivers. Depth and hydraulic radius The first assumption is that the channel is much wider than it is deep, and the equations can be solved as if the channel were infinitely wide. This means that side-wall effects can be ignored, and that the hydraulic radius, , can be assumed to be equal to the channel depth, . where is the cross sectional area of flow and is wetted perimeter. For a semicircular channel, the hydraulic radius would simply be the true radius. For an approximately rectangular channel (for simplicity in the mathematics of the explanation of the assumption), , where is the width (breadth) of the channel, and . For b>>h, , and therefore . Formally, this assumption can generally be held to hold when the width is greater than about 20 times the height; the exact amount of error accrued can be found by comparing the height to the hydraulic radius. For channels with a lower width-to-depth ratio, a better solution can be found by using the hydraulic radius instead of the above simplification. Pressure The total stress on the bed of an open channel of infinite width is given by the hydrostatic pressure acting on the bed. For a fluid of density , an acceleration due to gravity , and a flow depth , the pressure exerted on the bed