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2 TECHNIQUES OF WATER.RESOURCES INVESTIGATIONS TRANQUIL FLOWS e quently assumed. The assumption of stead iness is justified by the fact that at peak flows the discharge hydrograph flattens out and flow approximates steady conditions. High-water marks are left along the channel by these rela- tively steady, peak flows. Uniformity offlow is achieved by dividing channels into shorter lengths that are considered reasonably uni- form between cross sections, both on the basis of high-water marks and on channel geometry. When a steady discharge flows in a long channel of uniform hydraulic characteristics, it flows at a constant depth, called the normal ,depth, But adjacent subreaches are not identi- cal in channel dimensions. roughness, or bed slope, and so the normal depth in each is dif- ferent. A natural water-surface profile, there- fore. is a series of curves relating the normal depth in one subreach to the normal depth in the next. The normal depth line in a long stretch of river is, thus. rarely a long, smooth curve. Backwater curves The water-surface profiles resulting from nonuniform-flow conditions are called back- water curves. Various types of such profiles for gradually varied flows are described by Chow (1959) and Woodward and Posey (1941). Of the many backwater curves possible, those for subcritical flows on mild slopes (fig. 1), and those for supercritical flows on steep slopes (fig. 2), are generally of most concern, In fig- ures 1 and 2. the dashed lines represent the critical depth, y" and the straight, solid lines represent the normal depth, y,. The heavy, solid, curves lines represent the water-surface profiles from a control point to the normal- depth profile. (The expression "control point" or "control section" is defined as any section at which the depth of flow is known or can be controlled to a required stage.) Curve Ml in figure lA could be the result of a dam or a constriction downstream, or it could represent the backwater in a tributary which is flowing into a flooding main stream. Curve M2 could resultfrom a dropoff(falls or riffles) farther downstream where the water surface drops below the critical-depth line. The flow then passes into the supercritical regime. For both the Ml and M2 curves, the control is Yn>Yc :::::: M, ,/ - - - - - - - A - Normal depth line Watersurfaee Orawdown curve -, Yn Ya B HORIZONTAL SCALE IS CONDENSED Figure I.-Water-surface profiles on mild slopes. A. Tranquil flows. showing relations among Ml. 'II'll' and 11~ curves; B. Sketch showing typical instancesofM curves. e downstream; if the channel were long enough, both curves would asymptotically approach the normal-depth line upstream. Typical ex- amples M curves are shown in figure 18. In figure 2A, curve 83 could represent the profile of flow downstream from a sluice gate. Curve 82 could be the profile of flow which has just passed into the critical regime from a milder slope upstream, as might occur at a riffle. The control point for both these curves is upstream; if the channel were long enough, both curves would asymptotically converge toward the normal-depth line downstream. Typical examples of the 8 curves are shown in figure 28. Examination offigures lA and 2A will show that the lowest possible Ml curve and the highest possible M2 curve will each coincide with the normal-depth curve, which represents e