The slope of the water surface dy/dx depends on the slope of the bed So, the ratio of the normal
depth yo to the actual depth y and the ratio of the critical depth yc to the actual depth y. The
difference between flow resistance for steady uniform flow no to flow resistance for steady
nonuniform flow n is small and the ratio is taken as 1.0. With n = no, there are twelve types of
water surface profiles. These are illustrated in Figure 2.28 and summarized in Table 2.3.
When y → yc, the assumption that acceleration forces can be neglected no longer holds.
Equations 2.171 and 2.172 indicate that dy/dx is perpendicular to the bed slope when y → yc. For
cross-sections close to the cross-section where the flow is critical (a distance from [3 to 15 m (10
to 50 ft)]), curvilinear flow analysis and experimentation must be used to determine the actual
values of y.
Figure 2.28. Classification of water surface profiles.
When analyzing long distances 30 to 300 m (100 to 1,000 ft) or longer one can assume
qualitatively that y reaches yc. In general, when the flow is rapid (Fr ≥ 1), the flow cannot become
tranquil without a hydraulic jump occurring. In contrast, tranquil flow can become rapid (cross the
critical depth line). This is illustrated in Figure 2.29.
When there is a change in cross-section or slope or an obstruction to the flow, the qualitative
analysis of the flow profile depends on locating the control points, determining the type of curve
upstream and downstream of the control points, and then sketching the backwater curves. It
must be remembered that when flow is rapid (Fr > 1), the control of the depth is upstream and
the backwater proceeds in the downstream direction. When flow is tranquil (Fr < 1), the depth
control is downstream and the computations must proceed upstream. The backwater curves that
result from a change in slope of the bed are illustrated in Figure 2.29.