CHANGES OF GRADE IN STRAIGHT CHANNELS 85 



in the analysis of the flow profile is to compute the critical and normal 

 depths for the given discharge. Since the cross section of the channel 

 remains the same through the change of grade, the critical depth 

 remains the same. The normal depth will be different on each side of 

 the change of grade, and its value relative to the value of the critical 

 depth furnishes the basis for classification. 



The first case shown at Fig. 801 (a) illustrates the change from a mild 

 slope to a flatter mild slope. The normal depth is greater for the 

 flatter mild slope, and both normal depths are greater than the critical. 

 An Ml backwater curve forms above the mild slope at the left. It 

 joins the greater normal depth above the break of grade. No curve 

 may form above the flatter slope because (1) none of the backwater 

 curves in channels of mild slope (Fig. 602) become tangent to the 

 normal depth in the downstream direction, and (2) the flow is deeper 

 than the critical throughout, so that the break in grade can affect the 

 profile in the upstream direction only. 



Figure 801 {b) shows a break in grade from a mild slope to a steeper 

 mild slope. An M2 curve forms upstream from the change of grade, 

 and joins the normal depth line of the steeper slope. This is also true 

 for Fig. 801 (c) where the change of grade is from mild slope to critical 

 slope. Flow at or near the critical slope, however, is especially subject 

 to chance fluctuations, so that the profile in Fig. 801 (c) cannot be pre- 

 dicted with as much assurance as that in Fig. 801(6). 



Figure 801 (c?) shows a transition from a mild slope to a steep slope. 

 An M2 curve forms above the mild slope, and an S2 curve forms above 

 the steep slope. The flow passes through the critical depth over the 

 change of grade. According to the ordinary theory, both of the back- 

 water curves become vertical as they approach the critical depth. 

 Actually, the profile does not cross the critical depth vertically, because 

 of the influence of the vertical components of velocity, which are neg- 

 lected in the theory of the backwater curves. For the same reason, 

 the critical depth may not occur precisely above the change of grade. 

 If it is necessary to know the exact shape of the profile in the imme- 

 diate neighborhood of the break in grade, recourse may be had to 

 model study. A short distance away on each side, slopes become flat 

 and the backwater curves apply with good accuracy. 



Figure 801(e) shows a change in grade from critical slope to mild 

 slope. A CI curve forms over the critical slope, connecting with the 

 normal depth in the channel of mild slope. This represents a passing 

 transition between the case of Fig. 801 (a), in which the backwater curve 

 extends a long distance upstream, and that of Fig. 801(g), in which the 

 curve extends downstream (if the tailwater is shallow). 



