70 BACKWATER CURVES — INTRODUCTORY 



Cases SI, S2, and S3 correspond somewhat to cases Ml, M2, and 

 M3, respectively. It will be noticed, however, that some of the slopes 

 and directions are reversed. The Ml , M2, and M3 curves change into 

 SI, S2, and S3 when the grade is increased until the normal depth is 

 less than the critical depth. The same part of the table from which 

 the Ml curve is calculated gives SI and S2. The M2, M3, and S3 

 curves are obtained from another part of the table. 



Cases CI and C3, formed of straight lines, represent the limiting 

 conditions between the Ml, M2, and M3 curves, on the one hand, and 

 the SI, S2, and S3 curves, on the other. The condition necessary for 

 the formation of the CI and C3 curves is that the slope be just sufficient 

 to compensate for friction when the normal depth and critical depth 

 become equal, a condition seldom encountered because of the delicate 

 balance of slope and roughness required. 



Cases H2 and H3 represent the limiting shapes approached by the 

 M2 and M3 curves as the slope of the channel bottom approaches a 

 true level. Under these circumstances the normal depth, yn, has 

 become infinite. Making use of the relations developed previously, 

 equation (606) simplifies to 



[609] 



When the bottom slopes upward, the slope may be considered nega- 

 tive, yn becomes negative, and hence z is negative. In using equation 

 (608) with Part 3 of Table 601, care must be taken to see that none of 

 these negative signs is overlooked. 



Point of control. In deriving Bresse's equation for the backwater 

 curves, no mention was made of the constant of integration. It was 

 omitted because there is no advantage in fixing the mathematical 

 origin of the x's at any particular point along the curve; the origin 

 might as well have whatever arbitrary location results from using the 

 equation in its most convenient form, without any constant of inte- 

 gration. The location of the curve along the stream is not arbitrary, 

 however. Its position is fixed by a point of control at one end or the 

 other of the backwater curve. For example, the longitudinal position 

 of the Ml curve is fixed by the requirement that the right-hand branch 

 of the curve must meet the water surface in the reservoir into which 

 the stream is discharging. 



