CHAPTER IX 

 STEP METHODS FOR BACKWATER CURVES 



In natural watercourses or artificial channels with frequent changes 

 of section and grade, step methods become more convenient than the 

 integration methods discussed in Chapters VI and VII. The stretch 

 to be studied is divided into short reaches. Trial-and-error compu- 

 tations are made for each reach, based upon the data for the reach and 

 the result of the computation for the preceding reach. This neces- 

 sitates carrying the computations step by step from one end of the 

 stretch to the other. The reaches need to be short enough to reduce 

 to within permissible limits the error in approximating the true water- 

 surface slope through the reach by the average of the surface slopes 

 at each end, or by the slope corresponding to the average of the hydraulic 

 properties in the reach. Since the radius of curvature of many back- 

 water curves increases continuously in one direction, the error may be 

 systematic, but by using short reaches it can be made much smaller 

 than the error that would be introduced by considering the channel to 

 be uniform. 



The amount of departure introduced by lack of uniformity in the 

 channel can be considerable. Figure 901 shows the effect of different 

 degrees of non-uniformity upon what would be a smooth curve in a 

 uniform channel. The integration methods described in Chapters 

 VI and VII (or a special direct step method described in this chapter) 

 can be used for the uniform channel, but for the irregular channels a 

 trial-and-error step method is necessary if the most accurate results are 

 to be obtained. 



The step computations for the water-surface profile of Fig. 901 (&) 

 are started at the dam and carried upstream. For the profile of Fig. 

 901(c), the computations cannot be carried through the reach from 

 end to end, but have to be figured upstream and downstream from the 

 point of control, and upstream from the dam. Where the depth of flow 

 exceeds the critical, the step computations should be carried upstream, and 

 where the depth of flow is less than the critical, the computations should 

 proceed downstream. This simple rule serves as a useful guide, but it 

 needs to be supplemented by a knowledge of the methods presented 

 in Chapter VIII if the most complicated cases are to be solved expe- 

 ditiously. 



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