BRESSE'S BACKWATER CURVES 63 



tances sufficiently short for the effect of friction to be negligible. The 

 approximation would be poor near the critical depth in any case, for in 

 deriving the equation the effect of vertical components of velocity was 

 neglected. These become important when the value of dy/dx is large, 

 as it is near the critical depth. 



Another branch of the curve representing the mathematical equation 

 lies below the bottom of the channel. It is omitted because it has no 

 real meaning in this connection. 



PROBLEMS 



601. Prove, from the equation of the curve of Fig. 601, that branch II has a 

 horizontal asymptote at the location shown on the graph. 



602. What determines whether water flowing at the critical depth will follow 

 branch I or II? 



Bresse's backwater curves. Theoretical backwater curves including 

 the effect of friction as well as that of the change of kinetic energy were 

 first treated by Bresse, who considered only an infinitely wide rectangu- 

 lar channel, and used the Chezy formula for the evaluation of the effect 

 of friction.^ The Chezy formula is now outmoded, and it is known that 

 the shape of the channel may have an appreciable effect, so that Bresse's 

 curves cannot be used if the greatest accuracy is desired. However, 

 they are by far the most convenient for computation, and find use when 

 the available data or the results desired do not justify a more accurate 

 procedure. Where the channel is wide and flat, and the constant in 

 Chezy's formula does not vary over too wide a range, the results from 

 Bresse's method are as close to the actual observed values as are results 

 from the best of the other methods. The simplicity of Bresse's method 

 makes it useful in studying the various cases of the backwater curves. 

 Curves plotted from the tables prepared by Bresse are similar in general 

 shape to those plotted from tables of the more complicated backwater 

 functions. 



Bresse's method makes use of Chezy's formula, which is intended to 

 apply only to steady uniform flow, when the water surface is parallel to 

 the bottom. In that case the slope of the bottom, the slope of the water 

 surface, and the slope representing the rate at which head is being used 

 up in overcoming friction are all the same. If the flow is non-uniform, 

 all three slopes may be different. At a given section of a stream in 

 which the flow is non-uniform, we assume that the friction slope is 

 given correctly by Chezy's formula, so that if the channel is assumed to 



»J- A. C. Bresse, "M^chanique Appliquee," v. 2, Mallet-Bachdier, Paris, 1860. 



