72 BACKWATER CURVES — INTRODUCTORY 



an infinitely wide rectangular cross section are still present. In some 

 common cases the errors due to this assumption and to using a constant 

 value of C tend to offset each other. If more accuracy is desired than 

 can be obtained by Bresse's method with C constant, the methods 

 described in later chapters should be used. 



Flow in precipitous channels. In extremely steep channels, such as 

 those which are sometimes built in mountainous country, the condition 

 of uniform flow at normal depth is not stable. Instead, the water tends 

 to flow in a series of pulsations, " roll waves," or " slugs," with the 

 deepest part much deeper than the depth which would be indicated by 

 the standard formulas for uniform flow. This may easily result in over- 

 topping, which is always undesirable, and which may be dangerous 

 where ground slopes are steep. Although no complete analysis of the 

 phenomenon has been made which will allow the shape and velocity of 

 the waves to be predicted, something is known of the conditions under 

 which they will form. If the slope is long enough and steep enough, the 

 roll waves will form without any perceptible disturbance to start them. 

 They will start much sooner, however, and may build up to maximum 

 size in a fairly short channel if there is some initial disturbance such as a 

 side obstruction or irregularity near the upper end of the channel, or a 

 train of waves in the reservoir at the head of the channel. 



A value for the minimum slope that will sustain pulsating flow of this 

 type has been obtained by Jeffreys and by Thomas, who used different 

 methods, but who both assumed that Chezy's formula could be used to 

 evaluate the effect of friction, and that the channel had an infinitely 

 wide rectangular shape.^'* They obtain the same result, which is, in 

 our notation, 4g/C^, or four times the ordinary critical slope. Using 

 Manning's formula to evaluate friction, and assuming an infinitely wide 

 rectangular channel, Keulegan and Patterson obtain as a result a slope 

 of 2j times the ordinary critical slope. ^ Observations, however, indicate 

 that serious roll-wave disturbances seldom form except on steeper slopes 

 than indicated by either of these criteria. This may be because a very 

 great length would be required for the waves to build up, at slopes near 

 the minimum, or because the neglected frictional effect of the channel 

 sides is appreciable. 



Backwater curves, as described in this book, cannot exist in pre- 



^ " The Flow of Water in an Inclined Channel of Rectangular Section," by Harold 

 Jeffreys, Phil. Mag., [6] 49 (1925), 793. 



* " The Propagation of Waves in Steep Prismatic Channels," by Harold A. Thomas, 

 Proceedings of Hydraulics Conference, Studies in Engineering, Bulletin 20, State 

 University of Iowa, Iowa City. 



* " A Criterion for Instability of Flow in Steep Channels," by G. H. Keulegan and 

 G, W. Patterson, Transctctions American Geophysical Union, 1940, Part II, p. 594. 



