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EM 1110-2-1601 1 Jul 91 be special C:lSeS where it would be more economical to provide sufficient wall height to confme the waves rather than modify the botIDm slope. (c) Flow conditions resulting with Froude numbers ne:lt 1.0 have been studied by Boussinesq and Fawer. The results of their studies pertaining to wave height with unstable flow have been summarized by Jaeger (1957. pp 127-131), including an expression for approx- imating the wave heighL The subject is tre:lled in more detail in paragraph 4-3d below, Detennination of the critical depth instlbility region involves the proper selection of high and low resistance coefficients. This is demonstrated by the example shown in Plate 6 in which the depths are taken as nonnal depths and the hydraulic radii are equal to depths. Using the suggested equivalent roughness design values of k = 0.007 ft and k = 0,002 ft . bottom slope values of So = 0.00179 and So = 0,00143 . respectively. are required at critical depth. For the criteria ID avoid the region of instability (0.9<1" < d < 1.ldJ. use of the smaller k value for trnnquil flow with the bottom slope adjusted so that d ~ I.ldc will obviate incre:lSed wall heights for wave action, For rapid flow, use of the larger k value with the bottom slope adjusted so that d ~ 0.9dc will obviate increased wall heights should the actual surface be smoother. Thus, the importance of equivalent roughness and slope relative to stable flow is emphasized. These stability criteria should be observed in both unifonn and nonunifonn flow design. (2) Pulsating rapid flow. Another type of flow instability occurs at Froude numbers substantially greater than I. This type of flow is characterized by the fonnation of slugs particularly notice:lble on steep slopes with shn!low flow depth. A Manning's n for pulsating rapid flow can be computed from 0.0453R'/. n ( F )'/3 : 4,04 - log,o F. (2-10) The limiting Froude number F, for use in this equation was derived by Escoffier and Boyd (1962) and is given by F : ~ . .;g (3/' (1 . Z() (2-11) where ;. the flow function, is given by 2-4 ~: 0 b'I' where Q is the total discharge and ~,the depth-width ratio, is given by d ( : b where b is the botIDm width. Plate 7 shows the cW'Ves for a rectangular channel and tr.Ipezoidal channels with side slopes Z of I, 2, and 3. (3) Varied flow proftles. The flow profiles discussed herein relate to prismatic channels or unifonn cross sec- tion of boundary. A complete classification includes bottom slopes that are horizontal. less than critical, equal to critical. greater than criticn!. and adverse. However, the problems commonly encounteted in design are mild slopes that are less than criticn! slope and steep slopes that are greater than critical slope. The three types of profiles in each of these two classes are illustrated in HDC 010-1. Chow (1959) gives a well-<locumented discussion of all classes of varied flow profiles. It should be noted that trnnquil-flow profiles are computed proceed- ing upstre= and rapid-flow profiles downstream. Flow profiles computed in the wrong direction result in diver- gences from the correct profile, Varied,flow computl- tions used for general desi gn should not pass through critical depth, Design procedures fall into two basic Cate- gories: unifonn and nonunifonn or varied flow. Many graphs and tables have been published ID facilitate compu- tation of unifortn flow. Brater and King (1976) have specially prepared tables for trapezoidal channels based on the Manning equation. HDC 610-1 through 610-4{1-1 give graphs that afford rapid solution for the nonnn! depth in trapezoid channels. Nonunifonn or varied flow in prismatic channels can be solved rapidly by use of the varied flow function. (It should be noted that different authors have used the tenns "nonuni- fonn" flow and "varied" flow to mean the same thing; "varied flow" is used in litis manual.) Varied flow in nonprismatic channels. such as those with a gr,;,jually contracting or a gradually exp:mding cross section, is usu- ally handled by "step methods." It should be noted that short. rapidly contracting or expanding cross sections are =red in this manun! as transitions.