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3,000 I -1 c ~ lJj II: w .. 12 ~ :Ii <.> iil ::> <.> :! ui 1,000 ~ '" &l is 500 1'\..., I , " \ ~ WAVE ENVELOPE I Y (Qt - Ql + Qw) I \ I \ I \ I Ow \ I \ I \ , \ , \ , \ , \ , \ , \ , \ , \ , \ 2 3 FLOW DURATION, IN HOURS FIG. 2. Estimated Hydrographs for Gradually and Rapidly Varied Flow for Bronco Creek neer Wlldeup, Arizona; Wavs Envslope Os- plctelnatantaneoua Dlachargs for Approxlmataly 30 1l'analatory Waves at 4- to 5-Mln Intervals Overriding Bass Discharge on Basis of Fanchar's Report 2.500 2,000 1,500 o o '1 marson, hydrologist, USGS, written communication, 1971); and (3) at the slope area site indicate wave formation and growth in a downstream direction (Fig. I), Waves most likely were formed and continued to propagate in the smoother wider sand channel reaches downstream from the boulder and bed- rock channels. Design Implications Translatory waves can develop in steep channels of super- critical slope and therefore are a concern to the designer of engineered structures on these channels. Should pulsating flow occur in a channel designed for stable flow, the channel ca- pacity may be inadequate at a discharge that is much smaller than the design flow. For example, the flood of July 9, 1988, was adequately conveyed through a highway culvert at Al- buquerque, New Mexico, until pulsating waves at least 1 m high crashed into the headwall and splashed onto the roadway and ou passing automobiles (film on file at the Albuquerque Metropolitan Arroyo Flood ContrOl Authority). Another ex- ample is the Tonopah Canal, part of the Central Arizona Project, has produced pulsating translatory waves that ex- ceeded design capacity for the channel. Holmes (1936) de- scribes an example of large translatory waves that decreased a channel's capacity to convey flow by a factor of 7-8 when compared to the theoretical capacity for stable flow compu- tations. Also, the U.S. Highway 93 bridge over Bronco Creek was not designed for unstable flow conditions (translatory waves)~ however, results presented in this note suggest waves are possible in the reach upstream from the bridge for flow rates as low as 142 m'/s. The design capacity of U.S. Highway 93 bridge is 481 m'/s. As witnessed by Fancher, the base dis- charge was adequately conveyed through the bridge opening until the waves approached at high velocities and broke over the bridge. SUMMARY AND CONCWSIONS Application of free-surface instability criteria developed by Koloseus and Davidian (1966) show that, at n = 0.030, roll 574/ JOURNAL OF HYDRAULIC ENGINEERING / JUNE 1997 Qt 1 TOTAL INSTANTANEOUS DISCHARGE Ql BASE DISCHARGE FOR DURATION OF FLOW Ow INSTANTANEOUS WAVE DISCHARGE NOTE: Volume of runoff assocIaI8d with Ow 18 utmated ID be abou1 one percent of IOIal volume of rmoff. 0/.01 durlngmOltofftowdurallon. - 4 5 waves were possible for a wide range of discharge. The com- puted instantaneous peak discharge of 2,740 m'/s is comprised of Qlmu = 799 m'/s and Qwmu = 1,943 m'/s. The value of Ql mu is from estimates by Carmody (1980) and House and Pearthree (1995). The use of the standard step method, ou the basis of Manning's equation, produced values of DI and- VI' The value of Qw~, is obtained from the product of cross- section area and velocity of the largest waves computed using the equation by Brater and King (1954) for a large translatory wave. The results of these hydraulic computations, including the duration of the wave occurrences, the wave velocity, and the wave height are in close agreement with the observations of Ernest Fancher. Pulsating flow is a matter of concern in the design of en- gineered structures on steep channels. If potentially hazardous translatory waves occur at high stages in a channel designed for stable flow, the capacity of the channel may be inadequate. Roll waves and pulsating flow possibly are more common than previously thought and may have been overlooked in deter- mining peak-flow rates for some floods. Application of translatory wave techniques needs verification by additioual experiments. observations, and research. 1 ACKNOWLEDGMENTS This report is funded in part by the Flood Control District of Maricopa County. The writers also wish to thank Ernest Fancher. retired from ADOT, for his patience in allowing us to record his detailed eyewitness account of the flood. APPENDIX. REFERENCES Aldridge, B. N. (1972). "Investigation of floods from small drainage basins in Arizona," Proc., 21s1 Annu. Conf. on Roads and Streets, University of Arizona, Tucson, Ariz., 107-126. Belcher, J. (1976). "OcotilJo digs out-Flood leaves town battered but undefeated." Los Angeles Times. (Sept. 12), 1. Brater, E. P., and King, H. W. (1954). Handbook of hydraulics. McGraw- Hill Book Co., Inc., New York. Brock. R. B. (1969). "Development of roll-wave trains in open chan- nels:' J. Hydr. Engrg., ASCE. 95(4),1401-1427.