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EM 1110-2-1601 1 Jul 91 b. Jump characteristics. (I) The momentum equation for the hydraulic jump is derived by setting the hydrodynamic force plus momen- tum flux at the sections before and after the jump equal. as foDows: A + Q2 - A2Y-2 lYl 9Al- Q2 + ~ (4-3) where y is the depth to the center of gravity of the stre:lm cross section from the water surface. For a rectan. gular channel the following jump height equation can be obtained from Equation 4-3: Y2 = 2:. (V1 + 8Fi - 1) Yl 2 (4-4) where the subscripts 1 and 2 denote sections upStre:lm and downSlream of the jump, respectively. Equation 4-3 also gives good agreement for trapezoidal channels as shown by tests reponed by Posey and Hsing (1938). However. flood channels should not be designed with jumps in lrapezoidal sections because of complex flow patterns and increased jump lengths. (2) The energy loss in the hydraulic jump can be obtained by use of the energy equalion and the derived jump height relation (Chow 1959). This results in an equation that is a function only of the upsa-eam Froude number. The relations between the Froude number, the jump height (Equation 4-4). and the energy loss (Equation 15.1. Brater and King 1976) are presented in Plate 51. The relation between the Fronde number and the jump length, based on the data by Bradley and Peterka (1957) for rectangular channels, is also presented in this plate. c. Jump localion. (I) The location of the hydraulic jump is important in determining channel wall heights and in the design of bridge piers. junctions, or other channel SlruCtureS, as its location determines whether the flow is lranquil or rapid. The jump will occur in a channel with rapid flow if the initial and sequent depths satisfy Equation 4-3 (Equa- tion 4-4 for rectangular channels). The location of the jump is estimated by the sequent depths and jump length. 4-4 The mean location is found by making backwater compu- tations from UpSlre:llll and downStre:lm cona-ol points until Equation 4-3 or 4-4 is satisfied. With this mean jump location. a jump length can be obtained from Plate 51 and used for approximating the location of the jump limits. Because of the uncertainties of channel roughness, the jump should be located using pm::ticaI limits of cl1annel roughness (see paragraph 2-2c). A triaI-3JId.error procedure is illustrated on page 401 of Chow (1959). (2) The wall height required to confine the jump and the backwater downStre:lm should extend UpSlrealD and downStre:lm as determined by the assumed limits of chan- nel roughness. Studies also should be made on the height and location of the jump for discharges less than the design discharge to ensure that adequate wall heights extend over the full ranges of jump height and location. (3) In channels with relatively steep invert slopes, sequent depths are somewhat larger than for horizomal or mildly sloping channels and jump lengths are somewhat smaller than those given in Plate 51. Peterka (1957) sum- marizes the available knowledge of this subject. This reference and HDC 124-1 should be used for guidance when a jump wiD occur on channel slopes of 5 JlC"CCIlt or more. d. UnduJar jump. Hydraulic jumps with Froude numbers less than 1.7 are characterized as undular jumps (Bakhmeteff and Matzke 1936) (see Plate 52). In addi- tion. undulations will occur near critical depth if small disturbances are present in the channel. Jones (1964) shows that the ftrSt wave of the undular jump is consider. ably higher than given by Equation 4-4. The height of this solitary wave is given by a _ F2 _ 1 - - 1 Yl (4-5) where a is the unduJar wave height above initial depth Y l' Additional measurements were also made by Sandover and Zienkiewicz (1957) verifying Equatioa 4-5 and giving the length of the first undular wave. Other measurements with a theoretical analysis have heeD re- ported by Komura (1960). Fawer (Jaeger 1957) has also given a formula for the wavelength based on experimental data: Lemoine (Jaeger 1957) used small- amplitude wave theory to give the wavelength of the undulat jump. The results of these investigations are summarized in Pia 52. which gives the undular jump surge height. breaking surge height (Equation 4-4). and the wavelength of the ftrSt