7.3. In the Estuary problem given as Example, 7.1 determine algebraically (not graphically) from the simple wave method the time in hours required for the depth to drop to 6.50 ft at a distance upstream of x=25000 ft. Plot on the same axes the depth hydrographs at x=0 and x=25000 ft.

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318 CHAPTER 7: Govering Equations of Unsteady Flow locity are constant, with V. determined from ne of 1, = 3 hr deter at characteristic, and file are determined. For (7.35). Along each of these characteristics, both the depth and velocity are ce the depth, y, specified by the boundary condition and velocity, V. determ (7.33). The intersection of each Cl characteristic with the time line of t, = 3 mines the x position of the depth and velocity associated with that character thus the depth profile as well as the velocity along the depth profile are determin example, the characteristic that begins at r = 1 hr has a depth of 7.0 ft (2.1 m) 15.0 ft/s (4.58 m/s) from (7.35) and a slope dx/dt = -3- (2 x 16.05) + (3 x 15 =9.90 ft/s (3.02 m/s) from (7.34a). Its velocity V = -3- (2 x 16.05) + (2 X = -5.10 fus(-1.55 m/s) from (7.33). The intersection with the time liner, = 3 11 located at x = (dx/dr) x ( - 1) = 9.90 X 2 X 3600/5280 = 13.5 mi (21.7 km). So a location of 13.5 mi (21.7 km) upstream of the estuary mouth, the depth is 7.0 ft (2.1 m) and the velocity is 5.10 ft/s (1.55 m/s) at r = 3 hr. Dam-Break Problem As another application of the method of characteristics applied to the simple wave, we consider next the sudden removal of a vertical plate behind which a known depth of water is at rest. The simple-wave solution of this problem, which Stoker (1957) referred to as the breaking of a damn, is oversimplified in comparison to the solution of a realistic dam break discussed in more detail in the next chapter. How- ever, it illustrates the application of a velocity boundary condition and the forma- tion of a surge in a submerged downstream river bed, and provides further insight into the unsteady development of a negative wave as interpreted by the method of characteristics. The next two examples are presented following more closely the practical approach of Henderson (1966), who related the dam-break problem to sluice gate operation and hydroelectric load acceptance in a headrace, than the mathematical treatment by Stoker (1957). Example 7.2 vertical plate is fixed at time at x with a constant denth of water upstream equal to y, while the channel downstream of the gate is dry, as shown in Figure 7.11 The water upstream of the plate initially is at rest. Att > 0, the . Also, the plate suddenly is accel- erated to the left to a constant speed V, Determine the simple simple wave profile for this case and also for the case of the plate being removed instantane Solution. The physical and characteristical 318 CHAPTER 7: Govering Equations of Unsteady Flow locity are constant, with V. determined from ne of 1, = 3 hr deter at characteristic, and file are determined. For (7.35). Along each of these characteristics, both the depth and velocity are ce the depth, y, specified by the boundary condition and velocity, V. determ (7.33). The intersection of each Cl characteristic with the time line of t, = 3 mines the x position of the depth and velocity associated with that character thus the depth profile as well as the velocity along the depth profile are determin example, the characteristic that begins at r = 1 hr has a depth of 7.0 ft (2.1 m) 15.0 ft/s (4.58 m/s) from (7.35) and a slope dx/dt = -3- (2 x 16.05) + (3 x 15 =9.90 ft/s (3.02 m/s) from (7.34a). Its velocity V = -3- (2 x 16.05) + (2 X = -5.10 fus(-1.55 m/s) from (7.33). The intersection with the time liner, = 3 11 located at x = (dx/dr) x ( - 1) = 9.90 X 2 X 3600/5280 = 13.5 mi (21.7 km). So a location of 13.5 mi (21.7 km) upstream of the estuary mouth, the depth is 7.0 ft (2.1 m) and the velocity is 5.10 ft/s (1.55 m/s) at r = 3 hr. Dam-Break Problem As another application of the method of characteristics applied to the simple wave, we consider next the sudden removal of a vertical plate behind which a known depth of water is at rest. The simple-wave solution of this problem, which Stoker (1957) referred to as the breaking of a damn, is oversimplified in comparison to the solution of a realistic dam break discussed in more detail in the next chapter. How- ever, it illustrates the application of a velocity boundary condition and the forma- tion of a surge in a submerged downstream river bed, and provides further insight into the unsteady development of a negative wave as interpreted by the method of characteristics. The next two examples are presented following more closely the practical approach of Henderson (1966), who related the dam-break problem to sluice gate operation and hydroelectric load acceptance in a headrace, than the mathematical treatment by Stoker (1957). Example 7.2 vertical plate is fixed at time at x with a constant denth of water upstream equal to y, while the channel downstream of the gate is dry, as shown in Figure 7.11 The water upstream of the plate initially is at rest. Att > 0, the . Also, the plate suddenly is accel- erated to the left to a constant speed V, Determine the simple simple wave profile for this case and also for the case of the plate being removed instantane Solution. The physical and characteristical