The kinematic wave model under the assumption of balanced gravity and friction forces has been applied in open channel hydraulics and surface hydrology. There persists a severe misunderstanding that a discontinuity of a kinematic wave occurs due to a discontinuity of input and then dissipates. This study clarifies that a discontinuity can develop without dissipation under the smoothness of all input. The theory of first-order quasilinear partial differential equations shows that Cauchy problems for the kinematic wave model have unique measurable and bounded solutions, which are possibly discontinuous. Numerical examples are presented to visualize the fundamental properties of discontinuous kinematic waves.

  In the scope of dam rehabilitation to manage floods increase or to increase storage, the Piano Key Weir is a good solution for concrete dams. The efficiency of Piano Key Weirs is now well demonstrated through various experimental studies. Until now most of the Piano Key Weir prototypes close to the Piano Key Weir Type A using symmetrical up- and downstream overhangs for structural reasons. The position of the overhangs influences the inlet and outlet key slopes and cross section as well as the side crest length influenced by these slopes and cross sections. On this paper, the influence of these main parameters has been studied on several models considering variation of the overhang lengths.

  There are two types of mean flow velocity formulas for pipes, the Darcy-Weisbach equation and the Hazen-Williams equation. In the process of deriving the Hazen-Williams equation from the Darcy-Weisbach equation, it was clarified that the friction loss coefficient f is expressed by the exponential function of f ＝ aRe^-b. Next, the Hazen-Williams equation C was obtained by eliminating the effect of the Reynolds number Re. It was confirmed that only the constant a remained in the equation C and there was almost no influence of water temperature and viscosity, and the meanings of the constants a and b were considered. When designing the pipeline, the influence of the relative roughness k/D on the inner surface of the pipe was evaluated using the Colebrook equation in order to set the transition section in which both the effects of viscosity and roughness are effective as the design range. The relationship between these hydraulic constants f, C, Re and k/D was verified by the Moody chart, and hydraulic issues were raised in the design.

  In the Hazen-Williams formula that describes the friction loss of a pipeline, the friction loss coefficient f and the Reynolds number Re have an exponential function f ＝ aRe^-b, and b ＝ 0.15 is set to eliminate the effect of the Reynolds number. The coefficient C has the advantage that it is not a function of the Reynolds number, and at present, the coefficient C is set using the Hazen-Williams formula (b ＝ 0.15) from the results of hydraulic experiments. However, except for some pipe types, the f to Re function is b ＞ 0.15, so the coefficient C tends to increase in tandem as the flow velocity increases, even when experiments are performed using the same pipe. As a result, the value of the coefficient C produces a width between the maximum and the minimum depending on the flow velocity. In this report, a and b are set for each type of inner lining based on the results of hydraulic experiments.