Direct solution to problems of static sharp waves in shallow-water

The study of hydrodynamic canals is the first step in canals design, sediment transport, erosion, dissemination of pollution and other phenomena related to canals. When in canals, depth or flow rate is suddenly changed, the sharp wave is generated. When the position and characteristics of the sharp wave remain constant, after the steady flow, it is called the static-sharp wave. So generated jump hydraulic and sharp waves in transitions due to cross-obstacle in flow path at supercritical flows are classified as the static sharp waves. Since the equations governing the dynamics of sharp waves are the same as the flow equation in shallow-water, this paper uses linear solve for the equations governing shallow-water and specifying boundary conditions in shallow-water with the static wave and by using Taylor series, an equation for the bottom and two equations based on kinematic and dynamic boundary for the surface boundary were extracted. Also considering the frequency of the surface waves, frequency boundary conditions considering introduction of the dimensionless parameter  based on wave number and angular frequency, were extracted and other boundary conditions were rewritten based on the dimensionless parameter . Next, based on the velocity potential and the Laplace equation, generated the differential equations solved by Euler equation, which leads to generate potential velocity () and wave height () is the canal length.

Keywords:

The static sharp wave , Shallow-water , Taylor series , Euler's equation , Differential equations , Potential speed , Wave height