Abstract
The asymptotic behavior of small disturbances as they evolve spatially from boundary conditions in a flat inclined channel is determined. These small disturbances develop into traveling waves called roll waves, first discussed by Dressler in 1949. Roll waves exist if the Froude number F exceeds 2, which consist of a periodic pattern of bores, or discontinuities. After confirming the instability condition for F > 2 for the linearized equations in the boundary value case, the nonlinear boundary value problem for the weakly unstable region of F slightly larger than 2 is studied. Multiple scales and the Fredholm alternative theorem are applied to determine the evolution of the solution in space. It is found that the solution is dominated by the evolution of the disturbance along one characteristic. The shock conditions governing the asymptotic solution are determined and these conditions are used to determine the approximate shape of the resulting traveling wave from the solution. Both asymptotic and numerical results for periodic disturbances are presented.
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