An averaging method for the weakly unstable shallow water equations in a flat inclined channel

The asymptotic behavior of small disturbances as they evolve in space from boundary conditions in a flat inclined channel under weakly unstable conditions is determined using a method of averaging. The shallow water equations are first transformed to a normal form using a variation of parameters technique, and then a new method of averaging is defined. The method of averaging developed differs from averaging methods for ordinary differential equations and other partial differential equations by averaging along the characteristics of the linearized equations rather than along the time variable. It is found that the solution is dominated by the evolution of the solution along one characteristic. Both asymptotic and numerical results for periodic disturbances are presented. In addition, connections to dynamical systems concepts are described.