Free Surface Shear Flow Over a Wavy Bed

An analytical treatment of free surface shear flow over a wavy bed of regular sinusoidal form is developed from the one-dimensional energy equation. The effects of curvilinearity on the velocity and piezometric heads are examined in some detail. The equation is expanded into a series in dimensionless terms to obtain a systematic ordering of the magnitudes of the various terms and associated physical quantities. The third-order equation is linearized and solved to obtain expressions for the phase shift between and amplitude ratio of the bed waves and depth variations. The second-order nonlinear equation is solved numerically for a particular flow, and the profiles are presented graphically. The various flow configurations predicted by the linearized and the nonlinear formulations are discussed in some detail and the occurrence of a bi-stable flow of moderate and high Froude numbers is explained.