Abstract
Surface waves in the rapids of the river show the complicated patterns which are influenced by the bottom topography. Two-dimensional shallow water equations with dispersion terms are used to discuss the instability of steady nonuiform flows over sinusoidal bed. Floquet's theory shows that quasi-periodic eigenfunctions with respect to spacial variables can be introduced to solve the equation for an infinitesimal disturbance. Here, two physically important spatial patterns, such as Peak-valley structure and Staggered pattern, are investigated and their unstable regions are plotted in the phase plane of Froude number and the wavenumber of the bottom surface. Furthermore, the frequency of the waves observed at the waterside in the rapids will be discussed.
