Long waves on a thin fluid layer flowing down an inclined plane

Abstract

Long nonlinear waves on a thin fluid layer flowing down an inclined plane are investigated. For the condition of the Weber number of order one, the equation for the free surface is computed to the third‐order accuracy of the shallow water parameter. The development of a long monochromatic wave is analyzed by use of this equation. Near the upper branch of the neutral curve, the original steady flow is found to be supercritically stable and the amplitude of the monochromatic wave is determined. The stretching of the eigenfrequency by nonlinearity has no definite sign. For a comparatively large wavenumber, the wave velocity can be smaller than that of the linear wave. The nonexistence of the periodic wave in the region far from the upper branch of the neutral curve is discussed in connection with the temporally growing solution of the equation in order to determine the second harmonic.