Nonlinear Gravity Waves in a Thin Sheet of Viscous Fluid

Abstract

In the study of long gravity waves of finite amplitude, the main body of the existing theories has been built upon the simplifying assumption that the viscosity is either totally negligible, or adequately described by an empirical law. To date very little systematic account of the viscosity effect has appeared that is based on the Navier-Stokes' equations of motion. Thus, in the important problems of flood waves in rivers, Chezy's formula and a variety of empirical laws of hydraulics have been used to replace the viscous stress terms, and this is the approach taken in most of the hydraulic studies on open channel flows. Among theoretical contributions along this line, one may mention the book by Stoker (1957), the works of Dressler (1949), and of Lighthill and Whitham (1955, I). Dressler developed a rigorous theory of roll waves. In particular he obtained a discontinuous periodic solution in the case of relatively large amplitudes and a continuous periodic (cnoidal) solution in the case of small amplitudes. General flood movement in long rivers has been masterfully investigated by Lighthill and Whitham (1955, I), as a type of kinematic waves. Their method of predicting the transient motion of large amplitude waves with discontinuities (or shocks) is especially noteworthy.