Steady Deep-Water Waves on a Linear Shear Current

摘要

This study concerns one-dimensional, periodic, symmetric water waves of permanent form, propagating on an otherwise vertically varying uniform shear current. It is restricted to the case of deep water in a constant gravitational field, where both the surface tension of the water and the density of the upper fluid (air) are neglected. The theory is inviscid, and so the shear is assumed to have been produced by external effects. Numerical solutions for all wave-heights, up to and including the limited ones, are computed from a formulation which involves only the wave profile (parametrized in a natural way) and some constants of the motion. It is found that for some shear currents the highest waves are not necessarily those waves with sharp crests known as extreme waves. Furthermore a certain nonuniqueness in the sense of a fold is shown to exist, and a new type of limiting wave is discovered. For both small-amplitude waves and extreme waves the numerical results are compared with theoretical predictions. (Reprints)