Differential equation for a stratified fluid and its particular solutions

Abstract

The plane steady motion of a stratified ideal incompressible fluid in a gravity field is examined. Considering that the parameter characterizing the fluid particles — their density ρ0 — is constant along the streamline, it is convenient to take the stream function as one of the independent variables and, in view of the presence of the gravity force, the Cartesian coordinate as the other. In this study, on the basis of Lavrent'eva's equation [1, 2, 3], the differential equation is derived for a single unknown function — the vertical displacement of the streamline y(y0, x), where y0 is its initial position and x is the horizontal coordinate. The particular solutions corresponding to a wave guide, cnoidal and solitary waves and, moreover, waves of the type corresponding to a smooth ascent to a new level are presented. A similar coordinate system was used in [4], but there the problem was reduced to a system of partial differential equations.