The Lagrangian formulation for wave motion with a shear current and surface tension

TL;DR

This paper develops a Lagrangian formulation for wave motion in fluids with shear currents and surface tension, deriving model equations for various wave regimes, including well-known and novel nonlinear equations.

Contribution

It introduces a Lagrangian density for wave motion with shear currents using Hamiltonian formulation, extending the theoretical framework to complex wave regimes.

Findings

Reproduces the KdV equation in the long-wave regime

Derives nonlinear, nonlocal equations for short- and intermediate waves

Provides a unified Lagrangian approach for wave dynamics with shear and surface tension

Abstract

The Lagrangian formulation for the irrotational wave motion is straightforward and follows from a Lagrangian functional which is the difference between the kinetic and the potential energy of the system. In the case of fluid with constant vorticity, which arises for example when a shear current is present, the separation of the energy into kinetic and potential is not at all obvious and neither is the Lagrangian formulation of the problem. Nevertheless, we use the known Hamiltonian formulation of the problem in this case to obtain the Lagrangian density function, and utilising the Euler-Lagrange equations we proceed to derive some model equations for different propagation regimes. While the long-wave regime reproduces the well known KdV equation, the short- and intermediate long wave regimes lead to highly nonlinear and nonlocal evolution equations.