ABSTRACT
The traditional way of the mathematical description of nonlinear surface waves is based on Euler’s method. Another approach to the description of these waves is based on the method of Lagrange. Generally speaking, the Lagrangian approach is less convenient than the Eulerian approach if hydrodynamical problems are considered. However, it is completely true only for three-dimensional problems which are not considered here. When studying one- and two-dimensional problems, the Lagrangian approach simplifies the analysis and the solution of difficult nonlinear wave problems. For example, Airy has written down a strongly nonlinear wave equation for “long” waves using the Lagrangian approach.
Here, it is assumed that the law of the vertical movement of water particles is known. The equations for surface waves are derived using the averaging operation along the thickness of a liquid layer. The choice of the Lagrangian approach together with suitable assumptions allows us to derive new strongly nonlinear equations for ocean surface waves. In particular, examples of three- and two-speed equations for surface waves are derived. The equations take into account the various physical effects influencing the development of a wave: gravity, dispersion, nonlinearity, surface friction, and the vertical movement of water particles.
