Two Ways to Generalize Gerstner Waves in the Theory of Waves in Deep Water

By convention, water waves are studied under the assumption of their potentiality. This approximation is not always valid in natural conditions. The vorticity is introduced by shear currents, which are ubiquitous in the ocean. It is also generated in the near-surface layer as a result of wind action. When these factors are taken into account, the models developed for potential waves require refinement and generalization. This paper is devoted to a review of advances in the field of analytical description of surface vortical waves in deep water. The presentation is based on the Lagrangian approach. The focus is on the Gerstner wave, a particular exact solution of the Euler equation. Two ways of its generalization are discussed. The first suggests consideration of weakly nonlinear steady waves with a more general vorticity distribution (Gouyon waves). The second way is to construct exact solutions for waves with inhomogeneous and non-stationary pressure distribution on a free surface (generalized Gerstner waves).