Doubly periodic progressive permanent waves in deep water

Abstract

The Stokes wave is generalized to progressive waves in deep water which are periodic in two orthogonal directions, and are steady relative to a frame of reference moving in one of these directions. These doubly periodic waves are nonlinear at their lowest approximation, and are calculated from the nonlinear equations for irrotational motion in deep water. It is shown how doubly periodic waves of small but finite wave slope may be calculated also from the nonlinear Schrödinger equation. The three-dimensional paths of particles on the free surface of a doubly periodic wave are found, and the interesting property is demonstrated that the mean particle paths differ from the direction of advance of the wave crests. The upper boundary of occurrence of doubly periodic waves at the smaller wavelength ratios is identified with the stability boundary for Stokes waves. The investigation aims to provide a closer approximation than Stokes waves to local wave structures on the ocean.