On the particle paths and the stagnation points in small-amplitude deep-water waves

TL;DR

This paper provides analytic solutions for particle paths beneath small-amplitude deep-water waves, revealing diverse trajectory shapes and discussing stagnation points, enhancing understanding of wave particle dynamics.

Contribution

It introduces explicit analytic solutions for particle trajectories in small-amplitude deep-water waves, including peakon-like and elliptic function-based paths, and analyzes stagnation points.

Findings

Particle paths are not closed curves.

Trajectories include peakon-like and elliptic function solutions.

Remarks on stagnation points in wave motion.

Abstract

In order to obtain quite precise information about the shape of the particle paths below small-amplitude gravity waves travelling on irrotational deep water, analytic solutions of the nonlinear differential equation system describing the particle motion are provided. All these solutions are not closed curves. Some particle trajectories are peakon-like, others can be expressed with the aid of the Jacobi elliptic functions or with the aid of the hyperelliptic functions. Remarks on the stagnation points of the small-amplitude irrotational deep-water waves are also made.