Exact solutions for small-amplitude capillary-gravity water waves

TL;DR

This paper derives explicit analytical solutions for particle motion beneath small-amplitude capillary-gravity water waves using elliptic integrals and Abel differential equations.

Contribution

It provides the first explicit solutions for particle trajectories in small-amplitude capillary-gravity waves, involving advanced mathematical functions.

Findings

Explicit solutions for particle trajectories are obtained.

The solutions involve elliptic integrals and Abel differential equations.

The results enhance understanding of wave-particle interactions in capillary-gravity waves.

Abstract

We present explicit solutions for the ordinary differential equations system describing the motion of the particles beneath small-amplitude capillary-gravity waves which propagate on the surface of an irrotational water flow with a flat bottom. The required computations involve elliptic integrals of first kind, the Legendre normal form and a solvable Abel differential equation of the second kind.