A Kinematic and Geometric Analysis of Trochoidal Waves

TL;DR

This paper provides a detailed geometric and kinematic analysis of Gerstner's water wave model, focusing on the cycloidal, curtate, and prolate trochoid profiles and their properties.

Contribution

It introduces new geometric characterizations of trochoidal wave profiles and analyzes how Galilean transformations influence particle trajectories and accelerations.

Findings

Profiles' cusp, inflection, and self-intersection points are characterized.

Conditions for equal arc lengths in prolate and curtate profiles are derived.

Galilean transformations impact particle acceleration and wave geometry.

Abstract

To study the geometry of Gerstner's water wave model, we analyse the velocity of his fluid particles in a reference frame that moves with the wave. Gerstner wave profiles are cycloidal, curtate (flattened) trochoids, or prolate (extended) trochoids. We derive both the height of each profile's characterising point (cusp, inflection, or self-intersection), as well as a condition under which the arc lengths of prolate and curtate profiles coincide over a single wave cycle. We conclude with a discussion of how Galilean transformations affect particle acceleration and the geometry of their trajectories.