A Kinematic and Geometric Analysis of Trochoidal Waves

TL;DR

The article analyzes Gerstner’s gravity-wave model by examining particle velocities in a frame co-moving with the wave, revealing how cycloidal, curtate, and prolate trochoidal wave profiles arise from the relative motion of particles. By decomposing velocity into tangential and normal components and deriving curvature, inflection, cusp, and node conditions, the work links geometric features to the parameter product $rG$ and demonstrates how arc lengths differ across trochoidal forms, including exact cycloid results. A key contribution is the explicit characterization of when prolate and curtate trochoids share equal arc length and how Galilean transformations redistribute acceleration without altering total acceleration. The findings offer a rigorous geometric-kinematic framework for understanding wave profiles, stability thresholds ($rG<1$), and the interpretation of self-intersections and cusps in a physically meaningful, frame-dependent context.

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.

Figures (9)

• Figure 1: Simulation of Flick. Surface governed by (\ref{['xandy']}) and $y = y$ at time $t$ (top). Cross sections defined by displaying surface only for integer values of $y$ (bottom). Gerstner's model (see (\ref{['xandy']})) describes the $y = 0$ cross section.
• Figure 1: Sample of surface particles (green) at time $t$ (top) and an eighth of a cycle later (bottom). In the $xz-$plane, a particle subtends an angle $\theta$ at the centre of a fixed orbit (orange). Here, we employ (\ref{['xandy']}) using $\omega > 0$, $r < 1 / 2$, and only integer values of $u$.
• Figure 1: Our simulation of Emma Phillips' animation in Phillips. One hundred and sixty nine green dots at time $t$ (top) and later instant (bottom). Collectively, dots generate waves which travel in directions of decreasing $x$ and decreasing $z$.
• Figure 1: Sample of Phillips' dots at three moments spanning a third of a period ($t$ increases top to bottom). In a fixed frame (upper images), dots trace circular paths with velocity D and wave moves leftward. In a wave cycle's frame (lower images), the same dots have velocity R, moving rightward as they trace a (green) trochoidal cycle ($r G < 1$ here).
• Figure 1: Trajectory and associated velocity of fluid particle illustrated: in fixed frame (left), and in frame of the wave (bottom right) ($\textbf{R} = \textbf{D} - \textbf{P}$, where P defines the (phase) velocity of the wave). Paths traced by particles and wave peak marked in green. Positions of particles and peaks marked at time $t$ (cyan) and a fraction of a cycle later (blue).

Theorems & Definitions (3)

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