Gerstner Ocean Waves

František Gerstner's exact nonlinear trochoidal ocean wave equation, exhibiting physically accurate sharpened wave crests and broad, flat troughs.

Playground

60 FPS • Canvas 2D

Click + Drag to interact with field

Live Parameters

Wave Swell Velocity 1.2

Ocean current propagation rate

Trochoidal Steepness (Q) 0.65

Crest sharpness & horizontal pinching

Mathematical Formulation

85 chars medium

Analytical Equation

\mathbf{x} = \mathbf{x}_0 - \sum_{i=1}^{N} \frac{\mathbf{k}_i}{k_i} Q_i A_i \sin(\mathbf{k}_i \cdot \mathbf{x}_0 - \omega_i t), \quad y = y_0 + \sum_{i=1}^{N} A_i \cos(\mathbf{k}_i \cdot \mathbf{x}_0 - \omega_i t)

Compact Formula

x = x0 - (dx/k)*Q*sin(phase), y = y0 - A*cos(phase), phase = k*(dx*x0 + dy*y0) - ωt

Renderer Logic

const q = steepness / (k * wv.amp * waves.length);
displacedX -= (wv.dx / k) * (q * Math.sin(phase));
displacedY -= wv.amp * Math.cos(phase);

Open in Playground Zero Dependencies • Standalone

Mathematical Tags

#ocean #gerstner #waves #hydrodynamics #nature #water #physics

Author: Math Art Core Target: 60 FPS

Gerstner Ocean Waves

Live Parameters

Mathematical Formulation

Mathematical Tags

Export & Embed: Gerstner Ocean Waves

4K PNG Snapshot

WebM Video (5s Loop)

HTML Iframe Embed