Ocean Gerstner Waves - 2D cross-section

Sea state from a spectrum

The sea is a sum of 14 wave components. Their amplitudes follow the Pierson-Moskowitz fully developed-sea spectrum S(w) = (a g2/w5) exp(-1.25 (wp/w)4), rescaled so that the significant wave height is Hs = 4 sqrt(m0) with m0 = Σ ai2/2. Phases are random but fixed, directions are spread around the mean direction.

Dispersion

Each component obeys w2 = g k tanh(k h). Deep water (kh > π): c = g/w, long waves outrun short ones. Shallow water (kh < 0.31): c = sqrt(g h) independent of wavelength. The group speed is cg = (c/2)(1 + 2kh/sinh 2kh), which is where the energy travels; wave power per metre of crest is P = E cg with E = ρ g Hs2/16.

Gerstner (trochoidal) surface

Water points are labelled by their rest position (x₀, z₀). Each moves on a circle or ellipse: x = x0 - Σ ah,i cosφi sinθi, y = Σ ai cosθi with θ = k·x0 - w t + φ. Real crests are sharper than a sine and troughs flatter, and a Gerstner surface reproduces that. For finite depth the horizontal amplitude is a/tanh(kh) (an ellipse) and under the surface the excursions decay as sinh/cosh(k(h+z))/sinh(kh), which the fourth graph shows against ekz.

Why no loops

The area element of the map is the Jacobian J. It must stay positive; the sum of ah k must stay below 1 or crests form cusps and loops. The choppiness slider is clamped to 0.98 of that limit, and foam in the 3D view appears where J is small (steep, nearly breaking crests). Check 3 verifies mean J = 1 and J > 0.

Honest simplifications

This is a kinematic model, not a solution of the Navier-Stokes or Boussinesq equations: waves do not interact, do not break, do not shoal (height does not grow as depth decreases, only the wavelength and speed change), and Gerstner waves have no mass transport (Stokes drift) although real waves do. The Pierson-Moskowitz spectrum describes a fully developed open sea. The boat and buoys follow the surface height and slope; they do not alter the waves. Wave energy and power numbers use linear theory.