# Modelling Ocean surface waves

Trying to model Ocean surface waves, I have played using a linear superposition of sinusoids and also on the fact that phase speed (following a wave's crest) is twice as fast as group speed (following a group of waves). Finally, I more recently used such generation of a model of the random superposition of waves to model the wigggling lines formed by the refraction (bendings of the trajectory of a ray at the interface between air and water) of light, called caustics...

Observing real-life ocean waves instructed me that if a single wave is well approximated by a sinusoïd, each wave is qualitatively a bit different. Typical for these surface gravity waves are the sharp crests and flat troughs. As a matter of fact, modelling ocean waves is on one side very useful (Ocean dynamics and its impact on climate, modelling tides, tsunamis, diffraction in a bay to predict coastline evolution, ...) but quite demanding despite a well known mathematical model. Starting with the Navier-Stokes equations to an incompressible fluid (water) in a gravitational field leads to Luke's variational principle in certain simplifying conditions. Further simplifications lead to the approximate solution given by Stokes which gives the following shape as the sum of different harmonics:

This seems well enough at the moment and I will capture this shape in this notebook and notably if that applies to a random mixture of such waves...

(WORK IN PROGRESS)

The current situation is that these slutions seem to not fit what is displayed on the wikipedia page and that I do not spot the bug I may have introduced...