The main two regimes in formation of (deep) Water Waves are gravity and capillary. Gravity waves\swells are generated when the forces of gravity and buoyancy tries to restore equilibrium disturbed (mainly) by blowing wind. Capillary waves are those ripples we’re familiar to observe generated by light wind in open water (a.k.a "cat’s paws").
When waves travel into areas of shallow water, the free orbital motion of the water is disrupted by the ocean bottom. As the water becomes shallower, the swell becomes higher and steeper.
The approximated dispersion relation for ω (the wave’s frequency by time) as function of k (the wave’s spatial frequency - wave number) is given by:
where h is the depth, g gravity constant, γ is the surface constant tension and ρ is the water density.
The fact that the dispersion as function of k is a monotonic (yet non-linear) relation, enables us to easily find k as function of ω and particularly the suitable k numbers for gravity and capillary waves for given disturbance i.e. wind velocity.
The Airy wave is the theoretical description of non-linear swells:
Finite Elements method
The domain of the system (bounded area of water of non-constant depth) is divided into small sub-domains, on each of which we can apply inertia laws. The calculation is made for each of the sub-domains. Later the flux between elements is calculated and the whole (location space) data visualized.
A lot of more useful information about wave physics and modeling can be found in: