The mathematical study of water waves began with the derivation of the
basic mathematical equations of any fluid by Euler in 1752. Later, water
waves, which have a free boundary at the air interface, played a central role
in the work of Poisson, Cauchy, Stokes, Levi-Civita and many others.
In the last quarter century it has become a particularly active mathematical
research area.
I will limit my discussion to classical 2D traveling water waves with vorticity.
By means of local and global bifurcation theory using topological degree,
we now know that there exist many such waves. They are exact smooth
solutions of the Euler equations with the physical boundary conditions.
Numerical computations provide insight into their properties. I will mention
a number of properties that are the subjects of current research such as:
their heights, their steepness, the possibility of self-intersection, and
their stability or instability.