  The classical equations of irrotational water waves have recently been
reformulated as a system of two equations, one of which is an explicit
non-local equation for the wave height and for the velocity potential evaluated
on the free surface. Here, in the two dimensional case, (a) we generalise the
relevant formulation to the case of constant vorticity, as well as to the case
where the free surface is described by a multi-valued function; (b) in the case
of travelling waves we derive an upper bound for the free surface; (c) in the
case of constant vorticity we construct a sequence of nearly Hamiltonian
systems which provide an approximation in the asymptotic limit of certain
physical small parameters. In particular, the explicit dependence of the
vorticity on the coefficients of the KdV equation is clarified. Also, in the
irrotational case we extend the formalism to n>2 dimensions and analyse
rigorously the linear limit of these equations.
