  We consider Euler's equations for free surface waves traveling on a body of
density stratified water in the scenario when gravity and surface tension act
as restoring forces. The flow is continuously stratified, and the water layer
is bounded from below by an impermeable horizontal bed. For this problem we
establish three equivalent classical formulations in a suitable setting of
strong solutions which may describe nevertheless waves with singular density
gradients. Based upon this equivalence we then construct two-dimensional
symmetric periodic traveling waves that are monotone between each crest and
trough. Our analysis uses, to a large extent, the availability of a weak
formulation of the water wave problem, the regularity properties of the
corresponding weak solutions, and methods from nonlinear functional analysis.
