  In this paper we consider two-dimensional, stratified, steady water waves
propagating over an impermeable flat bed and with a free surface. The motion is
assumed to be driven by capillarity (that is, surface tension) on the surface
and a gravitational force acting on the body of the fluid. We prove the
existence of global continua of classical solutions that are periodic and
traveling. This is accomplished by first constructing a 1-parameter family of
laminar flow solutions, $\mathcal{T}$, then applying bifurcation theory methods
to obtain local curves of small amplitude solutions branching from
$\mathcal{T}$ at an eigenvalue of the linearized problem. Each solution curve
is then continued globally by means of a degree theoretic theorem in the spirit
of Rabinowitz. Finally, we complement the degree theoretic picture by proving
an alternate global bifurcation theorem via the analytic continuation method of
Dancer.
