  The existence of periodic waves propagating downstream on the surface of a
two-dimensional infinitely deep water under gravity is established for a
general class of vorticities. When reformulated as an elliptic boundary value
problem in a fixed semi-infinite strip with a parameter, the operator
describing the problem is nonlinear and non-Fredholm. A global connected set of
nontrivial solutions is obtained via singular theory of bifurcation. Each
solution on the continuum has a symmetric and monotone wave profile. The proof
uses a generalized degree theory, global bifurcation theory and Wyburn's lemma
in topology, combined with the Schauder theory for elliptic problems and the
maximum principle.
