  We prove the Benjamin and Lighthill conjecture for all two-dimensional steady
water waves with an arbitrary vorticity distribution. We show that the flow
force constant of an arbitrary smooth wave is bounded by the corresponding flow
force constants for conjugate laminar flows. We prove these inequalities
without any assumptions on the geometry of the surface profile and put no
restrictions on wave's amplitude. Furthermore, we give a complete description
of cases when equalities can occur. Our results are new already for Stokes
waves with vorticity, while the case of equalities is new even in the
irrotational setting. Beside proving the Benjamin and Lighthill conjectrure, we
establish sharp bounds for the surface profile, extending previous results on
two-dimensional steady water waves.
