  In the study of surface waves in the presence of a shear current, a useful
and much studied model is that in which the shear flow has constant vorticity.
Recently it was shown by Constantin [Eur. J. Mech. B/Fluids 30 (2011) 12-16]
that a flow of constant vorticity can only permit waves travelling exactly
upstream or downstream, but not at oblique angles to the current, and several
proofs to the same effect have appeared thereafter. Physical waves cannot
possibly adhere to such a restriction, however. We resolve the paradox by
showing that an oblique plane wave propagating atop a current of constant
vorticity according to the linearized Euler equation carries with it an
undulating perturbation of the vorticity field, hence is not prohibited by the
Constantin theorem since vorticity is not constant. The perturbation of the
vorticity field is readily interpreted in a Lagrangian perspective as the wave
motion gently shifting and twisting the vortex lines as the wave passes. In the
special case of upstream or downstream propagation, the wave advection of
vortex lines does not affect the Eulerian vorticity field, in accordance with
the theorem. We conclude that the study of oblique waves on shear currents
requires a formalism allowing undulating perturbations of the vorticity field,
and the constant vorticity model is helpful only in certain 2D systems.
