  Previous results on the scattering of surface waves by vertical vorticity on
shallow water are generalized to the case of dispersive water waves. Dispersion
effects are treated perturbatively around the shallow water limit, to first
order in the ratio of depth to wavelength. The dislocation of the incident
wavefront, analogous to the Aharonov-Bohm effect, is still observed. At short
wavelengths the scattering is qualitatively similar to the nondispersive case.
At moderate wavelengths, however, there are two markedly different scattering
regimes according to wether the capillary length is smaller or larger than
$\sqrt{3}$ times depth. The dislocation is characterized by a parameter that
depends both on phase and group velocity. The validity range of the calculation
is the same as in the shallow water case: wavelengths small compared to vortex
radius, and low Mach number. The implications of these limitations are
carefully considered.
