  There has been significant recent interest in the study of water waves
coupled with non-zero vorticity. We derive analytical approximations for the
exponentially-small free-surface waves generated in two-dimensions by one or
several submerged point vortices when driven at low Froude numbers. The
vortices are fixed in place, and a boundary-integral formulation in the
arclength along the surface allows the study of nonlinear waves and strong
point vortices. We demonstrate that for a single point vortex, techniques in
exponential asymptotics prescribe the formation of waves in connection with the
presence of Stokes lines originating from the vortex. When multiple point
vortices are placed within the fluid, trapped waves may occur, which are
confined to lie between the vortices. We also demonstrate that for the
two-vortex problem, the phenomenon of trapped waves occurs for a countably
infinite set of values of the Froude number. This work will form a basis for
other asymptotic investigations of wave-structure interactions where vorticity
plays a key role in the formation of surface waves.
