  Trapped surface waves have been observed in a swimming pool trapped by, and
rotating around, the cores of vortices. To investigate this effect, we have
numerically studied the free-surface response of a Lamb--Oseen vortex to small
perturbations. The fluid has finite depth but is laterally unbounded. The
numerical method used is spectrally accurate, and uses a novel non-reflecting
buffer region to simulate a laterally unbounded fluid. While a variety of
linear waves can arise in this flow, we focus here on surface gravity waves. We
investigate the linear modes of the vortex as a function of the perturbation
azimuthal mode number and the vortex rotation rate. We find that at low
rotation rates, linear modes decay by radiating energy to the far field, while
at higher rotation rates modes become nearly neutrally stable and trapped in
the vicinity of the vortex. While trapped modes have previously been seen in
shallow water surface waves due to small perturbations of a bathtub vortex, the
situation considered here is qualitatively different owing to the lack of an
inward flow and the dispersive nature of non-shallow-water waves. We also find
that for slow vortex rotation rates, trapped waves propagate in the opposite
direction to the vortex rotation, whereas, above a threshold rotation rate,
waves co-rotate with the flow.
