this ring model).

One of the main limitations in @sparke1984 and @sparke1988 is that the halo is treated as a static potential, whereas live haloes can respond to the time-evolving gravitational field of the disc. In addition, the disc will set up wakes in the halo, which then act back on the disc via dynamical friction [@chandra1943]. @bertin1980 argued that the halo-disc interaction can actually excite bending waves that are inside their co-rotation radius, though waves outside co-rotation are damped. Their analysis made a number of approximations, most notably that the halo comprised a spatially uniform Maxwellian distribution of particles. Also, they did not consider the halo contribution to the vertical restoring force. @nelson1995 re-examined these claims in the context of more realistic halo models and concluded that the halo is more likely to damp rather than excite bending waves. However, this conclusion itself was called into question by @binney1998 who carried out numerical experiments that incorporated a ring model for the disc into a conventional $N$-body simulation of a dark halo. They found that when the disc is initialized to the Sparke-Casertano tilt-mode, the warp rapidly winds up while its energy remains constant or even increases. @binney1998 interpret these 