results in terms of time-scales: the inner halo responds to the disc on a time-scale comparable to the precession period and therefore the potential that the disc precesses in will be markedly different from the one assumed in the static-potential analysis of @sparke1984 and @sparke1988. They go on to speculate that there will be true modes of the disc-live halo system, qualitatively similar to but quantitatively different from the Sparke-Casertano modes, that nevertheless avoid the rapid damping predicted in @nelson1995.

In this work, we show that a Fourier and spectral analysis of bending waves yields a spectrum that is broadly consistent with both the eigenmode and WKB analysis. Moreover, there is a low-frequency branch of modes with little differential precession. Waves along this branch are therefore long-lived and thus candidates for the bending waves seen in the Galaxy.

Our work is based on $N$-body simulations of two isolated Milky Way-like galaxy models, which are distinguished by the relative contributions of the disc and halo to the rotation curve, and thus by their susceptibility to the formation of a bar. The models and simulations are described in Section \[sec:simulations\]. In Section \[sec:bendingwaveanalysis\], we analyse the vertical bending waves that emerge in 