our simulated galaxies. In particular, we decompose the waves according to their azimuthal symmetry and present a spectral analysis that is reminiscent of the classic studies of density waves [see @sellwoodathanassoula1986]. In Section \[sec:linearringmodel\] we show that the qualitative features seen in our spectral analysis are in good agreement with an eigenmode analysis of the linear ring model as well as the dispersion relation derived in the WKB approximation. Section \[sec:externalwaves\] gives a brief description of how we might observe vertical bending waves in external galaxies. We discuss our results in Section \[sec:discussion\] and conclude in Section \[sec:conclusions\].

Simulations {#sec:simulations}
===========

![Circular speed curves for the initial conditions of Model 1 (top) and Model 2 (bottom). The solid black curve shows the total circular speed of the model. Also shown are the contributions to the circular speed curve from the bulge (blue, short-dashed), the disc (red, dotted), and the dark matter halo (magenta, long-dashed). Model 1 corresponds to the most stable Milky Way-like model of @widrow2008, and possesses a disc with enough self-gravity in the inner region to trigger bar formation. Model 2 was constructed with a disc mass 60% of that in Model 1, while the bulge and halo 