Z\left (R,\,\phi,\,t\right ) = \Sigma\left (R,\,\phi,\,t\right )^{-1}
  \int \tilde{f}\left ({\bf r},\,{\bf v},\,t\right
  ) \, z \, d^3\!v \,dz~.$$

In Fig. \[fig:zmap\_vmap\_bar\_nobar\], we show mean vertical displacement and vertical bulk velocity maps for the two models at the same five epochs shown in Fig. \[fig:surfacedensity\]. During the first Gyr, displacements of order 100 pc and bulk vertical motions of a few ${\rm km\,s}^{-1}$ are found across the discs in both models. We interpret these displacements and bulk motions as bending waves. By 4 Gyr, the presence of the bar is clearly evident in Model 1. Interestingly enough, by this epoch, the vertical displacements are stronger in Model 2 and extend further in toward the centre of the disc, as compared with Model 1. The implication is that the bar and/or more massive nature of the disc in Model 1 has a damping effect on bending waves.

The waves have a strong $m=1$ component, where $m$ is the azimuthal mode number. Moreover, these waves appear to be leading in the sense that the radius of the peak vertical displacement ridge increases with increasing $\phi$, that is, in the direction of rotation. The $m=1$ waves are tightly wound at intermediate radii but transition 