that extend almost the entire frequency range in Fig. \[fig:powertimeseries\]. More specifically, there is a band present in all time intervals at $R < 0.25 \, {\rm kpc}$, as well as weaker bands at larger radii that come and go across the various time intervals. These bands arise when the local surface density term in the vertical displacement Fourier coefficients ($\Sigma\left (R_j,\,\phi_j\right )$ in equation \[eq:bendmodefouriercoef\]) for one or more particles is computed to be nearly zero. This situation is an artefact of our scheme to estimate the local surface density as a truncated Fourier series.

Overlaid in all panels of Fig. \[fig:powertimeseries\] are the $m =1$ vertical resonances (analogous to the Lindblad resonances for density waves), $\omega = \Omega \pm \nu_x$, where $\Omega$ is the circular frequency of the disc and $\nu_x = \sqrt{\nu_{b}^{2} +
  \nu_{h}^{2}}$ is the vertical force oscillation frequency due to the bulge and halo potential, of the initial conditions. Power in the $m =1$ bending waves collects along the vertical resonances and tends to lie outside of the forbidden region between them, as predicted by the WKB approximation [@binney2008 Section 6.6.1, but also see Section \[sec:linearringmodel\]]. Furthermore, agreement between our power spectra and the vertical 