corresponding to $\omega_{k= \pm N/2}$. The two-sided power spectrum is then computed as

$$P_m (R,\omega_k) = \frac{1}{W} \, |\mathcal{Z}_m (R,\omega_k)|^2 ~,$$

where $W = N \, \sum_{j = 0}^{N-1} w(j)$ is the window function normalization.

In Fig. \[fig:powertimeseries\], we explore the time evolution of the $m=1$ bending wave power spectra for Model 1. To do so, we divide the simulation into five $\sim 2 \, {\rm Gyr}$ intervals and calculate $P_1$ using the method outlined above. Each interval has $N=200$ snapshots with a time resolution of $\Delta_t \approx 9.8 \, {\rm Myr}$, which gives a frequency resolution of $\sim 3.14 \, {\rm km \, s^{-1} \, kpc^{-1}}$. Prominent bands of power develop at early times and persist over the entire simulation. Much of the power lies along two main branches, which, as we describe below, follow the two lowest-order vertical resonances from linear theory. Power along the lower branch, which correponds to slowly counter-rotating waves, is seen across the disc at all times. On the other hand, power along the upper branch (more rapidly rotating waves) diminshes with time. In general, over time power migrates to larger radii.

We note the presence of spurious bands of power at fixed radii 