is also present in Model 2 though not so pronounced, suggests that warps may arise from the outward migration of $m=1$ bending waves.

Spectral Analysis {#sec:spectralanalysis}
-----------------

To connect our 3D simulations with the linear analysis of Section \[sec:linearringmodel\], we perform a spectral analysis of bending waves in our simulated discs. Our method uses the techniques from @sellwoodathanassoula1986 (see also @roskar2012 for a succinct description). We calculate the Fourier coefficients $Z_m (R,t)$ (see Section \[sec:fourieranalysis\], equation \[eq:bendmodefouriercoef\]) for the $t_j=j \, \Delta_t +t_0$ snapshots where $\Delta_t$ is the time between snapshots, $j=0\dots N-1$, and $N$ is even. We then perform a discrete Fourier transform in time to obtain the two-sided spectral coefficients

$$\label{eq:freqfouriercoefs}
\mathcal{Z}_{m} (R,\omega_k) = \sum_{j = 0}^{N-1} Z_{m} (R, t_j) \, w(j) \, e^{2 \pi i j k/N}~,$$

[@press2007 Section 13.4]. Here, $w(j)$ is a Gaussian window function with a standard deviation of $N/2^{5/2}$, which is introduced to mitigate spectral leakage from high frequencies. We note that using similarly shaped window functions, such as the Hamming window, does not significantly alter the resulting spectra or our conclusions. The discrete frequencies[^3] are given by

$$\omega_k = \frac{2 \pi}{m} \frac{k}{N \Delta_t}~,$$

with $k = -N/2\dots N/2$, and the Nyquist frequency 