55\%$ in the surface density for $R  > 22 \, {\rm kpc}$, which develops at early times, is likely due to a relaxation in the initial conditions.

The disc in Model 1 forms a bar at $\sim\!2.5\,{\rm Gyr}$, which eventually extends to $R \sim 5 \, {\rm kpc}$. In addition, a prominent ring (again, an enhancement relative to the initial exponential profile) develops at a radius of $\sim 10 - 15\,{\rm kpc}$. This feature is likely a result of mass redistribution due to the bar. On the other hand, the disc in Model 2 forms flocculent spiral structure but no bar.

Analysis of Bending Waves {#sec:bendingwaveanalysis}
=========================

Fourier Analysis {#sec:fourieranalysis}
----------------

The surface density shown in Fig. \[fig:surfacedensity\] represents the zeroth moment, with respect to $z$, of the coarse-grained stellar DF, $\tilde f$:

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

(The true DF for an $N$-body system is a sum of six-dimensional $\delta$-functions at the positions of the particles. In practice, coarse-graining amounts to calculating an average surface density over a small patch of the disc.) Likewise, the mean vertical displacement of the disc is given by the first moment with respect to $z$:

$$\label{eq:zdisplacement}
  