smoothly to a warp at the edge of the disc. In general, the amplitude of the waves in the outer disc grows over time while the amplitude decreases in the inner disc. We return to this point below.

The bulk vertical motions follow the same general pattern of the displacements except that they are out of phase, as one would expect for wave-like motion [see @gomez2013; @gomez2016; @gomez2017]. From the amplitudes of the velocity and height perturbations, we infer a characteristic pattern speed for the bending waves of $\sim 10-20\,{\rm km\,s}^{-1}\,{\rm kpc}^{-1}$.

In the study of density waves, it has proved instructive to write the surface density as a Fourier series in $\phi$ [see, for example, the seminal work of @sellwoodathanassoula1986]. Formally, one has

$$\label{eq:surfdens}
\Sigma\left (R,\,\phi,\,t\right ) = {\rm Re}\left \{ \sum_m \Sigma_m\left (R,\,t\right
) e^{-im\phi}\right \}~,$$

where

$$\Sigma_m\left (R,\,t\right ) = \frac{1}{2\pi}\int_0^{2\pi}
\Sigma\left (R,\,\phi,\,t\right
) e^{im\phi} \, d \phi~.$$

In practice, the $\Sigma_m$ are calculated over a ring of finite radius (essentially, coarse graining in $R$). For a ring at radius $R_\alpha$ and width $\Delta R_\alpha$ we have

$$\Sigma_m\left (R,\,t\right ) = A_\alpha^{-1} \sum_{j\in\alpha} m_j \,
e^{im\phi_j}~,$$

where $A_\alpha = 2\pi R_\alpha \Delta R_\alpha$ is the area of 