resonances of the initial conditions endures for many billions of years.

We can then ask the question as to why higher frequency waves subside over time, or conversely, why low-frequency waves persist? To this end, we consider a form of the tip-Line-of-Nodes (LON) plot [see @briggs1990] for waves of various pattern speeds. A tip-LON plot is constructed by calculating, for a series of concentric rings, the tilt, or tip away from the midplane, of the disc and the position of the LON. To do this, we construct an effective vertical displacement as a function of $R$ and $\phi$ for a narrow range in frequency, $\Delta\omega_k$, using only the $m=1$ spectral coefficient of the $t - \omega_k$ Fourier transform:

$$Z_1\left (R,\,\phi,\,\omega_k\right ) = 
\int_{\omega_k - \Delta\omega_k/2}^{\omega_k+\Delta\omega_k/2} {\rm Re} \bigg\{ \mathcal{Z}_1\left
  (R,\,\omega_k\right )e^{i\phi} \bigg\} \, d\omega_k~.$$

The function $Z_1\left (R,\,\phi,\,\omega_k\right )$ is essentially our series of rings from which the tilt and position angle for the LON as a function of $R$ can be easily determined.

Tip-LON plots are shown in Fig \[fig:tip-LON\_2\_to\_4Gyr\]. For illustrative purposes we only consider the time interval $2 \la t \la 4 \, {\rm Gyr}$ for Model 1 (i.e. Fig. \[fig:powertimeseries\], top middle panel) and show the 