Editing Appendix/Ramblings/PatrickMotl (section)

==January 7, 2021 (from Joel)==

<table border="0" align="right" cellpadding="15"><tr><td align="center">[[File:Motl n5 eigenfunction3.25.png|200px|Patrick's Eigenfunction for \tilde\xi = 3.25]]</td></tr></table>

In an [[#Initial_Comments|earlier paragraph, above]], I pointed to Patrick's "time-dependent eigenfunction" animation (one frame, again, reprinted here on the right) and commented on how it should relate to the '''blue segment''' of the eigenfunction plots that I derived from linear perturbation theory (see [[#Fig6|Figure 6, above]]).  The data that I have used to generate the '''blue segments''' for the models with <math>~\tilde\xi = 3.25</math> and  <math>~\tilde\xi = 3.50</math> is provided in the pair of scrollable tables that I have added to the figure shown immediately above.  For either model, the referenced '''blue segment''' can be generated by plotting x(renorm) vs. xi &#8212; i.e., using the first and third columns of either scrollable table &#8212; or by plotting x vs. xi &#8212; using the first and second columns of data.  The only difference is that, in the x(xi) plot, x is normalized to unity at the surface of the model; in the x(renorm) vs. xi plot, x is normalized to unity at the center of the model.

Recommendation:  
<ol>
  <li>Patrick could simply re-create the same linear-linear "time-dependent eigenfunction" movie, and add to every movie frame Joel's (time independent) '''blue segment''' eigenfunction; I guess, in this case, Joel's function should be normalized to unity at the center.  Make sure that the normalization of the radius (horizontal axis) is the same in Patrick's time-dependent plots as it is in Joel's time-independent dataset.</li>
  <li>Patrick could generate a new eigenfunction movie in which the vertical axis is log-amplitude; Joel's time-independent eigenfunction should be added to every movie frame but, of course, in this case it should also be plotted as log-amplitude versus radius.</li>
  <li>Better yet, turn Joel's eigenfunction into a time-dependent eigenfunction!  Using the evolutionary time that corresponds to each frame of the movie, multiply every x(xi) value by <math>~d_c \cdot e^{\sqrt{\omega^2}t}</math> with, I suppose, d_c = 1.5e-3.  Using a semi-log diagram, plot this time-evolving curve on top of Patrick's numerically generated x(x_0,t) values and let's see how well they match over time!</li>
</ol>


{{ SGFfooter }}