Editing Appendix/Ramblings/ToHadleyAndImamura (section)

==Additional Comments==
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<li>I was delighted to be able to find a single function, such as <math>f_\ln = \ln[f_1(\varpi)]</math>, that can pretty faithfully represent, not just one, but a set of HI11's observed eigenfunctions by simply adjusting one parameter, <math>~f_\mathrm{green}</math>.  I was quite happy with this finding and, originally, had no expectation that the function, <math>~f_\ln</math>, would in any way relate to the functional behavior of the corresponding "constant phase loci." After all, it seemed to me that, in general, one should expect that an eigenvector's "amplitude" and "phase" functions will be totally independent of one another.</li>
<li>I was extraordinarily pleased &#8212; and stunned! &#8212; to find that the same function, <math>~f_\ln</math>, also could be used to provide a reasonably faithful representation of the phase function, <math>~\phi(\varpi)</math>.</li>
<li>In retrospect, it seems clear that the pairs of "amplitude" and "phase" plots published by the [[#See_Also|Imamura &amp; Hadley collaboration]] for various unstable eigenmodes do exhibit a strong degree of interdependence:
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<li>The radial location, <math>~r_\mathrm{min}</math>, at which the "amplitude" plot exhibits a minimum (identified, for example, by the red, vertical, dashed line in our Figure 2a animation) also appears to be the radial location at which the "phase" plot exhibits a rapid phase swing (identified by the red, dashed circle in our Figure 2a animation).</li>
<li>The degree to which a given "constant phase locus" exhibits a rapid phase swing appears to correlate with the steepness of the <math>~f_\ln(\varpi)</math> function.  If the "amplitude" plot exhibits a sharp, well-defined minimum, then the "phase" plot exhibits a sharp phase swing; conversely, if the "amplitude" plot is rather smooth and featureless, then the "phase" plot exhibits milder phase swings.
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<table border="1" align="right"><tr><td align="center">[[File:ImamuraPaper2Fig4.png|center|200px|Figure 4c from Hadley et al.'s (2014) Paper 2]]</td></tr></table>
Then, of course, there are examples such as the one displayed here, on the right, taken from Figure 4 in Paper 2 ([http://adsabs.harvard.edu/abs/2014Ap%26SS.353..191H Hadley et al. 2014]) in which the number of times the "constant phase locus" plot swings through a full <math>~2\pi</math> radians correlates with the number of local minima exhibited by the corresponding "amplitude" plot.
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It has occurred to me that each local minimum in an "amplitude" plot may be representing a radial node of the underlying (Lagrangian) radial displacement function, <math>~\delta r/r</math>.  Related thoughts:
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<li>The density fluctuation may flip its sign &#8212; going from a positive to a negative fluctuation, for example &#8212; each time our traditional "amplitude" plot passes through a local minimum, in which case our traditional "amplitude" plot is really presenting (in some sense) the absolute value of the density fluctuation.</li>
<li>This idea is easier to swallow when we recognize that our traditional "amplitude" plot is a semi-log plot; on a linear scale, the minima indicate that the function is dropping close to zero, so it is not unreasonable to propose that the fluctuation is ''passing through zero'' at these radial locations.
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<li>This would also help explain why, during my empirical construction of each "constant phase locus" plot, I presently have to manually flip the sign on the phase function, <math>~\phi_1(\varpi)</math>, when crossing the radial location of a local minimum, <math>~r_\mathrm{min}</math>.
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