Editing Appendix/Ramblings/ToHadleyAndImamura (section)

===Generic Formulation===
As is explicitly defined in [[User:Tohline/Appendix/Ramblings/Azimuthal_Distortions#Figure1|Figure 1 of our accompanying detailed notes]], we have chosen to represent the spatial structure of an eigenfunction in the equatorial-plane of toroidal-like configurations via the expression,
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  <td align="right">
<math>~\biggl[ \frac{\delta\rho}{\rho_0}\biggr]_\mathrm{spatial} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl\{ f_m(\varpi)e^{-im\phi} \biggr\}  \, .</math>
  </td>
</tr>
</table>
</div>
In general, we should assume that the function that delineates the radial dependence of the eigenfunction has both a real and an imaginary component, that is, we should assume that,
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<math>~f_m(\varpi)</math>
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<math>~=</math>
  </td>
  <td align="left">
<math>~\mathcal{A}(\varpi) + i\mathcal{B}(\varpi) \, ,</math>
  </td>
</tr>
</table>
</div>
in which case the square of the modulus of the function is,
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<math>~|f_m|^2 \equiv f_m \cdot f^*_m </math>
  </td>
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<math>~=</math>
  </td>
  <td align="left">
<math>~\mathcal{A}^2 + \mathcal{B}^2 \, .</math>
  </td>
</tr>
</table>
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We can rewrite this complex function in the form,
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  <td align="right">
<math>~f_m(\varpi)</math>
  </td>
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<math>~=</math>
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<math>~|f_m|e^{-i[\alpha(\varpi) + \pi/2]} \, ,</math>
  </td>
</tr>
</table>
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if the angle, <math>~\alpha(\varpi)</math> is defined such that,
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<math>~\sin\alpha = \frac{\mathcal{A}}{\sqrt{\mathcal{A}^2 + \mathcal{B}^2}}</math>
  </td>
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&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~\cos\alpha = \frac{\mathcal{B}}{\sqrt{\mathcal{A}^2 + \mathcal{B}^2}}</math>
  </td>
</tr>

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<math>~\Rightarrow ~~~~ \alpha</math>
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<math>~\equiv</math>
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<math>~\tan^{-1}\biggl(\frac{\mathcal{A}}{\mathcal{B}}\biggr) 
= \tan^{-1}\biggl[ \frac{\mathrm{Re}(f_m)}{\mathrm{Im}(f_m)} \biggr] 
\, .</math>
  </td>
</tr>
</table>
</div>

Hence, the spatial structure of the eigenfunction can be rewritten as,
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<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl[ \frac{\delta\rho}{\rho_0}\biggr]_\mathrm{spatial} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~|f_m(\varpi)|e^{-i[\alpha(\varpi) + \pi/2+ m\phi]} \, . </math>
  </td>
</tr>
</table>
</div>

From this representation we can see that, at each radial location, <math>~\varpi</math>, the phase angle(s) at which the fractional perturbation exhibits its maximum amplitude, <math>~|f_m|</math>, is identified by setting the exponent of the exponential to zero.  That is,
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<math>~\phi = \phi_\mathrm{max}(\varpi)</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
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<math>~-\frac{1}{m}\biggl[\alpha(\varpi) +\frac{\pi}{2}\biggr] 
= -\frac{1}{m}\biggl\{ \tan^{-1}\biggl[ \frac{\mathrm{Re}(f_m)}{\mathrm{Im}(f_m)} \biggr] +\frac{\pi}{2} \biggr\}
\, .</math>
  </td>
</tr>
</table>
</div>

An equatorial-plane plot of <math>~\phi_\mathrm{max}(\varpi)</math> should produce the "constant phase locus" referenced, for example, in recent papers from the [[User:Tohline/Appendix/Ramblings/To_Hadley_and_Imamura#Summary_for_Hadley_.26_Imamura|Imamura &amp; Hadley collaboration]].


It should be noted that the leading (negative) sign that appears on the right-hand side of this expression for <math>~\phi_\mathrm{max}</math> is rather arbitrary, as is the additional <math>~\pi/2</math> phase shift that appears in that right-hand side expression.  Henceforth, for simplicity, we will omit both and use, instead,
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<math>~m\phi_\mathrm{max}(\varpi)</math>
  </td>
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<math>~\equiv</math>
  </td>
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<math>~\tan^{-1}\biggl[ \frac{\mathrm{Re}(f_m)}{\mathrm{Im}(f_m)} \biggr] 
\, ,</math>
  </td>
</tr>
</table>
</div>
unless and until the sign and/or a global phase shift is needed to adjust the orientation of a "constant phase locus" plot to facilitate comparison with published figures.