Editing Appendix/Ramblings/ForCohlHoward (section)

===Riemann S-Type Ellipsoids===

Notice that Eq. (25) of {{ Lebovitz89ahereafter }} assumes that the three-dimensional, time-dependent perturbation, <math>\boldsymbol\xi</math>, of the Riemann S-Type ellipsoid is of the form,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\boldsymbol\xi(\mathbf{x},t)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\sum_{k=1}^n a_k(t) \xi_k(\mathbf{x})\, .</math>
  </td>
</tr>
</table>
Notice that time-dependence and spatial-dependence have been separated.  And I presume that the time-dependent coefficients, <math>a_k(t)</math>, will include factors of <math>e^{i\omega t}</math>.  Notice as well that, in {{ LL96hereafter }}, <font color="green">"the basic equation"</font> appears in the form,

<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right"><math>0</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\boldsymbol{\xi}_{tt} +
A {\boldsymbol\xi}_t + B \boldsymbol\xi + \rho^{-1} \nabla ( \Delta p ) \, .
</math>
  </td>
</tr>

<tr>
  <td align="center" colspan="3">
{{ LL96hereafter }}, &sect;3.1, p. 701, Eq. (10)
  </td>
</tr>
</table>
The subscript <math>tt</math> means that the perturbation will be twice differentiated in time, while the subscript <math>t</math> means a single time-differentiation.  The oscillation frequency (or growth rate) probably will appear in the basic perturbation equation via these two terms.