Editing Appendix/Ramblings/ForCohlHoward (section)

==Determining Oscillation Frequencies and Growthrates==

6 February 2022: &nbsp; Howard asked me to clarify how the solution to the "basic linear perturbation equation" provides values for the growth rate of an unstable mode.

===Spherically Symmetric Situations===

In a [[SSC/Perturbations#The_Eigenvalue_Problem|separate chapter]] I have explained how the eigenvalue problem is set up for one-dimensional (spherically symmetric) configurations.  Traditionally, the assumption is that you are starting from an equilibrium configuration for which you know how pressure <math>(P_0)</math>, density <math>(\rho_0)</math>, and radius <math>(r_0)</math> vary with mass shell <math>(m)</math>,
everywhere inside (and on the surface) of the configuration. As the following three equations illustrate, you "perturb" the configuration by assuming that there are low-amplitude fluctuations to each variable, namely, <math>P_1</math>, <math>\rho_1</math>, and <math>r_1</math>, and that these fluctuations each vary with spatial location &#8212; that is, with mass shell &#8212; inside the configuration as well a s with time. 

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~P(m,t)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~P_0(m) + P_1(m,t) = P_0(m) \biggl[1 + p(m) e^{i\omega t}  \biggr] \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\rho(m,t)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\rho_0(m) + \rho_1(m,t) = \rho_0(m) \biggl[1 + d(m) e^{i\omega t}  \biggr] \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~r(m,t)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~r_0(m) + r_1(m,t) = r_0(m) \biggl[1 + x(m) e^{i\omega t}  \biggr] \, ,</math>
  </td>
</tr>
</table>
</div>
The convention is to assume, as shown, that the spatial portion of the variation can be separated from the time-dependent portion, and that the time-dependent portion is of the form, <math>e^{i\omega t}</math>.  Then, when these quantities are introduced into the governing linear perturbation equation, any term involving <b>one</b> partial time-derivative &#8212; for example, <math>\partial r/\partial t</math> &#8212; will generate a term of the form, <math>i\omega \cdot e^{i\omega t}</math>; while any term involving <b>a second</b> partial time-derivative will generate a term of the form, <math>-\omega^2 \cdot e^{i\omega t}</math>.  Usually, every term in the governing linear perturbation equation will contain a factor of <math>e^{i\omega t}</math>, so it can be divided out.  This leaves you with a perturbation equation that has no explicit time dependence; it only contains spatial derivatives of variables along with a few <math>\omega</math> or <math>\omega^2</math> terms.

The ''very nature'' of an eigenvalue problem is to see if the perturbation equation can be solved to give you the square of the characteristic oscillation frequency, <math>\omega^2</math> for one (or, hopefully more) mode ''along with'' a specification of how the spatial part of the mode varies with location inside the configuration.  If <math>\omega^2</math> is positive, then the exponent of <math>e^{i\omega t}</math> is imaginary, which identifies a periodic oscillation; if <math>\omega^2</math> is negative, then the exponent of <math>e^{i\omega t}</math> is real, which identifies an exponentially growing (or, formally as well, decaying) mode.

===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.