Editing SSC/Stability/BiPolytropes/HeadScratching (section)

===Numerical Integration===
====Through the Core====
Throughout the core, the governing LAWE is,

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

<tr>
  <td align="right">
<math>0</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{d^2x}{d\xi^2} + \frac{\mathcal{H}}{\xi} \frac{dx}{d\xi} 
+ 
\biggl(\frac{1}{4\pi}\biggr)\biggl[5\sigma_c^2 \mathcal{K}_1 + 2 \mathcal{K}_2\biggr]  x \, ,
</math>
  </td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\mathcal{K}_1</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{2\pi }{3}\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{1 / 2} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\biggl(\frac{\rho^*}{P^*}\biggr)\frac{ M_r^*}{(r^*)}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
2 \xi^2 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-1} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\mathcal{H}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
4 - 2 \xi^2 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-1} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\mathcal{K}_2</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl(\frac{4\pi}{3}\biggr) \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-1} \, .
</math>
  </td>
</tr>
</table>

Now, using the [[Appendix/Ramblings/NumericallyDeterminedEigenvectors#General_Approach|general finite-difference approach described separately]], we make the pair of substitutions,

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

<tr>
  <td align="right">
<math>x_i' = \frac{dx}{d\xi}</math>
  </td>
  <td align="center">
<math>\approx</math>
  </td>
  <td align="left">
<math>
\frac{x_+ - x_-}{2 \Delta_\xi}   \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
x_i'' = \frac{d^2x}{d\xi^2}
</math>
  </td>
  <td align="center">
<math>\approx</math>
  </td>
  <td align="left">
<math>\frac{x_+ - 2x_i + x_-}{\Delta_\xi^2} \, ,</math>
  </td>
</tr>
</table>
which will provide an approximate expression for <math>~x_+ \equiv x_{i+1}</math>, given the values of <math>~x_- \equiv x_{i-1}</math> and <math>~x_i</math>.  Specifically, if the center of the configuration is denoted by the grid index, <math>~i=1</math>, then for zones, <math>~i = 2 \rightarrow N</math>,

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

<tr>
  <td align="right">
<math>\biggl[ \frac{x_+ - 2x_i + x_-}{\Delta_\xi^2}  \biggr]</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- \frac{\mathcal{H}}{\xi} \biggl[ \frac{x_+ - x_-}{2 \Delta_\xi} \biggr]
- 
\biggl(\frac{1}{4\pi}\biggr)\biggl[5\sigma_c^2 \mathcal{K}_1 + 2 \mathcal{K}_2\biggr]  x 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \biggl[ \frac{x_+ }{\Delta_\xi^2}  \biggr] + \frac{\mathcal{H}}{\xi} \biggl[ \frac{x_+ }{2 \Delta_\xi} \biggr]</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{2x_i - x_-}{\Delta_\xi^2}  \biggr]
+ \frac{\mathcal{H}}{\xi} \biggl[ \frac{x_-}{2 \Delta_\xi} \biggr]
- 
\biggl(\frac{1}{4\pi}\biggr)\biggl[5\sigma_c^2 \mathcal{K}_1 + 2 \mathcal{K}_2\biggr]  x_i 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ x_+ \biggl\{ 1 + \frac{\mathcal{H}}{\xi} \biggl[ \frac{\Delta_\xi }{2 } \biggr] \biggr\}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
x_i\biggl\{2
- 
\biggl(\frac{\Delta_\xi^2}{4\pi}\biggr)\biggl[5\sigma_c^2 \mathcal{K}_1 + 2 \mathcal{K}_2\biggr] \biggr\} 
-  x_-\biggl\{1 - \frac{\mathcal{H}}{\xi} \biggl[ \frac{\Delta_\xi}{2 } \biggr] \biggr\} \, .
</math>
  </td>
</tr>
</table>

<table border="1" align="center" width="80%" cellpadding="8">
<tr><td align="left">
<div align="center">'''Check Against Independent Derivation'''</div>
We have dealt with this identical LAWE in connection with our [[SSC/Stability/Polytropes#Numerical_Integration_from_the_Center.2C_Outward|analysis of the stability of pressure-truncated n = 5 Polytropic configurations]].  Let's see whether that derivation matches our current one.  In that case, we found,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>x_+ \biggl\{ 2\theta +\frac{4\Delta_\xi \theta}{\xi} - \Delta_\xi (n+1)(- \theta^')\biggr\} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
x_i\biggl\{4\theta - \frac{\Delta_\xi^2(n+1)}{3}\biggl[ \frac{\sigma_c^2}{\gamma_g}  - 
2\alpha \biggl(- \frac{3\theta^'}{\xi}\biggr) \biggr]  \biggr\}
- x_- \biggl[2\theta - \frac{4\Delta_\xi \theta}{\xi} + \Delta_\xi (n+1)(- \theta^') \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ x_+ \biggl\{ 1 + \frac{\Delta_\xi }{2\xi} 
\biggl[ 4 - \frac{6\xi (- \theta^')}{\theta} \biggr]\biggr\}2\theta </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
x_i\biggl\{2 - \frac{\Delta_\xi^2}{\theta}\biggl[ \frac{5\sigma_c^2}{6}  + 
\frac{2}{3} \biggl(- \frac{3\theta^'}{\xi}\biggr) \biggr]  \biggr\}2\theta
- x_- \biggl\{ 1 - \frac{\Delta_\xi}{2\xi}\biggl[ 4 - \frac{6\xi (- \theta^')}{\theta}\biggr] \biggr\} 2\theta
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ x_+ \biggl\{ 1 + \frac{\Delta_\xi }{2\xi} 
\biggl[ \mathcal{H} \biggr]\biggr\}2\theta </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
x_i\biggl\{2 - \frac{\Delta_\xi^2}{4\pi}\biggl[ 5\sigma_c^2 \cdot \frac{4\pi}{6\theta}  + 
\frac{8\pi}{3} \biggl(- \frac{3\theta^'}{\xi \theta}\biggr) \biggr]  \biggr\}2\theta
- x_- \biggl\{ 1 - \frac{\Delta_\xi}{2\xi}\biggl[ \mathcal{H}\biggr] \biggr\} 2\theta \, .
</math>
  </td>
</tr>
</table>
Given that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{4\pi}{6\theta} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{2\pi}{3}\biggl(1 + \frac{1}{3}\xi^2 \biggr)^{1 / 2}
~~\rightarrow ~~ \mathcal{K}_1
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and &nbsp;&nbsp; &nbsp; </td>
  <td align="right">
<math>\frac{4\pi}{3}\biggl(-\frac{3\theta'}{\xi\theta}\biggr)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{4\pi}{3}\biggl(1 + \frac{1}{3}\xi^2 \biggr)^{-1}
~~\rightarrow ~~ \mathcal{K}_2 \, ,
</math>
  </td>
</tr>
</table>
we can confirm that the two expressions are identical.
</td>
</tr>
</table>

====Through the Envelope====
Throughout the envelope &#8212; that is, for <math>\eta_i \le \eta \le \eta_s</math> &#8212; the governing LAWE is,

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

<tr>
  <td align="right">
<math>0</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{d^2x}{d\eta^2} 
+ 
\frac{\mathcal{H}}{\eta} \frac{dx}{d\eta} 
+ 
\frac{1}{2\pi \theta_i^4}\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2}  \biggl[\biggl(\frac{\sigma_c^2}{2}\biggr) \mathcal{K}_1 -  \mathcal{K}_2\biggr]  x\, ,
</math>
  </td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathcal{H}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\biggl\{ 4 -\biggl(\frac{\rho^*}{P^*}\biggr)\frac{ M_r^*}{(r^*)}\biggr\}
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; , &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>~\mathcal{K}_1</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\frac{2\pi }{3}\biggl(\frac{\rho^*}{ P^* } \biggr) 
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>~\mathcal{K}_2</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\biggl(\frac{\rho^*}{ P^* } \biggr)\frac{M_r^*}{(r^*)^3} \, .
</math>
  </td>
</tr>
</table>

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
Keep in mind that, once <math>\mu_e/\mu_c</math> and <math>\xi_i</math> have been specified, other [[SSC/Structure/BiPolytropes/Analytic51#Parameter_Values|parameter values at the interface]] are:
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\theta_i</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl( 1 + \frac{1}{3}\xi^2_i \biggr)^{-1 / 2} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\eta_i</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl(\frac{\mu_e}{\mu_c}\biggr) \sqrt{3}~\theta_i^2 \xi_i \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Lambda_i</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{\eta_i} + \biggl( \frac{d\phi}{d\eta}\biggr)_i 
= 
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \frac{1}{\sqrt{3} \xi_i \theta_i^2} - \frac{\xi_i}{\sqrt{3}} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>A</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\eta_i(1+\Lambda_i^2)^{1 / 2} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>B</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\eta_i - \frac{\pi}{2} + \tan^{-1}(\Lambda_i) \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\eta_s</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
B + \pi \, .
</math>
  </td>
</tr>
</table>
</td></tr></table>

From a related discussion of [[SSC/Stability/BiPolytropes#Profile|interior structural profiles]], we appreciate that throughout the envelope we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\alpha_g</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
+~1
\, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\rho^*}{P^*}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{-1}_i \phi(\eta)^{-1}
\, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{M_r^*}{r^*}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-1}_i \biggl( \frac{2}{\pi} \biggr)^{1/2} \biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr)
\biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \theta^{-2}_i (2\pi)^{-1/2}\eta \biggr]^{-1}
=
2 \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \frac{\theta_i}{\eta} \biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr)
\, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{1}{(r^*)^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2\pi \biggl(\frac{\mu_e}{\mu_c}\biggr)^2 \theta_i^4 \eta^{-2}
\, .
</math>
  </td>
</tr>
</table>

Hence,

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

<tr>
  <td align="right">
<math>\mathcal{K}_1</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>~
\frac{2\pi }{3}\biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{-1}_i \phi(\eta)^{-1} 
=
\frac{2\pi }{3}\biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{-1}_i \biggl[ \frac{\eta}{A\sin(\eta-B)}\biggr] \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\biggl( \frac{\rho^*}{P^*}\biggr)\frac{M_r^*}{r^*}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- 2\cdot \frac{d\ln \phi}{d\ln \eta}
=
2\biggl[1 - \frac{\eta}{\tan(\eta-B)}\biggr] \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\mathcal{H}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>~
4 -\biggl(\frac{\rho^*}{P^*}\biggr)\frac{ M_r^*}{(r^*)}
=
4 -2\biggl[1 - \frac{\eta}{\tan(\eta-B)}\biggr]
=
2\biggl[1 + \frac{\eta}{\tan(\eta-B)}\biggr] \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\mathcal{K}_2</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
4\pi \biggl(\frac{\mu_e}{\mu_c}\biggr)^2 \theta_i^4 \biggl[1 - \frac{\eta}{\tan(\eta-B)}\biggr]\frac{1}{\eta^2} 
\, .
</math>
  </td>
</tr>
</table>

Finally, restructuring the radially dependent coefficient of the linear term in the LAWE, we have,

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

<tr>
  <td align="right">
<math>\frac{d^2x}{d\eta^2} 
+ 
\frac{\mathcal{H}}{\eta} \frac{dx}{d\eta} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{ 
\frac{1}{2\pi \theta_i^4}\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2}  \mathcal{K}_2
-
\biggl(\frac{\sigma_c^2}{2}\biggr)\frac{1}{2\pi \theta_i^4}\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2}   \mathcal{K}_1 
\biggr\} x
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{ 
\frac{1}{2\pi \theta_i^4}\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2}\cdot
4\pi \biggl(\frac{\mu_e}{\mu_c}\biggr)^2 \theta_i^4 \biggl[1 - \frac{\eta}{\tan(\eta-B)}\biggr]\frac{1}{\eta^2}  
-
\biggl(\frac{\sigma_c^2}{2}\biggr)\frac{1}{2\pi \theta_i^4}\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \cdot
\frac{2\pi }{3}\biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{-1}_i \biggl[ \frac{\eta}{A\sin(\eta-B)}\biggr]  
\biggr\} x
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{ 
\biggl[1 - \frac{\eta}{\tan(\eta-B)}\biggr]\frac{2}{\eta^2}  
-
\biggl(\frac{\sigma_c^2}{6}\biggr)\frac{1}{\theta_i^5}\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} 
\biggl[ \frac{\eta}{A\sin(\eta-B)}\biggr]  
\biggr\} x
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{ 
\biggl[4 - \mathcal{H}\biggr]\frac{1}{\eta^2}  
-
\biggl(\frac{\sigma_c^2}{6}\biggr)\frac{1}{\theta_i^5}\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} 
\biggl[ \frac{\eta}{A\sin(\eta-B)}\biggr]  
\biggr\} x \, .
</math>
  </td>
</tr>
</table>

Again, using the [[Appendix/Ramblings/NumericallyDeterminedEigenvectors#General_Approach|general finite-difference approach described separately]], we make the pair of substitutions,

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

<tr>
  <td align="right">
<math>x_i' = \frac{dx}{d\eta}</math>
  </td>
  <td align="center">
<math>\approx</math>
  </td>
  <td align="left">
<math>
\frac{x_+ - x_-}{2 \Delta_\eta}   \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
x_i'' = \frac{d^2x}{d\eta^2}
</math>
  </td>
  <td align="center">
<math>\approx</math>
  </td>
  <td align="left">
<math>\frac{x_+ - 2x_i + x_-}{\Delta_\eta^2} \, .</math>
  </td>
</tr>
</table>
For the envelope, we will integrate from the surface, into the core-envelope interface.  So, this time these "finite-difference" expressions will provide an approximate expression for <math>x_- \equiv x_{i-1}</math>, given the values of <math>x_+ \equiv x_{i+1}</math> and <math>x_i</math>.  If the surface of the configuration is denoted by the grid index, <math>i=N</math>, then for zones, <math>i = (N-1) \rightarrow ??</math>,

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

<tr>
  <td align="right">
<math>
\frac{x_+ - 2x_i + x_-}{\Delta_\eta^2} 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- 
\frac{\mathcal{H}}{\eta} \biggl[ \frac{x_+ - x_-}{2 \Delta_\eta}  \biggr]
+ 
\biggl\{ 
\biggl[4 - \mathcal{H}\biggr]\frac{1}{\eta^2}  
-
\biggl(\frac{\sigma_c^2}{6}\biggr)\frac{1}{\theta_i^5}\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} 
\biggl[ \frac{\eta}{A\sin(\eta-B)}\biggr]  
\biggr\} x_i  
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow~~~
\biggl[ \frac{x_-}{\Delta_\eta^2} \biggr]
-
\frac{\mathcal{H}}{\eta} \biggl[ \frac{x_- }{2 \Delta_\eta}  \biggr]
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-
\biggl[ \frac{x_+ - 2x_i }{\Delta_\eta^2} \biggr] 
-
\frac{\mathcal{H}}{\eta} \biggl[ \frac{x_+}{2 \Delta_\eta}  \biggr]
+ 
\biggl\{ 
\biggl[4 - \mathcal{H}\biggr]\frac{1}{\eta^2}  
-
\biggl(\frac{\sigma_c^2}{6}\biggr)\frac{1}{\theta_i^5}\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} 
\biggl[ \frac{\eta}{A\sin(\eta-B)}\biggr]  
\biggr\} x_i  
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow~~~x_- \biggl\{
1 - \frac{\mathcal{H}\Delta_\eta}{2\eta}
\biggr\}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{ 2 +
\biggl[4 - \mathcal{H}\biggr]\frac{\Delta_\eta^2 }{\eta^2}  
-
\biggl(\frac{\sigma_c^2}{6}\biggr)\frac{\Delta_\eta^2 }{\theta_i^5}\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} 
\biggl[ \frac{\eta}{A\sin(\eta-B)}\biggr]  
\biggr\} x_i  
-
\biggl[1+ \frac{\mathcal{H}\Delta_\eta}{2\eta}\biggr] x_+  \, .
</math>
  </td>
</tr>
</table>

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
<div align="center">'''[[#Surface_Boundary_Condition|Surface Boundary Condition]]'''</div>

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

<tr>
  <td align="right">
<math>\frac{d\ln x}{d\ln \eta}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-1
</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; at, &nbsp; &nbsp; &nbsp; <math>\eta = \eta_s</math>.</td>
</tr>
</table>
Hence,

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

<tr>
  <td align="right">
<math>
\biggl[\frac{x_+ - x_-}{2 \Delta_\eta}\biggr]_s
</math>
  </td>
  <td align="center">
<math>\approx</math>
  </td>
  <td align="left">
<math>\biggl[\frac{dx}{d\eta} \biggr]_s = -\frac{x_s}{\eta_s}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~
\biggl[\frac{x_{N+1} - x_{N-1}}{2 \Delta_\eta}\biggr]
</math>
  </td>
  <td align="center">
<math>\approx</math>
  </td>
  <td align="left">
<math>-\frac{x_N}{\eta_s}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~
x_{N+1}
</math>
  </td>
  <td align="center">
<math>\approx</math>
  </td>
  <td align="left">
<math>x_{N-1} - \frac{2\Delta_\eta x_N}{\eta_s} \, .</math>
  </td>
</tr>
</table>
Inserting this expression for "<math>x_+</math>" in the finite-difference representation of the envelope's LAWE allows us to determine the value for <math>x_- = x_{N-1}</math>.  Specifically,

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

<tr>
  <td align="right">
<math>x_{N-1} \biggl\{
1 - \frac{\mathcal{H}\Delta_\eta}{2\eta}
\biggr\}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{ 2 +
\biggl[4 - \mathcal{H}\biggr]\frac{\Delta_\eta^2 }{\eta^2}  
-
\biggl(\frac{\sigma_c^2}{6}\biggr)\frac{\Delta_\eta^2 }{\theta_i^5}\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} 
\biggl[ \frac{\eta}{A\sin(\eta-B)}\biggr]  
\biggr\}_s x_N  
-
\biggl[1+ \frac{\mathcal{H}\Delta_\eta}{2\eta}\biggr] \biggl[ x_{N-1} - \frac{2\Delta_\eta x_N}{\eta_s} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ x_{N-1} \biggl\{
1 - \frac{\mathcal{H}\Delta_\eta}{2\eta}
\biggr\}
+
\biggl[1+ \frac{\mathcal{H}\Delta_\eta}{2\eta}\biggr] x_{N-1} 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{ 2 +
\biggl[4 - \mathcal{H}\biggr]\frac{\Delta_\eta^2 }{\eta^2}  
-
\biggl(\frac{\sigma_c^2}{6}\biggr)\frac{\Delta_\eta^2 }{\theta_i^5}\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} 
\biggl[ \frac{\eta}{A\sin(\eta-B)}\biggr]  
\biggr\}_s x_N  
+
\biggl[1+ \frac{\mathcal{H}\Delta_\eta}{2\eta}\biggr]_s \biggl[ \frac{2\Delta_\eta }{\eta_s} \biggr] x_N
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ x_{N-1} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{ 1 
+
\biggl[ \frac{\Delta_\eta }{\eta_s} \biggr]
+
2\biggl[ \frac{\Delta_\eta^2 }{\eta^2_s}\biggr]  
+
\biggl(\frac{\sigma_c^2}{2^2\cdot 3}\biggr)\frac{\Delta_\eta^2 }{\theta_i^5}\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} 
\biggl[ \frac{\eta_s}{A}\biggr]  
\biggr\} x_N  \, .
</math>
  </td>
</tr>
</table>
Note that, in the last term of this last expression, we have acknowledged that, <math>(\eta_s - B) = \pi ~~\Rightarrow ~~ \sin(\eta_s - B) = -1</math>.

</td></tr></table>


<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
<div align="center">'''Slope at the Interface'''</div>

We will need to determine the slope that is associated with the envelope's eigenfunction, <math>[dx/d\eta]_\mathrm{env}</math>, precisely at the interface. While the envelope's eigenfunction does not actually exist on the "core" side of the interface, we can ''project'' what its value at <math>x_-</math> ''would'' be if the envelope's eigenfunction were to continue smoothly just one small step beyond the interface, then use this ''projected'' value to determine the function's slope ''at'' the interface location.  Labeling the interface at <math>i = J</math>, first we have, 

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

<tr>
  <td align="right">
<math>\biggl[
1 - \frac{\mathcal{H}\Delta_\eta}{2\eta}
\biggr]_J\biggl[x_-\biggr]_\mathrm{project} 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{ 2 +
\biggl[4 - \mathcal{H}\biggr]\frac{\Delta_\eta^2 }{\eta^2}  
-
\biggl(\frac{\sigma_c^2}{6}\biggr)\frac{\Delta_\eta^2 }{\theta_i^5}\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} 
\biggl[ \frac{\eta}{A\sin(\eta-B)}\biggr]  
\biggr\}_J x_J  
-
\biggl[1+ \frac{\mathcal{H}\Delta_\eta}{2\eta}\biggr]_J x_{J+1}  \, .
</math>
  </td>
</tr>
</table>
Then we conclude that,

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

<tr>
  <td align="right">
<math>[x_\mathrm{env}^']_J \equiv \biggl[ \frac{dx_\mathrm{env}}{d\eta} \biggr]_J</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{x_{J+1} }{2 \Delta_\eta}
- 
\frac{1}{2 \Delta_\eta} \biggl[ x_- \biggr]_\mathrm{project} \, .
</math>
  </td>
</tr>
</table>

</td></tr></table>