Editing Appendix/Ramblings/NumericallyDeterminedEigenvectors (section)

==Conclusions==
Initially, I was surprised to find that, by employing our above-described numerical algorithm, we were able to solve the combined/matched LAWEs for a continuum set &#8212; rather than a discrete set &#8212; of dimensionless oscillation frequencies, <math>~\sigma_c^2</math>.  After all, I have been taught to believe that radial oscillation modes are obtained by solving an eigenvalue problem.  After a bit of thought, I recognized that the continuum set of solutions has been obtained in the absence of a specified ''surface'' boundary condition.  I suspect that the continuum of solutions can only be relevant to a real ''astrophysical'' problem after a physically meaningful surface boundary condition has been imposed; for example, a specification of the ''slope'' of the eigenfunction at the equilibrium configuration's surface.  This should naturally reduce the continuum set to a discrete set of eigenvectors.

Watching the animation sequence reveals, for example, that as the value of <math>~\sigma_c^2</math> is reduced, the number of nodes inside the configuration is reduced, in a predictable, quantized fashion.  At the same time &#8212; between each drop in the integer number of nodes &#8212; the slope of the eigenfunction at the surface <math>~(r/R = 1)</math> varies between large positive, and large negative values.  Hence, we should be able to find a matched solution whose slope at the surface ''also'' matches any reasonably specified boundary condition.