deletions | additions       diff --git a/subsection_Mode_Visibility_Here_we__.tex b/subsection_Mode_Visibility_Here_we__.tex  index 9b347db..8ceb3da 100644  --- a/subsection_Mode_Visibility_Here_we__.tex  +++ b/subsection_Mode_Visibility_Here_we__.tex 
...
\label{eqn:kr}  k_r^2 = \frac{ \big(L_\ell^2 - \omega^2 \big) \big(N^2 - \omega^2 \big) }{v_s^2 \omega^2} \, .  \end{equation}  {\bf Deep within an evanescent region where $N^2 \ll \omega \ll L_\ell^2$, equation \ref{eqn:kr} evaluates to $k_r \sim \sqrt{\ell (\ell +1)}/r$, and equation \ref{eqn:integral} yields the expression for $T$ in equation \ref{eqn:integral2}.} Thesecond ethe which evaluates approximately to the expression in equation \ref{eqn:integral2}. The  wave crossing time for acoustic waves is \begin{equation}  t_{\rm cross} = \int^R_{r_2} \frac{dr}{v_s} \, .  \end{equation} 
...