deletions | additions       diff --git a/subsection_Mode_Visibility_Here_we__.tex b/subsection_Mode_Visibility_Here_we__.tex  index 2a687ae..9efd52f 100644  --- a/subsection_Mode_Visibility_Here_we__.tex  +++ b/subsection_Mode_Visibility_Here_we__.tex 
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\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 $k_r \sim \sqrt{\ell (\ell +1)}/r$. In this case, equation \ref{eqn:integral} evaluates exactly to yields the expression for $T$ in  equation \ref{eqn:integral2}.} The second ethe which evaluates approximately to the expression in equation \ref{eqn:integral2}. The wave crossing time for acoustic waves is  \begin{equation} 
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