deletions | additions       diff --git a/subsection_bf_Wave_Leakage_Time__.tex b/subsection_bf_Wave_Leakage_Time__.tex  index 26ec21d..bb698e4 100644  --- a/subsection_bf_Wave_Leakage_Time__.tex  +++ b/subsection_bf_Wave_Leakage_Time__.tex 
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\end{equation}  {\bf We have calculated the leakage timescales for waves at frequencies near $\omega_{\rm max} = 2 \pi \nu_{\rm max}$ for stellar models on the lower RGB. For this computational technique, the energy $E_{\rm ac}$ contained within the acoustic cavity peaks at the p mode frequencies of the stellar model. The energy loss rate $\dot{E}_{\rm leak}$ also peaks at the mode frequencies, so that the value of $t_{\rm leak}$ is essentially independent of wave frequency.}  {\bf Figure \ref{fig:DipoleTime} shows the exact value of $t_{\rm leak}$ calculated from equation \ref{eqn:tleak2}, and $t_{\rm leak}$ approximated from equation \ref{eqn:tleak}, with $T$ calculated via equations \ref{eqn:integral2} and \ref{eqn:integral}. Clearly, evaluating $t_{\rm leak}$ via equation \ref{eqn:tleak} with $T$ calculated from equation \ref{eqn:integral} is a very good approximation, accurate to within $\sim \! 10 \%$ for our stellar models. However, using the approximation of equation \ref{eqn:integral2} is not very accurate, and generally produces a value of $t_{\rm leak}$ too large by a factor of $\sim \! 2$. We conclude that we may accurately estimate mode visibilities using equation \ref{eqn:vis}, so long as the value of $T$ is calculate calculated  with an integral over the evanescent region as in equation \ref{eqn:integral}. The approximation of $T$ in equation \ref{eqn:integral2} should not be used for visibility calculations, although it is still useful because it demonstrates the scaling of $T$ with wave angular degree $\ell$ and the size of the evanescent region.}      
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