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\subsection{{\bf Wave Leakage Time}}  {\bf The wave leakage time scale $t_{\rm leak}$ on which wave energy tunnels from the acoustic cavity into the stellar core can be calculated several ways. As shown above, above,}  \begin{equation}  \label{eqn:tleak}  t_{\rm leak} \simeq \big( \Delta \nu T^2 \big)^{-1} \,,   \end{equation}  {\bf  where $T^2$ is the fraction of the wave energy that leaks into the core each time the wave hits the outer edge of the evanescent region at $r_2$. Using a WKB approximation, the value of $T$ can be calculated from the first part of equation \ref{eqn:integral}, or a rough estimate can be made using equation \ref{eqn:integral2}.} {\bfTo compute a more precise estimate, we solve the adiabatic linearized hydrodynamic (non-magnetic) wave equations for our stellar models, assuming all wave energy that tunnels into the outer core is lost within the inner core. To do this, we place the inner boundary of our computational grid at a radius $r/R = 0.01$, which is always within the stably stratified regions of our red giant models. We then impose a radiative inner boundary condition (see e.g., \cite{Fuller_2012}). On the outer boundary, we impose a forcing/normalization condition on the real part of the wave displacement vector ${\boldmath{\xi}$, i.e., we set ${\rm Re} \big( \xi_r \big) =1$.       
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