deletions | additions       diff --git a/subsection_bf_Wave_Leakage_Time__.tex b/subsection_bf_Wave_Leakage_Time__.tex  index 35b8135..bfb7403 100644  --- a/subsection_bf_Wave_Leakage_Time__.tex  +++ b/subsection_bf_Wave_Leakage_Time__.tex 
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After solving the wave equations, we compute the leakage time of the wave energy contained within the acoustic cavity at radii $r_2 < r < R$. The wave energy contained within the acoustic cavity is simply  \begin{equation}  \label{eqn:ecav}  E_{\rm ac} = \int^{R}_{r_2} dr \, \rho r^2 \omega^2 \bigg( | \xi_r |^2 + \ell(\ell+1) | \xi_\perp |^2 \bigg) \, ,  \end{equation}  where $\xi_\perp$ is the horizontal component of the wave displacement vector. The rate at which energy leaks through the inner boundary is  \begin{equation}  \label{eqn:eleak}  \dot{E}_{\rm leak} = \int d \Omega \, \rho r^3 \omega^3 \bigg[ {\rm Re}\big( \xi_\perp \big) {\rm Im}\big( \xi_r \big) - {\rm Re}\big( \xi_r \big) {\rm Im} \big( \xi_\perp \big) \bigg] \,  . \end{equation}       
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