Red giant oscillation modes are standing waves that are driven by stochastic energy input from turbulent near-surface convection \cite{Goldreich_1977,Dupret_2009}. Waves excited near the stellar surface propagate downward as acoustic waves until their angular frequency $\omega$ is less than the local Lamb frequency for waves of angular degree $\ell$, i.e., until $\omega = L_{\ell} = \sqrt{\ell(\ell+1)} v_s/r$, where $v_s$ is the local sound speed and $r$ is the radial coordinate. At this boundary, part of the wave flux is reflected, and part of it tunnels into the core.

The wave resumes propagating inward as a gravity wave in the radiative core where $\omega < N$, where $N$ is the local buoyancy frequency. In normal red giants, wave energy that tunnels into the core eventually tunnels back out to produce the observed oscillation modes. We show here that suppressed modes can be explained if wave energy leaking into the core never returns back to the stellar envelope.

The degree of wave transmission between the core and envelope is determined by the tunneling integral through the intervening evanescent zone. The transmission coefficient is

\begin{equation}
\label{eqn:integral2}
T \sim \bigg( \frac{r_1}{r_2} \bigg)^{\sqrt{\ell(\ell+1)}} \, ,
\end{equation}
where $r_1$ and $r_2$ are the lower and upper boundaries of the evanescent zone, respectively. The fraction of wave energy transmitted through the evanescent zone is $T^2$. For waves of the same frequency, larger values of $\ell$ have larger values of $r_2$, thus Eqn. \ref{eqn:integral2} demonstrates that high $\ell$ waves have much smaller transmission coefficients through the evanescent zone.

The visibility of stellar oscillations depends on the interplay between driving and damping of the modes \cite{Dupret_2009,Benomar_2014}. To estimate the reduced mode visibility due to energy loss in the core, we assume that all mode energy which leaks into the g mode cavity is completely lost. The mode then loses a fraction $T^2$ of its energy in a time $2 t_{\rm cross}$, where $t_{\rm cross}$ is the wave crossing time of the acoustic cavity. Due to the larger energy loss rate, the mode has less energy $E_{\rm ac}$ within the acoustic cavity and produces a smaller luminosity fluctuation $V$ at the stellar surface, whose amplitude scales as $V^2 \propto E_{\rm ac}$. We show (supplementary material) that the ratio of visibility between a suppressed mode $V_{\rm sup}$ and its normal counterpart $V_{\rm norm}$ is

\begin{equation}
\label{eqn:vis}
\frac{V_{\rm sup}^2}{V_{\rm norm}^2} = \bigg[1 + \Delta \nu \, \tau \, T^2 \bigg]^{-1} \, ,
\end{equation}
where $\Delta \nu \simeq (2 t_{\rm cross})^{-1}$ \cite{Chaplin_2013} is the large frequency separation between acoustic overtone modes, and $\tau$ is the damping time of a radial mode with similar frequency. We evaluate $T^2$ from our stellar models using Eqn. \ref{eqn:integral}, whereas $\tau \approx 5 \! - \! 10 \, {\rm days}$  \cite{Dupret_2009,Corsaro_2012,Grosjean_2014,Corsaro_2015} for stars ascending the RGB.

Most observed modes  are near the frequency  $\nu_{\rm max}$, which is determined by the evolutionary state of the star. On the RGB, more evolved stars generally have smaller $\nu_{\rm max}$. Fig. \ref{fig:moneyplot} compares our estimate for suppressed dipole mode visibility (Eqn. \ref{eqn:vis}) with {\it Kepler} observations \cite{Mosser_2011,Garcia_2014}. The objects identified by \cite{Mosser_2011} as depressed dipole mode stars lie very close to our estimate. The striking agreement holds over a large baseline in $\nu_{\rm max}$ extending from the very early red giants KIC8561221 \cite{Garcia_2014} and KIC9073950 at high $\nu_{\rm max}$ to near the luminosity bump at low $\nu_{\rm max}$. The observations are consistent with nearly total wave energy loss in the core, as partial energy loss would create stars with less depressed modes, which seem to be rare.

We conclude that the cores of stars with depressed dipole modes efficiently trap or disrupt waves tunneling through the evanescent region. This is further supported by their normal $\ell=0$ mode visibility, because radial modes do not propagate within the inner core and because much larger field strengths are required to alter acoustic waves. The absence (or perhaps smaller degree) of  depression observed for $\ell=2$ modes \cite{Mosser_2011} occurs because quadrupole modes have a smaller transmission coefficient $T$, and less of their energy leaks into the core.

An additional consequence is that the larger effective damping rate for depressed modes will lead to larger line widths in the oscillation power spectrum. The linewidth of a depressed dipole mode is $\tau^{-1} + \Delta \nu T^2_{\ell}$ and is generally much larger than that of a normal mode. The depressed dipole modes in KIC8561221 \cite{Garcia_2014} indeed have much larger linewidths than normal dipole modes in similar stars.