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\section{}

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JCD, (3.1)--(3.3):
\begin{equation}
\rho_0(r)\frac{\partial^2\xxi}{\partial t^2}=-\nab p'+\rho'\grav
\end{equation}
\begin{equation}
\rho'=-\rho_0(r)\nab\cdot\xxi
\end{equation}
\begin{equation}
p'=\cs^2\rho'-\left(\frac{\dd p_0}{\dd r}-\cs^2\frac{\dd\rho_0}{\dd r}\right)\xi_r
\end{equation}

\section{}

\subsection{Equilibrium solution}

\begin{equation}
p_0=p_{00}e^{-z/H}
\end{equation}
\begin{equation}
\rho_0=\rho_{00}e^{-z/H}
\end{equation}

\subsection{Perturbations}

Standard approach, using $\DD s/\DD t=0$, and therefore also
$\DD\ln p/\DD t=\gamma\DD\ln\rho/\DD t=-\gamma\nab\cdot\uu$, so
\begin{equation}
\frac{\partial\uu}{\partial t}+\frac{1}{\rho_0}\nab p'+\frac{1}{\rho_0} \rho' g\zzz=0,
\end{equation}
and EOS:
\begin{equation}
\frac{\rho'}{\rho_0}=\frac{p'}{\gamma p_0}-\frac{s'}{c_p}.
\end{equation}
gives
\begin{equation}
\frac{\partial\uu}{\partial t}+\frac{1}{\rho_0}\nab p'+
\frac{p'}{\gamma p_0}g\zzz-\frac{s'}{c_p}g\zzz =0,
\end{equation}
along with
\begin{equation}
\frac{\partial s}{\partial t}+u_z\frac{\dd s_0}{\dd z}=0.
\end{equation}
\begin{equation}
\frac{\partial p'}{\partial t}+\gamma p_0 \frac{u_z}{H_p}+\gamma p_0 \nab\cdot\uu=0.
\end{equation}
Matrix formulation, 2-D
\begin{equation}
\pmatrix{
-\ii\omega & 0 & 0 & \ii k_x/\rho_0 \cr
0 & -\ii\omega & -g/c_p & [(\gamma H_p)^{-1}+\ii k_z]/\rho_0 \cr
0 & \dd s_0/\dd z & -\ii\omega & 0 \cr
\ii k_x \gamma p_0 & (-H_p^{-1}+\ii k_z) \gamma p_0 & 0 & -\ii\omega }
\pmatrix{u_x \cr u_z \cr s' \cr p'}=0.
\end{equation}
Formulate as $\partial_t\qq+\MMMM\qq=0$, with
\begin{equation}
\MMMM=\pmatrix{
0 & 0 & 0 & \ii k_x/\rho_0 \cr
0 & 0 & -g/c_p & \rho_0^{-1}[(\gamma H_p)^{-1}+D_z] \cr
0 & \dd s_0/\dd z & 0 & 0 \cr
\ii k_x \rho_0\cs^2 & \rho_0\cs^2 [-(\gamma H_p)^{-1}+D_z] & 0 & 0 }
\end{equation}
and
\begin{equation}
\qq=\pmatrix{u_x \cr u_z \cr s' \cr p'}
\end{equation}

\section{Isentropic stratification}

No entropy fluctuations; just p-modes:
\begin{equation}
\frac{\partial\uu}{\partial t}+\frac{1}{\rho_0}\nab p'+
\frac{p'}{\gamma p_0}g\zzz=0,
\end{equation}
along with
\begin{equation}
\frac{\partial p'}{\partial t}-p_0 \frac{u_z}{H}+\gamma p_0 \nab\cdot\uu=0.
\end{equation}
Formulate as $\partial_t\qq+\MMMM\qq=0$, with
\begin{equation}
\MMMM=\pmatrix{
0 & \rho_0^{-1}[(\gamma H_p)^{-1}+D_z] \cr
\gamma p_0 \left[-(\gamma H_p)^{-1}+D_z\right] & 0 }
\end{equation}
and
\begin{equation}
\qq=\pmatrix{u_z \cr p'}
\end{equation}

\subsection{Analytic solution}

\begin{equation}
u_z=\cs\, \tilde{u} e^{\ii kz+z/2H}
\end{equation}
\begin{equation}
p'=\gamma p_{00}\, \tilde{p} e^{\ii kz-z/2H}
\end{equation}
so the tilded quantities are dimensionless.
Insert, using $\cs^2=\gamma p_{00}/\rho_{00}=\gamma H g=\const$,
\begin{equation}
\cs \frac{\partial \tilde{u}}{\partial t}+\left(\ii k-\frac{1}{2H}\right) \tilde{p}\cs^2+\tilde{p} g=0,
\end{equation}
along with
\begin{equation}
\gamma p_{00} \frac{\partial\tilde{p}}{\partial t}-p_{00} \frac{\tilde{u}}{H}+\gamma p_{00} \left(\ii k+\frac{1}{2H}\right)\tilde{u}=0.
\end{equation}
Using $g/\cs^2=1/(\gamma H)$, we have
\begin{equation}
\frac{1}{\cs} \frac{\partial \tilde{u}}{\partial t}+\left(\ii k+\frac{1}{\tilde{H}}\right) \tilde{p}=0,
\end{equation}
along with
\begin{equation}
\frac{1}{\cs} \frac{\partial\tilde{p}}{\partial t}+\left(\ii k-\frac{1}{\tilde{H}}\right)\tilde{u}=0.
\end{equation}
where $\tilde{H}^{-1}=(1-2/\gamma)\,H^{-1}$.
Dispersion relation is $\omega^2=\cs^2\,(\tilde{H}^{-2}+k^2)$.

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\url{X}
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%\begin{table}[htb]\caption{
%}\vspace{12pt}\centerline{\begin{tabular}{lccccccc}
%$R$ & $T$ & $\kf$ & $\omf$ & $\EEGW$ & $\hrms$ & $k_{\rm peak}$ \\
%\hline
%\label{Ttimescale}\end{tabular}}\end{table}

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%\bibitem[Biskamp \& M\"uller(1999)]{BM99}
%Biskamp, D., \& M\"uller, W.-C.\yprl{1999}{83}{2195}

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