\appendix

\section{Linearized Evolution Equations}

The hydrodynamic equations are linearized using Eulerian 
perturbations of the form
\begin{equation}
\rho = \rho_o+\rho_1(\varpi,z,t){\rm exp}(im\phi),
\end{equation}
\begin{equation}
v_{\varpi}=v_{\varpi,1}(\varpi,z,t){\rm exp}(im\phi),
\end{equation}
\begin{equation}
v_{\phi} = \Omega\varpi+v_{\phi,1}(\varpi,z,t){\rm exp}(im\phi),
\end{equation}
\begin{equation}
v_{z}= v_{z,1}(\varpi,z,t){\rm exp}(im\phi),
\end{equation}
where $\rho_o$ and $\Omega$ are the equilibrium density and angular 
velocity, respectively. The linearized hydrodynamic equations are then
\begin{eqnarray}
\frac{\partial\rho_1}{\partial t}&=&-im\Omega\rho_1-
\rho_o\frac{v_{\varpi,1}}{\varpi}- v_{\varpi,1}\frac{\partial
\rho_o}{\partial\varpi}- v_{z,1}\frac{\partial\rho_o}{\partial 
z}-\rho_o(\frac{\partial
 v_{\varpi,1}}{\partial\varpi} 
 +\frac{im}{\varpi} v_{\phi,1}
 +\frac{\partial v_{z,1}}{\partial z}) \\
\frac{\partial v_{\varpi,1}}{\partial t}&=&-im\Omega v_{\varpi,1}
+ 2\Omega v_{\phi,1}
-\frac{\gamma P_o}{\rho_o^2}\frac{\partial
\rho_1}{\partial\varpi}-(\gamma-2)\frac{\rho_1}{\rho_o^2}
\frac{\partial P_o}
{\partial\varpi} 
-\frac{\partial\Phi_1}{\partial\varpi},\\
\frac{\partial v_{\phi,1}}{\partial t}&=&-im\Omega v_{\phi,1}
-\frac{v_{\varpi,1}}{\varpi}\frac{\partial(\Omega\varpi^2)}{\partial\varpi}
-\frac{im}{\varpi}\frac{\gamma P_o}{\rho_o^2}\rho_1-\frac{im}{\varpi}\Phi_1,
\end{eqnarray}
and
\begin{eqnarray}
\frac{\partial v_{z,1}}{\partial t}&=&-im\Omega v_{z,1}-\frac{\gamma P_o}{\rho_o^2}
\frac{\partial\rho_1}{\partial z}-(\gamma-2)\frac{\rho_1}{\rho_o^2}
\frac{\partial P_o}
{\partial z}-\frac{\partial\Phi_1} {\partial z}.
\end{eqnarray}
In the above, we suppressed the $``(\varpi,z,t)$'' from the 
notation describing the perturbations, and we let $P_o$ = 
$K\rho_o^{\gamma}$ where $\gamma$ = 1+1/$n$. The linearized equations are 
complex. We solve them as pairs of real equations by substituting 
\begin{equation}
\rho_1=\eta_R+i\eta_I,
v_{\varpi,1}=T_R+iT_I,
v_{\phi,1}=V_R+iV_I,
v_{z,1}=W_R+iW_I,
\end{equation}
and
\begin{equation}
\Phi_1=\Phi_R+i\Phi_I
\end{equation}                                                 
into the linearized equations. We find 
\begin{eqnarray}
\frac{\partial\eta_R} {\partial t} &=&m\Omega\eta_I-T_R
\frac{\rho_o} {\varpi} (1+\frac{\varpi}{\rho_o}
\frac{\partial\rho_o} {\partial\varpi})
-W_R\frac{\partial\rho_o} {\partial z}-\rho_o(\frac{\partial T_R} {\partial\varpi}
-\frac{m} {\varpi}
V_I+\frac{\partial W_R} {\partial z}),\\
\frac{\partial\eta_I} {\partial t} &=&-m\Omega\eta_R-T_I\frac{\rho_o}{\varpi}(1+
\frac{\varpi}{\rho_o}\frac{\partial\rho_o}{\partial\varpi}
-W_I\frac{\partial\rho_o} {\partial z}-\rho_o(\frac{\partial T_I}{\partial\varpi}
+\frac{m}{\varpi}
V_R+\frac{\partial W_I}{\partial z}),\\
\frac{\partial T_R}{\partial t}&=&m\Omega T_I
+ 2\Omega V_R-\gamma \frac{P_o}{\rho_o^2}\frac{\partial
\eta_R}{\partial\varpi}
-(\gamma-2)\frac{\eta_R}{\rho_o^2}\frac{\partial P_o}
{\partial\varpi}-\frac{\partial\Phi_R}{\partial\varpi},\\
\frac{\partial T_I}{\partial t}&=&-m\Omega T_R
+ 2\Omega V_I-\gamma \frac{P_o}{\rho_o^2}\frac{\partial
\eta_I}{\partial\varpi}
-(\gamma-2)\frac{\eta_I}{\rho_o^2}\frac{\partial P_o}
{\partial\varpi}-\frac{\partial\Phi_I}{\partial\varpi},\\
\frac{\partial V_R}{\partial t}&=&m\Omega V_I
-\frac{T_R}{\varpi}\frac{\partial(\Omega\varpi^2)}{\partial\varpi}
+\frac{m}{\varpi}[\frac{\gamma P_o}{\rho_o^2}\eta_I+\Phi_I],\\
\frac{\partial V_I}{\partial t}&=&-m\Omega V_R
-\frac{T_I}{\varpi}\frac{\partial(\Omega\varpi^2)}{\partial\varpi}
-\frac{m}{\varpi}[\frac{\gamma P_o}{\rho_o^2}\eta_R+\Phi_R],\\
\frac{\partial W_R}{\partial t}&=&m\Omega W_I-\frac{\gamma P_o}{\rho_o^2}
\frac{\partial\eta_R}{\partial z}-(\gamma-2)\frac{\eta_R}{\rho_o^2}
\frac{\partial P_o}
{\partial z}-\frac{\partial\Phi_R}{\partial z},
\end{eqnarray}
and
\begin{eqnarray}
\frac{\partial W_I}{\partial t}&=&-m\Omega W_R-\frac{\gamma P_o}{\rho_o^2}
\frac{\partial\eta_I}{\partial z}-(\gamma-2)\frac{\eta_I}{\rho_o^2}
\frac{\partial P_o}
{\partial z}-\frac{\partial\Phi_I}{\partial z}.
\end{eqnarray}              

The Poisson equation for the gravitational perturbations is 
\begin{equation}
\nabla^2\Phi_1 = 4\pi G\rho_1,
\end{equation}
and so the $\Phi_R$ and $\Phi_I$ may be calculated from $\eta_R$ and 
$\eta_I$ individually.  We use a Poisson equation solver taken from the 
numerical hydrodynamics code of J. Tohline (Tohline 1980) to
calculate $\Phi_R$ and $\Phi_I$.