e., for any quantity $X$, the perturbation is $\delta{\tilde X} = \delta X \exp\{{i(k_rr+k_z z)-i\omega t}\}$), we will suppose that the wavevectors obey three conditions: that $k = (k_z^2 + k_r^2)^{1/2} \ll \kappa \rho$; that $k_z \gg 1/h$; and that $k_r \gg 1/r$. The first limit means that the diffusion approximation applies, i.e. ${\bf p_r} = p_r{\bf I}$, so that $E = 3p_r$. The second is the WKB approximation, as applied to variations in both the radial and vertical directions. Note that we further restrict our attention to axisymmetric perturbations. The condition $k_z h \gg 1$ also means that we can ignore any gradients in the gravity or equilibrium radiation flux. In addition, assuming $k_r\gg 1/r$ allows us to neglect the terms in vector divergences arising from cylindrical geometry. For example, after Fourier-transforming, $\nabla\cdot\delta\vec v$ becomes $ik_r\delta v_r +\delta v_r/r + 
ik_z\delta v_z \simeq ik_r\delta v_r + ik_z\delta v_z$. We then find: $$\begin{aligned}
-i\omega \delta\rho & + & \rho i\vec k \cdot \delta \vec v = 0 \\
-i\omega \rho \delta v_r & = & (\kappa \rho /c) \delta {\cal F}_r +
         2\rho\Omega\delta v_\phi \\
-i\omega\rho \delta v_z & = & (\kappa \rho/c) \delta {\cal F}_z \\
-i\omega\rho \delta v_\phi & = & (\kappa\rho/c)\delta {\cal F}_\phi - 
         (1/2)\rho\Omega \delta v_r \\
-3i\omega \delta p_r  -  3 \rho g \delta v_z & + & ik_z \delta {\cal F}_z + 
ik_r\delta{\cal F}_r  + 4p_r\left(i k_z \delta v_z + i k_r \delta v_r\right) 
-i{2\omega\over c^2}\delta v_z F_z  =
 \delta Q \\
-i(\omega/c)\delta {\cal F}_r & + & ik_r c \delta p_r  +  i k_z (g/\kappa) 
\delta v_r
- 4i(\omega/c)p_r \delta v_r  =  -\kappa\rho \delta {\cal F}_r \\
-i\left({\omega\over c}\right)\delta {\cal F}_z  + ik_z c \delta p_r  &
+ & {g\over\kappa}(2ik_z \delta v_z + ik_r \delta v_r) - 4i{\omega\over c}p_r \delta v_z =
     -cg\delta\rho - \kappa\rho \delta{\cal F}_z
%-{g\over z\kappa}\delta v_z 
\\
-i(\omega/c)\delta {\cal F}_\phi &+ & i(k_z/c){\cal F}_z\delta v_\phi  -  
4i(\omega/c)p_r\delta 