e modes increases by factors of several when $G$ falls from $10^{-2}$ to $10^{-4}$. When $\tau \simlt 10^4$, and $G \sim 10^{-4}$, a second mode, in addition to the convective mode, becomes unstable at short wavelengths. This mode is weakly oscillatory, and its very low phase velocity ($\sim 2.5\times10^{-2}$ – $1.6\times 10^{-3}$ the free fall velocity) behaves similarly to that of damped radiation sound waves at the same wavelengths. Analysis of the corresponding amplitudes of the various physical quantitities’ perturbations shows that this mode has magnitude and especially phase relationships between perturbations in pressure, density and velocity components that are intermediate between those of convective modes and those characteristic of radiation sound waves. It is important to note, however, that if the canonical disk solution applies, the product $$G\tau = 2 \left({z \over h}\right)^{1/2} \alpha^{-1} {R_T R_z^{1/2} \over R_R} .$$ In other words, the product $G\tau$ should be very nearly independent of radius (if $\alpha$ is). Even where the relativistic corrections are substantial, $G\tau$ hardly changes because $R_R$ and $R_T$ are very nearly proportional, and $R_z$ has a total range of at most a factor of two (Krolik 1999). Thus, the criterion for which scaled wavelengths should be substantially affected by photon diffusion (i.e., $k_* > \sqrt{G\tau}$) is almost independent of radius, but does depend somewhat on $z/h$.

Given our estimate of $G$ (equation 22), we can also verify our expectation that $\omega/c\kappa\rho \ll 1$. Rewritten in terms of our dimensionless parameters, this ratio is $\omega_* G/\tau$. As we have just estimated, $G \ll 1$ and $\tau \gg 1$ in radiation pressure-supported $\alpha$-disks. Consequently, unless $\omega_*$ is extremely large, $\omega/c\kappa\rho \ll 1$ is a very safe assumption.

As already pointed out, ${\cal Q}_p$ is much like a local version of the vertically-averaged quantity $\alpha$, the ratio of the $r$–$\phi$ stress to the total pressure. On this basis 