  In this paper, we study small-scale fluctuations (baryon pressure sound
waves) in the baryon fluid during recombination. In particular, we look at
their evolution in the presence of relative velocities between baryons and
photons on large scales ($k \sim 10^{-1} \ {\rm Mpc}^{-1}$), which are
naturally present during the era of decoupling. Previous work concluded that
the fluctuations grow due to an instability of sound waves in a recombining
plasma, but that the growth factor is small for typical cosmological models.
These analyses model recombination in an inhomogenous universe as a
perturbation to the parameters of the homogenous solution. We show that for
relevant wavenumbers $k\gtrsim 10^3\ {\rm Mpc}^{-1}$ the dynamics are
significantly altered by the transport of both ionizing continuum ($h\nu>13.6$
eV) and Lyman-$\alpha$ photons between crests and troughs of the density
perturbations. We solve the radiative transfer of photons in both these
frequency ranges and incorporate the results in a perturbed three-level atom
model. We conclude that the instability persists at intermediate scales. We use
the results to estimate a distribution of growth rates in $10^{7}$ random
realizations of large-scale relative velocities. Our results indicate that
there is no appreciable growth; out of these $10^7$ realizations, the maximum
growth factor we find is less than $\approx 1.2$ at wavenumbers of $k \approx
10^{3} \ {\rm Mpc}^{-1}$. The instability's low growth factors are due to the
relatively short duration of the recombination epoch during which the electrons
and photons are coupled.
