  An ultra-cold Bose gas of two-level atoms can be thought of as a spin-1/2
Bose gas. It supports spin-wave collective modes due to the exchange mean
field. Such collective spin oscillations have been observed in recent
experiments at JILA with ${}^{87}$Rb atoms confined in a harmonic trap. We
present a theory of the spin-wave collective modes based on the moment method
for trapped gases. In the collisionless and hydrodynamic limits, we derive
analytic expressions for the frequencies and damping rates of modes with dipole
and quadrupole symmetry. We find that the frequency for a given mode is given
by a temperature independent function of the peak density $n$, and falls off as
$1/n$. We also find that, to a very good approximation, excitations in the
radial and axial directions are decoupled. We compare our model to the
numerical integration of a one dimensional version of the kinetic equation and
find very good qualitative agreement. The damping rates, however, show the
largest deviation for intermediate densities, where one expects Landau damping
-- which is unaccounted for in our moment approach -- to play a significant
role.
