N^{\frac{1}{3}} < k_{B}T < \frac{1}{m a^{2}_{12}}$, where $\langle \omega \rangle = (\omega_{x}\omega_{y}\omega_{z})^{1/3}$ is the average trap frequency [@pethick]. Although this is typically a wide temperature range $10^{-7} < T < 10^{-3}$K, diffusive relaxation damps out spin waves at higher temperatures.

A two-component gas has a longitudinal and a transverse spin mode. The longitudinal mode has a linear dispersion, but is strongly Landau damped [@bashkin]. By contrast, the transverse mode is weakly damped and propagates with dispersion $\omega = \frac{k^{2}v^{2}_{T}\tau_{ex}}{2}\left(1 - i \frac{\tau_{ex}}{\tau_{D}}\right)$, where $\frac{\tau_{ex}}{\tau_{D}} \sim \frac{a_{12}}{\Lambda_{T}} \le 1$ [@levy]. These weakly damped spin waves have been observed in the context of spin polarized Hydrogen by Johnson *et al.* [@cornell; @du; @lee].

Spin waves in the spin-$1$ gas
==============================

In this section we linearize Eq.(\[eq:2\]) about stationary states. In Sec.IV we consider more general dynamics.

Excitations about the $m_{F}=1$ state
-------------------------------------

We begin by linearizing about the $m_{F} = 1$ state. We start with a homogenous gas of particles with a Maxwellian velocity distribution and initial density $n_{0}$. The self-interaction between two particles is proportional to $a_{2}$, and so we define $\Omega_{int} = (c_{0} + c_{2})n_{0}$. We begin by considering the collisionless limit, $1 \ll \Omega_{int}\tau_{D}$.

(150, 160)(10, 10) (-38, 160)[(a)]{} 