in the normal state, with a wavelength comparable to typical cloud sizes.

From a theoretical perspective, the source of novel physics in a dilute spin-$1$ Bose gas (such as $^{87}$Rb, $^{23}$Na) is the structure of the interactions, described by *two* coupling constants: $c_{0}$ and $c_{2}$, representing spin-independent and spin-dependent contact interactions. The interaction Hamiltonian density takes the form $H_{\rm int} = c_0 n^2/2 + c_2 \langle \vec{\textbf{S}} \rangle^2/2$, where $n$ and $\vec{\bf S}$ are the local density and spin density respectively [@jason]. The coefficient $c_{2}$ has no corresponding analog in the spin-$\frac{1}{2}$ case. This interaction gives rise to spin mixing collisions, where two atoms in the $m_{F} = 0$ hyperfine sub-level can combine to form atoms in the $m_{F} = \pm 1$ states [@ketterle]. Another important consideration is the quadratic Zeeman effect, which arises from the hyperfine interaction and the difference in the coupling between the electronic and nuclear spins. The linear Zeeman effect can be neglected in the Hamiltonian as the total spin of the atoms is a conserved quantity.

We begin our analysis by setting up the problem, and reviewing spin waves in a spin-$\frac{1}{2}$ gas. This allows us to highlight the differences with the spin-$1$ case. Next, 