Alfvén wave

Remarks

Check for $\mathbf{k} = k \mathbf{e}_z$ and $\mathbf{U} = U \mathbf{e}_x$: the linearised equations are $\rho_0 \partial_t \mathbf{u} = -\nabla p^* + (B_0/\mu_0) \partial_z \mathbf{b}$ and $\partial_t \mathbf{b} = B_0 \partial_z \mathbf{u}$, where $p^*$ is the total pressure perturbation. With $p^* = 0$ and $\mathbf{b} = b \, \mathbf{e}_x \cos(kz - \omega t)$, the second gives $\omega b = -k B_0 U$, and the first gives $\rho_0 \omega U = -k B_0 b / \mu_0$; together $\omega^2 = k^2 B_0^2 / (\mu_0 \rho_0)$. For $\omega = k v_A$ this gives $\mathbf{b} = -\sqrt{\mu_0 \rho_0} \, \mathbf{u}$. The group velocity is $\pm v_A \mathbf{e}_z$ for every $\mathbf{k}$, the sign being that of $\omega / (\mathbf{k} \cdot \mathbf{e}_z)$, so energy always travels along the background field. Non-example: if $B_0 = 0$, then $v_A = 0$ and there is no oscillation; the perturbation does not propagate. In compressible MHD there are also the fast and slow magnetosonic waves, which compress the fluid and are different from Alfvén waves.