Alfvén wave in 1D plasma

What you are seeing: a driver at the coronal base shakes a bundle of magnetic field lines; the transverse kink runs outward along $B_0\hat x$ at the Alfvén speed $v_A = B_0 / \sqrt{\mu_0 \rho}$, with magnetic tension as the restoring force. Plasma parcels are frozen into the lines (ideal MHD) and ride them transversely with $v_y$, illustrating the Walén relation $v_y = \mp b_y / \sqrt{\mu_0 \rho}$ (cyan $v_y$ and orange $b_y$ arrows are antiphase). This is how Alfvén waves carry energy up coronal field lines into the solar wind. The lower strip plots $b_y(x,t)$ and $v_y(x,t)$. Raising $B_0$ or lowering the density speeds the wave up.