Snapshots of a Gaussian profile using Eq. (22) for the torsional Alfvén wave (solid lines), and Eq. (24) for the shear Alfvén wave (dashed lines), where B_ϕ/B_z0 and B_y/B_z0 are plotted against the running coordinate ξ/L normalised to the initial length of the pulse. The value of the plasma-β is stated on each panel and the characteristic time is . The top, left panel represents the times and the top, right panel the times . The middle left panel is for the times , and the middle right panel the times . The bottom left panel is for the times , and the bottom right panel the times at .