A new mechanism is proposed for predicting and interpreting the motion of auroral arcs observed in the nightside ionosphere during the expansion phase of a substorm. This mechanism is centred on the idea that such arcs act as visible manifestations of the arrival of earthward-propagating shock waves in the near-Earth magnetosphere. These shock waves are generated at a near-Earth X-line, and propagate at the local Alfvén speed. Because of the non-uniform nature of the magnetised plasma in the magnetotail, dispersion results in a change in the shape of the wave fronts as the shocks propagate towards the ionosphere. Theoretical analysis shows that a variety of arc motions can occur as a result of this dispersion, depending on factors such as the reconnection rate, the location of the re-connection site, and gradients in the magnetic field strength and plasma density.