It is common to describe the field-aligned currents associated with auroral arcs in terms of an Alfven wave propagating earthward. Taking account of convection, the wave can be stationary relative to the earth with its normal nearly perpendicular to the field. Far enough ahead of the wave linear theory is valid and here theory is restricted to both the linear and cold plasma approximations, but is sufficient to yield a significant result. This predicts that, as the wave propagates towards the earth, the wave normal becomes more nearly perpendicular to the field and the characteristic thickness decreases.