The wave motion resulting from a line load moving through an unbounded elastic solid is considered. The line load suddenly appears in the body, and then moves in a fixed direction with nonuniform speed. Exact expressions are derived for the displacement potential functions for all places in the body, for all time, by Laplace transform methods. The wavefronts, particularly the Mach waves trailing the moving load, are determined by integrating the bicharacteristic equations. The details of the wavefront shapes are illustrated and discussed for a load accelerating or decelerating in a prescribed way. Finally, a sample first-motion calculation is carried out in which the mean normal stress is determined immediately behind the dilatational wavefront by asymptotic methods.