Models for high Mach number, isothermal, pressure-matched jets ejected from sources with variable velocities are presented. It is found that the relaxation of the assumption of a time-independent source allows the complex position-velocity diagrams and multiple bow shock structures observed in some stellar jets to be explained in a straightforward way. Analytic and numerical models are presented for jets with variable velocity sources. A comparison with previously published observations of the HH 46/47 jet is performed which shows how the velocity variations observed along a jet can be used to reconstruct the past time variability of the source. This technique is promising for future studies of the mechanism by which stellar jets are produced. 41 refs.