We investigate free-surface oscillating jets with elliptical cross section, focusing on behavior associated with decaying surface tension. Previous one-dimensional equations for an oscillating jet are extended to allow variable surface tension on short space and time scales relevant for surfactant mixtures. We presume the decay of surface tension as a function of surface age, and derive the resulting jet behavior. Three plausible forms of decay are studied: an exponential decay, a diffusion model derived in Brazee et al. (1994), and an algebraic form due to Hua and Rosen (1991). Our simulations suggest both experimental regimes, and measurable jet features in these regimes, which may be exploited in an inverse formulation to deduce the unknown rapid surface tension decay of a given surfactant mixture. In particular, we establish numerical relationships between the amplitude and the wavelength of either a sustained far-field oscillation or oscillation at a fixed downstream location and the entire history of surface tension decay. These numerical relationships are ideal for the inverse formulation, in that the complete surface tension evolution may be deduced solely from far-field or downstream jet measurements, away from the confined part of the jet where the surface tension is rapidly changing.