We study stress wave propagation in two impulsively forced split Hopkinson bar systems: one with a prestressed interface and one with a frictional interface. We first consider only primary wave transmission and reflection, allowing for reduction of the problem to a first-order, strongly, nonlinear ordinary differential equation. A high-order finite element model is then developed and used to validate the results of the primary-pulse model. A spring that hardens with increasing preload is used to model the prestressed interface while an Iwan element is used to model the frictional interface. Using the primarywave propagation model, we perform nonlinear system identification by matching simulation and experiment results and identify the nonlinear hardening characteristics for the prestressed interface and Iwan parameters for the frictional interface. These parameters are then used in the finite element model to compare the experimentally measured secondary effects with the simulated effects. Our results demonstrate that the primary-wave propagation model can be used as a reduced order model for nonlinear system identification at a fraction of the computational cost of higher-order models.