Elastic wave propagation in the split Hopkinson pressure bar (SHPB) is discussed with an emphasis on the origin and nature of the oscillations that often trail the leading edge of the pressure wave. We show that in the conditions of the SHPB test the pressure bars vibrate in the fundamental mode and that elastic wave propagation can be fully described mathematically. Excellent agreement is found between experimental results and predictions of the mathematical treatment. This suggests that dispersion effects in the pressure bars can be removed from the strain gage records, which reduces the magnitude of the oscillations in the resulting stress strain curve.