Fluid flow in the volute of a turbocharger turbine can be decidedly unsteady due to the pulsating nature of the exhaust gas in the manifolds of an internal combustion engine. Despite this it is conventional to use a quasi-steady or "filling and emptying" technique to model the turbine in one-dimensional turbocharged engine simulations. Depending on the inherent level of unsteadiness, this approach may be insufficient to capture the true turbine operation since neither method is able to resolve unsteady effects due to the presence of any wave action in the flow. Building on previously reported work, this paper aims to establish a measure of unsteadiness that takes account of the attributes of engine exhaust gas flow that give rise to gas dynamic unsteadiness. This characterization is achieved by decomposing the pulse into its constituent frequencies using Fourier analysis. A one-dimensional wave action code, featuring a bespoke boundary condition that permits application of a pressure pulse in Fourier series form, is used to investigate the effect of the contributing variables for some simplified cases. This allows the construction of the correct form of dimensionless parameter. Finally, the new dimensionless measures, the Fourier series Strouhal and acoustic Strouhal numbers (FSt and FaSt respectively), are evaluated at different test conditions to establish criteria for the transition from a filling and emptying mode to gas dynamic operation. The analysis suggests limiting values of FSt < 0.15, and FaSt < 0.02, to be used as an approximate guide for turbine model selection.