A Comparison of a One-Dimensional Finite Element Method and the Transfer Matrix Method for the Computation of Wind Music Instrument Impedance

Abstract

This work presents a computation tool for the calculation of wind instrument input impedance in the context of linear planar wave propagation with visco-thermal losses. The originality of the approach lies in the usage of a specific and simple 1D finite element method (FEM). The popular Transfer Matrix Method (TMM) is also recalled and a seamless formulation is proposed which unifies the cases cylinders vs. cones. Visco-thermal losses, which are natural dissipation in the system, are not exactly taken into account by this method when arbitrary shapes are considered. The introduction of an equivalent radius leads to an approximation that we quantify using the FEM method. The equation actually solved by the TMM in this case is exhibited. The accuracy of the two methods (FEM and TMM) and the associated computation times are assessed and compared. Although the TMM is more efficient in lossless cases and for lossy cylinders, the FEM is shown to be more efficient when targeting a specific precision in the realistic case of a lossy trumpet. Some additional features also exhibit the robustness and flexibility of the FEM over the TMM. All the results of this article are computed using the open-source python toolbox OpenWind.