A comparison of modeled and measured impedance of brass instruments and their mouthpieces and bells

Several techniques for modeling and measuring the impedances of mouthpieces and flaring bells of brass instruments (in this case, a trumpet) are presented. A method for isolating the measured instrument impedance without a mouthpiece is described. The ultimate goal is to be able to use transfer matrix techniques to model a full brass instrument. Additionally, methods for characterizing mouthpieces from both model and measurement are outlined, along with a parameterization of the inner geometry of the mouthpiece. Results of transfer matrix and finite element calculations for isolated mouthpieces are found to match each other well, though there is room for improvement regarding the agreement between calculation and measurement, most likely due to uncertainty in the geometry. The calculation of transfer matrix elements from impedance measurements is promising, but is also affected by uncertainty in the geometry of the end condition. Several different impedance calculations have been compared with a measurement of a 3D-printed scale model of a trumpet bell. There is very good agreement between calculation and measurement using the finite element technique, and the transfer matrix technique is found to match best when pulsating sphere radiation and segment lengths corresponding to spherical wave propagation are taken into account.