Comparison between acoustic measurements of brass instruments and one-dimensional models with curved wavefronts and transformed axial coordinates.

Abstract

A progressive spherical or spheroidal wavefront approximation has previously been found to be a necessary step for a more accurate application of Webster's wave equation to rapidly flaring horns. This leads to a necessary transformation of the horn area function, from the usual flat cross-sectional area in terms of the axial coordinate, into a curved cap-like wavefront area as a function of either the axial coordinate, the arc-length coordinate along the horn profile, the leading curved wavefront coordinate, or still other possible longitudinal coordinates. In this article, horn functions, and related frequency potential functions are calculated from the measured horn profiles of a trombone and a trumpet for several of the above parameterizations. From them, cutoff frequencies and effective lengths are determined. A comparison is drawn between theoretical results using different parameterizations, results calculated via transfer-matrix models, and experimental measurements of the acoustical input impedance and reflection function of both instruments. Results indicate that one-dimensional models accurately predict the effective lengths, and consequently the fundamental resonance frequency of the instruments within ±25 cents, but fail noticeably in predicting cutoff frequencies, leading to what is probably an inaccurate representation of perceived timbre.