Abstract
This paper investigates how two-port network theory as a means for system identification can be applied to the analysis of brass instruments. A special focus is placed on the energy conversion efficiency as this is limited by inner damping, which receives much attention by expert players and makers of brasses. Theory suggests that a reconstruction of the 2 × 2 matrix representing the network requires input impedance and transfer function for two different boundary conditions. Besides the normal case of free sound radiation, instruments are also analyzed with the bell closed by a spherical cap. For this purpose, a customized 3D-printed spherical cap was fabricated and attached to the bell. Four measured spectra and the passivity condition over-determine the set of system equations. It is shown how to take advantage of this freedom when analyzing wind instruments. Measurements and simulations of a trumpet and a trombone are presented and compared.
Highlights
This paper investigates how two-port network theory as a means for system identification can be applied to the analysis of brass instruments
Brass instruments can be modeled as wave guides, in which sound waves move back and forth between the two more or less reflective terminations when excited by an oscillating sound flow at the mouthpiece
This research is based on the already mentioned paper by Elliott et al (1982), who applied electrical network theory to wind instruments by treating trumpets or trombones as acoustical two-port networks. They described such instruments in terms of their input impedance, when terminated by an impedance corresponding to a radiation load at the bell, and by the pressure transfer function, which corresponds to the voltage gain of an electrical circuit
Summary
Brass instruments can be modeled as wave guides, in which sound waves move back and forth between the two more or less reflective terminations when excited by an oscillating sound flow at the mouthpiece. The four matrix elements of each elementary two-port network are complex functions of the angular frequency x, geometric parameters such as bore diameters and length of the slice, and physical parameters such as air density and viscosity, speed of sound, and temperature. The input impedance spectrum of a wind instrument contains information about frequency, strength, and quality factor Q of all air column resonances that can be excited by a player. The transfer function TpðxÞ between the sound pressure at the excitation point p1ðxÞ and the sound pressure at the open mouth of the bell p2ðxÞ deserves specific attention Without this knowledge, it is possible neither to predict the radiated sound nor to assess the power efficiency of brass wind instruments—both being important for musicians and makers when discussing quality aspects
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