A pressure-based transfer matrix method and measurement technique for studying resonances in flutes and other open-input resonators.

Abstract

This paper provides an alternative formulation of a transfer matrix method (TMM) long used for modeling the passive resonances of wind instruments. In the conventional, impedance-based TMMs, the quantities being evaluated are the acoustic pressure P and the acoustic flow U; in the present pressure-based TMM (a variation of the Heavens matrix method developed for thin film optics), the quantities being evaluated are the amplitudes of the forward-going and backward-going pressure waves, P+ and P-. Power spectra (|P++P-|2 vs frequency) for several types of branched and non-branched open-input resonators are computed and then compared to experimental spectra obtained from a Fourier analysis of the acoustic pressure at a position inside the resonator recorded during excitation by a localized sound source placed just outside the resonator. Agreement is good for both one-end-closed polyvinyl chloride chimney pipes and a modern flute set up with a variety of lip, keywork, and open/closed tonehole configurations, although fits for very short straight tubes appeared to require unphysically high wall losses. Input impedance derived from the computed values of P+ and P- at the flute's embouchure hole appears consistent with directly measured input impedance data in the literature.