Theory of the single woodwind tone hole

Abstract

A single tone hole may be generally represented in the transmission-line description of a woodwind musical instrument by a T-section circuit. This circuit element is comprised of a shunt and series impedance whose values differ depending on whether the hole is open or closed. The imaginary parts of these impedances have here been calculated as functions of frequency in terms of the main bore diameter, and the tone-hole chimney height and diameter. The loss-free wave equation is solved for a single tone hole placed through the wall of a cylindrical main bore. Green function expansions are made in both the main air column and tone hole. These two expansions must be equal on the surface of intersection between the main bore and tone hole, and this condition may be expressed by an integral equation over the surface. An approximate solution to the integral equation is obtained by Schwinger’s variational formulation, and the impedance parameters in the T section are thereby computed and displayed in a form for use by musical acousticians. The series impedance for both open and closed holes is a negative inertance. The shunt impedance of a closed hole is a compliance which is simply related to the closed-hole volume. The shunt impedance of an open hole is written in terms of the open-hole effective length and expressed as a function of the tone-hole proportions.