### Abstract

A single tone hole may be generally represented in the transmission-lines description of a woodwind musical instrument by a T-section circuit. This circuite 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 equationis 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 there by computed and displayed in a form for use by musical acousticians. The series impedance for both open and closed holes is a negativeinertance. The shunt impedance of a closed hole is a compliance which is simply relatedto 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.

Original language | English (US) |
---|---|

Pages (from-to) | 676-687 |

Number of pages | 12 |

Journal | Journal of the Acoustical Society of America |

Volume | 72 |

Issue number | 3 |

DOIs | |

State | Published - Jan 1 1982 |

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### ASJC Scopus subject areas

- Acoustics and Ultrasonics
- Arts and Humanities (miscellaneous)

### Cite this

**Theory Of The Single Woodwind Tone Hole.** / Keefe, Douglas H.

Research output: Contribution to journal › Article

*Journal of the Acoustical Society of America*, vol. 72, no. 3, pp. 676-687. https://doi.org/10.1121/1.388248

}

TY - JOUR

T1 - Theory Of The Single Woodwind Tone Hole

AU - Keefe, Douglas H

PY - 1982/1/1

Y1 - 1982/1/1

N2 - A single tone hole may be generally represented in the transmission-lines description of a woodwind musical instrument by a T-section circuit. This circuite 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 equationis 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 there by computed and displayed in a form for use by musical acousticians. The series impedance for both open and closed holes is a negativeinertance. The shunt impedance of a closed hole is a compliance which is simply relatedto 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.

AB - A single tone hole may be generally represented in the transmission-lines description of a woodwind musical instrument by a T-section circuit. This circuite 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 equationis 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 there by computed and displayed in a form for use by musical acousticians. The series impedance for both open and closed holes is a negativeinertance. The shunt impedance of a closed hole is a compliance which is simply relatedto 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.

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U2 - 10.1121/1.388248

DO - 10.1121/1.388248

M3 - Article

VL - 72

SP - 676

EP - 687

JO - Journal of the Acoustical Society of America

JF - Journal of the Acoustical Society of America

SN - 0001-4966

IS - 3

ER -