A Transfer Matrix for the Input Impedance of weakly tapered Cones as of Wind Instruments

TL;DR

This paper introduces a transfer matrix for calculating the input impedance of weakly tapered conical wind instrument horns, improving resonance frequency predictions by explicitly incorporating viscous and thermal losses.

Contribution

It presents a novel impedance transmission matrix for conical waveguides that accounts for losses and aligns better with detailed models, without requiring axis discretization.

Findings

Resonance frequencies match more comprehensive models.

The matrix effectively models long, slightly tapered cones.

No axis discretization needed for accurate results.

Abstract

A formula for the local acoustical admittance in a conical waveguide with viscous and thermal losses given by Nederveen [(1969) Acoustical Aspects of Woodwind Instruments (Frits Knuf, Amsterdam)] is rewritten as an impedance transmission matrix. Based on a self-consistent approximation for the cone, it differs from other one-dimensional transmission matrices used in musical acoustics, which implicitly include the loss model of a cylinder. The resonance frequencies of air columns calculated with this transmission matrix are in better agreement with more comprehensive models. Even for long cones with a slight taper, there is no need to discretize along the axis.