Oscillation thresholds for "strinking outwards" reeds coupled to a resonator

TL;DR

This paper analyzes the oscillation thresholds of a 'striking outwards' reed coupled to a resonator, providing analytical and numerical insights into frequency ratios and blowing pressures relevant to musical acoustics.

Contribution

It extends previous models by analytically deriving frequency ratio bounds and numerically confirming the influence of reed quality factor on blowing pressure.

Findings

Frequency ratio between threshold and reed resonance is between 1 and √3.

Smallest blowing pressure is proportional to reed quality factor.

Numerical results align with previous studies by Cullen et al.

Abstract

This paper considers a "striking outwards" reed coupled to a resonator. This expression, due to Helmholtz, is not discussed here : it corresponds to the most common model of a lip-type valve, when the valve is assumed to be a one degree of freedom oscillator. The presented work is an extension of the works done by Wilson and Beavers (1974), Tarnopolsky (2000). The range of the playing frequencies is investigated. The first results are analytical : when no losses are present in the resonator, it is proven that the ratio between the threshold frequency and the reed resonance frequency is found to be necessarily within the interval between unity and the square root of 3. This is a musical sixth. Actually the interval is largely smaller, and this is in accordance with e.g. the results by Cullen et al.. The smallest blowing pressure is found to be directly related to the quality factor of the reed. Numerical results confirm these statements, and are discussed in comparison with previous ones by Cullen et al (2000).