A mathematical study about the sustaining phenomenon of overtone in Flageolet harmonics on bowed string instruments

Abstract

Flageolet harmonics is a playing technique, in which a player lightly touches a nodal point on a string with their finger. Previous studies have reported that the harmonic sound sustains for a short time after the finger is removed from the string while flageolet harmonics is performed on bowed string instruments. However, the mechanism of this harmonics-sustaining phenomenon and the parameter dependency of its sustaining time remains unclear. The purpose of this study is to mathematically investigate the dependence of sustaining time on parameters related to violin playing and string characteristics, and thereby, to elucidate the mechanism of the harmonics-sustaining phenomenon. To this end, a mathematical model was devised by incorporating the effects of bowing and touching into a one-dimensional wave equation. Subsequently, numerical simulation were performed to analyze the behavior of the model. The devised model successfully reproduced the harmonics-sustaining phenomenon in which the parameter dependence of sustaining time was qualitatively consistent with the author’s empirical observations. It was found that the parameter dependence of sustaining time follows the power law. Furthermore, dimensional analysis was performed, yielding a formula that expresses the relationship between the sustaining time and the maximum and minimum bow force required to generate Helmholtz motion.