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 when the finger is removed from the string after flageolet harmonics is performed on bowed string instruments. However, little has been understood about this harmonics-sustaining phenomenon and the parameter dependency of its sustaining time. We devise a mathematical model by incorporating the effects of bowing and touching into a one-dimensional wave equation. The devised model reproduces the harmonics-sustaining phenomenon, and the parameter dependence of sustaining time is qualitatively consistent with empirical observations. We find that the parameter dependence of sustaining time follows the power law, which gives rise to a formula that expresses the relationship between the sustaining time and the maximum and minimum bow force required to generate Helmholtz motion.

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