Helmholtz vibrations in bowed strings

For almost 160 years, it has been known that Helmholtz oscillations, unique to vibrating strings in bowed instruments (violin, cello, etc.), have two distinct regimes: "slip" and "stick." During the slip regime, the force at the bow-string interaction is attributed to friction between the sliding bow hair and the vibrating string, with a friction coefficient that decreases with increasing relative velocity. Yet the hair-string interaction during the stick regime is less understood. We propose that the interaction force during the stick regime is proportional to the product of the longitudinal acoustic impedance of the bow hair to the relative bow-string velocity. We validate this hypothesis by solving the string's differential equation of motion, including an enhanced formulation to avoid parasitic high-frequency oscillations. This physical model enables us to analyze, in real time, the characteristics of the Helmholtz oscillations, including the string shape, excitation of harmonics, Schelleng ripples, and string energy, showing that the bowed string gains energy during the stick regime and loses energy during the slip regime.