On Minimum Bow Force for Bowed Strings

A famous theoretical prediction of the minimum bow force to maintain Helmholtz motion of a bowed string is re-examined to take account of effects associated with resonances of the instrument body. Starting from a more robust assumption of an ideal stick-slip velocity waveform at the bowing point rather than a perfect sawtooth-shaped excitation force at the bridge, the analysis predicts that the minimum bow force, and the force waveform exciting the instrument bridge, can depend in a complicated way on the position of the bow on the string. Also, the frequency of maximum wolfiness of an instrument like a cello is predicted to shift away from that of the strong body resonance causing a wolf note. Simulations are used to evaluate the new formulation. For the simple case in which the string vibrates only in a single polarisation, the results are accurately confirmed. However, simulation also reveals that string vibration in the second polarisation can change the detailed response. Further simulations are used to investigate the influence on minimum bow force of some physical details of the model, especially torsional string motion and the presence of sympathetic strings.