The Wolf and the Cello: Modelling and design of multiple resonance suppressors in large string instruments

The wolf note is an acoustic instability that occurs in large bowed string instruments when a strong body resonance interacts with the vibrating string, producing amplitude modulation and loss of tonal control. Various wolf suppressors - tuned mass dampers attached to the string or to the instrument body - are used in practice to mitigate this effect. In this paper, we propose a mathematical model describing the coupled dynamics of a string and a two-dimensional body equipped with one or two wolf suppressors. Both string and body include elastic (second-order) and stiffness (fourth-order) contributions and can be excited either by plucking or bowing. Three performance indicators are introduced: The first one perceives the wolf-tone appearance, the second one quantifies the attenuation of the notes possibly caused by the wolf suppressor, and the third one measures the acoustic fidelity (in terms of spectrum) with respect to the original instrument. The proposed numerical tests give insights about optimal tuning and placement of one or two suppressors, achieving effective wolf-note suppression while preserving as much as possible the global tonal balance.