The dynamics of Tibetan singing bowls

Tibetan bowls have been traditionally used for ceremonial and meditation purposes, but are also increasingly being used in contemporary music-making. They are handcrafted using alloys of several metals and produce different tones, depending on the alloy composition, their shape, size and weight. Most important is the sound-producing technique used - either impacting or rubbing, or both simultaneously - as well as the excitation location, the hardness and friction characteristics of the exciting stick (called puja). Recently, researchers became interested in the physical modelling of singing bowls, using waveguide synthesis techniques for performing numerical simulations. Their efforts aimed particularly at achieving real-time synthesis and, as a consequence, several aspects of the physics of these instruments do not appear to be clarified in the published numerical formulations and results. In the present paper, we extend to axi-symmetrical shells - subjected to impact and friction-induced excitations - our modal techniques of physical modelling, which were already used in previous papers concerning plucked and bowed strings as well as impacted and bowed bars. We start by an experimental modal identification of three different Tibetan bowls, and then develop a modelling approach for these systems. Extensive nonlinear numerical simulations were performed, for both impacted and rubbed bowls, which in particular highlight important aspects concerning the spatial patterns of the friction-induced bowl vibrations. Our results are in good agreement with preliminary qualitative experiments.