Modeling the Longitudinal Vibration of Piano Strings

This study is motivated by the physical modeling of the longitudinal string vibrations in the piano. Informal listening tests show that the longitudinal vibrations play an important role in the attack of the sound, and are responsible for the metallic character of low notes. First, a simple mathematical model is developed for qualitative understanding. Detailed analysis is given for sinusoidal transversal displacement with non-rigid termination, clarifying the generation of phantom partials. To investigate how these effects develop in more natural circumstances, finite-difference string and hammer models are used, with parameters taken from real pianos. For real-time sound synthesis purposes, an efficient modeling approach is presented. The model extends the digital waveguide string model by implementing two additional string models, one for the phantom partials and one for the longitudinal modes.