Physical Modeling by Directly Solving Wave PDE

We are currently developing a physical modeling method for string instruments based upon the direct solution of the string equation, rather than the simulation of travelling waves as is instead done in the Waveguide method. Although computationally heavier in comparison with Waveguides, our approach enables any physical parameter to be time-varied during playing, obtaining thus interesting effects from a musical viewpoint. Particular attention has been paid in the modeling of the bow, which is the main responsible of the peculiar timbre of bowed strings. The rosin thermal behaviour has been modelled by thresholds. Moreover, the bowing noise is simulated by adding noise to the bow force profile. The whole model runs in real time on today Wintel Workstations for bowed string of the violin or viola class. It produces good sounds not only in sustained, gently bowed articulations, but also in harsh articulations like e.g. "strappato", as well as outside the physically possible parameters zone. The model has been actually used with a primary role by the Italian Composer Michelangelo Lupone in two newly composed electronic and mixed music pieces. Glissandos of string parameters – like f.i. the internal damping and displacements of the bowing point play a main expressive role in these works. The novelty in the string equation we used is a partial third-order derivative mixed term describing the internal friction of the string, which is responsible of the frequency-dependent damping in actual strings. The free-motion analytic solutions of the corresponding PDE is discussed, together to the numerical integration method developed (based on a combination of the sinus transform, and of the well-known Stoermer rule), whose results are analysed from a timbre point of view. We also show a short comparison with Waveguides, and future improvements to continuously varying bowing point and left-hand finger position, and to reduce both asymptotic and actual computational complexity.