Multirate Simulations of String Vibrations Including Nonlinear Fret-String Interactions Using the Functional Transformation Method

• Published in: EURASIP journal on advances in signal processing 2004-06, Vol.2004 (7), p.949-963
• Main Authors: R. Rabenstein, L. Trautmann
• Format: Article
• Language: English
• Subjects: functional transformation; multidimensional system; multirate approach; nonlinear; partial differential equation; vibrating string
• ISSN: 1687-6172; 1687-6180
• DOI: 10.1155/S1687617204312059

The functional transformation method (FTM) is a well-established mathematical method for accurate simulations of multidimensional physical systems from various fields of science, including optics, heat and mass transfer, electrical engineering, and acoustics. This paper applies the FTM to real-time simulations of transversal vibrating strings. First, a physical model of a transversal vibrating lossy and dispersive string is derived. Afterwards, this model is solved with the FTM for two cases: the ideally linearly vibrating string and the string interacting nonlinearly with the frets. It is shown that accurate and stable simulations can be achieved with the discretization of the continuous solution at audio rate. Both simulations can also be performed with a multirate approach with only minor degradations of the simulation accuracy but with preservation of stability. This saves almost 80% of the computational cost for the simulation of a six-string guitar and therefore it is in the range of the computational cost for digital waveguide simulations.