Improved frequency-dependent damping for time domain modelling of linear string vibration

Abstract

Lossy linear stiff string vibration plays an important role in musical acoustics. Experimental studies have demonstrated the dependence of decay time on frequency, confirmed by detailed modelling of dissipated power in linear strings. Losses at a particular frequency can be expressed in terms of the physical parameters defining the system; damping due to air viscosity is predominant at low frequencies, whereas internal friction prevails at higher frequencies. Such a frequency domain characterisation is well-suited to simulation methods based on, e.g., modal decomposition. However, more general string models might include features difficult to realise with such models, in particular nonlinear effects. In this case, it is useful to approach modelling directly in the space-time domain. This work is concerned with the translation of the frequency domain damping characteristics to a space-time domain framework, leading to a coupled system of partial differential equations. Such a system can be used as a starting point for a time-stepping algorithm; an important constraint to ensure numerical stability is then that of passivity. Candidate loss terms are characterised in terms of positive real functions, as a starting point for optimisation procedures. Simulation results are presented for a variety of linear strings.