This is the companion page for the paper:
C.Desvages, S.Bilbao, M.Ducceschi, Improved frequency-dependent damping for time domain modelling of linear string vibration
which was presented at the 22nd International Congress on Acoustics, held in Buenos Aires, Argentina, on September 8, 2016.
Abstract: Lossy linear stiff string vibration plays an important role in musical acoustics. Experimental studies have demonstrated the complex dependence of decay time with frequency, confirmed by detailed modelling of dissipated power in linear strings. Losses at a particular frequency can be expressed as a function of the physical parameters defining the system; damping due to air viscosity is predominant at low frequencies, whereas internal friction prevails in the higher frequency range.
Such a frequency domain characterisation is clearly well-suited to simulation methods based on, e.g., modal decompositions, for experimental comparison or sound synthesis. 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, ultimately, 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, or dissipativity. 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.
Physical model and numerical resolution
A time-domain system of partial differential equations describing the evolution of the transverse displacement of a string is discretised into an energy-balanced, dissipative finite difference scheme. A two-step implicit recurrence is derived, allowing to compute the state of the string at a given time step using results at the two previous time steps. The theoretical loss profile is first computed from the physical parameters of the string and air; then, an optimisation routine is performed, yielding the appropriate damping coefficients; finally, these coefficients are injected into the scheme, and the recurrence is implemented, calculating the string displacement step by step, at audio sample rates. The string position is initialised with a triangular function whose peak lies at the desired plucking location; the output signal is read out as the displacement of the last moving grid point before the boundary.
We present here a few synthesised plucked string sounds, including a violin, cello, guitar, and piano string. The results are computed for an isolated string, simply supported at its boundaries, without any body/soundboard effects.
Synthesised sound examples
The most eloquent demonstration of the need of a realistically frequency-dependent loss profile for the synthesis of musical strings would probably be the comparison of a simple, frequency-independent loss model (constant across the spectrum), with the refined, much more realistic model, on the same violin A string:
A few other different strings were also simulated:
Cello D string:
Steel guitar B string:
Piano E2 string, initialised with a local raised cosine velocity distribution to simulate a basic strike (note: nonlinear effects are not part of the model):