The wave digital piano hammer
Stefan Bilbao, Julius O. Smith, Julien Bensa, Richard Kronland-Martinet · The Journal of the Acoustical Society of America · 2002
For sound synthesis purposes, the vibration of a piano string may be simply modeled using bidirectional delay lines or digital waveguides which transport traveling wavelike signals in both directions. Such a digital wave-type formulation, in addition to yielding a particularly computationally efficient simulation routine, also possesses other important advantages. In particular, it is possible to couple the delay lines to a nonlinear exciting mechanism (the hammer) without compromising stability; in fact, if the hammer and string are lossless, their digital counterparts will be exactly lossless as well. The key to this good property (which can be carried over to other nonlinear elements in musical systems) is that all operations are framed in terms of the passive scattering of discrete signals in the network, the sum of the squares of which serves as a discrete-time Lyapunov function for the system as a whole. Simulations are presented.