Reducing computational complexity in struck string physical models

Abstract

Chaigne and Askenfelt recently published a paper in which they described a physical model of a struck string [J. Acoust. Soc. Am. 95, 1112– 1118 (1994)]. The model is based on a finite difference approximation to a particular partial differential equation, and involves updating transversal displacement at discrete locations along a string. A nonlinear hammer is also incorporated into the model. It is shown in this paper that, if it is desired to know the output only at one particular point (or perhaps a weighted linear combination of points) along the string, then the number of operations required in order to compute the synthesized output can be reduced. This is done by expressing Chaigne and Askenfelt’s model in state space form, and by then performing appropriate coordinate transformations to ‘‘modal‘‘ coordinates. What is more, computational complexity in the modal coordinates will be independent of the number of spatial derivatives modeled, so that accurate modeling of dispersion can be achieved at no extra cost. The coordinate-changing method applies equally well to the case of spatially varying media, and is simply extended to models which are higher order in time.