Computational Modeling of Stiff Piano Strings Using Digital Waveguides and Finite Differences

As is well-known, digital waveguides offer a computationally efficient, and physically motivated means of simulating wave propagation in strings. The method is based on sampling the traveling wave solution to the ideal wave equation and linearly filtering this solution to simulate dispersive effects due to stiffness and frequency-dependent loss; such digital filters may terminate the waveguide or be embedded along its length. For strings of high stiffness, however, dispersion filters can be difficult to design and expensive to implement. In this article, we show how high-quality time-domain terminating filters may be derived from given frequency-domain specifications which depend on the model parameters. Particular attention is paid to the problem of phase approximation, which, in the case of high stiffness, is strongly nonlinear. Finally, in the interest of determining the limits of applicability of digital waveguide techniques, we make a comparison with more conventional finite difference schemes, in terms of computational cost and numerical dispersion, for a set of string stiffness parameters.