Nonlinear transverse string vibration: Model comparisons and numerical methods

Abstract

There has recently been an increase in activity in developing digital models of nonlinear string vibration, for sound synthesis purposes; in a musical setting, such behavior is of interest under large-amplitude plucked string conditions. The usual starting point for such a model is a one-dimensional time-dependent second-order partial differential equation, which is discretized to yield a recursion; finite difference schemes are perhaps the most straightforward approach. There are, however, many suitable models; a general nonlinear transverse wave equation is taken as a starting point, and several approximations, namely the Kirchhoff–Carrier equation, digital waveguide type (i.e., tension-modulated) formulations, and the linear wave equation, are examined. The various solutions are compared and contrasted, both in terms of travelling wave propagation and the evolution of frequency content. The possibilities of and problems inherent in developing well-behaved numerical methods are discussed on a case-by-case basis, calling particular attention to the use of energetic quantities in discrete time as a means of controlling stability.