Energy-conserving finite difference schemes for tension-modulated strings

The timbre of certain stringed instruments is strongly dependent on large-amplitude vibration, in which case linear models (such as the 1D wave equation), which are often used for sound synthesis purposes, are unsatisfactory. We discuss here a nonlinear generalization of the wave equation, sometimes called the Kirchhoff-carrier equation, which models large amplitude vibration through a modulation of string tension. In particular, we look at a finite difference scheme for the Kirchhoff-carrier equation which is both efficient, and has excellent stability properties (this is often difficult to ensure for nonlinear difference schemes). The key to this stability property is the close attention paid to the energetic behavior of the model and its analogue in the finite difference scheme; such a difference scheme is capable of discrete energy conservation to machine precision. Implementation details are discussed, and simulation results are presented.