Modelling the Mechanical Response of the Reed-mouthpiece-lip System of a Clarinet. Part I. The Finite-difference Approach

The motion of a clarinet reed that is clamped to a mouthpiece and supported by a lip is simulated in the time domain using finite difference methods. The reed is modelled as a bar with non-uniform cross section, and is described using a one-dimensional, fourth-order partial differential equation. The interactions with the mouthpiece lay and the player's lip are taken into account by incorporating conditional contact forces in the bar equation. The model is completed by clamped-free boundary conditions for the reed. An implicit finite difference method is used for discretising the system, and values for the physical parameters are chosen both from laboratory measurements and by accurate tuning of the numerical simulations. The accuracy of the numerical system is assessed through analysis of frequency warping effects and of resonance estimation. Finally, the mechanical properties of the system are studied by analysing its response to external driving forces. In particular, the effects of reed curling are investigated.