The Nonlinear String Wave Equation: Properties, Approximations and Resolution. Application to the Piano
The Nonlinear String Wave Equation
This seminar is the third of a series of three presentations devoted to the piano. The first presentation on Monday was aimed at presenting the “piano project” at ENSTA with the ultimate goal to model a piano in the time domain from the initial blow of the hammer to the radiated sound. The second presentation on Tuesday was mainly focused on the measurements program which is necessary for obtaining the parameters to insert in the equations, and for helping in determining the salient features of the physical aspects of the instrument.
Today’s presentation is more theoretical. It will be devoted to the study of the nonlinear string wave equation. As shown in the first lecture, taking the geometrical nonlinearity of the string into account is mandatory for an appropriate modeling of the precursor and phantom partials which are important parts of the piano tones, especially during the attack transients. The exact nonlinear string equation will be presented as well as some of current approximations. It will be shown to what extent such approximations are acceptable or not in the context of numerical modeling. Finally, the complete set of hammer-string-bridge equations used for the modeling of the piano will be presented and discussed.