| Résumé |
We present an exact method to solve a one-dimensional nonlinear transport equation in a dissipative non homogeneous media when the damping is frequency-independent. This work was motivated by the case of brass musical instruments whose functioning at high sound levels implies nonlinear propagation. Though in that latter case, the medium is homogeneous, our approach is more general. Usually, the wave propagation in musical wind instruments is justifiably considered to be linear. A well-known counter-example is the case of brass instruments at high sound level. In this case, the nonlinear effects become dominant. They account for the graduated waveshape distortion due to their cumulative nature which eventually leads to the arrival of shock-waves. For the class of propagation models under study in this paper, we derive an exact method which allows to recover an input-output formalism and an efficient algorithm in the time domain. The method is based on three key points: (1) a change of function which turns the original problem into a conservative problem of hyperbolic type, (2) the adaptation of the standard "characteristics method" from which all possible solutions can be deduced, and (3) the introduction of an easily computable criterion which naturally selects the "physically meaningful" solution (this latter point provides a generalization of the "potential function" proposed by Hayes [Hayes,1969]. This approach operates for regular and continuous solutions as well as shocks and multiple shocks. Finally, a fast algorithm is deduced and proposed for real-time sound synthesis issues. |
