Input impedance computation for wind instruments based upon the Webster-Lokshin model with curvilinear abscissa
Résumé
This work addresses the computation of acoustics immittances of axisymmetric waveguides, the shape of which is C1-regular (i.e. continuous and with a continuous derivative with respect to the space variable). With this intention, a refined version of the "Webster" horn equation is considered, namely, the "Webster-Lokshin equation with curvilinear abscissa", as well as simplified models of mouthpieces and well-suited radiation impedances. The geometric assumptions used to derive this uni-dimensional model (quasi-sphericity of isobars near the wall) are weaker than the usual ones (plane waves, spherical waves or fixed wavefronts). Moreover, visco-thermal losses at the wall are taken into account. For this model, exact solutions of the acoustic waves can be derived in the Laplace or the Fourier's domains for a family of parametrized shapes. An overall C1-regular bore can be described by connecting such pieces of shapes under the constrain that junctions are C1-regular. In this case and if the length of the bore is fixed, a description with N pieces precisely has 2N+1 degrees of freedom. An algorithm which optimizes those parameters to obtain a target shape has been built. It yields accurate C1-regular descriptions of the target even with a few number of pieces. A standard formalism based on acoustic transfer matrices (deduced from the exact acoustic solutions) and their products make the computation of the input impedance, the transmittance (and other immittances) possible. This yields accurate analytic acoustic representations described with a few parameters. The paper is organized as follows. First, some recalls on the history of the "Webster" horn equation and of the modeling of visco-thermal losses at the wall are given. The "Webster-Lokshin model" under consideration is established. Second, the family of parametrized shapes is detailed and the associated acoustic transfer matrices are given. Third, the algorithm which estimates the optimal parameters of the C1-regular model of target shapes is presented. Finally, input impedances obtained using this algorithm (and the Webster-Lokshin model) are compared to measured impedances (e.g. that of a trombone) and to results of other methods based on the concatenation of straight or conical pipes.