Inverse problem for wind musical instruments

Playing wind historical instruments is often conflicting with conservation and protection of museum collections. Some musical and acoustical properties of these instruments therefore remain unknown. Theoretical models can predict some of these properties from the only knowledge of the instruments geometry. A collaboration between Makutu Inria Bordeaux Sud Ouest team, Musée de la Musique - Philharmonie de Paris, Institut Technologique Européen des Métiers de la Musique (ITEMM) at Le Mans and the Centre de Recherche et de Restauration des Musées de France (C2RMF), lead to a procedure hereby applied to the museum collection Besson, which was a leading manufacture of wind instruments, and more specifically to several natural trumpets dating from the early 20th century. The geometry of the instruments has been measured non invasively by means of X-ray tomography in C2RMF. After extracting their bore (evolution of the inner diameter), their entry impedance has been computed. For one specific trumpet (E.0925), simulated data have been compared to measurements with excellent agreement. Despite many uncertainties on the way these instruments were played, simulated sounds can be computed. An « acoustical facsimile » of E.0925 has been crafted from the X-ray and impedance data, and played by a professional natural trumpet player. This allows to compare simulated sounds with human played sounds.

The internal geometry of a wind instrument can be estimated from acoustic measurements. For woodwind instruments, this involves characterizing the inner shape (bore) but also the side holes (dimensions and location). In this study, the geometric parameters are recovered by a gradient-based optimization process, which minimizes the deviation between simulated and measured linear acoustic responses of the resonator for several fingerings through an observable function. The acoustic fields are computed by solving a linear system resulting from the 1D spectral finite elements spatial discretization of the wave propagation equations (including thermo-viscous effects, radiation and side holes). The “full waveform inversion” (FWI) technique exploits the fact that the gradient of the cost function can be computed by solving the same linear system as that of the direct problem but with a different source term. The gradient is computed with better accuracy and less additional cost than with finite-difference. The dependence of the cost function on the choice of the observed quantity, the frequency range and the fingerings used, is first analyzed. Then, the FWI is used to reconstruct, from measured impedances, an elementary instrument with 14 design variables. The results, obtained in about 1 minute on a laptop, are in excellent agreement with the direct geometric measurements.