This paper presents two models of sound production in flute-like instruments that allow time-domain simulations. The models are based on different descriptions of the jet flow within the window of the instrument. The jet-drive model depicts the jet by its transverse perturbation that interacts with the labium to produce sound. The discrete-vortex model depicts the jet as two independent shear layers along which vortices are convected and interact with the acoustic field within the window. The limit of validity between both models is usually discussed according to the aspect ratio of the jet W/h, with W the window length and h the flue channel height. The present simulations, compared with experimental data gathered on a recorder, allow to extend the aspect ratio criterion to the notion of dynamic aspect ratio defined as λ/h where λ is the hydrodynamic wavelength that now accounts for geometrical properties, such as W/h, as well as for dynamic properties, such as the Strouhal number. The two models are found to be applicable over neighboring values of geometry and blowing pressure.

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Computational optimization algorithms coupled with acoustic models of wind instruments provide instrument makers an opportunity to explore new designs. Specifically, they give the possibility to automatically find geometries exhibiting desired resonance characteristics. In this paper, the design optimization of woodwind instruments with complex geometrical features (e.g., non-cylindrical bore profile and side holes with various radii and chimney heights) is investigated. Optimal geometric characteristics are searched to obtain specific target frequencies or amplitude characteristics. However, woodwind instruments exhibit complex input impedance whose features might change drastically for a small variation of the geometry, thus hampering gradient-based optimization. For this reason, this paper introduces new formulations of the impedance characteristics (resonance frequencies and amplitudes). The approach is applied to an illustrative instrument subjected to geometric constraints similar to the ones encountered by manufacturers (a key-less pentatonic clarinet with two-registers). Three optimization problems are considered, demonstrating a strategy to simultaneously adjust several impedance characteristics on all the fingerings.

The propagation in tubes with varying cross section and wall visco-thermal effects is a classical problem in musical acoustics. To treat this aspect, the first method is the division in a large number of short cylinders. The division in short conical frustums with uniform averaged wall effects is better, but remains time consuming for narrow tubes and low frequencies. The use of the WKB method for the transfer matrix of a truncated cone without any division is investigated. In the frequency domain, the equations due to Zwikker and Kosten are used to define a reference result for a simplified bassoon by considering a division in small conical frustums. Then expressions of the transfer matrix at the WKB zeroth and the second orders are derived. The WKB second order is good at higher frequencies. At low frequencies, the errors are not negligible, and the WKB zeroth order seems to be better. This is due to a slow convergence of the WKB expansion for the particular case: the zeroth order can be kept if the length of the missing cone is large compared to the wavelength. Finally, a simplified version seems to be a satisfactory compromise.

The musician’s ability to modify the pitch of his instrument is a common feature for most instruments from the flute family. Musicians can alter a given note by adjusting the airjet velocity or by changing the resonator impedance through the modification of the embouchure (opening or closing the open end of the resonator where the embouchure is). A trained musician can adapt to a wide variety of instruments and even correct the pitch of poorly built instruments. Flute makers, on the other hand, propose a pitch structure (diapason) by placing the tone holes, adapting their size and height and by choosing the bore’s internal geometry. The choices made by the maker take into account the musician’s evaluation, creating a loop that through centuries of iterations has produced optimal instruments following a range of cultural and technological constraints. Throughout this article, the relationship between flute manufacture and the musician’s control will be discussed and analyzed. Identifying, modeling and quantifying the possibilities the musician has to control the instrument as well as the parameters the flute maker can modify to close this optimization loop, we propose a methodology to determine the geometry of the instrument for a given control strategy and vice versa.