Response of an artificially blown clarinet to different blowing pressure profiles

Using an artificial mouth with an accurate pressure control, the onset of the pressure oscillations inside the mouthpiece of a simplified clarinet is studied experimentally. Two time profiles are used for the blowing pressure: in a first set of experiments the pressure is increased at constant rates, then decreased at the same rate. In a second set of experiments the pressure rises at a constant rate and is then kept constant for an arbitrary period of time. In both cases the experiments are repeated for different increase rates. Numerical simulations using a simplified clarinet model blown with a constantly increasing mouth pressure are compared to the oscillating pressure obtained inside the mouthpiece. Both show that the beginning of the oscillations appears at a higher pressure values than the theoretical static threshold pressure, a manifestation of bifurcation delay. Experiments performed using an interrupted increase in mouth pressure show that the beginning of the oscillation occurs close to the stop in the increase of the pressure. Experimental results also highlight that the speed of the onset transient of the sound is roughly the same, independently of the duration of the increase phase of the blowing pressure.Comment: 14 page

Resonance modes in a 1D medium with two purely resistive boundaries: calculation methods, orthogonality and completeness

Studying the problem of wave propagation in media with resistive boundaries can be made by searching for "resonance modes" or free oscillations regimes. In the present article, a simple case is investigated, which allows one to enlighten the respective interest of different, classical methods, some of them being rather delicate. This case is the 1D propagation in a homogeneous medium having two purely resistive terminations, the calculation of the Green function being done without any approximation using three methods. The first one is the straightforward use of the closed-form solution in the frequency domain and the residue calculus. Then the method of separation of variables (space and time) leads to a solution depending on the initial conditions. The question of the orthogonality and completeness of the complex-valued resonance modes is investigated, leading to the expression of a particular scalar product. The last method is the expansion in biorthogonal modes in the frequency domain, the modes having eigenfrequencies depending on the frequency. Results of the three methods generalize or/and correct some results already existing in the literature, and exhibit the particular difficulty of the treatment of the constant mode

Interaction of reed and acoustic resonator in clarinetlike systems

Sound emergence in clarinetlike instruments is investigated in terms of instability of the static regime. Various models of reed-bore coupling are considered, from the pioneering work of Wilson and Beavers ["Operating modes of the clarinet", J. Acoust. Soc. Am. 56, 653--658 (1974)] to more recent modeling including viscothermal bore losses and vena contracta at the reed inlet. The pressure threshold above which these models may oscillate as well as the frequency of oscillation at threshold are calculated. In addition to Wilson and Beavers' previous conclusions concerning the role of the reed damping in the selection of the register the instrument will play on, the influence of the reed motion induced flow is also emphasized, particularly its effect on playing frequencies, contributing to reduce discrepancies between Wilson and Beavers' experimental results and theory, despite discrepancies still remain concerning the pressure threshold. Finally, analytical approximations of the oscillating solution based on Fourier series expansion are obtained in the vicinity of the threshold of oscillation. This allows to emphasize the conditions which determine the nature of the bifurcation (direct or inverse) through which the note may emerge, with therefore important consequences on the musical playing performances

Contribution to harmonic balance calculations of self-sustained periodic oscillations with focus on single-reed instruments

International audienceThe harmonic balance method ͑HBM͒ was originally developed for finding periodic solutions of electronical and mechanical systems under a periodic force, but has been adapted to self-sustained musical instruments. Unlike time-domain methods, this frequency-domain method does not capture transients and so is not adapted for sound synthesis. However, its independence of time makes it very useful for studying any periodic solution, whether stable or unstable, without care of particular initial conditions in time. A computer program for solving general problems involving nonlinearly coupled exciter and resonator, HARMBAL, has been developed based on the HBM. The method as well as convergence improvements and continuation facilities are thoroughly presented and discussed in the present paper. Applications of the method are demonstrated, especially on problems with severe difficulties of convergence: the Helmholtz motion ͑square signals͒ of single-reed instruments when no losses are taken into account, the reed being modeled as a simple spring

Modal analysis of the input impedance of wind instruments. Application to the sound synthesis of a clarinet

International audienceThis paper investigates the modal analysis of wind instruments as seen from the input of their air column. Beside the treatment of analytical models, a particular emphasis is given to the analysis of measured input impedances. This requires special care because the measurements cover only a limited frequency band and are affected by some unknown errors. This paper describes how the Prony analysis and the Least Squares Complex Exponential (LSCE) classical techniques can be used in this context and how the main pitfalls can be avoided in their application. A physically acceptable method of reconstruction of the low frequency band is proposed. A technique using fictitious points in the high frequency range is described in order to ensure the passivity of the resonator in the whole frequency band. The principles of a real-time synthesis of clarinet sounds based on the modal representation of the resonator is given as an application, with a method to efficiently handle the modal representation during the transition between fingerings

Approximation of the acoustic radiation impedance of a cylindrical pipe

Useful approximation formulae for radiation impedance are given for the reflection coefficients of both infinitely flanged and unflanged rigid-walled cylindrical ducts. The expressions guarantee that simple but necessary physical and mathematical principles are met, like hermitian symmetry for the reflection coefficient (identical behaviour of positive and negative frequencies) and causality for the impulse response. A non causal but more accurate expression is also proposed that is suitable for frequency-domain applications. The formulae are obtained by analytical and numerical fitting to reference results from Levine & Schwinger for the unflanged case and extracted from the radiation impedance matrix given by Zorumski for the infinite flanged case.Comment: Journal of Sound and Vibration (2008) accepte

Iterated maps for clarinet-like systems

The dynamical equations of clarinet-like systems are known to be reducible to a non-linear iterated map within reasonable approximations. This leads to time oscillations that are represented by square signals, analogous to the Raman regime for string instruments. In this article, we study in more detail the properties of the corresponding non-linear iterations, with emphasis on the geometrical constructions that can be used to classify the various solutions (for instance with or without reed beating) as well as on the periodicity windows that occur within the chaotic region. In particular, we find a regime where period tripling occurs and examine the conditions for intermittency. We also show that, while the direct observation of the iteration function does not reveal much on the oscillation regime of the instrument, the graph of the high order iterates directly gives visible information on the oscillation regime (characterization of the number of period doubligs, chaotic behaviour, etc.)

Propagation of acoustic waves in two waveguides couples by perforations. I. Theory.

International audienceThe problem of propagation in two guides coupled by perforations, important for a perforated tube muffler, is discussed. At low frequencies, if the distance between perforations is sufficiently large, a discrete model can be used. An exact equivalent circuit for a perforation is obtained by using a modal theory and a matrix formalism. A series inductance due to the existence of antisymmetric field in the perforation is proven to exist, completing the perforation shunt impedance concept. This model is directly exploitable for lattice analysis. For homogeneous lattices (i.e. with identical propagation in the two guides), either regular or irregular, two modes exist: a planar mode and a "flute" mode, either propagating or evanescent. Cutoff frequencies of periodic lattices are found to depend on either the shunt inductance or the series inductance (the first cutoff depending on the shunt one). In homogeneous lattices, a new type of evanescent waves can exist, with non-zero energy flux, equal and opposite in sign in each guide. Finally, the effect of mean flow in such a lattice is discussed

Idealized digital models for conical reed instruments, with focus on the internal pressure waveform

International audienceTwo models for the generation of self-oscillations of reed conical woodwinds are presented. They use the fewest parameters (of either the resonator or the ex-citer), whose influence can be quickly explored. The formulation extends iterated maps obtained for loss-less cylindrical pipes without reed dynamics. It uses spherical wave variables in idealized resonators, with one parameter more than for cylinders: the missing length of the cone. The mouthpiece volume equals that of the missing part of the cone, and is implemented as either a cylindrical pipe (first model) or a lumped element (second model). Only the first model adds a length parameter for the mouthpiece and leads to the solving of an implicit equation. For the second model, any shape of nonlinear characteristic can be directly considered. The complex characteristics impedance for spherical waves requires sampling times smaller than a round trip in the resonator. The convergence of the two models is shown when the length of the cylindrical mouthpiece tends to zero. The waveform is in semi-quantitative agreement with experiment. It is concluded that the oscillations of the positive episode of the mouthpiece pressure are related to the length of the missing part, not to the reed dynamics