Jean Kergomard

Input impedance of brass musical instruments—Comparison between experiment and numerical models

We present a sequence of measured and calculated input impedance curves for tubes of increasing complexity, culminating in curves for a trombone and for a tuba. Measurements were made by the indirect method in a manner which afforded unprecedented accuracy. Calculations were done with a one‐dimensional model, including visco‐thermal losses, by a method of discretization, employing truncated cones. The correspondence between theory and experiment is sufficiently good that one can use model calculations to predict with some confidence the changes in input impedance caused by modifications to real instruments.

Calculation of the steady‐state oscillations of a clarinet using the harmonic balance technique

The harmonic balance technique is known as a time‐frequency simulation technique used for the study of large signal regimes of microwave circuits driven in forced oscillation. The technique can be adapted to self‐sustained oscillations, especially for wind musical instruments such as clarinets. The resonator (i.e., the instrument body) is the linear part, treated in the frequency domain, while the driving system (the reed) is the nonlinear part, treated in the time domain. The harmonic balance method is shown to connect the results of two known methods, so‐called weakly nonlinear (in frequency domain) and strongly nonlinear (in time domain). The advantages and disadvantages of the method are discussed.

General formulation of the dispersion equation in bounded visco-thermal fluid, and application to some simple geometries

Abstract The purpose of this paper is to give a new set of equations derived from the basic classical theory of the acoustic propagation in visco-thermal fluid and valid in the time domain, and to provide us with a general dispersion equation for harmonic waves in several boundary problems of interest. It is shown that this dispersion equation generalizes some known results as the equivalent specific impedance of plane boundaries and resonance frequencies of spherical resonators, and that it provides us with a new general equation giving the propagation constant of waves for all kind of modes in rigid walled cylindrical tubes.

Real-time synthesis of clarinet-like instruments using digital impedance models

A real-time synthesis model of wind instruments sounds, based upon a classical physical model, is presented. The physical model describes the nonlinear coupling between the resonator and the excitor through the Bernoulli equation. While most synthesis methods use wave variables and their sampled equivalent in order to describe the resonator of the instrument, the synthesis model presented here uses sampled versions of the physical variables all along the synthesis process, and hence constitutes a straightforward digital transposition of each part of the physical model. Moreover, the resolution scheme of the problem (i.e., the synthesis algorithm) is explicit and all the parameters of the algorithm are expressed analytically as functions of the physical and the control parameters.

Approximation formulae for 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 and Schwinger for the unflanged case and extracted from the radiation impedance matrix given by Zorumski for the infinite flanged case.

An analytical prediction of the oscillation and extinction thresholds of a clarinet

This paper investigates the dynamic range of the clarinet from the oscillation threshold to the extinction at high pressure level. The use of an elementary model for the reed-mouthpiece valve effect combined with a simplified model of the pipe assuming frequency independent losses (Ramans model) allows an analytical calculation of the oscillations and their stability analysis. The different thresholds are shown to depend on parameters related to embouchure parameters and to the absorption coefficient in the pipe. Their values determine the dynamic range of the fundamental oscillations and the bifurcation scheme at the extinction.

Some aspects of tuning and clean intonation in reed instruments

Abstract The influence of the first and second resonance frequencies on tuning, timbre (or tone colour) and ease of playing is investigated for reed instruments, such as the clarinet, alto saxophone and oboe. Theoretical analyses of the effects of the reed and the player embouchure (i.e. lip position and pressure on the reed) are reviewed, as well as the consequences of inharmonicity in the resonance frequencies. This review allows us to present interesting interpretations of the numerous experiments reported here. Three kinds of results are given: (1) comparison of playing frequencies and first resonance frequencies, for several fingerings with or without open register hole, leading to the definition of a frequency independent length correction for the embouchure; (2) examination of the effect of inharmonicity of the two first resonance frequencies on both tone colour and ease of playing, the causes coming from either the player embouchure or the instrument construction; (3) comparison between theory and experiment for the inharmonicity produced by the changes in conicity in oboes, leading to an interpretation of the maker s choices. The results show how judicious use of simple tools, such as calculations or measurements of input impedance or playing data obtained using an artificial mouth, can be of help to the understanding of instrument construction and to the instrument designer.

Boundary layer attenuation of higher order modes in waveguides

The attenuation of higher order modes in rectangular and circular tubes is treated here by using results for the boundary layer admittance for the respective normal modes. Comparison with results available in the literature for propagating modes is given. Results for evanescent modes and at the cut-off frequencies are discussed. Finally, the well-known Kirchhoff theory is extended to obtain a test of validity for the proposed calculations.

Analysis of higher order mode effects in an expansion chamber using modal theory and equivalent electrical circuits

Abstract Modal theory for propagation in an acoustic waveguide with variable cross-section, in particular in an expansion chamber, can lead to several different numerical or approximate analytical solutions. The advantages of these are compared, and one of their number is shown to be a generalization of the plane piston inlet/outlet approximation made by Ih and Lee. The results are expressed in the form of two kinds of equivalent electrical circuits, connecting the plane modes in the inlet and outlet. In the first one, the higher modes are described by two series inductances and one shunt inductance. In the second one, there is no shunt inductance, and series inductances are replaced by complex series impedances. The existence of resistive elements is discussed in terms of energy transport by higher modes, especially at low frequencies, when tunneling occurs. Approximate analytical, numerical and experimental results for the elements of circuits are given, with a study of coupling between the two step discontinuities.

Resonance modes in a one-dimensional 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 one-dimensional 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.