Time-Domain Numerical Modeling of Brass Instruments Including Nonlinear Wave Propagation, Viscothermal Losses, and Lips Vibration

Abstract

A time-domain numerical modeling of brass instruments is proposed. On one hand, outgoing and incoming waves in the resonator are described by the Menguy-Gilbert model, which incorporates three key issues: nonlinear wave propagation, viscothermal losses, and a variable section. The nonlinear propagation is simulated by a TVD scheme well-suited to non-smooth waves. The fractional derivatives induced by the viscothermal losses are replaced by a set of local-in-time memory variables. A splitting strategy is followed to couple optimally these dedicated methods. On the other hand, the exciter is described by a one-mass model for the lips. The Newmark method is used to integrate the nonlinear ordinary differential equation so-obtained. At each time step, a coupling is performed between the pressure in the tube and the displacement of the lips. Finally, an extensive set of validation tests is successfully completed. In particular, self-sustained oscillations of the lips are simulated by taking into account the nonlinear wave propagation in the tube. Simulations clearly indicate that the nonlinear wave propagation has a major influence on the timbre of the sound, as expected. Moreover, simulations also highlight an influence on playing frequencies, time envelopes and on the playability of the low frequencies in the case of a variable lips tension.

Separability of wave solutions in nonlinear brass instrument modelling

“…Wave propagation in brass instruments is bidirectional. In this case, one modeling approach is to make use of an uncoupled pair of equations 2,5 of the form of Eq. (7), in the two variables v þ and v À : .…”

“…The use of one-way wave equation models in brass instrument modeling is widespread. 2,3 However, it is clear that, due to variations in the bore cross-section, leading to incremental back-scattering along the length of the instrument, such a one-way model is incomplete, even in the linear regime. This short contribution is concerned with an examination of the validity of such one-way models.…”

Diffusive Approximation of a Time-Fractional Burger's Equation in Nonlinear Acoustics

“…It is only a reasonable choice, which can be sharpened depending on the application at hand. For instance, let us consider the simulation of resonators in musical acoustics [2]: then, the optimization range must be included in the range of interest lies in the audible spectrum [20 Hz, 20 kHz].…”

“…In this paper, we use optimization with constraints of positivity, which provides a great improvement of accuracy compared with the aforementionned quadrature methods. This type of optimization has already been used with success in the context of poroelasticity [5], viscoelasticity [7], and recently for Chester's equation describing nonlinear acoustic waves in a guide [2].…”

“…Compared with previous works on nonlinear waves with fractional derivatives [30,2], this paper introduces three novelties:…”