Acoustics

Basic Asumptions

From measurements (see [2]) and dimensionless analysis (see [5]), we show that nonlinear propagation (at high amplitude of playing) as well as visco-thermal boundary-layer effects must be taken into account to describe acoustic propagation in the slide of a trombone. Neglecting high order contributions (such as nonlinear interactions in the main flow or nonlinearities in the boundary-layers), the solution in the cylindrical pipe may be described as a linear combination of an incoming and an outgoing plane waves where both waves are submitted to the viscothermal losses and the nonlinear distortion during the propagation. Neglecting the viscothermal losses, the nonlinear distortion of a simple wave is described in [6] equation (Eq. 1) with the propagation speed c being a function of the fluid velocity u ( being the linear sound speed, the ratio of the specific heats for gases):

Simulations

  
Figure 7: Simulation model

Fig. 7 displays the model we used to demonstrate some of the effects that nonlinear propagation introduces in a waveguide system simulating a wind-instrument, such as a trombone. This model includes a one-mass lips model as nonlinear excitator, two linear/nonlinear delay-lines for propagating the incoming and outgoing waves, and a low-pass filter for the linear modelling of the bell (see [5]). Comparisons are made based on simulations of the two systems including linear or nonlinear propagation driven by the same parameters.

 

 

Figure 8: Comparison of the amplitudes of the first ten harmonics: linear propagation vs. nonlinear propagation

It appears clearly (Fig. 8) that the nonlinear propagation produces high frequency components which makes the sound brassier. It seems also that the nonlinear propagation plays a rather subtle role for low frequencies (here lower than 800Hz) corresponding to the reflexion function bandwidth of the bell. But the simulations show that, even if the motion of the lips is perturbed, the frequency-spectrum of the input incoming wave is not modified much. That would support [2] where it is supposed that the nonlinear propagation has a rather small effect on the self oscillation process at steady-state, compared to the effect on the radiated sound.