In [Rodet, 1995], we explained and demonstrated a simple model of trumpet-like instruments. The aim was to find the simplest structure which would exhibit the basic properties of this class of instruments. Furthermore, as long as we keep to a very simple model, we can study analytically and really understand the functioning of the corresponding system.
Knowing the exact conditions and characteristics of the oscillations is important for a musical use of the model of an instrument. Firstly, it provides the performer with parameter regions in which the model will sound. Moreover, it allows the performer to know in advance the behavior which the system will exhibit in this or that region.
In the present paper, we start with the analytical study of this model. For this we look for and examine the fixed point which corresponds to the non-oscillating position of the natural instrument. We find the stability condition of this fixed point. Then we look for a proof of the hypothesis that oscillation occurs as a Hopf bifurcation of the fixed point when a parameter, such as blowing pressure, becomes greater than a critical value.
The system which describes our model, is autonomous, nonlinear, and includes a delay and the corresponding type of equation is very complex to cope with.
As proposed in [Rodet, 1993] we show that the Hopf theorem can be applied to the bifurcation of our system. It is a powerful method for studying periodic solutions in nonlinear autonomous systems. It allows us to prove the existence, uniqueness and stability of an oscillation around the fixed point under study. Moreover, the frequency and amplitude of the oscillation can be forecast as a function of the bifurcation parameter value. The experimental results obtained by numerical simulations are found to be in good agreement with the theoretical predictions.
After the analysis of the behavior, we present several improvements of our basic model aimed at getting a better approximation of the natural instrument sound production and at improving its phrasing and articulation qualities and its expressivity. A more precise description of the bore is first taken into account. Then we design a better model of the forces acting on the lip. We also discuss the attack transients.
Improvement are done taking care not to eliminate the possibility of understanding the model and applying the Hopf theorem. Thus we can keep the behavior of the model under control.