240
5 Blow That Horn: An Elementary Model of Brass Playing
amplitude can be significantly different from f thr . The prediction of the threshold
frequency from linear stability analysis must therefore be used with caution as an
estimate of playing frequency even at very low amplitudes in the case of an inverse
Hopf bifurcation.
The bifurcation diagrams discussed above are generic examples capable of
describing the finite amplitude behaviour of many types of wind instrument model.
The study of the different regimes predicted by such models, including quasiperiodic states corresponding to multiphonics, falls within the general scope of
the area known as nonlinear dynamical systems (Guckenheimer and Holmes 1983;
Manneville 2010). Some aspects of this study are reviewed and applied to both reed
and brass instruments in the Going Further Sect. 5.4.
Various mathematical techniques have been developed in this field to calculate
bifurcation diagrams. A diagram corresponding to a direct bifurcation is explicitly
derived from the Van der Pol self-sustained oscillator in Sect. 5.4.1, using the
mathematical technique of continuation. Bifurcation diagrams corresponding to
direct and inverse bifurcations are calculated and discussed at the end of Sect. 5.4.5
in the context of an instrument air column having only two acoustic resonances close
to harmonicity (Gilbert et al. 2020), and in the context of brass instrument design
and analysis (Fréour et al. 2020).
5.4 Going Further: From Linear Stability Analysis to
Oscillation Regimes
The generation of oscillations by the destabilisation of a nonlinear dynamical system
is a major field of academic study, with practical applications ranging from the
explanation of snoring (Auregan and Depollier 1995) to the design of liquid fuel
valves in rocket engines (Chang 1994). Some useful mathematical techniques which
have been developed to study the nature of these oscillations, including linear
stability analysis and bifurcation diagrams, were outlined in Sects. 5.2 and 5.3. In
this section the mathematical background to these techniques is explained, and some
applications to the study of brass instruments are reviewed.
To introduce the mathematical treatment of self-sustained oscillations, we begin
in Sect. 5.4.1 with a discussion of the Van der Pol oscillator. The analytical tools
which will be used to study brass instrument behaviour are first demonstrated in
this mathematically simple case. State-space representations of the elementary brass
playing model are developed in Sect. 5.4.2 and used to carry out linear stability
analysis of equilibrium states of the model in Sect. 5.4.3. The threshold results
obtained from LSA, and the periodic regimes generated by numerical simulation
above the oscillation threshold, are discussed and compared with the experience
of brass instrument players. In Sect. 5.4.5 bifurcation diagrams of reed and brass
instruments are obtained using the mathematical technique of continuation. Discussion of these results offers a theoretical insight into the Bouasse-Benade prescription
5 Blow That Horn: An Elementary Model of Brass Playing
amplitude can be significantly different from f thr . The prediction of the threshold
frequency from linear stability analysis must therefore be used with caution as an
estimate of playing frequency even at very low amplitudes in the case of an inverse
Hopf bifurcation.
The bifurcation diagrams discussed above are generic examples capable of
describing the finite amplitude behaviour of many types of wind instrument model.
The study of the different regimes predicted by such models, including quasiperiodic states corresponding to multiphonics, falls within the general scope of
the area known as nonlinear dynamical systems (Guckenheimer and Holmes 1983;
Manneville 2010). Some aspects of this study are reviewed and applied to both reed
and brass instruments in the Going Further Sect. 5.4.
Various mathematical techniques have been developed in this field to calculate
bifurcation diagrams. A diagram corresponding to a direct bifurcation is explicitly
derived from the Van der Pol self-sustained oscillator in Sect. 5.4.1, using the
mathematical technique of continuation. Bifurcation diagrams corresponding to
direct and inverse bifurcations are calculated and discussed at the end of Sect. 5.4.5
in the context of an instrument air column having only two acoustic resonances close
to harmonicity (Gilbert et al. 2020), and in the context of brass instrument design
and analysis (Fréour et al. 2020).
5.4 Going Further: From Linear Stability Analysis to
Oscillation Regimes
The generation of oscillations by the destabilisation of a nonlinear dynamical system
is a major field of academic study, with practical applications ranging from the
explanation of snoring (Auregan and Depollier 1995) to the design of liquid fuel
valves in rocket engines (Chang 1994). Some useful mathematical techniques which
have been developed to study the nature of these oscillations, including linear
stability analysis and bifurcation diagrams, were outlined in Sects. 5.2 and 5.3. In
this section the mathematical background to these techniques is explained, and some
applications to the study of brass instruments are reviewed.
To introduce the mathematical treatment of self-sustained oscillations, we begin
in Sect. 5.4.1 with a discussion of the Van der Pol oscillator. The analytical tools
which will be used to study brass instrument behaviour are first demonstrated in
this mathematically simple case. State-space representations of the elementary brass
playing model are developed in Sect. 5.4.2 and used to carry out linear stability
analysis of equilibrium states of the model in Sect. 5.4.3. The threshold results
obtained from LSA, and the periodic regimes generated by numerical simulation
above the oscillation threshold, are discussed and compared with the experience
of brass instrument players. In Sect. 5.4.5 bifurcation diagrams of reed and brass
instruments are obtained using the mathematical technique of continuation. Discussion of these results offers a theoretical insight into the Bouasse-Benade prescription