Optimal control theory and design of wind instruments : oscillation regime optimization

It is generally accepted that harmonicity requirements for wind instruments lead either to cones or cylinders for continuous bores. It has been shown previously by the author that optimal control theory can be an valuable framework for theoretical studies about the shape that a wind instrument should have in order to satisfy some optimization criterion, inside a fairly general class, related to interesting physical/acoustical properties. The purpose of the present work is to deepen this approach with more precise physical considerations as regards the criterion to be optimized. In this setting, the Webster horn equation is regarded as a dynamical equation in the space variable. Pressure is the state, the control being the diameter variation of the bore. Then one looks for a control that optimizes a criterion related to the definition of an oscillation regime as the cooperation of several natural modes of vibration with the excitation, the playing frequency being the one that maximizes the total generation of energy, as exposed by A.H. Benade, following H. Bouasse.