Wind Instruments: Variable Cross Section and Toneholes

Abstract

Wind instruments are studied in two steps. In the first step, one-dimensional models are presented. The horn equation is equivalent to the heterogeneous string studied in Chap. 3 It can either be used for tubes with discontinuities, such as chimney pipes, flutes, or trumpet mouthpieces, or for tubes with a continuous change in cross section, such as conical instruments or bells of brass instruments. The natural modes are calculated with some approximations for several basic shapes of wind instruments. The geometry has an important and complex effect on eigenfrequencies and on amplitudes of input impedance peaks. The role of dissipation is more simple, because it can be averaged over the length of the instrument. However, it depends on the radius, and this yields non-proportional damping. In the second step, it is investigated how to reduce three-dimensional geometric elements, such as toneholes or bends, to lumped elements. For this purpose, the definition of duct modes is given, and the mode-matching method is presented. The basis is a general formulation of the junction of several waveguides at low frequencies. Useful formulas are given for many elements of this kind. The chapter ends with the use of the theory of periodic media in order to analyze the tonehole lattice, with the explanation of important features that distinguish baroque from modern instruments. Attempt is made to give the most recent formulas of use for designing the resonators of wind instruments.