Lattices of sound tubes with harmonically related eigenfrequencies

Abstract

International audienceThe only continuous acoustic horns (in plane wave approximation) known for having harmonic eigenfrequencies are conical and cylindrical. Because of this, these shapes have been widely used for woodwind musical instruments. Other, non continuous, shapes are shown here to have the same property: they consist of a succession of truncate cones (or cylinders) of equal length, which are defined by three initial values for the radii (e.g. the input and output radii of the first cone and the input radius of the second one). The recurrence relations are obtained in the frequency domain, the principle being to impose the existence of travelling waves at the nodes of the lattice: the successive reflections at discontinuities are cancelled at the nodes (but only there). Several kinds of boundary conditions are studied. For the "closed-open" conditions, the unique solution is based upon cylinders and the input impedance curves and its Fourier Transform are shown to have interesting properties. Measurements have been made and the agreement between experiment and theory is satisfactory