New lattices of sound tubes with harmonically related eigenfrequencies

In 1994, the first author of the present paper and J. Kergomard published in acta acustica a paper entitled : Latticies of sound tubes with harmonically related eigenfrequencies. These lattices are made of a succession of truncated cones with equal length that have to respect certain rules. When looking at the eigenmodes with closed-open boundary conditions (i.e. reed wind instruments case), the solution were found to be stepped cone, that is cones made with a succession of cylinders whith cross sections following the law a n = a 1 n(n + 1)/2. At the end of the paper, the authors wrote : Finally, it remains to be demonstrated rigorously that no other shapes of horn lattices have harmonically related eigenfrequencies. In the present paper we show instead that other stepped horns have this property.