A modelling approach for wind instruments with toneholes is presented. It de-
parts from the classical one, based on the transmission lines analogy with 1D prop-
agation in ducts, in which the impedance is a central concept. The key theoretical
idea is to describe the topology of a wind instrument through a convenient graph,
leading to a nonstandard vector boundary value problem for the 1D wave equation
on a graph. In a second step this last one is transformed into a standard matrix
one, giving an efficient numerical solution method as well as the possibility of for-
mal developments that can be useful for either analysis or design of instruments.
Results on elementary ducts with/without toneholes illustrate the method and are
also used for checking against the usual method. One feature of the approach is to
compute natural frequencies and the eigenmodes at once, even for geometries with
large discontinuities. A nice byproduct is the answer to a conjecture formulated 20
years ago by Dalmont and Kergomard for stepped cones.