Graph Modelling of Musical Wind Instruments: A Method for Natural Frequencies Computation: Modélisation des instruments à vent sur un graphe : une méthode de calcul des fréquences naturelles

Many aspects of the musical wind instruments behavior can be studied within the framework of 1D acoustic propagation. A common method is based on an analogy with electric transmission lines, in which the impedance is a central concept. The present work proposes a different approach, based on results from mathematical modelling and analysis of repetitive structures such as networks of strings or beams. The method, that keeps at the one dimensional level, uses concepts and methods from graph theory for modelling the duct of wind instruments. A key point is a special matrix reformulation of the original wave equation on a graph. The focus is on natural frequencies computations for complex, piecewise cylindrical ducts. Examples with or without toneholes illustrate the method and its potential through symbolic computations. In the case of closed-open piecewise cylindrical resonators, the approach answers in a practical way a conjecture formulated twenty years ago by Dalmont and Kergomard, which is then replaced in the general framework of inverse spectral problems. An assertion of Benade is checked on one woodwind example.