A Mathematical Model for Air Columns in a Class of Woodwinds

A mathematical description of the class of piecewise cylindrical woodwinds and their air column is developed, as a step towards modelling more realistic instruments. It is based on a recent modelling approach that describes the air column of musical wind instruments as a linear graph, on the edges of which the one dimensional linear wave propagation equation holds. Our previous studies dealt with the direct eigenvalue problem for this class, i.e. the numerical computation, for a given geometry, of natural frequencies for resonators and instruments with one to three toneholes. The present work considers the same family of wind instruments from a different point of view. Firstly, instruments having any number of toneholes are now considered in a systematic fashion: a set of equations is exhibited that is able to describe any fingering of more realistic instruments, generalizing the approach. The second important point is the formulation of an inverse eigenvalue problem for the considered model. It leads to the principle of a new design method for instruments in the class. An important feature of this approach is a closed-form relationship between the geometry of an instrument and the natural frequencies of all tonehole configurations, i.e a musical scale. This brings a flexible tool for the design process that we have in mind. No real world application is presented as the computations require further mathematical developments.