Wall losses in straight ducts with non-circular cross sections: a scaling rule of identical losses for wooden flue organ pipes

Viscothermal effects at the walls are the dominant sources of loss in the air columns of various types of wind instruments. The classical theory of viscous and thermal boundary layers gives an analytical result that allows for computing the wall losses of acoustical wave propagation in cylindrical ducts. Based on theoretical considerations, a simple formula is derived for straight ducts that allows for taking the shape of the cross section into account in the wall loss coefficient. This unidimensional model is compared to three-dimensional finite element computations of different geometries, and an excellent agreement is found. As an application of the theory, a revised scaling method for wooden flue organ pipes with rectangular cross sections is elaborated. In organ building practice, wooden pipes are often made narrower than the reference width because of space limitations. Organ builders reported an undesired change of timbre for narrow pipes, which may be explained by the increased amount of wall losses. The proposed scaling approach enables designing narrower wooden pipes with keeping the amount of wall losses the same as of a reference pipe. Two series of experimental organ pipes designed following the traditional and new scaling rules are examined and compared by means of acoustical measurements and sound analyses, proving the practical applicability of the proposed scaling method.