Corrections to the Plane-Wave Approximation in Rapidly Flaring Horns

Abstract

The input impedance of an air column (horn) with a slowly varying cross section can be calculated approximately by solving the wave equation assuming plane waves (Webster equation). In some cases analytical solutions are possible. For an arbitrary horn shape described as a succession of short cones the one-dimensional transmission-line (TL) method can be used. This fails for horns with a large flare, where transverse flow demands kinetic energy, effectively increasing the local inertance. The quantities determining the magnitude of this correction appear from a variational calculation based on Rayleigh's principle. The magnitude of these quantities was determined by numerically solving the wave equation at low frequency in a number of horn shapes by a finite difference method. Studied are conical, hyperboloid and catenoidal horns. Since horns, in particular those found in wind instruments, are found in many other shapes, general use of this method is not practical. In this paper a simple formula is proposed useful for any horn shape to calculate approximately the additional inertance. It is introduced as a correction in the TL method and is applicable to any shape, even for a stepwise diameter change and for a flange. Its accuracy for most geometries found on wind instruments is within the detection threshold of the ear, a 0.2% frequency shift. It is not applicable any more when cross-dimensions become comparable to the wavelength.