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## Multimodal propagation

In chapter 2 we describe how the input impedance may be calculated theoretically using multimodal decomposition. We present the background to this technique here and go on to explain why it is useful for describing the behaviour of musical instruments in the introduction to chapter 2.

The problem of wave propagation in a uniform duct has a well known solution in the form of a sum of modes. While the lowest order mode has planar wavefronts, the higher order modes have non-uniform pressure distributions on a plane perpendicular to the direction of propagation. In a uniform pipe the modes propagate independently while at a change in cross-section the modes couple . Each mode propagates with a different wavelength along the central axis for a given excitation frequency . Furthermore, each mode (with the exception of the planar component) has a cut-off frequency below which the wavelength is imaginary and propagation is not possible. In general, the wave equation in an acoustic duct with a non-uniform cross-section is not solvable analytically, meaning we must resort to numerical methods.

A numerical calculation is often performed ignoring the effects of the higher order modes [23,24]. The results are valid at low frequency because the higher order modes are non-propagating and at low rates of flare because the coupling between modes is minimised.

Higher order modes have been included in the theory of acoustic ducts by various authors. Individual discontinuities in tubes were probably first treated with higher modes included explicitly in papers by Miles [25,26,27]. The solution of the wave equation in ducts of varying cross-section was treated by Stevenson for acoustic horns  and for electromagnetic waves in conducting horns known as waveguides . In electromagnetic theory there are 6 components of the electric and magnetic vector fields whereas the acoustic pressure is a scalar field. The theory of acoustic horns followed as a simplification of the electromagnetic theory, hence tubular objects in acoustics are refered to as acoustic waveguides. The equations provided by Stevenson cannot be solved explicitly, however. In order to determine the behaviour of a duct of varying circular cross-section, the internal profile must be approximated by a series of concentric cylinders as described by Alfredson . The same may be done for horns of varying rectangular cross-section by using concentric oblongs.

Input impedances have been calculated by Oie et al  and in papers by Pagneux et al [32,33] which form the basis for the theoretical work of chapter 2. Part of the procedure inevitably involves calculation of the radiation properties at the mouth of the horn.

If we assume that the highest frequency of interest is sufficiently low that only the plane mode may propagate in the duct, the radiation condition may be obtained from Levine and Schwinger  for a cylindrical pipe of zero wall thickness or from Ando et al. [35,36] for a cylindrical pipe of a certain wall thickness. We wish to keep the applicability of our method at high frequencies. There is no general expression currently available for the radiation condition without assuming that an infinite baffle is present around the opening of the horn. This radiation condition is due to Zorumski  for a cylindrical pipe and due to Kemp et al.  for a rectangular duct. Chapter 3 reviews the analysis of the radiation condition. The input impedance calculation method is then implemented in chapter 4.

In chapter 6 a multimodal method for calculating the reflections of a pressure wave from a tubular object is discussed. Initial work is shown for the case of a single discontinuity in two infinite tubes with acoustic pressure waves incident from only one side. This theory was initially formulated by Miles  in a summation notation and no calculations were presented. We will show how the situation may be easily represented in the matrix notation of Pagneux et al.  and calculations performed will show the response in both the frequency and time domains.

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Jonathan Kemp 2003-03-24