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## Aim 1

The first aim was to study and develop the theory of multimodal propagation in acoustic horns, in order to enable the influence of higher mode propagation to be assessed.

The theory of propagation of modes in pipes has been reviewed. Plane wave propagation was discussed, followed by the theory of multimodal propagation. This work centred on a method of calculation of the input impedance of an acoustic horn. This quantity gives information on the resonance properties. Also discussed was a method for studying the pressure field inside the horn. New work presented includes the treatment of ducts of rectangular cross-section in addition to the existing theory for circular cross-section. In order to perform calculations, the properties of radiation of sound from the open end must be characterised by calculation of the radiation impedance.

An expression for the multimodal radiation impedance of both cylindrical and rectangular ducts terminated in an infinite baffle has been derived. From the initial quadruple integral expression the problem was reduced to one dimension in cylindrical geometry and two dimensions in rectangular geometry with the singularity removed to allow practical numerical integration. Results are presented highlighting the difference between direct and coupled impedances and comparisons made between the geometries.

Numerical calculations of the input impedance and pressure field of the bell section of a trumpet then followed. The inclusion of just one extra mode made a large difference in the input impedance, modelling the absorption of energy from plane wave propagation by mode conversion. Inclusion of each successive extra mode made changes of ever decreasing magnitude to the input impedance calculation.

The reflection of sound from a single step between two infinite tubes is trivial if the plane wave approximation is used. We went on to discuss the reflection from such a geometry when multimodal propagation is used. In this case the reflected amplitude of a sinusoidal pressure wave is frequency dependent. Calculations of the frequency response were presented. The results demonstrate how the reflected amplitude increases with frequency in the low frequency range where only the plane wave mode can propagate. The time domain input impulse response was then calculated by Fourier transform. If an ideal impulse is incident from one side of the geometry, the plane wave component of the reflection is a pulse with a wake consisting of oscillations. These oscillations are caused by a peak in the frequency domain reflection at the first cut-off frequency; only the plane wave mode is propagating below the cut-off frequency but the plane wave mode and one higher mode may propagate above cut-off.

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Jonathan Kemp 2003-03-24