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We have shown how including higher modes within a layer peeling bore reconstruction algorithm may be possible and have highlighted the problems which must be addressed before it is achieved. It is also worthwhile presenting an alternative way of including multimodal effects in reconstructing the bore. This section discusses how a multimodal calculation of the input impulse response may provide the control for an iteration procedure designed to arrive at the correct bore.

Iterative bore reconstruction has been studied by Kausel [61] for the case of deducing the bore from measured input impedance. As a starting point, a fairly arbitrary starting bore was chosen and the iteration procedure used to minimise the difference between a plane wave calculation of the input impedance of the bore and that which was measured. We will refer to the result as the ``plane wave equivalent bore'' since it is the bore whose input impedance, according to the plane wave approximation, would match that which was measured experimentally.

Our aim is to find a bore where the multimodal method calculation of some acoustic variable matches that which was measured experimentally. We do not know the multimodal open end condition without assuming an infinite baffle so it may be helpful to use the time domain response as a control for iteration. This does not depend on the choice of end condition until the primary reflections from the open end arrive. Our starting point is to calculate the plane approximation bore reconstruction from the measured input impulse response. This is also a ``plane wave equivalent bore.''

In the experimental impulse response measurement some energy was lost to the non-planar modes within the instrument. The measured reflected amplitudes are therefore smaller than we would expect from plane wave approximation scattering. A slight under-prediction of the cross-section changes within the real instrument is therefore expected in a plane wave equivalent bore.

The impulse response of the plane wave equivalent bore may then be calculated by the multimodal method. This calculation will not match the experimentally measured input impulse response. Firstly, we expect the amplitude of the reflections to be reduced because the plane wave equivalent bore slightly under-predicts the cross-section changes within the real instrument. Secondly, as mentioned before, the correct radiation impedance is not known. We must assume that the instrument is terminated in an infinite baffle, so the impulse responses may differ slightly after the primary reflection from the open end returns to the input.

By comparing the impulse response of the plane wave equivalent bore with that measured experimentally, we have information on how the reconstructed bore needs to be changed to bring it in line with the experimental results. An iteration procedure could be used where we modify the reconstructed bore, calculate the impulse response including higher modes, compare the result with experiment and continue. The final result would be achieved when the difference between the multimodal calculation of the impulse response of the bore profile and the experimental impulse response is minimised. Obviously, the information in the impulse response after the primary reflection from the end condition would not be used in such a procedure because of the end condition problem.

In order to provide a guide for how the bore reconstruction deviates from the actual bore, the difference between the measured input impulse response and the calculated input impulse response of the bore reconstruction could be fed into the reconstruction algorithm. This would return an almost cylindrical bore which expands at points along the bore where the reconstruction is under-predicting and contracts where the bore is over-predicting. The iteration procedure should then proceed quickly towards a final answer.

The iteration procedure may be too computationally taxing if every point in the bore is varied separately by the iteration procedure. The bore profile could instead be represented by a polynomial fit to a number of points along the bore. This gives a number of degrees of freedom which can be changed by the iteration procedure before their effect on the response is calculated. In general, convergence will be faster if the number of degrees of freedom is reduced. Too low a number of degrees of freedom however will mean any improvement due to the modelling of multimodal effects will be counteracted by over simplification of the bore. Since a multimodal calculation must be performed at each step of the procedure, the computational power required would be large.

This thesis has moved to Jonathan Kemp Thesis at http://www.kempacoustics.com/thesis

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