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# Introduction

The reflectance is a quantity which gives the relative amplitudes of the forward and backward going waves at a particular place in an acoustic system. In the plane wave approximation it is a scalar ratio. For a simple discontinuity between two tubes, if the waves are incident from only one side, the reflectance does not vary with frequency. In general, however, the reflectance of an object is frequency dependent, equivalent to the frequency spectrum of the input impulse response.

In chapter 2 we presented a multimodal method of calculating the impedance throughout a tubular object provided the impedance at one end is known. Here we show how the impedance matrix can be used to derive the multimodal reflectance, a matrix relating the amount of each mode reflected due to each mode incident. Because the non-planar modes always have frequency dependent impedances, the reflectance matrix will be frequency dependent, even for the special case of a plane wave incident from only one side. Theoretical results follow for this situation, showing both the frequency spectrum of the reflectance and the input impulse response obtained by performing an inverse Fourier transform on the reflectance.

In order to accurately reconstruct the bore of an instrument with a rapidly flaring bell, the effects of higher modes should be included. However, the layer peeling algorithm discussed in chapter 5 assumed plane wave propagation. This chapter concludes with a discussion of two possible solutions to this problem and the difficulties involved.

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Jonathan Kemp 2003-03-24