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# Multimodal reflectance matrix

A formula relating the forward and backward components of the volume velocity amplitude vector in terms of the impedance matrix was quoted in Pagneux et al. [32] p.2046. Here we show the derivation for the pressure amplitude vector. The first step is to express the total pressure amplitude vector as the sum of the forward going () and backward going () components:

 (6.1)

Now the total volume velocity is expressed using the same notation for forward and backward components:
 (6.2)

Recalling the characteristic impedance of higher modes from equation (2.36), the ratio of the th element in the forward going pressure vector to the th element in the forward going volume velocity vector is . Using , the diagonal characteristic impedance matrix defined in equation (2.42):
 (6.3)

Similarly for the backward going waves,
 (6.4)

Defining the impedance matrix at a particular point as with we get
 (6.5)

which may be rearranged to give
 (6.6)

so the result is
 (6.7)

where is the reflectance matrix:
 (6.8)

Notice that this is a correction to the reflectance matrix quoted in [41]. The correction arises because in general, even when is a diagonal matrix. The correction only has an effect on the non-diagonal entries in . The graphs presented in [41] are of the element and are unaffected by the correction.

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Jonathan Kemp 2003-03-24