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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) |

(6.2) |

Similarly for the backward going waves,

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

(6.5) |

(6.6) |

where is the reflectance matrix:

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.

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

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