We denote the pressure field on surface 1 as and the pressure field on surface 2 as . In plane wave propagation we saw that the pressure is taken to be the same on both sides of the discontinuity. In the multimodal case the pressure field is matched at either side. For the case shown where the pressure is matched on the air they share, .

(B.2) |

We will use and to denote the complex mode amplitudes on surfaces 1 and 2 respectively. and are the corresponding mode profiles on surfaces 1 and 2. Combining equations (B.1) and B.3) gives

(B.4) |

(B.5) |

Using matrix notation, is a column vector whose entries are given by and

where F is a matrix with elements . We have now proved equations (2.79) and (2.80) from chapter 2.

We now have a formula giving the pressure on a smaller cross-section
at a discontinuity from the pressure on a larger cross-section on the other
side. Since and are assumed to not be separated by any distance
along the axis the formula holds whatever side the largest cross-section
is on. Consider if . Now section 1 is the larger cross-section
meaning that we just have to interchange the labels 1 and 2 in
equation (B.7):

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