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By way of summary, the equation for projection across a discontinuity is

The projection matrices are given by

where with and .

The equation for projection through a distance is

(4.7) |

is the wavenumber of the th mode along the tube neglecting the effect of losses; the corresponding expression for lossy propagation is given in section 2.4.1. is the th zero of the Bessel function as tabulated in appendix A.

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