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The equations for projection of the impedance matrix
were derived in section 2.6.
Remember that the labels
(0), (1) and (2) refer to planes 0, 1 and 2 in figure 4.1.
Figure 4.1:
Detail of a waveguide consisting of straight sections of length joined discontinuously

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

(4.3) 
where
,
are the crosssectional areas and

(4.4) 
The projection matrices are given by

(4.5) 
where
with and
.
The equation for projection through a distance is (N.B. This is a correction to the original version of my thesis
and D_{2} is given in equation (2.41))

(4.6) 
where

(4.7) 
Here

(4.8) 
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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Jonathan Kemp
20030324