Projection across a discontinuity

Using equations (2.79) and (2.83) we can show that the impedance matrix on surface 1 can be found from the impedance matrix on surface 2 at an expansion as follows:

$$Z^{(1)} = F Z^{(2)} F^{\mathrm{T}}, \mbox{\hspace{1cm}} S_1 < S_2.$$ (2.97)

The equivalent at at a contraction of the bore follows from equations (2.81) and (2.84):

$$Z^{(1)} = V^{-1} Z^{(2)} (V^{\mathrm{T}})^{-1}, \mbox{\hspace{1cm}} S_1 > S_2.$$ (2.98)