Method for calculation of input impedance

We have presented the equations for the pressure and axial velocity in a duct in terms of the unknowns $P_n(z)$ and $U_n(z)$. Now we define the left hand side (negative $z$ side) of a waveguide of arbitrary cross-section as the input end and the right hand side (positive $z$ side) as the output end. The waveguide is then approximated by a series of uniform sections as shown in figure 2.8 (cylinders in circular geometry and rectangular sections in rectangular geometry).

Figure 2.8: Horn approximated by a series of cylinders

In plane wave acoustics the impedance is defined as the ratio of the acoustic pressure and volume velocity. For the multimodal case we define the acoustic impedance matrix, $Z$, as follows:

$${\mathbf P} = Z {\mathbf U}$$ (2.95)

so that

$$P_n = \sum^\infty_{m=0} Z_{nm} U_m$$ (2.96)

and therefore $Z_{nm}$ is the factor of contribution to the pressure amplitude of the $n$th mode due to the volume velocity amplitude of the $m$th mode. The vectors and matrices are infinite but must be truncated before numerical implementation.