New insights into mode behaviours in waveguides with impedance boundary conditions

In this paper we investigate mode nonorthogonal properties and their effects on the sound power attenuation in a waveguide with impedance boundary conditions. By introducing two quantities: self-nonorthogonality $K_p$, which measures the nonorthogonality between left and right eigenfunctions of a mode, and mutual-nonorthogonality $S_{ij}$, which measures the nonorthogonality between modes $i$ and $j$, two opposite limiting cases are clearly identified in the boundary impedance $\mathbb Z$ plane: one is non-dissipation, i.e., acoustic rigid, pressure-release, and purely reactive impedance; the other is Cremer's optimum impedances which are exceptional points --- a subject has attracted much attention in recent years in different physical domains. Variations along an arbitrary path in the complex boundary impedance plane, $K_p$ and $S_{i,j}$ varies between the two opposite extremes. It is found that $K_p$ and $S_{i,j}$ play crucial roles in sound power attenuation.