  Optimal impedance for each mode is an important concept in an infinitely long
duct lined with uniform absorption material. However it is not valid for finite
length linings. This is because that the modes in lined ducts are not
power-orthogonal; the total sound power is not equal to the sum of the sound
power of each mode; cross-power terms may play important roles. In this paper,
we study sound propagation and attenuation in an infinite rigid duct lined with
a finite length of lining impedance. The lining impedance may be axial segments
and circumferentially non-uniform. We propose two new physical quantities Kp
and S to describe the self-overlap of the left eigenfunction and right
eigenfunction of one mode and the normalized overlap between modes,
respectively. The two new physical quantities describe totally the mode
behaviors in lined ducts.
