Mode propagation and attenuation in lined ducts
  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. 
Wenping BI and Vincent Pagneux
physics.class-ph