Conversion of acoustic energy by lossless liners

The Blokhintzev acoustic energy equation is applied to a two-dimensional duct containing a uniform flow with a finite length lining. It is shown that the difference of the incident and outgoing acoustic energy differs in general from the energy dissipated in the liner, the difference being related to the displacements at the liner's edges. It is shown that in the case of a locally reacting lossless liner for frequencies below the first cut-off frequency and for low Mach number acoustic energy is generated if the flow and the incident sound wave are in the same direction and is absorbed if these two directions are opposite unless special edge conditions are met. Furthermore it is shown under the same conditions that the ratio of the reflection coefficient at finite flow velocity to the reflection coefficient at vanishing velocity is to first order in Mach number independent of the liner characteristics. A numerical calculation confirms these predictions at least for mass-like linear admittance.