Adjointness properties for differential systems with eigenvalue-dependent boundary conditions, with application to flow-duct acoustics

Abstract

When the boundary conditions at the two endpoints of a second-order differential system depend explicitly upon the eigenvalues such that the system becomes non-self-adjoint, a generalized condition of orthogonality which includes endpoint terms can be developed. The generalized orthogonality condition is used to determine modal coefficients for the expansion of arbitrary functions in series of eigenfunctions. The method is applied to the particular case of acoustic wave propagation in a rectangular duct with a uniform mean-flow profile and walls with finite acoustic impedance. The ability of the eigenfunction expansion to converge to a plane-wave acoustic pressure profile is demonstrated under a variety of flow, frequency, and wall-impedance conditions.