Some links between dispersion equations and orthogonality relations, and an application to fluid-structure interaction

Abstract

Orthogonality and bi-orthogonality relations are derived and employed to solve a problem of wave propagation in an infinitely long thin elastic cylindrical shell with a uniform mean flow of an incompressible fluid inside. For this non-symmetric waveguide, links between dispersion equations and orthogonality relations in regular (direct flow) and reversed flow cases are derived. It is shown that a bi-orthogonality relation exists only for two solutions of the same (either regular or reversed flow) problem. Regimes of stable wave motion in the presence of mean flow are identified, Green’s matrix is derived using the bi-orthogonality relation, and partition of energy flux between alternative transmission paths is analysed.