On reciprocity, orthogonality and bi-orthogonality relations for non-symmetric waveguides: An elastic plate loaded by a fluid layer with uniform mean flow

Abstract

The reciprocity, orthogonality and bi-orthogonality relations are well established for symmetric waveguides, with the most known example being the Rayleigh–Lamb problem of wave propagation in an infinite elastic layer. In this paper, these relations are derived for non-symmetric waveguides using as an example the canonical problem of wave propagation in an infinite elastic plate loaded by a fluid layer of a finite height with uniform mean flow parallel to its surface. The links between formulations of dispersion equation, reciprocity/bi-orthogonality relations and energy flux are demonstrated and discussed. The Green’s function for stable regimes of wave motion in the presence of a flow is derived using the bi-orthogonality relation, and partition of the energy flux between transmission paths is analysed.