Journal of Sound and Vibration 417 (2018) 315–340 Contents lists available at ScienceDirect Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi Bi-orthogonality relations for fluid-filled elastic cylindrical shells: Theory, generalisations and application to construct tailored Green’s matrices Lasse S. Ledet * , Sergey V. Sorokin Department of Mechanical and Manufacturing Engineering, Aalborg University, Fibigerstraede 16, 9220 Aalborg, Denmark article info Article history: Received 9 June 2017 Accepted 6 December 2017 Available online XXX Keywords: Bi-orthogonality relations Modal decomposition Tailored Green’s matrices Symmetric waveguides Energy flow Convergence and error calculation abstract The paper addresses the classical problem of time-harmonic forced vibrations of a fluid-filled cylindrical shell considered as a multi-modal waveguide carrying infinitely many waves. The forced vibration problem is solved using tailored Green’s matrices formulated in terms of eigenfunction expansions. The formulation of Green’s matrix is based on special (bi- )orthogonality relations between the eigenfunctions, which are derived here for the fluid- filled shell. Further, the relations are generalised to any multi-modal symmetric waveguide. Using the orthogonality relations the transcendental equation system is converted into alge- braic modal equations that can be solved analytically. Upon formulation of Green’s matri- ces the solution space is studied in terms of completeness and convergence (uniformity and rate). Special features and findings exposed only through this modal decomposition method are elaborated and the physical interpretation of the bi-orthogonality relation is discussed in relation to the total energy flow which leads to derivation of simplified equations for the energy flow components. © 2017 Elsevier Ltd. All rights reserved. 1. Introduction In this paper we address the classical problem of time-harmonic wave propagation in a thin elastic fluid-filled cylindrical shell loaded by an inviscid compressible fluid without mean flow. This is a subject broadly covered in literature on applied mathematics, see e.g. Refs. [1–6]. Among other applications, this formulation is used to address transmission of vibro-acoustic energy which is of primary interest in e.g. the oil and gas industry as well as in larger pumping systems conveying waste water or distributing domestic water to inhabitants. While the analysis of free waves in such a waveguide is a well-established sub- ject, a forced response in various excitation conditions has not yet been fully explored. To cover arbitrarily distributed acoustic and structural sources it is convenient to derive Green’s matrices i.e. to study the response to an excitation modelled as delta- functions. In this formulation of the problem it is expedient, on the one hand, to consider detailed analysis of the energy redis- tribution and mode conversion in the near-field to gain additional physical insight. On the other hand, the mathematical issues of completeness and convergence need to be addressed. To understand the energy redistribution and mode conversion in the near-field e.g. from pump to pipe or across flange con- nections, an accurate coupled vibro-acoustic model of an infinite pipe needs to be formulated. In this paper we adopt the tailored Green’s function/matrices as introduced in Ref. [5]. These functions deviate from the canonical free-space Green’s function of acoustics, in that they satisfy additional boundary conditions – continuity at the fluid-structure interface. Here we consider only * Corresponding author. E-mail addresses: lsl@m-tech.aau.dk (L.S. Ledet), svs@m-tech.aau.dk (S.V. Sorokin). https://doi.org/10.1016/j.jsv.2017.12.010 0022-460X/© 2017 Elsevier Ltd. All rights reserved.