Background <p>Wave propagation and attenuation in waveguides with flexible components play a critical role in noise control applications such as HVAC systems, aero-engines, and gas turbines. Conventional analyses typically address either rigid-walled ducts, single-region impedance linings, or flexible shells in isolation, leaving the combined effects of shell flexibility, bifurcated chamber geometry, membrane-mediated coupling, and dissipative linings underexplored. The non-orthogonal eigenfunctions arising from shell flexibility further complicate standard mode-matching approaches.</p> Methods <p>A semi-analytical framework is developed for a thin, flexible cylindrical shell enclosing a bifurcated expansion chamber with concentric sub-regions coupled by membrane discs and an internal acoustic liner. The shell dynamics follow Donnell–Mushtari thin-shell theory, while the fluid is treated as compressible and inviscid. A Galerkin-augmented mode-matching formulation handles the non-orthogonal eigenfunctions and higher-order derivatives at flexible boundaries.</p> Results <p>Convergence and consistency are confirmed through energy-balance checks and reconstruction of the interface matching conditions. Resistive linings dissipate a significant portion of the incident acoustic energy and yield broadband transmission loss, whereas purely reactive linings produce sharp but narrow resonance peaks with negligible absorption. Parametric studies show that the inner radius, intermediate radius, outer radius, and cavity half-length act as independent handles for trading off reflection, absorption, and transmission.</p> Conclusions <p>The proposed framework reliably models coupled fluid-structure wave propagation in flexible, membrane-coupled, bifurcated waveguides with acoustic linings, providing practical design guidance for thin-walled waveguide structures in noise-control applications.</p>