<p>This study concentrates on developing an analytical mode-matching method, combined with the Galerkin approach, to investigate the scattering of radiation through a flexible shell within a finite, coaxial lined chamber connected by membrane discs at the interfaces. The modeled problem involves dynamical boundary and interface conditions, as well as various ring conditions. Eigenfunction expansions are derived in different regions with homogeneous media and boundary conditions, incorporating unknown coefficients. The associated eigenfunctions are non-orthogonal, forming non-Sturm-Liouville systems. The dynamical behavior of the membrane discs is formulated using the Galerkin approach. By applying field matching at the interfaces and utilizing generalized orthogonal properties, the differential systems are transformed into linear algebraic systems. Numerical solutions of these systems, after truncation, yield the unknown amplitudes. The validation of the matching conditions at the interfaces and the convergence of the computed powers strongly support the accuracy of the truncated solution and the algebraic procedures. The interaction between the flexible shell and the compressible fluid within it significantly influences the system’s vibration characteristics. Results on scattering powers and transmission loss across various parametric settings provide insights for optimizing the attenuation of fluid–structure coupled modes within the guiding channel.</p>