Enhanced Transmission Loss and Modal Coupling in Dual-Membrane Flexible-Shell Cylindrical Waveguides: A Rigorous Mode-Matching–Galerkin Framework

This paper develops an analytical treatment of vibro-acoustic wave propagation in a cylindrical waveguide containing two clamped elastic membranes and a central flexible-shell segment. The acoustic field obeys the time-harmonic Helmholtz equation, the shell motion is described by Donnell–Mushtari thin-shell theory under axisymmetric loading, and the membrane response is governed by classical membrane theory and incorporated through a tailored Galerkin scheme. The resulting coupled fluid–structure boundary-value problem is solved by the Mode-Matching Method: the acoustic potentials are expanded in orthogonal radial eigenfunctions within each subregion, and continuity of pressure, normal velocity, and structural displacement are enforced at every interface. The mirror symmetry of the configuration is exploited by an exact decomposition into symmetric and anti-symmetric sub-problems, each of which reduces to a truncated linear algebraic system of dimension $4N+4$ for the unknown modal amplitudes. Acoustic power-balance identities provide a quantitative consistency check on the numerical implementation and diagnose convergence with respect to the truncation order; structural damping is accommodated through complex-modulus substitutions for the shell and the membrane tension without altering the algebraic structure of the system. The numerical results demonstrate that the dual-membrane configuration delivers transmission-loss values exceeding $25\mathrm{dB}$ across the low-frequency band relevant to HVAC and automotive applications, with a representative plateau near $13\mathrm{dB}$ at the reference geometry, through resonance-driven modal coupling between the acoustic field and the compliant interfaces. Parametric studies identify the excitation frequency, the inner-membrane radius, the shell radius, and the chamber length as effective design parameters for tuning the attenuation. The formulation furnishes a unified and computationally efficient analytical tool for predicting and optimising noise attenuation in flexibly coupled cylindrical duct systems.