On the Acoustic Radiation from a Partially Lined and Perforated Semi-Infinite Duct

This study investigates acoustic wave propagation and radiation in a circular cylindrical duct featuring a partially lined segment followed by a perforated extension. The duct configuration consists of an infinitely long structure with a rigid inner surface for $$z < - l$$ , an acoustically absorbent lining over the interval $$ - l < z < 0$$ , and a perforated section for $$z > 0$$ . The outer wall remains rigid for $$z < 0$$ , with perforations introduced only in the $$z > 0$$ region. Assuming no mean flow, the linearized wave equation is solved subject to the relevant boundary conditions. The analysis utilizes a Fourier transform along the axial direction coupled with the Mode Matching Method to enforce continuity at geometric discontinuities. This hybrid approach leads to a scalar modified Wiener–Hopf equation whose solution involves an infinite set of coupled modal coefficients satisfying an infinite system of linear algebraic equations. Numerical solutions of these systems reveal the impact of key physical parameters, such as surface impedance $$Z$$ , Helmholtz number $$ka$$ , normalized lining length $$kl$$ , and perforation properties, on the far-field radiation and modal behavior. The findings offer significant insights for the acoustic design and optimization of duct systems aimed at noise control.