Acoustic Radiation from a Lined Flanged Duct at an Order-Two Exceptional Point: Mode Matching with an Improper-Integral Radiation Closure

Exceptional points are parameter values at which two eigenvalues and their corresponding eigenfunctions coalesce, rendering the wave operator defective. They arise widely in non-Hermitian wave physics and disrupt the modal expansions on which semi-analytic scattering methods rely. For lined acoustic waveguides, an augmented mode-matching ansatz that restores completeness at such a degeneracy—by adjoining the generalised eigenfunction obtained from the derivative of the parametrised duct mode with respect to its transverse spectral parameter—has been established for junctions between duct sections with discrete modal sets. This article extends that ansatz to an open, radiating configuration: a rigid feed duct communicates through an impedance-lined throat, tuned to an order-two exceptional point, with a half-space bounded by a rigid flange. The radiating mouth replaces the discrete modal closure by a continuous spectrum, so the augmented basis must be matched against an improper spectral integral. The half-space field is generated by the aperture velocity, which builds the rigid-flange condition into the representation exactly, and the resulting improper integrals are rendered analytic by branch-aware substitutions whose cutoff is tied to the retained modal content. The formulation is validated on the matching and boundary conditions themselves: pointwise continuity of pressure and of normal velocity at the internal junction, pointwise pressure continuity at the radiating mouth, the vanishing of the normal velocity on the rigid flange, and the recovery of the classical flanged-duct radiation problem in the rigid limit, cross-checked against an independent implementation. The full lined problem, including the defective case, has been further verified against an independent finite-volume solution of the same boundary-value problem, whose grid-converged fractions agree with the mode-matching values to within 8×10−5. A conserved-power identity is monitored as a necessary but not sufficient check. Numerical experiments confirm the known breakdown of the standard expansion at the exceptional point and the well-conditioned convergence of the augmented one in this radiating setting, and a scan of the complex admittance plane, refined by local optimisation and repeated across throat lengths and frequencies, shows that flange radiation detunes the absorption optimum away from the exceptional point, by an amount that grows with the radiated share of the power budget and vanishes as the throat lengthens.