Edge diffraction of in-duct modal waves by unflanged or flanged circular duct terminations

An asymptotic approach is presented for studying the diffraction problem of in-duct acoustic modes by the termination of a rigid, circular duct with negligible thickness, based on Keller’s (1957, 1958, 1962) geometrical theory of diffraction (GTD). The diffracted field is solved first for the unflanged duct case, followed by an extension to the flanged duct case for which no closed-form exact solutions are available. The GTD solution for the primary diffraction of unflanged ducts, which invokes the half-plane diffraction coefficient obtained from Sommerfeld’s exact solution to the half-plane diffraction problem, is shown to yield agreement with the leading term of the Wiener–Hopf solution, in which the split functions of the Wiener–Hopf kernel are replaced with their steepest-descent approximations. Despite being developed for high-frequency analysis, experimental data from an unflanged duct and the numerical solutions for a flanged duct, both including the radiation directivity and the reflection coefficient, indicate that GTD solutions perform reasonably well even for wavelengths smaller than the duct’s radius, provided the frequency does not approach the cutoff condition. A reciprocity relation, which couples the absorption and emission of the (un)flanged duct, is derived from the reciprocity principle and verified by the Wiener–Hopf (if available) and GTD solutions. Physical insights are supplied by the GTD to explain why, for example, only a plane wave would be excited within the duct by a plane wave incident normally from the exterior of the duct. In cases where the uniform flow is present, an extended GTD formulation is proposed by utilising the canonical solution to the half-plane diffraction problem. The resulting correction factor for the diffracted field of unflanged ducts that accounts for an arbitrary amount of shedding vortices is consistent with Rienstra’s (1984 J. Sound Vib. vol. 94 (2), pp. 267–288) Wiener–Hopf solution. Potential strategies for addressing variants and extensions of the current work are outlined.