Ray Method for Sound-Wave Reflection in an Open-Ended Circular Pipe

Abstract

Modal reflection and coupling coefficients are calculated for a semi-infinite flanged and unflanged hard-wall circular tube excited by one of the propagating modes. The calculation is performed by a ray method wherein the incident mode is decomposed into local plane waves impinging on the rim. Consequent diffraction effects establish the reflected waves and are represented equivalently in terms of radiation from a nonisotropic ring source having a radiation pattern determined by the local scattering properties of the rim. The rays emitted by the nonisotropic ring source and reflected repeatedly from the tube wall are summed into modal form, the quality of the resulting asymptotic mode series being improved by comparison with the exact solution for an isotropic ring source. Reflection and coupling coefficients are provided by the modal amplitudes in this representation and are determined for primary diffraction due to the incident mode only and also for multiple diffraction due to interaction across the mouth of the tube. Comparison with exact computations for the thin-walled tube shows that the multiple interaction ray-acoustical calculation is capable of high accuracy not only at high frequencies but even in the dominant mode regime. Data are also presented for flanged terminations for which no exact solution is available.