Approximate expressions for the reflection coefficient of ducts terminated by circular flanges

estimating the magnitude of the pressure reflection coefficient |r| and the end correction l at the open end of ducts is a critical procedure when designing or predicting the acoustic behavior of acoustical systems, such as exhausts, tailpipes, mufflers, loudspeaker enclosures and so on. for cylindrical ducts and plane waves, exact intricate solutions exist for two distinct open-end boundary conditions, namely for a thin-walled unflanged pipe and for a pipe terminated by an infinite flange. this work provides simple approximate expressions for |r| and l of cylindrical pipes terminated by circular flanges with finite radii. the expressions are obtained from a polynomial fit performed over the numerical results provided by a boundary element model, and is valid for helmholtz numbers in the range 0 < ka < 3.0, as well as for 0 < a/b < 1, where a and b are the pipe and flange radii, respectively. when compared with the exact solutions for both the unflanged and the infinite-flanged pipe, the approximate formulae provide a maximum error of ~2% at the upper frequency limit (ka →3.0).