Approximation formulae for the acoustic radiation impedance of a cylindrical pipe

Useful approximation formulae for radiation impedance are given for the reflection coefficients of both infinitely flanged and unflanged rigid-walled cylindrical ducts. The expressions guarantee that simple but necessary physical and mathematical principles are met, like Hermitian symmetry for the reflection coefficient (identical behaviour of positive and negative frequencies) and causality for the impulse response. A non-causal but more accurate expression is also proposed that is suitable for frequency-domain applications. The formulae are obtained by analytical and numerical fitting to reference results from Levine and Schwinger for the unflanged case and extracted from the radiation impedance matrix given by Zorumski for the infinite flanged case.