The acoustic impedance of a semi-infinite tube fitted with a conical flange

Abstract

The acoustic characteristics of a semi-infinite tube of constant cross sectionσ, fitted with a semi-infinite conical flange of solid angleω, are investigated. Let the acoustic potential in the tube be given by $$\varphi = e^{ - ikx} - Ae^{ikx}$$ where exp(−ikx) represents the incident wave travelling along the tube towards the open endx=0 and −A exp(ikx) represents the reflected wave. It is shown that for long waves,A can be written in the form $$A = \exp \{ - 2ik\alpha (\sigma /\omega )^{1/2} - 2k^2 \sigma /\omega \}$$ whereα is a parameter which depends only on the solid angleω. From existing work the value ofα is known forω=2π (the infinite plane flange) andω=4π (the unflanged tube). In this paper, the variation ofα for small values ofω is calculated. Because 1≦α≦1.2266 over the whole range 0≦ω≦4π, it is suggested that general values ofα can be interpolated from the existing results. An explicit interpolation formula is proposed.