Attenuation of sound in ducts with acoustic treatment—a generalized approximate equation

A generalized approximate equation for duct lining sound attenuation is presented. The specification of two parameters, the maximum possible attenuation and the optimum wall acoustic impedance, is shown to completely determine the sound attenuation for any acoustic mode at any selected wall impedance. The equation is based on the nearly circular shape of the constant attenuation contours in the wall acoustic impedance plane. For impedances far from the optimum the equation reduces to Morse's approximate expression. The equation can be easily used for initial acoustic liner design. Not least important is the illustrative nature of the solutions which provide an understanding of the duct propagation problem which is usually obscured in the exact calculations. An additional property of the approximate equation is that empiricism can be injected into the expression to account for phenomena which can not currently be handled by exact analysis. The peak and the bandwidth of the sound attenuation spectrum are presented by quite simple functions of the ratio of actual wall acoustic resistance to optimum resistance.