Development of a sound radiation model for a finite-length duct of arbitrary shape

Abstract

A new variational formulation by integral equations has been developed to solve Helmholtz's equation with mixed boundary conditions. Contrary to previous methods generally based on the Wiener-Hopf technique which are limited to the case of a circular semi-infinite duct, our method allows the computation of sound radiation from the duct with arbitrary shape and finite length. Experimental works have been conducted using a spinning mode synthesizer. Comparison between theoretical and experimental results of pressure reflection coefficients for two inlet shapes and directivity patterns shows a very good agreement. Differences between results for finite and semi-infinite length ducts are twofold: locations of principal lobe of radiation are not the same, and secondary lobes of radiation arise at a frequency under the cutoff frequency of the first radial mode. Nomenclature a = duct radius d = bilinear form D = boom radius / = velocity potential on source section SI f0 = frequency g = duct wall velocity G = Green's function k = wave number L = duct length m = azimuthal wave number n = radial number IP I = normalized linear amplitude of sound pressure, pco2 \<p \R I = pressure reflection coefficient modulus 57 = source section S2 = duct wall surface / = time A = Laplace's operator 0 = polar angle X = wavelength IJL = density of double-layer potential on the duct walls p = fluid density o = density of single-layer potential on the source section <p = velocity potential ty = cylindrical angle coordinate GO = angular frequency 0 = fluid domain