Sound radiation from a cylindrical duct. Part 2. Source modelling, nil-shielding directions, and the open-to-ducted transfer function

Abstract

This paper analyses the sound radiated from the front face of a hard-walled circular cylindrical duct in a subsonic mean flow when the duct contains acoustic sources typical of those in a ducted-fan aeroengine. Two main results are established for modes of any given frequency and circumferential order. The first result is that in certain easily calculated directions, called here the nil-shielding directions, the sound radiated by ducted sources is the same as the sound radiated by the corresponding open sources, i.e. by unducted sources of the same distribution and strength radiating into free space. Thus in these special directions the duct has no noise-shielding effect. The second result is that, in the Kirchhoff approximation, the sound radiated by the open sources in the nil-shielding directions determines the sound radiated by the ducted sources in all directions; i.e. the sound fields radiated by open and ducted sources are related by an open-to-ducted transfer function. This function is such that the sound radiated by the ducted sources is a linear combination of certain diffraction functions, in which the coefficients are given by the sound radiated by the open sources in the nil-shielding directions. The diffraction functions do not depend on the sources and are here calculated explicitly in terms of Bessel functions. The method used in the paper is Kirchhoff's approximation; within linear theory this gives the nil-shielding directions exactly, i.e. in agreement with the Wiener—Hopf solution, and gives the main beam of the radiated field, including the major side-lobes, to good accuracy. The results are relevant to the sound radiated into the forward arc by a ducted turbofan aeroengine.