Fundamental Investigation into the directivity function of multi-mode sound fields from ducts

Abstract

Multi-mode sound radiation from hard-walled semi-infinite ducts with uniform subsonic flow is investigated theoretically. An analytic expression, valid in the high frequency limit, is derived for the multi-mode directivity function in the forward arc of the duct for a general family of mode distribution function. The multi-mode directivity depends on the amplitude of each mode, and on the single mode directivity functions. The amplitude of each mode is expressed as a function of cut-off ratio for a uniform distribution of incoherent monopoles, a uniform distribution of incoherent axial dipoles and for equal power per mode. The single mode directivity functions are obtained analytically by applying a Lorentz Transformation to the zero flow solution. The analytic formula for the multi-mode directivity with flow is derived by assuming total transmission of power at the open-end of the duct. The high frequency formula is compared to exact numerical solutions from the Wiener Hopf technique and for a flanged duct. The agreement is shown to be excellent