Acoustic Theory of Axisymmetric Multisectioned DuctsEquations are developed for the acoustic field in a duct system which is made up of a number of connected circular and annular ducts. These equations are suitable for finding the acoustic field inside of and radiated from an aircraft turbofan engine. Acoustic modes are used as generalized coordinates in order to develop a set of matrix equations for the acoustic field. Equations for these modes are given for circular and annular ducts with uniform flow. Modal source equations are derived for point acoustic sources. General equations for mode transmission and reflection are developed and detailed equations are derived for ducts with multiple sections of acoustic treatment and for ducts with circumferential splitter rings. The general theory is applied to the special case of a uniform area circular duct with multisection liners and it is shown that the mode reflection effects are proportional to differences of the acoustic admittances of adjacent liners. A numerical example is given which shows that multisection liners may provide greater noise suppression than uniform liners. |
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Contents
SUMMARY | 1 |
Governing Equations and Boundary Conditions | 7 |
Flanged duct | 21 |
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Aa[Jm Ab[Jm acoustic admittances acoustic field acoustic modes acoustic treatment admittance changes aircraft turbofan engine amplitude eq annular duct axisymmetric boundary condition circular and annular circular duct continuity and momentum cylindrical coordinates dimensionless Dirac delta function duct section duct systems eigenfunction equation 25 equation 39a equation 69 equations 17 equations 42 equations are derived finite flange Green's function hard-walled identity matrix infinite duct Langley Research Center matrix equations matrix formulation matrix notation mode amplitudes momentum equations multisection liners noise source orthogonal planes zj Pm r,n problem radiation directivity factors radiation patterns radiation reflection reflection coefficients reflection effects reflection equations reflection matrices shown in figure splitters submatrices subscripts and superscripts theory transmission and reflection unflanged duct uniform duct uniform liner uniform multisectioned duct vector wall admittance wave amplitudes wave equation wave number Zorumski ημ ομν ΩΚ བ བ བ