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Ab[Ym acoustic admittances acoustic field acoustic modes acoustic treatment admittance discontinuity aircraft turbofan engine amplitude eq annular duct backward-moving wave Beckemeyer boundary condition circular and annular circular duct continuity and momentum cylindrical coordinates dimensionless Dirac delta function duct section duct systems e_lt eigenfunction equation 25 equation 39a equation 69 equations 17 equations 42 equations are derived finite ducts flange Green's function hard-walled identity matrix infinite Langley Research Center m/i v m/iy matrix equations matrix formulation matrix notation mean flow mode amplitudes mode number momentum equations multisection liners NASA noise source number of planes orthogonal problem radiation directivity factors Radiation Impedances Radiation of Sound radiation reflection reflection coefficients reflection effects reflection equations reflection matrices scaled complex acoustic shown in figure solution splitters submatrix subscripts and superscripts theory transmission and reflection unflanged duct uniform duct uniform flow uniform liner uniform multisectioned duct vector w+k-j wave amplitudes wave equation wave number Zorumski