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0.5i Exact admittance acoustic admittance acoustic field aeroacoustic axial propagation constant Beckemeyer boundary conditions governing boundary layer boundary value problem centerline Mach number coefficient matrices coordinate system Determining Acoustic-Liner Admittance eigenfunction F(y,z eigenvalue problem results equa equation 18 equation 25 equation 30 equation and boundary exact solutions exact values Excellent comparison finite-element method flow impedance test flow impedance tube flow profile governing the acoustic grazing flow impedance grazing-flow impedance tube impedance test laboratory integration involving shear jet engine Langley Research Center linear matrix measurements NASA node normal particle velocity number of points numerical method NY,NY+1 NY+I NY+1 O.OOi ONE-DIMENSIONAL SHEARED FLOW parameters partial differential equation particle displacement propagating mode sample calculation single cross-sectional direction solution to equation sound pressure level specified test sample test specimen tion two-dimensional sheared flow uniform flow unknown admittance value upper wall eigenfunction value problem wall pressure eigenfunction wave number z-direction