Explanation of Anomalous Behavior Observed in Impedance Eduction Techniques Using Measured Data

Several enhancements that improve the accuracy and robustness of an impedance eduction technique that use an automatic optimizer are presented. These enhancements are then used to launch an intensive investiga-tion into the cause of anomalous behavior that occurs for a small number of test conditions. This anomalous behavior is investigated for both a hardwall insert and a conventional liner. The primary conclusions of the study are that: (1) for the hard wall insert, the anomalies are due to narrow peaks in the objective function, (2) For the conventional liner, the anomalies are due to the presence of an extremely flat objective function, and (3) the anomalies appear to be triggered by inconsistencies between the duct propagation model and the measured data. At high frequencies, the duct propagation model may need to include the effects of higher-order duct modes, whereas at low frequencies, the effects of the mean boundary layer may have to be included. Nomenclature [A],{P},{F} = stiffness matrix, vector of unknown nodal coefficients, vector containing source effects c0,M0,ρ0 = speed of sound, uniform flow Mach number, mean flow density d, f,k, t = cavity depth, frequency of sound source, freespace wavenumber (ω/c0), time H,L,W = height of duct, length of duct, width of duct i,ω = unit imaginary number ( √−1), circular frequency (2pi f) N, p = number of lower wall microphones, acoustic pressure field L1,L2 = axial location of leading edge of test liner, axial location of trailing edge of test liner X,Y = unconstrained resistance variable, unconstrained reactance variable x,y,z = Cartesian coordinates β,ζ = dimensionless admittance of test liner, dimensionless impedance of test liner θ,χ = dimensionless resistance of test liner, dimensionless reactance of test liner φ(ζ), | | | | = objective function, complex absolute value Subscripts: Meas,s, t = a measured quantity, a source plane quantity, an exit plane quantity min,max = minimum value of a quantity, maximum value of a quantity I