## Optimal one-section and two-section circular sound-absorbing duct liners for plane-wave and monopole sources without flow |

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acoustic impedance acoustic particle velocity acoustic pressure acoustical energy admittance plane admittance properties admittance space admittance vector ascent vector based on modal Beckemeyer calculated circular duct CIRCULAR LINER FIG constant transmission loss Cremer's criterion defined DFP algorithm DFP method DFP solution duct liner equation frequencies kb frequency range hard-wall Infinite Duct intermediate frequency iteration Langley Research Center lined section liner properties loss function Luenberger matrix maximum insertion loss maximum transmission loss modal coefficients modal theory approach monopole source multisectioned duct Newton's method normalized frequency optimal admittance values optimal liner design OPTIMAL PROPERTIES optimal uniform liner plane wave PLANE-WAVE AND MONOPOLE plane-wave source positive-definite matrix propagation model Rice's second section shown in figure simplified sources source and termination source impedance source plane superposition Susceptance termination impedance effects tion transmission loss TL turbomachinery source two-section ducts uniform and two-section UNIFORM CIRCULAR LINER uniform pressure source velocity source