The Cremer Impedance for Double-lined Rectangular Ducts

The Cremer impedance refers to the locally reacting boundary condition that maximizes the propagation damping of a certain acoustic mode in a uniform waveguide with infinite length. Previously, the Cremer impedance for rectangular ducts with only one lined wall is obtained by setting the first order derivative of the waveguide eigenvalue equation to zero, which leads to the merging of the fundamental and the first non-plane mode. By symmetry, this solution (referred to as the 'double root' hereafter) can be used on two opposite walls, which is equivalent to a rectangular duct with twice the height. However, as suggested by Zorumski and Mason, it is alternatively possible to create conditions for two different impedances on opposite walls by requiring both the first and second order derivative to be zero. By this means that the second higher order mode will also merge with the two lower modes. In this paper, two such solutions ('triple root') are proposed and compared with the double root. Some improvement in damping is found in both the low and high frequency range for the new triple roots. Alternative ways to create double roots compared to the Cremer symmetric case based on symmetry-breaking are also discussed.