The Cremer impedance revisited: Investigating passivity at low frequencies for uniform and sheared mean flows

The Cremer impedance identifies the optimal wall treatment for maximizing the axial attenuation in a waveguide by enforcing the coalescence of the least-attenuated modes. At low frequencies, however, models employing the Ingard-Myers boundary condition often yield negative resistances. This study shows that these results are a mathematical consequence of combining the coalescence criterion with the impedance boundary condition, with neither enforcing acoustic passivity. As frequency decreases, the region of admissible impedances with positive resistance contracts significantly, while the unconstrained coalescence condition may drive the optimized solution outside this shrinking passive domain. In uniform flow, the classical continuity of velocity condition from Abom and Jacob [JASA Express Lett. 1, 022801 (2021)] restores passivity by forcing the resistance toward zero, but numerical evaluations using realistic no-slip shear profiles reveal that velocity gradients can still reintroduce negative resistances even when the wall velocity is zero. Together, these findings indicate that achieving strictly passive Cremer-optimized impedances remains a fundamental challenge in low-frequency inviscid duct acoustics.