Improved multimodal methods for the acoustic propagation in waveguides with a wall impedance and a uniform flow

We address the problem of acoustic propagation in waveguides in presence of a uniform flow and with, locally, a spatially varying impedance boundary condition on the walls. In time harmonic regime, the Myers impedance boundary condition is chosen. Classically, that is, in waveguides with rigid boundaries, multimodal formulations involve the expansion of the solution on the "rigid" transverse modes satisfying the Neumann boundary condition. Since these modes define a complete basis, the decomposition is still valid in segments with Myers boundary condition at the walls. However, because the boundary condition is not satisfied by the transverse modes, the series has poor convergence. This results in the appearance of Gibbs oscillations close to the treated wall. Our aim is to develop an improved multimodal methods, to remedy the problem of the low convergence of the series in the standard modal approach. In the case of waveguides with varying cross-section or of straight guides with impedance condition but without flow, it has been shown that this situation can be remedied by adding to the usual expansion an additional term (called supplementary mode). In this approach, the role of the supplementary mode is to restore asymptotically the right boundary condition. In the no flow case, the convergence of the scattering properties is increased from N∼l in the standard modal method to N∼5 in the formulation with a supplementary mode. The case of the limit toward a discontinuous impedance boundary conditions is also considered, leading to the constraint of an edge condition for the pressure at the discontinuities.