Impedance boundary conditions for acoustic time-harmonic wave propagation in viscous gases in two dimensions

We present impedance boundary conditions for the viscoacoustic equations for approximative models that are in terms of the acoustic pressure or in terms of the macroscropic acoustic velocity. The approximative models are derived by the method of multiple scales up to order 2 in the boundary layer thickness. The boundary conditions are stable and asymptotically exact, which is justified by a complete mathematical analysis. The models can be discretized by finite element methods without resolving boundary layers. In difference to an approximation by asymptotic expansion for which for each order 1 PDE system has to be solved, the proposed approximative are solutions to one PDE system only. The impedance boundary conditions for the pressure of first and second orders are of Wentzell type and include a second tangential derivative of the pressure proportional to the square root of the viscosity and take thereby absorption inside the viscosity boundary layer of the underlying velocity into account. The conditions of second order incorporate with curvature the geometrical properties of the wall. The velocity approximations are described by Helmholtz-like equations for the velocity, where the Laplace operator is replaced by backward difference div, and the local boundary conditions relate the normal velocity component to its divergence. The velocity approximations are for the so-called far field and do not exhibit a boundary layer. Including a boundary corrector, the so-called near field, the velocity approximation is accurate even up to the domain boundary. The results of numerical experiments illustrate the theoretical foundations.

Automated summary

This paper presents impedance boundary conditions for the viscoacoustic equations, with approximative models that can be discretized by finite element methods without resolving boundary layers. The conditions are stable and asymptotically exact, and the results of numerical experiments illustrate the theoretical foundations.