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

Abstract

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 ∇div0.3em, 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.

Propagation of Elastic Waves along a Cylindrical Rod of Micropolar Elastic Solid with Stretch under Impedance Boundary Conditions

“…A few literature available on using of impedance boundary conditions on wave propagation. Kersten Schmidt (7) discussed wave propagation with the use of impedance boundary conditions. Somaiah (8) studied the rotation and initial stress effects on Love wave propagation subject to impedance boundary conditions.…”