# Boundary conditions for wave propagation in viscous media

#### Exposé 1

**Kersten Schmidt, Institut fur Mathematik - TU Berlin.**

Local impedance boundary conditions for wave propagation in viscous gases

*Résumé*

In this talk we present Helmholtz like equations for the approximation of the time-harmonic wave propagation in gases with small viscosity (inner friction), which are completed with local boundary conditions on rigid walls. First, we show equations for the pressure, for which the boundary conditions relate the normal derivative of the pressure, the Neumann trace, to the pressure itself, the Dirichlet trace. The boundary condition is of Wentzel type due to a second tangential derivative of the pressure. Then, we introduce Helmholtz like equations for the velocity, where the Laplace operator is replaced by $\mathbf{grad} \mathrm{div}$, and the local boundary conditions relate the normal velocity component (Dirichlet trace) to its divergence (Neumann trace). We discuss the variational formulations for those symmetric local boundary conditions, on which finite elements approximations are based on.

The presented models are obtained by asymptotic expansions of the linearized compressible Navier-Stokes equations for gases in rest with no-slip boundary condition on rigid walls. This is a singularly perturbed partial differential equation (PDE) leading to boundary layers in the velocity. The well-known Helmholtz equation with Neumann boundary condition is the limit for vanishing viscosity. The incorporation of small viscosities is especially attractive if one is interesting in the "noise" absorption. The method of multiscale expansion is a systematic tool for small viscosity to decompose the solution in so called far fields, which are accurate approximations except in a small vicinity of the boundary, and boundary layer correctors. The far field terms can be defined for the pressure and velocity separately even so they cannot be separated the original equations. This enable us to derive approximative models for either the (far field) pressure or the (far field) velocity, for which the solution can be resolved by the finite element method without need of very fine meshes in the vicinity of the boundary. The models are approximative and error estimates in powers of the viscosity have been rigorously proved.

#### Exposé 2

**Adrien Semin, Institut fur Mathematik - TU Berlin.**

Conditions Transparentes dans des guides viscoacoustiques