  We present Helmholtz or Helmholtz like equations for the approximation of the
time-harmonic wave propagation in gases with small viscosity, which are
completed with local boundary conditions on rigid walls. We derived
approximative models based on the method of multiple scales for the pressure
and the velocity separately, both up to order 2. Approximations to the pressure
are described by the Helmholtz equations with impedance boundary conditions,
which relate its normal derivative to the pressure itself. The boundary
conditions from first order on 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 velocity approximations are described
by Helmholtz like equations for the velocity, where the Laplace operator is
replaced by $\nabla \operatorname{div}$, 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 boundary
conditions are stable and asymptotically exact, which is justified by a
complete mathematical analysis. The results of some numerical experiments are
presented to illustrate the theoretical foundation.
