  There is a qualitative difference between one-dimensional and
multi-dimensional solutions to the Euler equations: new features that arise are
vorticity and a nontrivial incompressible (low Mach number) limit. They present
challenges to finite volume methods. It seems that an important step in this
direction is to first study the new features for the multi-dimensional acoustic
equations. There exists an analogue of the low Mach number limit for this
system and its vorticity is stationary.
  It is shown that a scheme that possesses a stationary discrete vorticity
(vorticity preserving) also has stationary states that are discretizations of
all the analytic stationary states. This property is termed stationarity
preserving. Both these features are not generically fulfilled by finite volume
schemes; in this paper a condition is derived that determines whether a scheme
is stationarity preserving (or, equivalently, vorticity preserving) on a
Cartesian grid.
  Additionally, this paper also uncovers a previously unknown connection to
schemes that comply with the low Mach number limit. Truly multi-dimensional
schemes are found to arise naturally and it is shown that a multi-dimensional
discrete divergence previously discussed in the literature is the only possible
stationarity preserving one (in a certain class).
