  Some general properties of the advective-acoustic instability are described
and understood using a toy model which is simple enough to allow for analytical
estimates of the eigenfrequencies. The essential ingredients of this model, in
the unperturbed regime, are a stationary shock and a subsonic region of
deceleration. For the sake of analytical simplicity, the 2D unperturbed flow is
parallel and the deceleration is produced adiabatically by an external
potential. The instability mechanism is determined unambiguously as the
consequence of a cycle between advected and acoustic perturbations. The purely
acoustic cycle, considered alone, is proven to be stable in this flow. Its
contribution to the instability can be either constructive or destructive. A
frequency cut-off is associated to the advection time through the region of
deceleration. This cut-off frequency explains why the instability favours
eigenmodes with a low frequency and a large horizontal wavelength. The relation
between the instability occurring in this highly simplified toy model and the
properties of SASI observed in the numerical simulations of stellar
core-collapse is discussed. This simple set up is proposed as a benchmark test
to evaluate the accuracy, in the linear regime, of numerical simulations
involving this instability. We illustrate such benchmark simulations in a
companion paper.
