  Sand pile models are dynamical systems describing the evolution from $N$
stacked grains to a stable configuration. It uses local rules to depict grain
moves and iterate it until reaching a fixed configuration from which no rule
can be applied. Physicists L. Kadanoff {\em et al} inspire KSPM, extending the
well known {\em Sand Pile Model} (SPM). In KSPM($D$), we start from a pile of
$N$ stacked grains and apply the rule: $D\!-\!1$ grains can fall from column
$i$ onto columns $i+1,i+2,\dots,i+D\!-\!1$ if the difference of height between
columns $i$ and $i\!+\!1$ is greater or equal to $D$. Toward the study of fixed
points (stable configurations on which no grain can move) obtained from $N$
stacked grains, we propose an iterative study of KSPM evolution consisting in
the repeated addition of one grain on a heap of sand, triggering an avalanche
at each iteration. We develop a formal background for the study of avalanches,
resumed in a finite state word transducer, and explain how this transducer may
be used to predict the form of fixed points. Further precise developments
provide a plain formula for fixed points of KSPM(3), showing the emergence of a
wavy shape.
