  We develop a scheme to take into account the effects of short-range
nucleon-nucleon correlations in the nucleon-pair wave function by solving the
Bethe-Goldstone equation for a coarse grained delta shell potential in S-wave
configuration. The S-wave delta shell potential has been adjusted to reproduce
the $^1$S$_0$ phase shifts of the AV18 potential for this partial wave up to 2
GeV in the laboratory kinetic energy. We show that a coarse grained potential
can describe the high momentum tail of the back-to-back correlated pairs and
the $G$-matrix in momentum space. We discuss the easiness and robustness of the
calculation in coordinate space and the future improvements and utilities of
this model. This work suggests the possibility of using perturbation theory for
describing the short-range correlations, and related to this, to substitute the
$G$-matrix by an appropriate coarse-grained potential.
