  We simulate the low temperature behaviour of an elastic chain in a random
potential where the displacements $u(x)$ are confined to the {\it longitudinal}
direction ($u(x)$ parallel to $x$) as in a one dimensional charge density
wave--type problem. We calculate the displacement correlation function $g(x)=<
(u(x)-u(0))^2>$ and the size dependent average square displacement
$W(L)=<(u(x)-\bar{u})^2>$. We find that $g(x)\sim x^{2\eta}$ with
$\eta\simeq3/4$ at short distances and $\eta\simeq3/5$ at intermediate
distances. We cannot resolve the asymptotic long distance dependence of $g$
upon $x$. For the system sizes considered we find $g(L/2)\propto W\sim
L^{2\chi}$ with $\chi\simeq2/3$. The exponent $\eta\simeq3/5$ is in agreement
with the Random Manifold exponent obtained from replica calculations and the
exponent $\chi\simeq2/3$ is consistent with an exact solution for the chain
with {\it transverse} displacements ($u(x)$ perpendicular to $x$).The
distribution of nearest distances between pinning wells and chain-particles is
found to develop forbidden regions.
