  We consider a chain of $n$ coupled oscillators placed on a one-dimensional
lattice with periodic boundary conditions. The interaction between particles is
determined by a weakly anharmonic potential $V_n = r^2/2 + \sigma_nU(r)$, where
$U$ has bounded second derivative and $\sigma_n$ vanishes as $n \to \infty$.
The dynamics is perturbed by noises acting only on the positions, such that the
total momentum and length are the only conserved quantities. With relative
entropy technique, we prove for dynamics out of equilibrium that, if $\sigma_n$
decays sufficiently fast, the fluctuation field of the conserved quantities
converges in law to a linear $p$-system in the hyperbolic space-time scaling
limit. The transition speed is spatially homogeneous due to the vanishing
anharmonicity. We also present a quantitative bound for the speed of
convergence to the corresponding hydrodynamic limit.
