  This work deals with the numerical solution of systems of oscillatory
second-order differential equations which often arise from the
semi-discretization in space of partial differential equations. Since these
differential equations exhibit (pronounced or highly) oscillatory behavior,
standard numerical methods are known to perform poorly. Our approach consists
in directly discretizing the problem by means of Gautschi-type integrators
based on $\operatorname{sinc}$ matrix functions. The novelty contained here is
that of using a suitable rational approximation formula for the
$\operatorname{sinc}$ matrix function to apply a rational Krylov-like
approximation method with suitable choices of poles. In particular, we discuss
the application of the whole strategy to a finite element discretization of the
wave equation.
