In this paper we consider the stability of a linear time invariant system in feedback with a string equation. A new Lyapunov functional candidate is proposed based on the use of augmented states which enriches and encompasses the classical Lyapunov functional proposed in the literature. It results in a hierarchical tractable stability condition expressed in terms of linear matrix inequalities. This methodology follows from the application of the Bessel inequality together with Legendre polynomials. Two numerical examples illustrate the potential of our approach through two scenarious: a stable ODE perturbed by the PDE and an unstable ODE stabilized by the PDE.