We consider the semi-discrete space approximation of the one dimensional controlled linear wave equation with a potential. We analyze the conditions which ensure the convergence of the numerical scheme. We show that if the high frequencies of the discretization of the initial data are ltered out, then the convergence of the approximating family of controls can be ensured. Moreover, we prove that there exists a space of regular initial data which are directly uniformly controllable. The main diculty consist of the fact that the corresponding discrete operator does not posses an explicit family of eigenvalues, which is an important issue in our strategy (which reduces the controllability problem to a moment problem solved by constructing a biorthogonal sequence to a family of exponential functions).