We study the boundary observability of the 1-D homogeneous wave equation when using a Legendre-Galerkin semi-discretization method. It is already known that spurious high frequencies are responsible for its lack of uniformity with respect to the discretization parameter [4] which may prevent convergence in the approximation of the associated controllability problem. A classical remedy is to filter out the highest frequency components but this comes with a high computational cost in several space dimensions. We present here three remedies: a spectral filtering method, a mixed formulation (already used in the context of finite element method [14]) and a Nitsche's method. Our numerical results show that the uniform boundary observability inequalities are recovered. On the other hand, surprisingly, none of them seem to provide the trace (or direct) inequality uniformly, a property used to prove the convergence of the numerical controls [11]. However, our numerical tests suggest that convergence of the numerical controls is ensured when the uniform observability inequality holds.