An adaptive finite element algorithm is presented for the wave equation in two space dimensions. The goal of the adaptive algorithm is to control the error in the same norm as for parabolic problems, namely the L2(0,T;H1(\Omega)) norm, where T denotes the final time and Omega the computational domain. The mesh aspect ratio can be large whenever needed, thus allowing a given level of accuracy to be reached with fewer vertices than with classical isotropic meshes. The refinement and coarsening criteria are based on anisotropic, a posteriori error estimates and on an elliptic reconstruction. A numerical study of the effectivity index on non-adapted meshes confirms the sharpness of the error estimator. Numerical results on adapted meshes indicate that the error indicator slightly underestimates the true error. We conjecture that the missing information corresponds to the interpolation error between successive meshes. It is observed that the error indicator becomes sharp again when considering the damped wave equation with a large damping coefficient, thus when the parabolic character of the PDE becomes predominant.