The main objective of this paper consists in studying the dynamic and observation of a wave equation [1] in a bounded domain in the plan. This work is inscribed in the field of control of systems governed by partial differential equations (PDE). We consider the wave equation system with Dirichlet boundary condition whose dynamic evolves in an infinite-dimensional Hilbert space. We assume that velocity is measured on some subdomain along the boundary. An infinite-dimensional exponentially convergent Luenberger-like observer is presented to estimate the system state: displacement and velocity on the whole domain. The main contribution of the work consists in building a reliable numerical simulator based on the finite element method (FEM). We examine the influence of the gain on the convergence rate of the observer.