In this paper, we study the observer design problem for a one-dimensional wave equation with van der Pol type boundary condition, whose dynamics presents spatiotemporal chaotic behaviors. By introducing a linear error output injection on the boundary, we construct an observer via a method of characteristics. The main approach is to construct two one-dimensional mappings which can characterize the evolutionary dynamics of the system as well as the observer, and the global convergence of error dynamics is obtained in terms of a C-norm. The range of the observer gain is identified. Numerical simulations are provided to illustrate the theoretical outcomes.