A finite element approach to the calculation of nonlinear sound propagation is proposed. Under the assumption of a weak nonlinearity, a linearized one-dimensional equation is considered. The equation is discretized in space, and is then solved for time by using Newmark-β integration scheme, in which a numerical damping is devised. Some numerical demonstrations are made for the nonlinear sound propagation of a single-shot pulse in air. It is shown that the shock wave propagation is stably and accurately simulated by the introduction of the numerical damping.