In this paper, we investigate the nonlinear wave propagation in a series-connected discrete Josephson junction transmission line (DJTL). This structure consists of a superconductive coplanar waveguide (CPW), which is assisted by Al Josephson junctions (JJ) in a periodic fashion. Each junction is represented by the basic circuit model which leads to a nonlinear inductor element. Having a significant number of junctions per wavelength, the discrete transmission line (TL) can be considered as a uniform nonlinear transmission line. The nonlinear wave equations are solved numerically by finite difference time domain (FDTD) method. Features and characteristics such as cut-off propagation, dispersive behavior and shock wave formation, which are expected from wave propagation through the nonlinear DJTL, are discussed in this paper.