Formation and evolution of N (-like) waves is studied without the restriction of low amplitude, namely weak nonlinearity. To this end, the classical piston problem of gasdynamics is investigated, in which the wave is radiated by a piston executing a single cycle of harmonic oscillation into an inviscid perfect gas. The method of analysis is based on the simple-wave theory up to the shock formation time, and beyond that time on the numerical calculation by a high-resolution TVD upwind scheme. The initial sinusoid-like wave profile is rapidly distorted as the wave propagates, and this leads to the formation of head and tail shocks. The main effects of strong nonlinearity may be listed as follows: (i) entropy production at shock fronts, (ii) the existence of waves reflected from shocks, (iii) an asymmetric wave profile stemming from the boundary condition at the source of the strongly nonlinear problem. As the result, the strongly nonlinear wave possesses the following remarkable distinctive features, in contrast to its counterpart in the weakly nonlinear regime. The tail shock is not formed at the tail of the wave, and the expansion wave behind the head shock has non-uniform intensity. The N (-like) wave propagates with some excess mass. Thereby a region with low density, associated with the entropy production, appears in the vicinity of the source.