We report experimental results and analytically study unidemensional (1D) high- amplitude sound wave propagation in an acoustic metamaterial composed of an air- filled waveguide with periodic change of cross section. We propose a theoretical modelling of the periodic waveguide, which relies on a transmission-line approach, leading to a nonlinear and dissipative dynamical lattice model. In addition to the linear viscothermal and radiation losses, high amplitude sound waves at the locations of the change of cross section introduce nonlinear losses. Both dissipative effects are included in the transmission line model. We show that in the lossless regime, the latter representation allows for an analytical description of pulse-shaped soliton solutions of the Boussinesq and KdV equations, which we extend to the dissipative regime as well. We experimentally find that when considering the propagation of high amplitude pulses, nonlinear losses cannot be neglected. Despite this, our analytical results are in good agreement with the experiments. Finally, we propose a supercell representation of the periodic waveguide. The latter is in excellent agreement with the full 2D Navier Stokes equations and our experimental findings.