Nonlinear acoustic wave propagation through a stratified atmosphere is considered. The initial signal is taken to be an isolated N-wave, which is the disturbance that is generated some distance away from a supersonic body in horizontal flight. The effect of cylindrical spreading and exponential density stratification on the propagation of the disturbance is considered, with the shock structure controlled by molecular relaxation mechanisms and by thermoviscous diffusion. An augmented Burgers equation is obtained and asymptotic solutions are derived based on the limit of small dissipation and dispersion. For a single relaxation mode, the solution depends on whether relaxation alone can support the shock or whether a sub-shock arises controlled by other mechanisms. The resulting shock structures are known as fully dispersed and partly dispersed shocks, respectively. In this paper, the spatial location of the transition between fully dispersed and partly dispersed shocks is identified for shocks propagating above and below the horizontal. This phenomenon is important in understanding the character of sonic booms since the transition to a partly dispersed shock structure leads to the appearance of a shorter scale in the shock rise-time, associated with the embedded sub-shock.