Initially developed by Ed McDonald [B. E. McDonald, W. A. Kuperman, J. Acoust. Soc. Am. 81, 1406-1417, (1987)], the NPE model is a time-domain parabolic-equation (PE) method to describe the nonlinear propagation of compressional waves. Combining fluid momentum and mass continuity equations, substituting an assumed adiabatic equation of state, and transforming to a frame moving (Fig. 1) in the x direction at the speed of sound (Lagrangian coordinates) leads to a nonlinear wave equation (the NPE). In cartesian coordinates the NPE is

Fig. 1: Schematic representation of the spatial moving frame used in the NPE model.