| The nonlinear progressive wave equation (NPE) is a time-domain formulation of Euler's fluid equations designed to model low angle wave propagation using a wave-following computational domain. The standard formulation consists of four separate mathematical quantities that physically represent refraction, nonlinear steepening, radial spreading, and diffraction. The latter two of these effects are linear whereas the steepening and refraction are nonlinear. This formulation recasts pressure, density, and velocity into a single variable - a dimensionless pressure perturbation - which allows for greater efficiency in calculations. The wave-following frame of reference permits the simulation of long-range propagation that is useful in modeling the effects of blast waves in the ocean waveguide. Nonlinear effects such as weak shock formation are accurately captured with the NPE. The numerical implementation is a combination of two numerical schemes: a finite-difference Crank-Nicholson algorithm for the linear terms of the NPE and a flux-corrected transport algorithm for the nonlinear terms. In this work, an existing implementation is extended to allow for a penetrable fluid bottom. Range-dependent environments, characterized by sloping bathymetry, are investigated and benchmarked using a rotated coordinate system approach. |
