Nonlinear parabolic equation model for finite-amplitude sound propagation over porous ground layers

The nonlinear parabolic equation (NPE) is a time-domain method widely used in underwater sound propagation applications. It allows simulating weakly nonlinear sound propagation within an inhomogeneous medium. For this method to be suited for outdoor applications, it must account for the effects of an absorbing ground surface. The NPE being formulated in the time domain, complex impedances cannot be used. The ground layer is thus included in the computational system with the help of a second NPE model based on the Zwikker-Kosten model. A two-way coupling between these two layers (air and ground) is required for the whole system to behave correctly. Coupling equations are derived from linearized Euler equations. In the frame of a (small-angle) parabolic model, this two-way coupling only involves spatial derivatives, making its implementation easy. Several propagation examples are then presented, and the method is shown to give satisfactory results for a wide range of ground characteristics. Finally, the problem of including Forchheimer's nonlinearities in the two-way coupling is addressed.