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 simulation of weakly nonlinear sound propagation within an inhomogeneous medium. So that this method can be used for outdoor sound propagation 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 and, hence, the ground layer is included in the computational system with the help of a second NPE 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's equations. In the frame of a parabolic model, this two-way coupling only involves spatial derivatives, making its numerical implementation straightforward. Several propagation examples, both linear or nonlinear, are then presented. 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 and an approximate solution is proposed.