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

Abstract

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.

Computational Model for Long-Range Non-Linear Propagation over Urban Cities

“…A wave causes a vibration of air particles contained in the ground pores, while the ground frame does not vibrate. The model derivation and the applications to propagation over porous layers can be found in [23]. In this work, it is proposed to use this model to study long-range sound propagation over urban cities.…”

“…In this section, we summarise the NPE model for sound propagation in multiple media proposed in [23]. This propagation model is composed of three entities.…”

“…In a previous work, the original NPE model has been extended to propagation within and over porous ground layers [24,25]. In this model the porous layer is assumed to be equivalent to a continuous fluid medium.…”

“…The NPE model is well adapted for long-range sound propagation applications and it can account for most features of weakly non-linear sound propagation outdoors: geometrical spreading, weak non-linearities, refraction effects (see for example [27,6]), site topography [24], ground impedance [23] and thermoviscous effects [39]. Different models, such as the Fast Field Program (FFP, see [14,31]) or the (linear, frequency-domain) Parabolic Equation (PE, see [15]) could as well be used.…”

“…When a wave is reflecting on a plane surface, its amplitude is changed and a (possibly negative) delay is given to the reflected wave. Parameters (μ, α, γ, ) appearing in equations (11b) and (11c) could be used because the propagation model can recreate the behaviour of a porous ground layer [23] and could thus be used "as is" to control the wave reflection. However, in order to reduce the number of parameters in the mean model and consequently, to reduce the stochastic model complexity, another solution is proposed.…”