Time-domain impedance boundary condition for outdoor sound propagation numerical simulations

In the context of transportation noise, acoustic sources are usually broadband and in motion, and the propagation environment can be complex, with various types of ground, wind and temperature fluctuations, and topographic effects. Finite-difference time-domain methods are particularly well suited to deal with these different aspects. In this paper, time-domain boundary conditions (TDBCs) are considered for different impedance models classically used for outdoor grounds. These impedance models have usually been obtained in the frequency domain, and they don't necessarily meet the causality, reality and passivity conditions for a model to be translated into the time domain. Furthermore, when it is possible to derive a TDBC, a convolution must be solved, which is not efficient from a numerical point of view. The TDBC that is presented here is based on the approximation of the impedance as a sum of well chosen template functions. The approximation process can be performed in the frequency domain or in the time domain. Thanks to the forms of the template functions, the recursive convolution technique can be applied; this is a fast and computationally efficient method to calculate a discrete convolution. The TDBC is validated using a linearized Euler equations solver in one- and three-dimensional geometries; comparisons with analytical solutions in the time and frequency domains are presented. The methods used to identify the coefficients of the template functions are shown to be of great importance. Indeed, the numerical simulations become inaccurate when the values of the poles of the template functions are large with respect to the time step. Thus, a constraint on the values of the poles must be included in the coefficients identification methods to obtain accurate solutions.