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Finite difference time-domain methods are well suited to study sound propagation in the context of transportation noise. In this paper, time-domain boundary conditions are considered for impedance models classically used for outdoor grounds. These impedance models have usually been obtained in the frequency domain and cannot be translated directly into the time domain. The derivation of the time-domain boundary condition is based on the approximation of the impedance as a sum of well-chosen template functions. Because of 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 impedance boundary conditions are 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. Among the three methods described, the optimization method in the frequency domain can be recommended, because it can be applied to many impedance models and allows the values of the coefficients to be constrained, which is needed to obtain accurate numerical results. Nomenclature B= Gaussian half-width, m c 0 = speed of sound, m=s d L = porous layer thickness, m f = frequency, Hz Im = imaginary part j = imaginary unit k = complex wave number, m 1 p = pressure, Pa q = tortuosity Re = real part S = number of first-order systems in the impedance approximation s f = coefficient of the selective filter T = number of second-order systems in the impedance approximation t = time, s v = velocity component normal to impedance surface, m=s Z = complex impedance, kg=m 2 =s = ratio of specific heats L = sound pressure level relative to the free field, dB t = time step, s x = spatial mesh size, m 0 = air density, kg=m 3 0 , e = flow resistivity, Pa s=m 2 = porosity ! = angular frequency, rad=s

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The current work aims at developing a linearized Euler equations solver in curvilinear coordinates to account for the effects of topography on sound propagation. In applications for transportation noise, the propagation environment as well as the description of acoustic sources is complex, and time-domain methods have proved their capability to deal with both atmospheric and ground effects. First, equations in curvilinear coordinates are examined. Then time-domain boundary conditions initially proposed for a Cartesian coordinate system are implemented in the curvilinear solver. Two test cases dealing with acoustic scattering by an impedance cylinder in a two-dimensional geometry and by an impedance sphere in a three-dimensional geometry are considered to validate the boundary conditions. Accurate solutions are obtained for both rigid and impedance surfaces. Finally, the solver is used to examine a typical outdoor sound propagation problem. It is shown that it is well-suited to study coupled effects of topography, mixed impedance ground and meteorological conditions.

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