Time-Domain Impedance Boundary Conditions for Simulations of Outdoor Sound Propagation

Abstract

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

Time-Domain Simulations of Outdoor Sound Propagation with Suitable Impedance Boundary Conditions

“…Its implementation has been described in detail in Cotté et al [2] and will be summarized in Sec. II.B.…”

“…Following the original work of Tam and Auriault [8] and other works done in the community of duct acoustics to model the impedance of lining materials in the time domain [4,[9][10][11], others developed timedomain boundary conditions (TDBCs) from classical ground impedance models, such as the phenomenological Zwikker and Kosten model [12,13]. This former approach was used by Cotté et al [2], who proposed a TDBC for outdoor sound propagation based on a recursive convolution method originally developed in the context of electromagnetic propagation [14] and introduced in acoustics by Reymen et al [11]. It was shown that this TDBC is accurate and computationally efficient [15].…”

“…Cotté et al [2] compared different methods to identify the coefficients A k and k of Eq. (12) and proposed an optimization method in the frequency domain that ensures accurate numerical results [2,15]. Two impedance models will be investigated here:…”

“…Optimized finite difference schemes and selective filters over 11 points are used for spatial derivation and grid-to-grid oscillations removal, respectively. These schemes allow us to accurately calculate acoustic wavelengths down to five or six times the spatial mesh size [2,22]. For the interior points, which are the ones separated by at least five points from the boundary, the centered fourth-order finite difference scheme of Bogey and Bailly [18] and the centered sixth-order selective filter of Bogey et al [21] are used.…”

“…Introduction R ECENTLY, there has been an increased interest in the study of sound propagation in the atmosphere in connection with European regulations limiting aircraft noise in the proximity of airports. Besides, numerical solutions of the linearized Euler equations (LEEs) with finite difference time-domain (FDTD) techniques are becoming more and more popular [1,2] in the outdoor sound propagation community. Indeed, these methods have proved their ability to account for many effects [3], such as refraction due to mean wind or mean temperature profiles, scattering by turbulence, or effects of topography.…”