FDTD modelling of frequency dependent boundary conditions for room acoustics

FDTD has become a popular tool in acoustic modelling in recent years. A main attraction of FDTD is that it can be implemented in computer code easily using simple straightforward marching algorithms and finite difference equations. However, this simplicity comes at a price. Dispersion errors, source scattering, and frequency dependent boundary reflections are just a few of the problems that FDTD has to deal with. Generally the former two are artificial problems of the numerical scheme. The last one, however, is a key component in any room acoustics applications. In theory, the boundary condition can be presented as an impulse response to be convoluted with the FDTD update equations. Unfortunately, this is a rather time consuming process. There are various approximations that can be used to represent a frequency dependent boundary condition in the time domain to speed up the calculation, but their suitability for room acoustics applications has rarely been properly validated. In particular, a practical problem faced in real room acoustics application is that full bandwidth data on a boundary’s impedance value is rarely available. In fact, in most cases one may only have information on the absorption coefficient of the boundary in octave frequency bands. Hence it will be of interest to see if an approximation based on the absorption coefficient alone can be used in a FDTD scheme to produce acceptable results. The purpose of this paper is to compare different ways of modelling frequency dependent boundary condition in a FDTD scheme to predict the sound field in a room. In this study, the accuracy of these methods will be validated against calculations by the more accurate boundary element method to assess their applicability in terms of room acoustics criteria. INTRODUCTION Modelling of frequency dependent boundary conditions in the FDTD method has received much attention recently since it is a key feature for room acoustic simulation over others such as source scattering, dispersion problem and so on. To implement proper frequency dependent boundary conditions in a time domain method, we first require a representation of the acoustical properties of common wall surfaces such as porous materials, commonly defined in terms of impedance, reflection coefficient or absorption coefficient. Since most of these quantities are defined or measured in the frequency domain, we will need to determine how to approximate these data in the time domain, and to consider the various possibilities of implementing them in a time domain method. Here, we studied several ways of approximating and implementing different types of acoustical properties of porous materials [17]. Once the data of the materials are approximated, they are formulated as a digital filter in the z-domain using a bilinear transform. For the implementation, two different approaches were applied – one based on the impedance and the other based on the reflection coefficient of the surface. The FDTD implementations of the frequency dependent boundary conditions will be validated with results from a boundary element method (BEM). The spectrum of the point-received impulse response in a three dimensional setting will be used for the validation. The comparison between the different implementations will be discussed using both frequency responses and room acoustics parameters to determine their usability. THEORITICAL FORMULATIONS Basic Equaions In room acoustic simulation, a FDTD method models the sound propagation using a finite difference scheme in time and space. The first order Euler and Continuity differential equations are the basic equations for the conventional leapfrog FDTD method. 1 ,