Virtual room acoustics using finite difference methods. How to model and analyse frequency-dependent boundaries?

In this paper, we present new methods for constructing and analysing frequency-dependent boundaries in room acoustic modelling with the use of finite difference time domain (FDTD) techniques. Novel FDTD formulations of simple locally reacting wall models with complex impedance are proposed and analysed in terms of pressure wave reflectance for different wall impedances and angles of incidence. The analysis is done using both numerical experiments and analytic evaluation. For the numerical experiments, a compact implicit scheme of 4th-order accuracy is used for updating the room interior grid points, the results of which are analysed in both time and frequency domains. The simulation results show that the 2D frequency-dependent locally reacting wall models adhere well to their theoretical counterparts, particularly at low frequencies. Furthermore, they validate the analytic evaluation method, which paves the way for using either method as a tool for analysis of numerical reflectance.

I. Introduction

Finite Difference (FD) is a modelling technique that can be applied to room acoustic simulations [1]. Recent research efforts have focused on developing accurate FD approximations of realistic boundaries. Most of the boundary models available in the literature are based on a 1D approach [2], [3]. However, this leads to significant errors in the reflectance phase and amplitude [4]. For a physically more correct impedance boundary formulation, the boundary should be included in the medium, which can be obtained by combining the 1D boundary condition with the multi-dimensional (i.e. 2D or 3D) wave equation. Since frequency-dependent absorption occurs at realistic boundaries, the amplitude and phase of the reflected sound wave differ from those of the incident wave [5]. Instead of combining the FD implementation for the room interior with reflectance filters at boundaries [6], direct incorporation of the wall impedance in the boundary condition can be used. Alternative models rely on modelling the wave propagation in the wall [7]. However, previous studies [1] have suggested that there is no significant difference between the locally and non-locally reacting wall models used in FD room acoustic simulations.