Click here to purchase the entire book in PDF format. Physical models While the ray-trace and image models use mathematical descriptions of real acoustic spaces, they do not necessarily mimic the behaviour of an acoustic wavefront in those spaces. This is due largely to the fact that each considers only the propagation speed and direction of the expanding wavefront, and does not take into account such factors as the diffusion or diffraction [Kleiner et al., 1993]. As a result, although these models will provide excellent models of early specular reflection patterns, they do not produce accurate impulse responses for high orders of reflections. This is because the spatial spread of quantized radiation angles increases with distance from the sound source, and thus the order of the reflection. Consequently, it is likely that many reflections will not be recorded at the receiver's position. In order to achieve a higher level of accuracy, a model which is based on a larger set of physical rules is required. In this so-called physical model, the system is equipped with equations that describe the mechanical and acoustic behaviour of the various components of the physical system being modeled. In theory, the result is a system that mimics the behaviour of the physical counterpart [Roads, 1994]. The term ``physical model'' is widely-used in many fields to describe various systems. In music technology-based applications, a physical model is one where mathematical models of physical acoustics are used to produce the characteristics of a resonant instrument or enclosure [Roads, 1994]. This mathematical model can take various forms, however, the typical implementation involves a recursive delay including a filter in the feedback path, an example of which is the Karplus-Strong plucked string algorithm [Karplus and Strong, 1983]. Two general methods of physical models used in auralization and predictive acoustics software are known as the boundary element method (BEM), and the finite element method FEM. Boundary element method Using the boundary element method, surfaces are subdivided into discrete components, each with a particular set of reflective characteristics [Kleiner et al., 1993]. Each of these components is considered to be a sound source, re-radiating power into the enclosure with an individual contribution to the whole room impulse response. One of the principal problems with the boundary element method is the large number of calculations required in order to build a complete impulse response. This is particularly true with higher order reflections, since the increase in the number of interacting elements is exponentially proportional to the order of reflection. Finite element model The boundary element method is limited in that it only considers the characteristics of the surfaces which define a space while neglecting the behaviour of elements within the space itself. The finite element model corrects for this omission, subdividing the entire room into a collection of interconnected discrete elements arranged in a mesh. In this manner, the entire space is modeled, albeit with a rather high computational cost. One notable example of the finite element method is the digital waveguide mesh. Digital waveguide mesh Various systems have been developed and proposed which use the concept of digital waveguides to simulate the resonant and reverberant characteristics both of room acoustics and, on a smaller scale, of instrument acoustics. Initially proposed for room models by Crawford in 1968 [Roads, 1994], digital waveguides use bidirectional delay lines (and therefore recursion) to simulate the characteristics of acoustic waveguides with considerable efficiency. A notable development in the field of room acoustics modeling was the extension of the digital waveguide into a multi-dimensional mesh. This mesh is comprised of a number of discrete digital waveguides that are interconnected using scattering junctions. In its canonical form, the delay time of the individual bidirectional delays is 1 sample. The purpose of the scattering junctions is to re-distribute the wave energy into connected delays based on the relative acoustic impedances of the incoming and outgoing waveguides. For example, in the case of a scattering junction in the centre of a room, the impedances of the outputs of all connected delays would be equal, therefore any incoming energy is equally distributed among all their inputs. By comparison, if the scattering junction is located at a very reflective surface, then there is a mismatch in the amplitudes of the waveforms that are routed to the connected delays, sending more power in some directions than others. Typically, digital waveguide meshes are arranged in a two dimensional configuration as is represented in Figure 9.97. One excellent introductory description of this system is [Van Duyne and Smith III, 1993]. Figure 9.97: Two-dimensional digital waveguide mesh. Each circle denotes a scattering junction with four connected bidirectional delays[Van Duyne and Smith III, 1993]. There are a number of problems associated with the implementation of digital waveguide meshes such as dispersion and quantization error. One particular difficulty with using the discrete rather than a continuous representation of space that occurs in the mesh is frequency-dependent differences in wave propagation speed in different directions on the mesh. This is particularly noticeable in two-dimensional meshes with square layouts as shown in Figure 9.97. In such a configuration, diagonally-travelling waves have a frequency-independent propagation speed. Low-frequency waves travelling along either of the two axes will have an identical speed to their diagonal counterparts, however, high frequency information traveling along the axes has a speed of 0.707 that of lower frequencies. This results in a distortion of the wavefront in the form of a loss of transient information in some propagation directions. One solution to this problem is to increase the number of dimensions on the mesh. Although the system is still used to model a two-dimensional surface, it contains a larger number of axes and thus reduces differences in wave speed with direction. One example of such a system is shown in Figure 9.98 which displays a triangular mesh. Savioja [Savioja, 1999] provides an evaluation of various mesh topologies and the resulting wave speeds. Figure 9.98: Two-dimensional triangular digital waveguide mesh. Digital waveguide meshes also suffer from cumulative quantization errors due to the multiplicity of parallel and series combinations of gain modification in the junctions. One suggested solution to this issue is the re-distribution of the error into the spatial domain. In this topology, the quantization error is traded for a much less significant dispersion error on the mesh [Van Duyne and Smith III, 1993]. Unfortunately, although physical modelling reverberation systems can produce excellent results, the processing power required to simulate the characteristics of large three-dimensional spaces is presently prohibitive in real-time systems.