Theory of Banded Waveguides

This article describes banded waveguides, a way of synthesizing sounds made by solid objects and an alternative method for treating two- and three-dimensional objects. It belongs to the synthesis algorithms known as physical models, and in particular, it is a departure from waveguide synthesis.

Physical modeling of musical instruments is a synthesis technique that is well established in computer music. Physical models are historically related to computationally expensive algorithms (Ruiz 1969) but have become more efficient with faster methods such as waveguide synthesis (Smith 2003). Digital waveguide models provide discrete-time models of distributed media such as vibrating strings, bores, horns, and plates.

We begin by outlining related synthesis methods with emphasis on traditional waveguide synthesis, which motivated the creation of this new structure. To simulate sustained and transient excitations such as striking, bowing, and rubbing, different excitation models are also proposed in this article. Instruments that have been modeled using banded waveguides are discussed in a companion article (Essl et al. 2003).

Digital Waveguides

Figure 1 shows a one-dimensional digital waveguide. A lossless digital waveguide is a bidirectional delay line at some wave impedance, and each delay line element contains a sampled traveling-wave component (Smith 2003).

Efficient physical models of vibrating strings, wind instruments, and other quasi-harmonic systems have been implemented using digital waveguides. For a review of physical models using digital waveguides, see Smith (2003) and the references therein.

Digital Waveguide Strings

Using digital waveguides, it is easy to create a physical model of a vibrating string. The structure of this model is shown in Figure 2. In this case, the delay line represents the sampled string in which traveling waves propagate. The low-pass filter is used to model losses along the string and at the extremities. For simplification, we assume that the string is excited at one extremity. This corresponds to the original Karplus–Strong algorithm (Karplus and Strong 1983).

Figure 1.

A digital waveguide formed by two delay lines of N samples each with opposite wave propagation directions.

Advantages and Disadvantages of One-Dimensional Waveguides

Digital waveguides provide an efficient synthesis tool for quasi-harmonic resonators, which include vibrating strings when there is negligible or weak dispersion. In situations where stiffness is noticeable but not high, such as in piano strings, all-pass filters have been used to model the inharmonicity of overtones (Rocchesso and Scalcon 1996). The role of all-pass filters is to create a frequency-dependent propagation velocity, resulting in partials that are stretched in frequency.

Figure 2.

A simplified waveguide model of a vibrating string. The delay line of N samples represents the string, and "LP" represents the low-pass filter that accounts for losses.

For very stiff systems such as rigid bars, however, a combination of waveguides and all-pass filters provides a less efficient structure for sound synthesis. Some inharmonic structures such as bells (Karjalainen, Välimäki, and Esquef 2002) and higher-dimensional structures (Rocchesso and Dutilleux 2001) have recently been modeled using all-pass filters. In these cases, other synthesis techniques, such as spectral modeling synthesis (Serra 1986) or modal synthesis (Adrien 1991), have been used. Another approach to modeling complex resonators in higher dimensions is to use the waveguide mesh (van Duyne and Smith 1993), a generalization of the digital waveguide described in the following section.

The Digital Waveguide Mesh

Figure 3 shows a two-dimensional digital waveguide mesh. It is a regular array of digital one-dimensional waveguides arranged along each perpendicular dimension, interconnected at their crossings by scattering junctions J. In the figure, each waveguide is one sample long. In addition to the rectilinear mesh shown in Figure 3, other mesh geometries have been explored, such as triangular and tetrahedral.

Advantages and Disadvantages of Waveguide Meshes

Digital waveguide meshes are a synthesis technique usually adopted to generate large numbers of modes. They are especially useful for modeling the high frequency modes of complex resonators. The computational cost of the waveguide mesh, however, may not allow...