Wave Digital Elements

When modeling mechanical systems composed of masses, springs, and dashpots, it is best to begin with an electrical equivalent circuit. Equivalent circuits make clear the network-theoretic structure of the system, clearly indicating, for example, whether interacting elements should be connected in series or parallel. Each element of the equivalent circuit can then be replaced by a first-order wave digital element, and the elements are finally parallel or series connected by means of scattering-junction interfaces known as adaptors.

Wave digital elements may be derived from their describing differential equations (in continuous time) as follows:

• First express all physical quantities (such as force and velocity) in terms of traveling-wave components. The traveling wave components are called wave variables. For example, the force on a mass is decomposed as , where is regarded as a traveling wave propagating toward the mass, while is seen as the traveling component propagating away from the mass. A ``traveling wave'' view of force mediation is actually much closer to physical reality than any instantaneous model.

• Second, digitize the resulting traveling-wave system using the bilinear transform. The bilinear transform is equivalent in the time domain to the trapezoidal rule for numerical integration (see §L.4).

• Connect elementary units together by means of -port scattering junctions. There are two basic types of scattering junction, one for parallel, and one for series connection. (See §H.8 for the theory of scattering junctions.)

The next section will examine the above steps in greater detail.

An important benefit of introducing wave variables prior to bilinear transformation is the elimination of delay-free loops when connecting elementary building blocks. In other words, any number of elementary models can be interconnected, in series or in parallel, and the resulting finite-difference scheme remains explicit (free of delay-free loops).

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Subsections

• A Physical Derivation of Wave Digital Elements
  • Reflectance of a General Lumped Waveguide Termination
  • Reflectances of Elementary Impedances
  • Choosing Waveguide Impedance to Simplify Element Reflectance
  • Digitizing Elementary Reflectances via the Bilinear Transform
  
  
• Summary of Wave Digital Elements
• Wave Digital Mass
• Wave Digital Spring
• Wave Digital Dashpot
• Limiting Cases
• Unit Elements

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research rela