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 §7.3.2). - 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 §C.8 for the theory of scattering junctions.)

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).

- A Physical Derivation of Wave Digital Elements
- Reflectance of a General Lumped Waveguide Termination
- Reflectances of Elementary Impedances
- Capacitor Reflectance
- Inductor Reflectance
- Resistor Reflectance
- Choosing Impedance to Simplify Element Reflectance
- Digitizing Elementary Reflectances by Bilinear Transform

- Summary of Wave Digital Elements
- Wave Digital Mass
- Wave Digital Spring
- Wave Digital Dashpot
- Limiting Cases
- Unit Elements

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University