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 $f(n)$ on a mass is decomposed as $f(n) = f^{{+}}(n)+f^{{-}}(n)$ , where $f^{{+}}(n)$ is regarded as a traveling wave propagating toward the mass, while $f^{{-}}(n)$ 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 $N$ elementary units together by means of $N$ -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.)

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).