### Digital Waveguide Modeling Elements

As mentioned above, digital waveguide models are built out of digital delay-lines and filters (and nonlinear elements), and they can be understood as propagating and filtering sampled traveling-wave solutions to the wave equation (PDE), such as for air, strings, rods, and the like [433,437]. It is noteworthy that strings, woodwinds, and brasses comprise three of the four principal sections of a classical orchestra (all but percussion). The digital waveguide modeling approach has also been extended to propagation in 2D, 3D, and beyond [518,396,522,400]. They are not finite-difference models, but paradoxically they are equivalent under certain conditions (Appendix E). A summary of historical aspects appears in §A.9.

As mentioned at Eq.(1.1), the ideal wave equation comes directly from Newton's laws of motion (). For example, in the case of vibrating strings, the wave equation is derived from first principles (in Chapter 6, and more completely in Appendix C) to be

where

Defining
, we obtain the usual form of the PDE known as
the *ideal 1D wave equation*.

where is the string displacement at time and position . (We omit the time and position arguments when they are the same for all signal terms in the equation.) For example, can be the transverse displacement of an ideal stretched string or the longitudinal displacement (or pressure, velocity, etc.) in an air column. The independent variables are time and the distance along the string or air-column axis. The partial-derivative notation is more completely written out as

As has been known since d'Alembert [100], the 1D wave
equation is obeyed by arbitrary *traveling waves* at speed :

In digital waveguide modeling, the traveling-waves are *sampled:*

where denotes the time sampling interval in seconds, denotes the spatial sampling interval in meters, and and are defined for notational convenience.

An ideal string (or air column) can thus be simulated using a
*bidirectional delay line*, as shown in
Fig.1.13 for the case of an -sample
section of ideal string or air column. The ``'' label denotes its
*wave impedance* (§6.1.5) which is needed when connecting
digital waveguides to each other and to other kinds of computational
physical models (such as finite difference schemes). While
propagation speed on an ideal string is
, we will
derive (§C.7.3) that the wave impedance is
.

Figure 1.14 (from Chapter 6,
§6.3), illustrates a simple digital waveguide model for
rigidly terminated vibrating strings (more specifically, one
polarization-plane of transverse vibration). The traveling-wave
components are taken to be *displacement* samples, but the
diagram for velocity-wave and acceleration-wave simulation are
identical (inverting reflection at each rigid termination). The
output signal is formed by summing traveling-wave
components at the desired ``virtual pickup'' location (position
in this example). To drive the string at a particular point,
one simply takes the *transpose* [449] of the output sum,
*i.e.*, the input excitation is summed equally into the left- and
right-going delay-lines at the same position (details will be
discussed near Fig.6.14).

In Chapter 9 (example applications), we will discuss digital waveguide models for single-reed instruments such as the clarinet (Fig.1.15), and bowed-string instruments (Fig.1.16) such as the violin.

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General Modeling Procedure

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Wave Digital Filters