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
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
In digital waveguide modeling, the traveling-waves are sampled:
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  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).
General Modeling Procedure
Wave Digital Filters