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A (lossless) digital waveguide is defined as a bidirectional delay line at some wave impedance [430,433]. Figure 2.11 illustrates one digital waveguide.
As before, each delay line contains a sampled acoustic traveling wave. However, since we now have a bidirectional delay line, we have two traveling waves, one to the ``left'' and one to the ``right'', say. It has been known since 1747 [100] that the vibration of an ideal string can be described as the sum of two traveling waves going in opposite directions. (See Appendix C for a mathematical derivation of this important fact.) Thus, while a single delay line can model an acoustic plane wave, a bidirectional delay line (a digital waveguide) can model any one-dimensional linear acoustic system such as a violin string, clarinet bore, flute pipe, trumpet-valve pipe, or the like. Of course, in real acoustic strings and bores, the 1D waveguides exhibit some loss and dispersion^{3.4} so that we will need some filtering in the waveguide to obtain an accurate physical model of such systems. The wave impedance (derived in Chapter 6) is needed for connecting digital waveguides to other physical simulations (such as another digital waveguide or finite-difference model).