We now derive the digital waveguide formulation by *sampling* the
*traveling-wave* solution to the wave equation. It is easily
checked that the lossless 1D wave equation
is solved
by any string shape which travels to the left or right with speed
[6]. Denote
*right-going* traveling waves in general by
and
*left-going* traveling waves by
, where and
are assumed twice-differentiable. Then, as is well known, the
general class of solutions to the lossless, one-dimensional,
second-order wave equation can be expressed as

Sampling these traveling-wave solutions yields

where a ``'' superscript denotes a ``right-going'' traveling-wave component, and ``'' denotes propagation to the ``left''. This notation is similar to that used for acoustic-tube modeling of speech [17].

Figure 2 shows a signal flow diagram for the computational model of Eq. (6), which is often called a digital waveguide model (for the ideal string in this case) [25,29]. Note that, by the sampling theorem, it is an exact model so long as the initial conditions and any ongoing additive excitations are bandlimited to less than half the temporal sampling rate [27, Appendix G].

Note also that the position along the string, meters, is laid out from left to right in the diagram, giving a physical interpretation to the horizontal direction in the diagram, even though spatial samples have been eliminated from explicit consideration. (The arguments of and have physical units of time.)

The left- and right-going traveling wave components are summed to produce a physical output according to

In Fig. 2, ``transverse displacement outputs'' have been arbitrarily placed at and . The diagram is similar to that of well known ladder and lattice digital filter structures [17], except for the delays along the upper rail, the absence of scattering junctions, and the direct physical interpretation.

Download wgfdtd.pdf

Copyright ©

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

[Automatic-links disclaimer]