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
![]() |
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